:: Towards the construction of a model of Mizar concepts :: by Grzegorz Bancerek environ vocabularies NUMBERS, NAT_1, SUBSET_1, FUNCT_1, TARSKI, CARD_3, RELAT_1, XBOOLE_0, STRUCT_0, CATALG_1, PBOOLE, MSATERM, FACIRC_1, MSUALG_1, ZFMISC_1, ZF_MODEL, FUNCOP_1, CARD_1, FINSEQ_1, TREES_3, TREES_4, MARGREL1, MSAFREE, CLASSES1, SETFAM_1, FINSET_1, QC_LANG3, ARYTM_3, XXREAL_0, ORDINAL1, MCART_1, FINSEQ_2, ORDINAL4, PARTFUN1, ABCMIZ_0, FUNCT_2, FUNCT_4, ZF_LANG1, CAT_3, TREES_2, MSUALG_2, MEMBER_1, AOFA_000, CARD_5, ORDERS_2, YELLOW_1, ARYTM_0, LATTICE3, EQREL_1, LATTICES, YELLOW_0, ORDINAL2, WAYBEL_0, ASYMPT_0, LANG1, TDGROUP, DTCONSTR, BINOP_1, MATRIX_7, FUNCT_7, ABCMIZ_1, SETLIM_2; notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, XFAMILY, SUBSET_1, DOMAIN_1, SETFAM_1, RELAT_1, FUNCT_1, RELSET_1, BINOP_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCT_2, PARTIT_2, FUNCT_4, FUNCOP_1, XXREAL_0, ORDINAL1, XCMPLX_0, NAT_1, MCART_1, FINSET_1, CARD_1, NUMBERS, CARD_3, FINSEQ_1, FINSEQ_2, TREES_2, TREES_3, TREES_4, FUNCT_7, PBOOLE, MATRIX_7, XXREAL_2, STRUCT_0, LANG1, CLASSES1, ORDERS_2, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSUALG_1, MSUALG_2, MSAFREE, EQUATION, MSATERM, CATALG_1, MSAFREE3, AOFA_000, PRE_POLY; constructors DOMAIN_1, MATRIX_7, MSAFREE1, FUNCT_7, EQUATION, YELLOW_1, CATALG_1, LATTICE3, WAYBEL_0, FACIRC_1, CLASSES1, MSAFREE3, XXREAL_2, RELSET_1, PRE_POLY, PARTIT_2, XTUPLE_0, XFAMILY; registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, RELSET_1, FUNCT_1, FINSET_1, STRUCT_0, PBOOLE, MSUALG_1, MSUALG_2, FINSEQ_1, CARD_1, MSAFREE, FUNCOP_1, TREES_3, MSAFREE1, PARTFUN1, MSATERM, ORDERS_2, TREES_2, DTCONSTR, WAYBEL_0, YELLOW_1, LATTICE3, MEMBERED, RELAT_1, INDEX_1, INSTALG1, MSAFREE3, FACIRC_1, XXREAL_2, CLASSES1, FINSEQ_2, PARTIT_2, XTUPLE_0; requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL; definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, FINSEQ_2, LANG1, PBOOLE, TREES_3, MSUALG_1, WAYBEL_0, XTUPLE_0; equalities TARSKI, RELAT_1, FINSEQ_1, LANG1, LATTICE3, MSAFREE, MSAFREE3, CARD_3, MSUALG_1, ORDINAL1; expansions TARSKI, FUNCT_1, LANG1, LATTICE3, PBOOLE, TREES_3; theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, ZFMISC_1, FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, SUBSET_1, ENUMSET1, FUNCT_4, PROB_2, LANG1, MATRIX_7, NAT_1, MCART_1, PBOOLE, FINSET_1, RELAT_1, RELSET_1, ORDINAL3, CARD_1, CARD_3, CARD_5, CLASSES1, ORDINAL1, SETFAM_1, MSUALG_2, TREES_4, FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3, PARTFUN1, LATTICE3, YELLOW_0, WAYBEL_0, YELLOW_1, YELLOW_7, DTCONSTR, MSAFREE1, XXREAL_2, CARD_2, XTUPLE_0; schemes XBOOLE_0, FUNCT_1, NAT_1, FRAENKEL, PBOOLE, MSATERM, DTCONSTR, CLASSES1, FUNCT_2; begin :: Variables reserve i for Nat, j for Element of NAT, X,Y,x,y,z for set; theorem Th1: for f being Function holds f.x c= Union f proof let f be Function; x in dom f or not x in dom f; then f.x in rng f or f.x = {} by FUNCT_1:3,def 2; hence thesis by ZFMISC_1:74; end; theorem for f being Function st Union f = {} holds f.x = {} by Th1,XBOOLE_1:3; theorem Th3: for f being Function for x,y being object st f = [x,y] holds x = y proof let f be Function, x,y be object; assume A1: f = [x,y]; then A2: {x} in f by TARSKI:def 2; A3: {x,y} in f by A1,TARSKI:def 2; consider a,b being object such that A4: {x} = [a,b] by A2,RELAT_1:def 1; A5: {a} = {a,b} by A4,ZFMISC_1:5; A6: x = {a} by A4,ZFMISC_1:4; consider c,d being object such that A7: {x,y} = [c,d] by A3,RELAT_1:def 1; A8: x = {c} & y = {c,d} or x = {c,d} & y = {c} by A7,ZFMISC_1:6; then c = a by A5,A6,ZFMISC_1:4; hence thesis by A2,A3,A4,A5,A7,A8,FUNCT_1:def 1; end; theorem Th4: (id X).:Y c= Y proof let x be object; assume x in (id X).:Y; then ex y being object st [y,x] in id X & y in Y by RELAT_1:def 13; hence thesis by RELAT_1:def 10; end; theorem Th5: for S being non void Signature for X being non-empty ManySortedSet of the carrier of S for t being Term of S, X holds t is non pair proof let S be non void Signature; let X be non-empty ManySortedSet of the carrier of S; let t be Term of S, X; given x,y being object such that A1: t = [x,y]; (ex s being SortSymbol of S, v being Element of X.s st t.{} = [v,s]) or t.{} in [:the carrier' of S,{the carrier of S}:] by MSATERM:2; then (ex s being SortSymbol of S, v being Element of X.s st t.{} = [v,s]) or ex a,b being object st a in the carrier' of S & b in {the carrier of S} & t.{} = [a,b] by ZFMISC_1:def 2; then {{}} <> {{}, t.{}} by ZFMISC_1:5; then A2: [{}, t.{}] <> {x} by ZFMISC_1:5; {} in dom t by TREES_1:22; then [{}, t.{}] in t by FUNCT_1:def 2; then A3: [{}, t.{}] = {x,y} by A1,A2,TARSKI:def 2; x = y by A1,Th3; hence thesis by A2,A3,ENUMSET1:29; end; registration let S be non void Signature; let X be non empty-yielding ManySortedSet of the carrier of S; cluster -> non pair for Element of Free(S,X); coherence proof let e be Element of Free(S,X); e is Term of S, X (\/) ((the carrier of S) --> {0}) by MSAFREE3:8; hence thesis by Th5; end; end; theorem Th6: for x,y,z being set st x in {z}* & y in {z}* & card x = card y holds x = y proof let x,y,z be set such that A1: x in {z}* and A2: y in {z}* and A3: card x = card y; reconsider x, y as FinSequence of {z} by A1,A2,FINSEQ_1:def 11; A4: dom x = Seg len x by FINSEQ_1:def 3 .= dom y by A3,FINSEQ_1:def 3; now let i be Nat; assume A5: i in dom x; then A6: x .i in rng x by FUNCT_1:def 3; A7: y.i in rng y by A4,A5,FUNCT_1:def 3; thus x .i = z by A6,TARSKI:def 1 .= y.i by A7,TARSKI:def 1; end; hence thesis by A4,FINSEQ_1:13; end; definition let S be non void Signature; let A be MSAlgebra over S; mode Subset of A is Subset of Union the Sorts of A; mode FinSequence of A is FinSequence of Union the Sorts of A; end; registration let S be non void Signature; let X be non empty-yielding ManySortedSet of S; cluster -> DTree-yielding for FinSequence of Free(S,X); coherence proof let p be FinSequence of Free(S,X); let x be object; assume x in rng p; hence thesis; end; end; theorem Th7: for S being non void Signature for X being non empty-yielding ManySortedSet of the carrier of S for t being Element of Free(S,X) holds (ex s being SortSymbol of S, v being set st t = root-tree [v,s] & v in X.s) or ex o being OperSymbol of S, p being FinSequence of Free(S,X) st t = [o,the carrier of S]-tree p & len p = len the_arity_of o & p is DTree-yielding & p is ArgumentSeq of Sym(o, X(\/)((the carrier of S)-->{0})) proof let S be non void Signature; let X be non empty-yielding ManySortedSet of the carrier of S; let t be Element of Free(S,X); set V = X(\/)((the carrier of S)-->{0}); reconsider t9 = t as Term of S,V by MSAFREE3:8; defpred P[set] means $1 is Element of Free(S,X) implies (ex s being SortSymbol of S, v being set st $1 = root-tree [v,s] & v in X.s) or ex o being OperSymbol of S, p being FinSequence of Free(S,X) st $1 = [o,the carrier of S]-tree p & len p = len the_arity_of o & p is DTree-yielding & p is ArgumentSeq of Sym(o,V); A1: for s being SortSymbol of S, v being Element of V.s holds P[root-tree [v,s]] proof let s be SortSymbol of S; let v be Element of V.s; set t = root-tree [v,s]; assume A2: t is Element of Free(S,X); {} in dom t by TREES_1:22; then t.{} in rng t by FUNCT_1:3; then [v,s] in rng t by TREES_4:3; then v in X.s by A2,MSAFREE3:35; hence thesis; end; A3: for o being OperSymbol of S, p being ArgumentSeq of Sym(o,V) st for t being Term of S,V st t in rng p holds P[t] holds P[[o,the carrier of S]-tree p] proof let o be OperSymbol of S; let p be ArgumentSeq of Sym(o,V) such that for t being Term of S,V st t in rng p holds P[t]; set t = [o,the carrier of S]-tree p; assume t is Element of Free(S,X); then consider s being object such that A4: s in dom the Sorts of Free(S,X) and A5: t in (the Sorts of Free(S,X)).s by CARD_5:2; reconsider s as Element of S by A4; A6: the Sorts of Free(S,X) = S-Terms(X,V) by MSAFREE3:24; the_sort_of(Sym(o,V)-tree p) = the_result_sort_of o by MSATERM:20; then s = the_result_sort_of o by A5,A6,MSAFREE3:17; then rng p c= Union (S-Terms(X,V)) by A5,A6,MSAFREE3:19; then A7: p is FinSequence of Free(S,X) by A6,FINSEQ_1:def 4; len the_arity_of o = len p by MSATERM:22; hence thesis by A7; end; for t being Term of S,V holds P[t] from MSATERM:sch 1(A1,A3); then P[t9]; hence thesis; end; definition let A be set; func varcl A -> set means : Def1: A c= it & (for x,y st [x,y] in it holds x c= it) & for B being set st A c= B & for x,y st [x,y] in B holds x c= B holds it c= B; uniqueness proof let B1, B2 be set; assume A1: not thesis; then A2: B1 c= B2; B2 c= B1 by A1; hence thesis by A1,A2,XBOOLE_0:def 10; end; existence proof set F = {C where C is Subset of Rank the_rank_of A: A c= C & for x,y st [x,y] in C holds x c= C}; take D = meet F; A3: A c= Rank the_rank_of A by CLASSES1:def 9; A4: now let x,y; assume A5: [x,y] in Rank the_rank_of A; A6: {x} in {{x,y},{x}} by TARSKI:def 2; A7: {{x,y},{x}} c= Rank the_rank_of A by A5,ORDINAL1:def 2; A8: x in {x} by TARSKI:def 1; {x} c= Rank the_rank_of A by A6,A7,ORDINAL1:def 2; hence x c= Rank the_rank_of A by A8,ORDINAL1:def 2; end; Rank the_rank_of A c= Rank the_rank_of A; then A9: Rank the_rank_of A in F by A3,A4; hereby let x be object; assume A10: x in A; now let C be set; assume C in F; then ex B being Subset of Rank the_rank_of A st C = B & A c= B & for x,y st [x,y] in B holds x c= B; hence x in C by A10; end; hence x in D by A9,SETFAM_1:def 1; end; hereby let x,y; assume A11: [x,y] in D; thus x c= D proof let z be object; assume A12: z in x; now let X; assume A13: X in F; then A14: [x,y] in X by A11,SETFAM_1:def 1; ex B being Subset of Rank the_rank_of A st X = B & A c= B & for x,y st [x,y] in B holds x c= B by A13; then x c= X by A14; hence z in X by A12; end; hence thesis by A9,SETFAM_1:def 1; end; end; let B being set; assume that A15: A c= B and A16: for x,y st [x,y] in B holds x c= B; set C = B /\ Rank the_rank_of A; reconsider C as Subset of Rank the_rank_of A by XBOOLE_1:17; A17: A c= C by A3,A15,XBOOLE_1:19; now let x,y; assume A18: [x,y] in C; then [x,y] in B by XBOOLE_0:def 4; then A19: x c= B by A16; x c= Rank the_rank_of A by A4,A18; hence x c= C by A19,XBOOLE_1:19; end; then C in F by A17; then A20: D c= C by SETFAM_1:3; C c= B by XBOOLE_1:17; hence thesis by A20; end; projectivity; end; theorem Th8: varcl {} = {} proof A1: for x,y st [x,y] in {} holds x c= {}; for B being set st {} c= B & for x,y st [x,y] in B holds x c= B holds {} c= B; hence thesis by A1,Def1; end; theorem Th9: for A,B being set st A c= B holds varcl A c= varcl B proof let A, B be set such that A1: A c= B; B c= varcl B by Def1; then A2: A c= varcl B by A1; for x,y st [x,y] in varcl B holds x c= varcl B by Def1; hence thesis by A2,Def1; end; theorem Th10: for A being set holds varcl union A = union the set of all varcl a where a is Element of A proof let A be set; set X = the set of all varcl a where a is Element of A; A1: union A c= union X proof let x be object; assume x in union A; then consider Y such that A2: x in Y and A3: Y in A by TARSKI:def 4; reconsider Y as Element of A by A3; A4: Y c= varcl Y by Def1; varcl Y in X; hence thesis by A2,A4,TARSKI:def 4; end; now let x,y be set; assume [x,y] in union X; then consider Y being set such that A5: [x,y] in Y and A6: Y in X by TARSKI:def 4; ex a being Element of A st ( Y = varcl a) by A6; then A7: x c= Y by A5,Def1; Y c= union X by A6,ZFMISC_1:74; hence x c= union X by A7; end; hence varcl union A c= union X by A1,Def1; let x be object; assume x in union X; then consider Y being set such that A8: x in Y and A9: Y in X by TARSKI:def 4; consider a being Element of A such that A10: Y = varcl a by A9; A is empty or A is not empty; then a in A or a is empty by SUBSET_1:def 1; then a c= union A by ZFMISC_1:74; then Y c= varcl union A by A10,Th9; hence thesis by A8; end; scheme Sch14{A() -> set, F(set) -> set, P[set]}: varcl union {F(z) where z is Element of A(): P[z]} = union {varcl F(z) where z is Element of A(): P[z]} proof set Z = {F(z) where z is Element of A(): P[z]}; set X = {varcl F(z) where z is Element of A(): P[z]}; A1: union Z c= union X proof let x be object; assume x in union Z; then consider Y such that A2: x in Y and A3: Y in Z by TARSKI:def 4; A4: ex z being Element of A() st ( Y = F(z))&( P[z]) by A3; A5: Y c= varcl Y by Def1; varcl Y in X by A4; hence thesis by A2,A5,TARSKI:def 4; end; now let x,y be set; assume [x,y] in union X; then consider Y being set such that A6: [x,y] in Y and A7: Y in X by TARSKI:def 4; ex z being Element of A() st ( Y = varcl F(z))&( P[z]) by A7; then A8: x c= Y by A6,Def1; Y c= union X by A7,ZFMISC_1:74; hence x c= union X by A8; end; hence varcl union Z c= union X by A1,Def1; let x be object; assume x in union X; then consider Y being set such that A9: x in Y and A10: Y in X by TARSKI:def 4; consider z being Element of A() such that A11: Y = varcl F(z) and A12: P[z] by A10; F(z) in Z by A12; then Y c= varcl union Z by A11,Th9,ZFMISC_1:74; hence thesis by A9; end; theorem Th11: varcl (X \/ Y) = (varcl X) \/ (varcl Y) proof set A = the set of all varcl a where a is Element of {X,Y}; X \/ Y = union {X,Y} by ZFMISC_1:75; then A1: varcl (X \/ Y) = union A by Th10; A = {varcl X, varcl Y} proof thus now let x be object; assume x in A; then consider a being Element of {X,Y} such that A2: x = varcl a; a = X or a = Y by TARSKI:def 2; hence x in {varcl X, varcl Y} by A2,TARSKI:def 2; end; let x be object; assume x in {varcl X, varcl Y}; then x = varcl X & X in {X,Y} or x = varcl Y & Y in {X,Y} by TARSKI:def 2; hence thesis; end; hence thesis by A1,ZFMISC_1:75; end; theorem Th12: for A being non empty set st for a being Element of A holds varcl a = a holds varcl meet A = meet A proof let B be non empty set; set A = meet B; assume A1: for a being Element of B holds varcl a = a; now thus A c= A; let x,y; assume A2: [x,y] in A; now let Y; assume A3: Y in B; then A4: [x,y] in Y by A2,SETFAM_1:def 1; Y = varcl Y by A1,A3; hence x c= Y by A4,Def1; end; hence x c= A by SETFAM_1:5; end; hence varcl A c= A by Def1; thus thesis by Def1; end; theorem Th13: varcl ((varcl X) /\ (varcl Y)) = (varcl X) /\ (varcl Y) proof set A = (varcl X) /\ (varcl Y); now thus A c= A; let x,y; assume A1: [x,y] in A; then A2: [x,y] in varcl X by XBOOLE_0:def 4; A3: [x,y] in varcl Y by A1,XBOOLE_0:def 4; A4: x c= varcl X by A2,Def1; x c= varcl Y by A3,Def1; hence x c= A by A4,XBOOLE_1:19; end; hence varcl ((varcl X) /\ (varcl Y)) c= (varcl X) /\ (varcl Y) by Def1; thus thesis by Def1; end; registration let A be empty set; cluster varcl A -> empty; coherence by Th8; end; deffunc F(set,set) = {[varcl A, j] where A is Subset of $2, j is Element of NAT: A is finite}; definition func Vars -> set means : Def2: ex V being ManySortedSet of NAT st it = Union V & V.0 = the set of all [{}, i] where i is Element of NAT & for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite}; existence proof consider f being Function such that A1: dom f = NAT and A2: f.0 = the set of all [{}, i] where i is Element of NAT and A3: for n being Nat holds f.(n+1) = F(n,f.n) from NAT_1:sch 11; reconsider f as ManySortedSet of NAT by A1,PARTFUN1:def 2,RELAT_1:def 18; take Union f, V = f; thus Union f = Union V; thus V.0 = the set of all [{}, i] where i is Element of NAT by A2; let n be Nat; thus thesis by A3; end; uniqueness proof let A1, A2 be set; given V1 being ManySortedSet of NAT such that A4: A1 = Union V1 and A5: V1.0 = the set of all [{}, i] where i is Element of NAT and A6: for n being Nat holds V1.(n+1) = F(n,V1.n); given V2 being ManySortedSet of NAT such that A7: A2 = Union V2 and A8: V2.0 = the set of all [{}, i] where i is Element of NAT and A9: for n being Nat holds V2.(n+1) = F(n,V2.n); A10: dom V1 = NAT by PARTFUN1:def 2; A11: dom V2 = NAT by PARTFUN1:def 2; V1 = V2 from NAT_1:sch 15(A10,A5,A6,A11,A8,A9); hence thesis by A4,A7; end; end; theorem Th14: for V being ManySortedSet of NAT st V.0 = the set of all [{}, i] where i is Element of NAT & for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} for i,j being Element of NAT st i <= j holds V.i c= V.j proof let V be ManySortedSet of NAT such that A1: V.0 = the set of all [{}, i] where i is Element of NAT and A2: for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite}; defpred Q[Nat] means V.0 c= V.$1; A3: now let j; assume Q[j]; A4: V.(j+1) = {[varcl A, k] where A is Subset of V.j, k is Element of NAT: A is finite} by A2; thus Q[j+1] proof let x be object; assume x in V.0; then A5: ex i being Element of NAT st x = [{}, i] by A1; {} c= V.j; hence thesis by A4,A5,Th8; end; end; defpred P[Nat] means for i st i <= $1 holds V.i c= V.$1; A6: P[ 0 ] by NAT_1:3; A7: now let j be Nat; assume A8: P[j]; A9: V.j c= V.(j+1) proof per cases by NAT_1:6; suppose j = 0; hence thesis by A3; end; suppose ex k being Nat st j = k+1; then consider k being Nat such that A10: j = k+1; reconsider k as Element of NAT by ORDINAL1:def 12; A11: V.j = {[varcl A, n] where A is Subset of V.k, n is Element of NAT: A is finite} by A2,A10; A12: V .(j+1) = {[varcl A, n] where A is Subset of V.j,n is Element of NAT: A is finite} by A2; A13: V.k c= V.j by A8,A10,NAT_1:11; let x be object; assume x in V.j; then consider A being Subset of V.k, n being Element of NAT such that A14: x = [varcl A, n] and A15: A is finite by A11; A c= V.j by A13; hence thesis by A12,A14,A15; end; end; thus P[j+1] proof let i; assume i <= j+1; then i = j+1 or V.i c= V.j by A8,NAT_1:8; hence thesis by A9; end; end; for j being Nat holds P[j] from NAT_1:sch 2(A6,A7); hence thesis; end; theorem Th15: for V being ManySortedSet of NAT st V.0 = the set of all [{}, i] where i is Element of NAT & for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} for A being finite Subset of Vars ex i being Element of NAT st A c= V.i proof let V be ManySortedSet of NAT such that A1: V.0 = the set of all [{}, i] where i is Element of NAT and A2: for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite}; let A be finite Subset of Vars; A3: Vars = Union V by A1,A2,Def2; defpred P[object,object] means $1 in V.$2; A4: now let x be object; assume x in A; then consider Y such that A5: x in Y and A6: Y in rng V by A3,TARSKI:def 4; consider i being object such that A7: i in dom V and A8: Y = V.i by A6,FUNCT_1:def 3; reconsider i as object; take i; thus i in NAT & P[x,i] by A5,A7,A8; end; consider f being Function such that A9: dom f = A & rng f c= NAT and A10: for x being object st x in A holds P[x,f.x] from FUNCT_1:sch 6(A4); per cases; suppose A = {}; then A c= V.0; hence thesis; end; suppose A <> {}; then reconsider B = rng f as finite non empty Subset of NAT by A9,FINSET_1:8,RELAT_1:42; reconsider i = max B as Element of NAT by ORDINAL1:def 12; take i; let x be object; assume A11: x in A; then A12: f.x in B by A9,FUNCT_1:def 3; then reconsider j = f.x as Element of NAT; j <= i by A12,XXREAL_2:def 8; then A13: V.j c= V.i by A1,A2,Th14; x in V.j by A10,A11; hence thesis by A13; end; end; theorem Th16: the set of all [{}, i] where i is Element of NAT c= Vars proof consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = the set of all [{}, i] where i is Element of NAT and for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} by Def2; dom V = NAT by PARTFUN1:def 2; then V.0 in rng V by FUNCT_1:def 3; hence thesis by A1,A2,ZFMISC_1:74; end; theorem Th17: for A being finite Subset of Vars, i being Nat holds [varcl A, i] in Vars proof let A be finite Subset of Vars, i be Nat; consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = the set of all [{}, k] where k is Element of NAT and A3: for n being Nat holds V.(n+1) = {[varcl b, j] where b is Subset of V.n, j is Element of NAT: b is finite} by Def2; consider j being Element of NAT such that A4: A c= V.j by A2,A3,Th15; A5: V.(j+1) = {[varcl B, k] where B is Subset of V.j, k is Element of NAT: B is finite} by A3; i in NAT by ORDINAL1:def 12; then A6: [varcl A, i] in V.(j+1) by A4,A5; dom V = NAT by PARTFUN1:def 2; hence thesis by A1,A6,CARD_5:2; end; theorem Th18: Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT: A is finite} proof consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = the set of all [{}, i] where i is Element of NAT and A3: for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} by Def2; set X = {[varcl A, j] where A is Subset of Vars, j is Element of NAT: A is finite}; A4: dom V = NAT by PARTFUN1:def 2; defpred P[Nat] means V.$1 c= X; A5: P[ 0] proof let x be object; assume A6: x in V.0; A7: {} c= Vars; ex i being Element of NAT st x = [{}, i] by A2,A6; hence thesis by A7,Th8; end; A8: now let i be Nat; assume P[i]; A9: V.(i+1) = {[varcl A, j] where A is Subset of V.i, j is Element of NAT: A is finite} by A3; thus P[i+1] proof let x be object; assume x in V.(i+1); then consider A being Subset of V.i, j being Element of NAT such that A10: x = [varcl A, j] and A11: A is finite by A9; reconsider ii=i as Element of NAT by ORDINAL1:def 12; V.ii in rng V by A4,FUNCT_1:def 3; then V.i c= Vars by A1,ZFMISC_1:74; then A c= Vars; hence thesis by A10,A11; end; end; A12: for i being Nat holds P[i] from NAT_1:sch 2(A5,A8); now let x; assume x in rng V; then ex y being object st y in NAT & x = V.y by A4,FUNCT_1:def 3; hence x c= X by A12; end; hence Vars c= X by A1,ZFMISC_1:76; let x be object; assume x in X; then ex A being Subset of Vars, j being Element of NAT st x = [varcl A, j] & A is finite; hence thesis by Th17; end; theorem Th19: varcl Vars = Vars proof consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = the set of all [{}, i] where i is Element of NAT and A3: for n being Nat holds V.(n+1) = {[varcl A, j] where A is Subset of V.n, j is Element of NAT: A is finite} by Def2; defpred P[Nat] means varcl(V.$1) = V.$1; now let x,y; assume [x,y] in V.0; then ex i being Element of NAT st [x,y] = [{}, i] by A2; then x = {} by XTUPLE_0:1; hence x c= V.0; end; then A4: varcl (V.0) c= V.0 by Def1; V.0 c= varcl (V.0) by Def1; then A5: P[ 0] by A4,XBOOLE_0:def 10; A6: now let i; assume A7: P[i]; reconsider i9 = i as Element of NAT by ORDINAL1:def 12; A8: V.(i+1) = {[varcl A, j] where A is Subset of V.i, j is Element of NAT: A is finite} by A3; now let x,y; assume [x,y] in V.(i+1); then consider A being Subset of V.i, j being Element of NAT such that A9: [x,y] = [varcl A, j] and A is finite by A8; x = varcl A by A9,XTUPLE_0:1; then A10: x c= V.i by A7,Th9; V.i9 c= V.(i9+1) by A2,A3,Th14,NAT_1:11; hence x c= V.(i+1) by A10; end; then A11: varcl (V.(i+1)) c= V.(i+1) by Def1; V.(i+1) c= varcl (V.(i+1)) by Def1; hence P[i+1] by A11,XBOOLE_0:def 10; end; A12: P[i] from NAT_1:sch 2(A5,A6); A13: varcl Vars = union the set of all varcl a where a is Element of rng V by A1,Th10; thus now let x be object; assume x in varcl Vars; then consider Y such that A14: x in Y and A15: Y in the set of all varcl a where a is Element of rng V by A13,TARSKI:def 4; consider a being Element of rng V such that A16: Y = varcl a by A15; consider i being object such that A17: i in dom V and A18: a = V.i by FUNCT_1:def 3; reconsider i as Element of NAT by A17; varcl (V.i) = a by A12,A18; hence x in Vars by A1,A14,A16,A17,A18,CARD_5:2; end; thus thesis by Def1; end; theorem Th20: for X st the_rank_of X is finite holds X is finite proof let X; assume the_rank_of X is finite; then the_rank_of X in NAT by CARD_1:61; then A1: Rank the_rank_of X is finite by CARD_2:67; X c= Rank the_rank_of X by CLASSES1:def 9; hence thesis by A1; end; theorem Th21: the_rank_of varcl X = the_rank_of X proof A1: X c= Rank the_rank_of X by CLASSES1:def 9; set a = the_rank_of X; A2: a c= succ a by ORDINAL3:1; succ a c= succ succ a by ORDINAL3:1; then a c= succ succ a by A2; then A3: Rank a c= Rank succ succ a by CLASSES1:37; now let x,y; assume [x,y] in Rank the_rank_of X; then x in Rank a by A3,CLASSES1:45; hence x c= Rank the_rank_of X by ORDINAL1:def 2; end; then varcl X c= Rank a by A1,Def1; hence the_rank_of varcl X c= a by CLASSES1:65; X c= varcl X by Def1; hence thesis by CLASSES1:67; end; theorem Th22: for X being finite Subset of Rank omega holds X in Rank omega proof let X be finite Subset of Rank omega; deffunc F(object) = the_rank_of $1; consider f being Function such that A1: dom f = X and A2: for x being object st x in X holds f.x = F(x) from FUNCT_1:sch 3; A3: rng f c= NAT proof let y be object; assume y in rng f; then consider x being object such that A4: x in X and A5: y = f.x by A1,FUNCT_1:def 3; the_rank_of x in omega by A4,CLASSES1:66; hence thesis by A2,A4,A5; end; per cases; suppose X = {}; then the_rank_of X = 0 by CLASSES1:71; hence thesis by CLASSES1:66; end; suppose X <> {}; then reconsider Y = rng f as finite non empty Subset of NAT by A1,A3,FINSET_1:8,RELAT_1:42; reconsider mY = max Y as Element of NAT by ORDINAL1:def 12; set i = 1+mY; X c= Rank i proof let x be object; reconsider xx=x as set by TARSKI:1; assume A6: x in X; then A7: f.x in Y by A1,FUNCT_1:def 3; A8: f.x = the_rank_of xx by A2,A6; reconsider j = f.x as Element of NAT by A7; j <= mY by A7,XXREAL_2:def 8; then Segm j c= Segm mY by NAT_1:39; then A9: j in succ mY by ORDINAL1:22; succ Segm mY = Segm i by NAT_1:38; hence thesis by A8,A9,CLASSES1:66; end; then the_rank_of X c= i by CLASSES1:65; then A10: the_rank_of X in succ i by ORDINAL1:22; Segm(i+1) = succ Segm i by NAT_1:38; hence thesis by A10,CLASSES1:66; end; end; theorem Th23: Vars c= Rank omega proof consider V being ManySortedSet of NAT such that A1: Vars = Union V and A2: V.0 = the set of all [{}, i] where i is Element of NAT and A3: for n being Nat holds V.(n+1) = {[varcl a, j] where a is Subset of V.n, j is Element of NAT: a is finite} by Def2; let x be object; assume x in Vars; then consider i being object such that A4: i in dom V and A5: x in V.i by A1,CARD_5:2; reconsider i as Element of NAT by A4; defpred P[Nat] means V.$1 c= Rank omega; A6: P[ 0] proof let x be object; assume x in V.0; then consider i being Element of NAT such that A7: x = [{}, i] by A2; A8: Segm(i+1) = succ Segm i by NAT_1:38; A9: {} c= i; A10: i in i+1 by A8,ORDINAL1:6; A11: {} in i+1 by A8,A9,ORDINAL1:6,12; A12: the_rank_of {} = {} by CLASSES1:73; A13: the_rank_of i = i by CLASSES1:73; A14: {} in Rank (i+1) by A11,A12,CLASSES1:66; i in Rank (i+1) by A10,A13,CLASSES1:66; then A15: x in Rank succ succ (i+1) by A7,A14,CLASSES1:45; succ succ (i+1) c= omega; then Rank succ succ (i+1) c= Rank omega by CLASSES1:37; hence thesis by A15; end; A16: now let n be Nat such that A17: P[n]; A18: V.(n+1) = {[varcl a, j] where a is Subset of V.n, j is Element of NAT: a is finite} by A3; thus P[n+1] proof let x be object; assume x in V.(n+1); then consider a being Subset of V.n, j being Element of NAT such that A19: x = [varcl a, j] and A20: a is finite by A18; a c= Rank omega by A17,XBOOLE_1:1; then a in Rank omega by A20,Th22; then reconsider i = the_rank_of a as Element of NAT by CLASSES1:66; reconsider k = j \/ i as Element of NAT by ORDINAL3:12; A21: the_rank_of varcl a = i by Th21; A22: the_rank_of j = j by CLASSES1:73; A23: k in succ k by ORDINAL1:6; then A24: i in succ k by ORDINAL1:12,XBOOLE_1:7; A25: j in succ k by A23,ORDINAL1:12,XBOOLE_1:7; A26: succ Segm k = Segm(k+1) by NAT_1:38; then A27: varcl a in Rank (k+1) by A21,A24,CLASSES1:66; j in Rank (k+1) by A22,A25,A26,CLASSES1:66; then A28: x in Rank succ succ (k+1) by A19,A27,CLASSES1:45; succ succ (k+1) c= omega; then Rank succ succ (k+1) c= Rank omega by CLASSES1:37; hence thesis by A28; end; end; for n being Nat holds P[n] from NAT_1:sch 2(A6,A16); then V.i c= Rank omega; hence thesis by A5; end; theorem Th24: for A being finite Subset of Vars holds varcl A is finite Subset of Vars proof let A be finite Subset of Vars; A c= Rank omega by Th23; then A in Rank omega by Th22; then the_rank_of A in omega by CLASSES1:66; then the_rank_of varcl A is finite by Th21; hence thesis by Th9,Th19,Th20; end; registration cluster Vars -> non empty; correctness proof [{},0] in the set of all [{}, i] where i is Element of NAT; hence thesis by Th16; end; end; definition mode variable is Element of Vars; end; registration let x be variable; cluster x`1 -> finite for set; coherence proof x in Vars; then consider A being Subset of Vars, j being Element of NAT such that A1: x = [varcl A,j] and A2: A is finite by Th18; x`1 = varcl A by A1; hence thesis by A2,Th24; end; end; notation let x be variable; synonym vars x for x`1; end; definition let x be variable; redefine func vars x -> Subset of Vars; coherence proof x in Vars; then consider A being Subset of Vars, j being Element of NAT such that A1: x = [varcl A,j] and A2: A is finite by Th18; x`1 = varcl A by A1; hence thesis by A2,Th24; end; end; theorem [{}, i] in Vars proof i in NAT by ORDINAL1:def 12; then [{}, i] in the set of all [{}, j]; hence thesis by Th16; end; theorem Th26: for A being Subset of Vars holds varcl {[varcl A, j]} = (varcl A) \/ {[varcl A, j]} proof let A be Subset of Vars; A1: {[varcl A, j]} c= (varcl A) \/ {[varcl A, j]} by XBOOLE_1:7; A2: varcl A c= (varcl A) \/ {[varcl A, j]} by XBOOLE_1:7; now let x,y; assume [x,y] in (varcl A) \/ {[varcl A, j]}; then [x,y] in varcl A or [x,y] in {[varcl A, j]} by XBOOLE_0:def 3; then [x,y] in varcl A or [x,y] = [varcl A, j] by TARSKI:def 1; then x c= varcl A or x = varcl A by Def1,XTUPLE_0:1; hence x c= (varcl A) \/ {[varcl A, j]} by A2; end; hence varcl {[varcl A, j]} c= (varcl A) \/ {[varcl A, j]} by A1,Def1; A3: {[varcl A, j]} c= varcl {[varcl A, j]} by Def1; [varcl A, j] in {[varcl A, j]} by TARSKI:def 1; then varcl A c= varcl {[varcl A, j]} by A3,Def1; hence thesis by A3,XBOOLE_1:8; end; theorem Th27: for x being variable holds varcl {x} = (vars x) \/ {x} proof let x be variable; x in Vars; then consider A being Subset of Vars, j such that A1: x = [varcl A, j] and A is finite by Th18; varcl {x} = (varcl A) \/ {x} by A1,Th26; hence thesis by A1; end; theorem for x being variable holds [(vars x) \/ {x}, i] in Vars proof let x be variable; x in Vars; then consider A being Subset of Vars, j such that A1: x = [varcl A, j] and A is finite by Th18; A2: varcl {x} = (varcl A) \/ {x} by A1,Th26; A3: vars x = varcl A by A1; i in NAT by ORDINAL1:def 12; hence thesis by A2,A3,Th18; end; begin :: Quasi loci notation let R be Relation, A be set; synonym R dom A for R|A; end; definition func QuasiLoci -> FinSequenceSet of Vars means :Def3: for p being FinSequence of Vars holds p in it iff p is one-to-one & for i st i in dom p holds (p.i)`1 c= rng (p dom i); existence proof defpred P[object] means ex p being Function st p = $1 & p is one-to-one & for i st i in dom p holds (p.i)`1 c= rng (p|i); consider L being set such that A1: for x being object holds x in L iff x in Vars* & P[ x ] from XBOOLE_0:sch 1; L is FinSequenceSet of Vars proof let x be object; assume x in L; then x in Vars* by A1; hence thesis by FINSEQ_1:def 11; end; then reconsider L as FinSequenceSet of Vars; take L; let p be FinSequence of Vars; p in L iff p in Vars* & ex q being Function st q = p & q is one-to-one & for i st i in dom q holds (q.i)`1 c= rng (q|i) by A1; hence thesis by FINSEQ_1:def 11; end; correctness proof let L1, L2 be FinSequenceSet of Vars such that A2: for p being FinSequence of Vars holds p in L1 iff p is one-to-one & for i st i in dom p holds (p.i)`1 c= rng (p|(i qua set)) and A3: for p being FinSequence of Vars holds p in L2 iff p is one-to-one & for i st i in dom p holds (p.i)`1 c= rng (p|(i qua set)); thus now let x be object; assume A4: x in L1; then reconsider p = x as FinSequence of Vars by FINSEQ_2:def 3; A5: p is one-to-one by A2,A4; for i st i in dom p holds (p.i)`1 c= rng (p|(i qua set)) by A2,A4; hence x in L2 by A3,A5; end; let x be object; assume A6: x in L2; then reconsider p = x as FinSequence of Vars by FINSEQ_2:def 3; A7: p is one-to-one by A3,A6; for i st i in dom p holds (p.i)`1 c= rng (p|(i qua set)) by A3,A6; hence thesis by A2,A7; end; end; theorem Th29: <*>Vars in QuasiLoci proof reconsider p = <*>Vars as FinSequence of Vars; p is one-to-one & for i st i in dom p holds (p.i)`1 c= rng (p dom i); hence thesis by Def3; end; registration cluster QuasiLoci -> non empty; correctness by Th29; end; definition mode quasi-loci is Element of QuasiLoci; end; registration cluster -> one-to-one for quasi-loci; coherence by Def3; end; theorem Th30: for l being one-to-one FinSequence of Vars holds l is quasi-loci iff for i being Nat, x being variable st i in dom l & x = l.i for y being variable st y in vars x ex j being Nat st j in dom l & j < i & y = l.j proof let l be one-to-one FinSequence of Vars; thus now assume A1: l is quasi-loci; let i be Nat, x be variable such that A2: i in dom l and A3: x = l.i; let y be variable such that A4: y in vars x; vars x c= rng (l|(i qua set)) by A1,A2,A3,Def3; then consider z being object such that A5: z in dom (l dom i) and A6: y = (l dom i).z by A4,FUNCT_1:def 3; A7: dom (l dom i) = dom l /\ i by RELAT_1:61; reconsider z as Element of NAT by A5,A7; reconsider j = z as Nat; take j; A8: card Segm z = z; card Segm i = i; hence j in dom l & j < i & y = l.j by A5,A6,A7,A8,FUNCT_1:47,NAT_1:41 ,XBOOLE_0:def 4; end; assume A9: for i being Nat, x being variable st i in dom l & x = l.i for y being variable st y in vars x ex j being Nat st j in dom l & j < i & y = l.j; now let i; assume A10: i in dom l; then l.i in rng l by FUNCT_1:def 3; then reconsider x = l.i as variable; thus (l.i)`1 c= rng (l dom i) proof let y be object; assume y in (l.i)`1; then A11: y in vars x; then reconsider y as variable; consider j being Nat such that A12: j in dom l and A13: j < i and A14: y = l.j by A9,A10,A11; A15: card Segm i = i; card Segm j = j; then j in i by A13,A15,NAT_1:41; hence thesis by A12,A14,FUNCT_1:50; end; end; hence thesis by Def3; end; theorem Th31: for l being quasi-loci, x being variable holds l^<*x*> is quasi-loci iff not x in rng l & vars x c= rng l proof let l be quasi-loci, x be variable; A1: (l^<*x*>).(1+len l) = x by FINSEQ_1:42; A2: dom (l^<*x*>) = Seg (len l + len <*x*>) by FINSEQ_1:def 7 .= Seg (len l + 1) by FINSEQ_1:39; 1 <= 1+len l by NAT_1:11; then A3: 1+len l in dom (l^<*x*>) by A2; A4: dom l = Seg len l by FINSEQ_1:def 3; thus now assume A5: l^<*x*> is quasi-loci; thus not x in rng l proof assume x in rng l; then consider a being object such that A6: a in dom l and A7: x = l.a by FUNCT_1:def 3; reconsider a as Element of NAT by A6; A8: (l^<*x*>).a = x by A6,A7,FINSEQ_1:def 7; A9: a <= len l by A4,A6,FINSEQ_1:1; A10: len l < 1+len l by NAT_1:13; dom l c= dom (l^<*x*>) by FINSEQ_1:26; hence thesis by A1,A3,A5,A6,A8,A9,A10,FUNCT_1:def 4; end; thus vars x c= rng l proof let a be object; assume A11: a in vars x; then reconsider a as variable; consider j being Nat such that A12: j in dom (l^<*x*>) and A13: j < 1+len l and A14: a = (l^<*x*>).j by A1,A3,A5,A11,Th30; reconsider j as Element of NAT by ORDINAL1:def 12; A15: j <= len l by A13,NAT_1:13; j >= 1 by A2,A12,FINSEQ_1:1; then A16: j in dom l by A4,A15; then a = l.j by A14,FINSEQ_1:def 7; hence thesis by A16,FUNCT_1:def 3; end; end; assume that A17: not x in rng l and A18: vars x c= rng l; A19: (l^<*x*>) is one-to-one proof let a,b be object; assume that A20: a in dom (l^<*x*>) and A21: b in dom (l^<*x*>) and A22: (l^<*x*>).a = (l^<*x*>).b; reconsider a,b as Element of NAT by A20,A21; A23: a >= 1 by A2,A20,FINSEQ_1:1; A24: b >= 1 by A2,A21,FINSEQ_1:1; A25: a <= 1+len l by A2,A20,FINSEQ_1:1; A26: b <= 1+len l by A2,A21,FINSEQ_1:1; A27: a <= len l or a = 1+len l by A25,NAT_1:8; A28: b <= len l or b = 1+len l by A26,NAT_1:8; A29: a in dom l or a = 1+len l by A4,A23,A27; A30: b in dom l or b = 1+len l by A4,A24,A28; A31: a in dom l & l.a = (l^<*x*>).a & l.a in rng l or a = 1+len l by A29, FINSEQ_1:def 7,FUNCT_1:def 3; b in dom l & l.b = (l^<*x*>).b & l.b in rng l or b = 1+len l by A30, FINSEQ_1:def 7,FUNCT_1:def 3; hence thesis by A17,A22,A31,FINSEQ_1:42,FUNCT_1:def 4; end; now let i be Nat, z be variable; assume that A32: i in dom (l^<*x*>) and A33: z = (l^<*x*>).i; A34: i >= 1 by A2,A32,FINSEQ_1:1; i <= 1+len l by A2,A32,FINSEQ_1:1; then i <= len l or i = 1+len l by NAT_1:8; then A35: i in dom l or i = 1+len l & z = x by A4,A33,A34,FINSEQ_1:42; let y be variable; assume A36: y in vars z; thus ex j being Nat st j in dom (l^<*x*>) & j < i & y = (l^<*x*>).j proof per cases by A33,A35,FINSEQ_1:def 7; suppose A37: i = 1+len l & z = x; then consider k being object such that A38: k in dom l and A39: y = l.k by A18,A36,FUNCT_1:def 3; reconsider k as Element of NAT by A38; take k; A40: dom l c= dom (l^<*x*>) by FINSEQ_1:26; k <= len l by A4,A38,FINSEQ_1:1; hence thesis by A37,A38,A39,A40,FINSEQ_1:def 7,NAT_1:13; end; suppose i in dom l & z = l.i; then consider j being Nat such that A41: j in dom l and A42: j < i and A43: y = l.j by A36,Th30; take j; dom l c= dom (l^<*x*>) by FINSEQ_1:26; hence thesis by A41,A42,A43,FINSEQ_1:def 7; end; end; end; hence thesis by A19,Th30; end; theorem Th32: for p,q being FinSequence st p^q is quasi-loci holds p is quasi-loci & q is FinSequence of Vars proof let p,q be FinSequence; assume A1: p^q is quasi-loci; then A2: p is one-to-one FinSequence of Vars by FINSEQ_1:36,FINSEQ_3:91; now let i be Nat, x be variable such that A3: i in dom p and A4: x = p.i; let y be variable such that A5: y in vars x; A6: dom p c= dom (p^q) by FINSEQ_1:26; x = (p^q).i by A3,A4,FINSEQ_1:def 7; then consider j being Nat such that A7: j in dom (p^q) and A8: j < i and A9: y = (p^q).j by A1,A3,A5,A6,Th30; take j; A10: dom p = Seg len p by FINSEQ_1:def 3; dom (p^q) = Seg len (p^q) by FINSEQ_1:def 3; then A11: j >= 1 by A7,FINSEQ_1:1; i <= len p by A3,A10,FINSEQ_1:1; then j < len p by A8,XXREAL_0:2; hence j in dom p & j < i by A8,A10,A11; hence y = p.j by A9,FINSEQ_1:def 7; end; hence thesis by A1,A2,Th30,FINSEQ_1:36; end; theorem for l being quasi-loci holds varcl rng l = rng l proof let l be quasi-loci; now let x,y; assume A1: [x,y] in rng l; then reconsider xy = [x,y] as variable; consider i being object such that A2: i in dom l and A3: xy = l.i by A1,FUNCT_1:def 3; reconsider i as Nat by A2; A4: vars xy = x; thus x c= rng l proof let a be object; assume A5: a in x; then reconsider a as variable by A4; ex j being Nat st j in dom l & j < i & a = l.j by A2,A3,A4,A5,Th30; hence thesis by FUNCT_1:def 3; end; end; hence varcl rng l c= rng l by Def1; thus thesis by Def1; end; theorem Th34: for x being variable holds <*x*> is quasi-loci iff vars x = {} proof let x be variable; A1: <*x*> = (<*>Vars)^<*x*> by FINSEQ_1:34; A2: rng {} = {}; vars x c= {} implies vars x = {}; hence thesis by A1,A2,Th29,Th31; end; theorem Th35: for x,y being variable holds <*x,y*> is quasi-loci iff vars x = {} & x <> y & vars y c= {x} proof let x,y be variable; A1: rng <*x*> = {x} by FINSEQ_1:38; A2: <*x*> is quasi-loci iff vars x = {} by Th34; y in {x} iff y = x by TARSKI:def 1; hence thesis by A1,A2,Th31,Th32; end; theorem for x,y,z being variable holds <*x,y,z*> is quasi-loci iff vars x = {} & x <> y & vars y c= {x} & x <> z & y <> z & vars z c= {x,y} proof let x,y,z be variable; A1: rng <*x,y*> = {x,y} by FINSEQ_2:127; A2: <*x,y*> is quasi-loci iff vars x = {} & x <> y & vars y c= {x} by Th35; z in {x,y} iff z = x or z = y by TARSKI:def 2; hence thesis by A1,A2,Th31,Th32; end; definition let l be quasi-loci; redefine func l" -> PartFunc of Vars, NAT; coherence proof A1: dom (l") = rng l by FUNCT_1:33; rng (l") = dom l by FUNCT_1:33; hence thesis by A1,RELSET_1:4; end; end; begin :: Mizar Constructor Signature definition func a_Type -> set equals 0; coherence; func an_Adj -> set equals 1; coherence; func a_Term -> set equals 2; coherence; func * -> set equals 0; coherence; func non_op -> set equals 1; coherence; :: func an_ExReg equals 3; coherence; :: func a_CondReg equals 4; coherence; :: func a_FuncReg equals 5; coherence; end; definition let C be Signature; attr C is constructor means : Def9: the carrier of C = {a_Type, an_Adj, a_Term} & {*, non_op} c= the carrier' of C & (the Arity of C).* = <*an_Adj, a_Type*> & (the Arity of C).non_op = <*an_Adj*> & (the ResultSort of C).* = a_Type & (the ResultSort of C).non_op = an_Adj & for o being Element of the carrier' of C st o <> * & o <> non_op holds (the Arity of C).o in {a_Term}*; end; registration cluster constructor -> non empty non void for Signature; coherence; end; definition func MinConstrSign -> strict Signature means : Def10: it is constructor & the carrier' of it = {*, non_op}; existence proof set A = {a_Type, an_Adj, a_Term}; reconsider t = a_Type, a = an_Adj as Element of A by ENUMSET1:def 1; reconsider aa = <*a*> as Element of A*; set C = ManySortedSign(# A, {*, non_op}, (*, non_op) --> (<*a,t*>, aa), (*, non_op) --> (t, a) #); reconsider C as non void non empty strict ManySortedSign; take C; thus the carrier of C = {a_Type, an_Adj, a_Term} & {*, non_op} c= the carrier' of C; thus (the Arity of C).* = <*an_Adj, a_Type*> by FUNCT_4:63; thus (the Arity of C).non_op = <*an_Adj*> by FUNCT_4:63; thus (the ResultSort of C).* = a_Type by FUNCT_4:63; thus (the ResultSort of C).non_op = an_Adj by FUNCT_4:63; thus thesis by TARSKI:def 2; end; correctness proof let C1, C2 be strict Signature such that A1: C1 is constructor and A2: the carrier' of C1 = {*, non_op} and A3: C2 is constructor and A4: the carrier' of C2 = {*, non_op}; set A = {a_Type, an_Adj, a_Term}; A5: the carrier of C1 = A by A1; A6: the carrier of C2 = A by A3; A7: (the Arity of C1).* = <*an_Adj, a_Type*> by A1; A8: (the Arity of C2).* = <*an_Adj, a_Type*> by A3; A9: (the Arity of C1).non_op = <*an_Adj*> by A1; A10: (the Arity of C2).non_op = <*an_Adj*> by A3; A11: (the ResultSort of C1).* = a_Type by A1; A12: (the ResultSort of C2).* = a_Type by A3; A13: (the ResultSort of C1).non_op = an_Adj by A1; A14: (the ResultSort of C2).non_op = an_Adj by A3; A15: dom the Arity of C1 = {*, non_op} by A2,FUNCT_2:def 1; A16: dom the Arity of C2 = {*, non_op} by A4,FUNCT_2:def 1; A17: the Arity of C1 = (*, non_op) --> (<*an_Adj, a_Type*>, <*an_Adj*>) by A7 ,A9,A15,FUNCT_4:66; A18: the Arity of C2 = (*, non_op) --> (<*an_Adj, a_Type*>, <*an_Adj*>) by A8 ,A10,A16,FUNCT_4:66; A19: dom the ResultSort of C1 = {*, non_op} by A1,A2,FUNCT_2:def 1; A20: dom the ResultSort of C2 = {*, non_op} by A3,A4,FUNCT_2:def 1; the ResultSort of C1 = (*, non_op) --> (a_Type, an_Adj) by A11,A13,A19, FUNCT_4:66; hence thesis by A2,A4,A5,A6,A12,A14,A17,A18,A20,FUNCT_4:66; end; end; registration cluster MinConstrSign -> constructor; coherence by Def10; end; registration cluster constructor strict for Signature; existence proof take MinConstrSign; thus thesis; end; end; definition mode ConstructorSignature is constructor Signature; end; :: theorem ::? :: for C being ConstructorSignature holds the carrier of C = 3 :: by CONSTRSIGN,YELLOW11:1; definition let C be ConstructorSignature; let o be OperSymbol of C; attr o is constructor means : Def11: o <> * & o <> non_op; end; theorem for S being ConstructorSignature for o being OperSymbol of S st o is constructor holds the_arity_of o = (len the_arity_of o) |-> a_Term proof let S be ConstructorSignature; let o be OperSymbol of S such that A1: o <> * and A2: o <> non_op; reconsider t = a_Term as Element of {a_Term} by TARSKI:def 1; A3: len ((len the_arity_of o)|->a_Term) = len the_arity_of o by CARD_1:def 7; A4: the_arity_of o in {a_Term}* by A1,A2,Def9; (len the_arity_of o)|->t in {a_Term}* by FINSEQ_1:def 11; hence thesis by A3,A4,Th6; end; definition let C be non empty non void Signature; attr C is initialized means : Def12: ex m, a being OperSymbol of C st the_result_sort_of m = a_Type & the_arity_of m = {} & :: set the_result_sort_of a = an_Adj & the_arity_of a = {}; :: empty end; definition let C be ConstructorSignature; A1: the carrier of C = {a_Type, an_Adj, a_Term} by Def9; func a_Type C -> SortSymbol of C equals a_Type; coherence by A1,ENUMSET1:def 1; func an_Adj C -> SortSymbol of C equals an_Adj; coherence by A1,ENUMSET1:def 1; func a_Term C -> SortSymbol of C equals a_Term; coherence by A1,ENUMSET1:def 1; A2: {*, non_op} c= the carrier' of C by Def9; A3: * in {*, non_op} by TARSKI:def 2; A4: non_op in {*, non_op} by TARSKI:def 2; func non_op C -> OperSymbol of C equals non_op; coherence by A2,A4; func ast C -> OperSymbol of C equals *; coherence by A2,A3; end; theorem for C being ConstructorSignature holds the_arity_of non_op C = <*an_Adj C*> & the_result_sort_of non_op C = an_Adj C & the_arity_of ast C = <*an_Adj C, a_Type C*> & the_result_sort_of ast C = a_Type C by Def9; definition func Modes -> set equals [:{a_Type},[:QuasiLoci,NAT:]:]; correctness; func Attrs -> set equals [:{an_Adj},[:QuasiLoci,NAT:]:]; correctness; func Funcs -> set equals [:{a_Term},[:QuasiLoci,NAT:]:]; correctness; end; registration cluster Modes -> non empty; coherence; cluster Attrs -> non empty; coherence; cluster Funcs -> non empty; coherence; end; definition func Constructors -> non empty set equals Modes \/ Attrs \/ Funcs; coherence; end; theorem {*, non_op} misses Constructors proof assume not thesis; then consider x being object such that A1: x in {*, non_op} and A2: x in Constructors by XBOOLE_0:3; x in Modes \/ Attrs or x in Funcs by A2,XBOOLE_0:def 3; then x in Modes or x in Attrs or x in Funcs by XBOOLE_0:def 3; then consider Y,Z being set such that A3: x in [:Y,Z:]; A4: ex y,z being object st ( y in Y)&( z in Z)&( [y,z] = x) by A3,ZFMISC_1:def 2; reconsider x as set by TARSKI:1; x = * or x = non_op by A1,TARSKI:def 2; then the_rank_of x = 0 or the_rank_of x = 1 by CLASSES1:73; then the_rank_of x c= 1; then the_rank_of x in succ succ {} by ORDINAL1:6,12; then x in Rank succ succ {} by CLASSES1:66; hence thesis by A4,CLASSES1:29,45; end; definition let x be Element of [:QuasiLoci, NAT:]; redefine func x`1 -> quasi-loci; coherence by MCART_1:10; redefine func x`2 -> Element of NAT; coherence by MCART_1:10; end; notation let c be Element of Constructors; synonym kind_of c for c`1; end; definition let c be Element of Constructors; redefine func kind_of c -> Element of {a_Type, an_Adj, a_Term}; coherence proof c in Modes \/ Attrs or c in Funcs by XBOOLE_0:def 3; then c in Modes or c in Attrs or c in Funcs by XBOOLE_0:def 3; then c`1 in {a_Type} or c`1 in {an_Adj} or c`1 in {a_Term} by MCART_1:10; then c`1 = a_Type or c`1 = an_Adj or c`1 = a_Term by TARSKI:def 1; hence thesis by ENUMSET1:def 1; end; redefine func c`2 -> Element of [:QuasiLoci, NAT:]; coherence proof c in Modes \/ Attrs or c in Funcs by XBOOLE_0:def 3; then c in Modes or c in Attrs or c in Funcs by XBOOLE_0:def 3; hence thesis by MCART_1:10; end; end; definition let c be Element of Constructors; func loci_of c -> quasi-loci equals c`2`1; coherence; func index_of c -> Nat equals c`2`2; coherence; end; theorem for c being Element of Constructors holds (kind_of c = a_Type iff c in Modes) & (kind_of c = an_Adj iff c in Attrs) & (kind_of c = a_Term iff c in Funcs) proof let x be Element of Constructors; A1: x in Modes \/ Attrs or x in Funcs by XBOOLE_0:def 3; A2: x in Modes implies x`1 in {a_Type} by MCART_1:10; A3: x in Attrs implies x`1 in {an_Adj} by MCART_1:10; x in Funcs implies x`1 in {a_Term} by MCART_1:10; hence thesis by A1,A2,A3,TARSKI:def 1,XBOOLE_0:def 3; end; definition func MaxConstrSign -> strict ConstructorSignature means : Def24: the carrier' of it = {*, non_op} \/ Constructors & for o being OperSymbol of it st o is constructor holds (the ResultSort of it).o = o`1 & card ((the Arity of it).o) = card o`2`1; existence proof set S = {a_Type, an_Adj, a_Term}; set O = {*, non_op} \/ Constructors; deffunc F(Element of Constructors) = (len loci_of $1)|->a_Term; consider f being ManySortedSet of Constructors such that A1: for c being Element of Constructors holds f.c = F(c) from PBOOLE:sch 5; deffunc G(Element of Constructors) = kind_of $1; consider g being ManySortedSet of Constructors such that A2: for c being Element of Constructors holds g.c = G(c) from PBOOLE:sch 5; reconsider t = a_Type, a = an_Adj, tr = a_Term as Element of S by ENUMSET1:def 1; reconsider aa = <*a*> as Element of S*; set A = f+*(*, non_op)-->(<*a,t*>, aa); set R = g+*(*, non_op)-->(t, a); A3: dom (*, non_op)-->(<*a,t*>, aa) = {*, non_op} by FUNCT_4:62; A4: dom (*, non_op)-->(t, a) = {*, non_op} by FUNCT_4:62; A5: dom f = Constructors by PARTFUN1:def 2; A6: dom g = Constructors by PARTFUN1:def 2; A7: dom A = O by A3,A5,FUNCT_4:def 1; A8: dom R = O by A4,A6,FUNCT_4:def 1; rng f c= S* proof let y be object; assume y in rng f; then consider x being object such that A9: x in Constructors and A10: y = f.x by A5,FUNCT_1:def 3; reconsider x as Element of Constructors by A9; y = (len loci_of x)|->tr by A1,A10; hence thesis by FINSEQ_1:def 11; end; then A11: rng f \/ rng (*, non_op)-->(<*a,t*>, aa) c= (S*) \/ (S*) by XBOOLE_1:13; rng g c= S proof let y be object; assume y in rng g; then consider x being object such that A12: x in Constructors and A13: y = g.x by A6,FUNCT_1:def 3; reconsider x as Element of Constructors by A12; y = kind_of x by A2,A13; hence thesis; end; then A14: rng g \/ rng (*, non_op)-->(t, a) c= S \/ S by XBOOLE_1:13; rng A c= rng f \/ rng (*, non_op)-->(<*a,t*>, aa) by FUNCT_4:17; then reconsider A as Function of O, S* by A7,A11,FUNCT_2:2,XBOOLE_1:1; rng R c= rng g \/ rng (*, non_op)-->(t, a) by FUNCT_4:17; then reconsider R as Function of O, S by A8,A14,FUNCT_2:2,XBOOLE_1:1; reconsider Max = ManySortedSign(# S, O, A, R #) as non empty non void strict Signature; Max is constructor proof thus the carrier of Max = {a_Type, an_Adj, a_Term}; thus {*, non_op} c= the carrier' of Max by XBOOLE_1:7; A15: * in {*, non_op} by TARSKI:def 2; A16: non_op in {*, non_op} by TARSKI:def 2; thus (the Arity of Max).* = ((*, non_op)-->(<*a,t*>, aa)).* by A3,A15,FUNCT_4:13 .= <*an_Adj, a_Type*> by FUNCT_4:63; thus (the Arity of Max).non_op = ((*, non_op)-->(<*a,t*>, aa)).non_op by A3,A16,FUNCT_4:13 .= <*an_Adj*> by FUNCT_4:63; thus (the ResultSort of Max).* = ((*, non_op)-->(t, a)).* by A4,A15,FUNCT_4:13 .= a_Type by FUNCT_4:63; thus (the ResultSort of Max).non_op = ((*, non_op)-->(t, a)).non_op by A4,A16,FUNCT_4:13 .= an_Adj by FUNCT_4:63; let o be Element of the carrier' of Max; assume that A17: o <> * and A18: o <> non_op; A19: not o in {*, non_op} by A17,A18,TARSKI:def 2; then reconsider c = o as Element of Constructors by XBOOLE_0:def 3; reconsider tr as Element of {a_Term} by TARSKI:def 1; (the Arity of Max).o = f.c by A3,A5,A19,FUNCT_4:def 1 .= (len loci_of c)|->tr by A1; hence (the Arity of Max).o in {a_Term}* by FINSEQ_1:def 11; end; then reconsider Max as strict ConstructorSignature; take Max; thus the carrier' of Max = {*, non_op} \/ Constructors; let o being OperSymbol of Max; assume that A20: o <> * and A21: o <> non_op; A22: not o in {*, non_op} by A20,A21,TARSKI:def 2; then reconsider c = o as Element of Constructors by XBOOLE_0:def 3; thus (the ResultSort of Max).o = g.c by A4,A6,A22,FUNCT_4:def 1 .= o`1 by A2; thus card ((the Arity of Max).o) = card (f.c) by A3,A5,A22,FUNCT_4:def 1 .= card F(c) by A1 .= card o`2`1 by CARD_1:def 7; end; uniqueness proof let it1, it2 be strict ConstructorSignature such that A23: the carrier' of it1 = {*, non_op} \/ Constructors and A24: for o being OperSymbol of it1 st o is constructor holds (the ResultSort of it1).o = o`1 & card ((the Arity of it1).o) = card o`2`1 and A25: the carrier' of it2 = {*, non_op} \/ Constructors and A26: for o being OperSymbol of it2 st o is constructor holds (the ResultSort of it2).o = o`1 & card ((the Arity of it2).o) = card o`2`1; set S = {a_Type, an_Adj, a_Term}; A27: the carrier of it1 = S by Def9; A28: the carrier of it2 = S by Def9; A29: now let c be Element of Constructors; reconsider o1 = c as OperSymbol of it1 by A23,XBOOLE_0:def 3; reconsider o2 = o1 as OperSymbol of it2 by A23,A25; assume that A30: c <> * and A31: c <> non_op; A32: o1 is constructor by A30,A31; A33: o2 is constructor by A30,A31; A34: card ((the Arity of it1).o1) = card c`2`1 by A24,A32; A35: card ((the Arity of it2).o2) = card c`2`1 by A26,A33; A36: (the Arity of it1).o1 in {a_Term}* by A30,A31,Def9; (the Arity of it2).o2 in {a_Term}* by A30,A31,Def9; then reconsider p1 = (the Arity of it1).o1, p2 = (the Arity of it2).o2 as FinSequence of {a_Term} by A36,FINSEQ_1:def 11; A37: dom p1 = Seg len p1 by FINSEQ_1:def 3; A38: dom p2 = Seg len p2 by FINSEQ_1:def 3; now let i be Nat; assume A39: i in dom p1; then A40: p1.i in rng p1 by FUNCT_1:def 3; A41: p2.i in rng p2 by A34,A35,A37,A38,A39,FUNCT_1:def 3; p1.i = a_Term by A40,TARSKI:def 1; hence p1.i = p2.i by A41,TARSKI:def 1; end; hence (the Arity of it1).c = (the Arity of it2).c by A34,A35,A37,A38; end; now let o be OperSymbol of it1; o in {*, non_op} or not o in {*, non_op}; then o = * or o = non_op or o in Constructors & o <> * & o <> non_op by A23,TARSKI:def 2,XBOOLE_0:def 3; then (the Arity of it1).o = <*an_Adj,a_Type*> & (the Arity of it2).o = <*an_Adj,a_Type*> or (the Arity of it1).o = <*an_Adj*> & (the Arity of it2).o = <*an_Adj*> or (the Arity of it1).o = (the Arity of it2).o by A29,Def9; hence (the Arity of it1).o = (the Arity of it2).o; end; then A42: the Arity of it1 = the Arity of it2 by A23,A25,A27,A28,FUNCT_2:63; now let o be OperSymbol of it1; reconsider o9 = o as OperSymbol of it2 by A23,A25; not o in {*, non_op} or o in {*,non_op}; then o = * or o = non_op or o in Constructors & o is constructor & o9 is constructor by A23,TARSKI:def 2,XBOOLE_0:def 3; then (the ResultSort of it1).o = a_Type & (the ResultSort of it2).o = a_Type or (the ResultSort of it1).o = an_Adj & (the ResultSort of it2).o = an_Adj or (the ResultSort of it1).o = o`1 & (the ResultSort of it2).o = o`1 by A24,A26,Def9; hence (the ResultSort of it1).o = (the ResultSort of it2).o; end; hence thesis by A23,A25,A27,A28,A42,FUNCT_2:63; end; end; registration cluster MinConstrSign -> non initialized; correctness proof given m, a being OperSymbol of MinConstrSign such that the_result_sort_of m = a_Type and A1: the_arity_of m = {} and the_result_sort_of a = an_Adj and the_arity_of a = {}; the carrier' of MinConstrSign = {*, non_op} by Def10; then m = * or m = non_op by TARSKI:def 2; hence contradiction by A1,Def9; end; cluster MaxConstrSign -> initialized; correctness proof set m = [a_Type, [{}, 0]], a = [an_Adj, [{}, 0]]; A2: a_Type in {a_Type} by TARSKI:def 1; A3: an_Adj in {an_Adj} by TARSKI:def 1; A4: [<*> Vars, 0] in [:QuasiLoci, NAT:] by Th29,ZFMISC_1:def 2; then A5: m in Modes by A2,ZFMISC_1:def 2; A6: a in Attrs by A3,A4,ZFMISC_1:def 2; A7: m in Modes \/ Attrs by A5,XBOOLE_0:def 3; A8: a in Modes \/ Attrs by A6,XBOOLE_0:def 3; A9: m in Constructors by A7,XBOOLE_0:def 3; A10: a in Constructors by A8,XBOOLE_0:def 3; the carrier' of MaxConstrSign = {*, non_op} \/ Constructors by Def24; then reconsider m,a as OperSymbol of MaxConstrSign by A9,A10,XBOOLE_0:def 3 ; A11: m is constructor; A12: a is constructor; take m, a; thus the_result_sort_of m = m`1 by A11,Def24 .= a_Type; len the_arity_of m = card m`2`1 by A11,Def24 .= card [{}, 0]`1 .= 0; hence the_arity_of m = {}; thus the_result_sort_of a = a`1 by A12,Def24 .= an_Adj; len the_arity_of a = card a`2`1 by A12,Def24 .= card [{}, 0]`1 .= 0; hence thesis; end; end; registration cluster initialized strict for ConstructorSignature; correctness proof take MaxConstrSign; thus thesis; end; end; registration let C be initialized ConstructorSignature; cluster constructor for OperSymbol of C; existence proof consider m, a being OperSymbol of C such that A1: the_result_sort_of m = a_Type and A2: the_arity_of m = {} and the_result_sort_of a = an_Adj and the_arity_of a = {} by Def12; take m; thus m <> * by A2,Def9; thus thesis by A1,Def9; end; end; begin :: Mizar Expressions definition let C be ConstructorSignature; A1: the carrier of C = {a_Type, an_Adj, a_Term} by Def9; func MSVars C -> ManySortedSet of the carrier of C means : Def25: it.a_Type = {} & it.an_Adj = {} & it.a_Term = Vars; uniqueness proof let V1,V2 be ManySortedSet of the carrier of C such that A2: V1.a_Type = {} and A3: V1.an_Adj = {} and A4: V1.a_Term = Vars and A5: V2.a_Type = {} and A6: V2.an_Adj = {} and A7: V2.a_Term = Vars; now let x be object; assume x in the carrier of C; then x = a_Type or x = an_Adj or x = a_Term by A1,ENUMSET1:def 1; hence V1.x = V2.x by A2,A3,A4,A5,A6,A7; end; hence thesis; end; existence proof deffunc F(object) = IFEQ($1, a_Term, Vars, {}); consider V being ManySortedSet of the carrier of C such that A8: for x being object st x in the carrier of C holds V.x = F(x) from PBOOLE:sch 4; take V; A9: IFEQ(a_Type, a_Term, Vars, {}) = {} by FUNCOP_1:def 8; A10: IFEQ(an_Adj, a_Term, Vars, {}) = {} by FUNCOP_1:def 8; A11: IFEQ(a_Term, a_Term, Vars, {}) = Vars by FUNCOP_1:def 8; A12: a_Type in the carrier of C by A1,ENUMSET1:def 1; A13: an_Adj in the carrier of C by A1,ENUMSET1:def 1; a_Term in the carrier of C by A1,ENUMSET1:def 1; hence thesis by A8,A9,A10,A11,A12,A13; end; end; :: theorem :: for C being ConstructorSignature :: for x being variable holds :: (C variables_in root-tree [x, a_Term]).a_Term C = {x} by MSAFREE3:11; registration let C be ConstructorSignature; cluster MSVars C -> non empty-yielding; coherence proof take a_Term; the carrier of C = {a_Type, an_Adj, a_Term} by Def9; hence a_Term in the carrier of C by ENUMSET1:def 1; thus thesis by Def25; end; end; registration let C be initialized ConstructorSignature; cluster Free(C, MSVars C) -> non-empty; correctness proof set X = MSVars C; consider m, a being OperSymbol of C such that A1: the_result_sort_of m = a_Type and A2: the_arity_of m = {} and A3: the_result_sort_of a = an_Adj and A4: the_arity_of a = {} by Def12; A5: root-tree [m, the carrier of C] in (the Sorts of Free(C, X)).a_Type by A1,A2,MSAFREE3:5; A6: root-tree [a, the carrier of C] in (the Sorts of Free(C, X)).an_Adj by A3,A4,MSAFREE3:5; set x = the variable; A7: a_Term C = a_Term; (MSVars C).a_Term = Vars by Def25; then A8: root-tree [x, a_Term] in (the Sorts of Free(C, X)).a_Term by A7,MSAFREE3:4; assume the Sorts of Free(C, X) is not non-empty; then {} in rng the Sorts of Free(C, X) by RELAT_1:def 9; then consider s being object such that A9: s in dom the Sorts of Free(C, X) and A10: {} = (the Sorts of Free(C, X)).s by FUNCT_1:def 3; s in the carrier of C by A9; then s in {a_Type, an_Adj, a_Term} by Def9; hence thesis by A5,A6,A8,A10,ENUMSET1:def 1; end; end; definition let S be non void Signature; let X be non empty-yielding ManySortedSet of the carrier of S; let t be Element of Free(S,X); attr t is ground means Union (S variables_in t) = {}; attr t is compound means : Def27: t.{} in [:the carrier' of S, {the carrier of S}:]; end; reserve C for initialized ConstructorSignature, s for SortSymbol of C, o for OperSymbol of C, c for constructor OperSymbol of C; definition let C; mode expression of C is Element of Free(C, MSVars C); end; definition let C, s; mode expression of C, s -> expression of C means : Def28: it in (the Sorts of Free(C, MSVars C)).s; existence proof set t = the Element of (the Sorts of Free(C, MSVars C)).s; dom the Sorts of Free(C, MSVars C) = the carrier of C by PARTFUN1:def 2; then t in Union the Sorts of Free(C, MSVars C) by CARD_5:2; hence thesis; end; end; theorem Th41: z is expression of C, s iff z in (the Sorts of Free(C, MSVars C)).s proof A1: dom the Sorts of Free(C, MSVars C) = the carrier of C by PARTFUN1:def 2; (the Sorts of Free(C, MSVars C)).s c= Union the Sorts of Free(C, MSVars C) by A1,CARD_5:2; hence thesis by Def28; end; definition let C; let c such that A1: len the_arity_of c = 0; func c term -> expression of C equals [c, the carrier of C]-tree {}; coherence proof the_arity_of c = {} by A1; then A2: root-tree [c, the carrier of C] in (the Sorts of Free(C, MSVars C)).the_result_sort_of c by MSAFREE3:5; dom the Sorts of Free(C, MSVars C) = the carrier of C by PARTFUN1:def 2; then root-tree [c, the carrier of C] in Union (the Sorts of Free(C, MSVars C)) by A2,CARD_5:2; hence thesis by TREES_4:20; end; end; theorem Th42: for o st len the_arity_of o = 1 for a being expression of C st ex s st s = (the_arity_of o).1 & a is expression of C, s holds [o, the carrier of C]-tree <*a*> is expression of C, the_result_sort_of o proof let o be OperSymbol of C such that A1: len the_arity_of o = 1; set X = MSVars C; set Y = X (\/) ((the carrier of C)-->{0}); let a be expression of C; given s being SortSymbol of C such that A2: s = (the_arity_of o).1 and A3: a is expression of C, s; reconsider ta = a as Term of C,Y by MSAFREE3:8; A4: dom <*ta*> = Seg 1 by FINSEQ_1:38; A5: dom <*s*> = Seg 1 by FINSEQ_1:38; A6: the_arity_of o = <*s*> by A1,A2,FINSEQ_1:40; A7: the Sorts of Free(C, X) = C-Terms(X, Y) by MSAFREE3:24; now let i be Nat; assume i in dom <*ta*>; then A8: i = 1 by A4,FINSEQ_1:2,TARSKI:def 1; let t be Term of C, Y; assume A9: t = <*ta*>.i; A10: the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by A7,PBOOLE:def 18; A11: t = a by A8,A9,FINSEQ_1:40; A12: (the Sorts of Free(C, X)).s c= (the Sorts of FreeMSA Y).s by A10; t in (the Sorts of Free(C, X)).s by A3,A11,Th41; hence the_sort_of t = (the_arity_of o).i by A2,A8,A12,MSAFREE3:7; end; then reconsider p = <*ta*> as ArgumentSeq of Sym(o, Y) by A4,A5,A6,MSATERM:25 ; A13: variables_in (Sym(o, Y)-tree p) c= X proof let s be object; assume s in the carrier of C; then reconsider s9 = s as SortSymbol of C; let x be object; assume x in (variables_in (Sym(o, Y)-tree p)).s; then consider t being DecoratedTree such that A14: t in rng p and A15: x in (C variables_in t).s9 by MSAFREE3:11; A16: C variables_in a c= X by MSAFREE3:27; A17: rng p = {a} by FINSEQ_1:38; A18: (C variables_in a).s9 c= X.s9 by A16; t = a by A14,A17,TARSKI:def 1; hence thesis by A15,A18; end; set s9 = the_result_sort_of o; A19: the_sort_of (Sym(o, Y)-tree p) = the_result_sort_of o by MSATERM:20; (the Sorts of Free(C, X)).s9 = {t where t is Term of C,Y: the_sort_of t = s9 & variables_in t c= X} by A7,MSAFREE3:def 5; then [o, the carrier of C]-tree <*a*> in (the Sorts of Free(C, X)).s9 by A13 ,A19; hence thesis by Th41; end; definition let C,o such that A1: len the_arity_of o = 1; let e be expression of C such that A2: ex s being SortSymbol of C st s = (the_arity_of o).1 & e is expression of C, s; func o term e -> expression of C equals : Def30: [o, the carrier of C]-tree<*e*>; coherence by A1,A2,Th42; end; reserve a,b for expression of C, an_Adj C; theorem Th43: (non_op C)term a is expression of C, an_Adj C & (non_op C)term a = [non_op, the carrier of C]-tree <*a*> proof A1: the_result_sort_of non_op C = an_Adj C by Def9; A2: the_arity_of non_op C = <*an_Adj C*> by Def9; then A3: len the_arity_of non_op C = 1 by FINSEQ_1:40; A4: (the_arity_of non_op C).1 = an_Adj C by A2,FINSEQ_1:40; then (non_op C)term a = [non_op, the carrier of C]-tree <*a*> by A3,Def30; hence thesis by A1,A3,A4,Th42; end; theorem Th44: (non_op C)term a = (non_op C)term b implies a = b proof assume (non_op C)term a = (non_op C)term b; then [non_op, the carrier of C]-tree <*a*> = (non_op C)term b by Th43 .= [non_op, the carrier of C]-tree <*b*> by Th43; then <*a*> = <*b*> by TREES_4:15; hence thesis by FINSEQ_1:76; end; registration let C,a; cluster (non_op C)term a -> compound; coherence proof (non_op C)term a = [non_op, the carrier of C]-tree <*a*> by Th43; then ((non_op C)term a).{} = [non_op C, the carrier of C] by TREES_4:def 4; hence ((non_op C)term a).{} in [:the carrier' of C, {the carrier of C}:] by ZFMISC_1:106; end; end; registration let C; cluster compound for expression of C; existence proof set a = the expression of C, an_Adj C; (non_op C)term a is compound; hence thesis; end; end; theorem Th45: for o st len the_arity_of o = 2 for a,b being expression of C st ex s1,s2 being SortSymbol of C st s1 = (the_arity_of o).1 & s2 = (the_arity_of o).2 & a is expression of C, s1 & b is expression of C, s2 holds [o, the carrier of C]-tree <*a,b*> is expression of C, the_result_sort_of o proof let o be OperSymbol of C such that A1: len the_arity_of o = 2; set X = MSVars C; set Y = X (\/) ((the carrier of C)-->{0}); let a,b be expression of C; given s1,s2 being SortSymbol of C such that A2: s1 = (the_arity_of o).1 and A3: s2 = (the_arity_of o).2 and A4: a is expression of C, s1 and A5: b is expression of C, s2; reconsider ta = a, tb = b as Term of C,Y by MSAFREE3:8; A6: dom <*ta,tb*> = Seg 2 by FINSEQ_1:89; A7: dom <*s1,s2*> = Seg 2 by FINSEQ_1:89; A8: the_arity_of o = <*s1,s2*> by A1,A2,A3,FINSEQ_1:44; A9: the Sorts of Free(C, X) = C-Terms(X, Y) by MSAFREE3:24; now let i be Nat; assume i in dom <*ta,tb*>; then A10: i = 1 or i = 2 by A6,FINSEQ_1:2,TARSKI:def 2; let t be Term of C, Y; assume A11: t = <*ta,tb*>.i; A12: the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by A9,PBOOLE:def 18; A13: i = 1 & t = a or i = 2 & t = b by A10,A11,FINSEQ_1:44; A14: (the Sorts of Free(C, X)).s1 c= (the Sorts of FreeMSA Y).s1 by A12; A15: (the Sorts of Free(C, X)).s2 c= (the Sorts of FreeMSA Y).s2 by A12; i = 1 & t in (the Sorts of Free(C, X)).s1 or i = 2 & t in (the Sorts of Free(C, X)).s2 by A4,A5,A13,Th41; hence the_sort_of t = (the_arity_of o).i by A2,A3,A14,A15,MSAFREE3:7; end; then reconsider p = <*ta,tb*> as ArgumentSeq of Sym(o, Y) by A6,A7,A8, MSATERM:25; A16: variables_in (Sym(o, Y)-tree p) c= X proof let s be object; assume s in the carrier of C; then reconsider s9 = s as SortSymbol of C; let x be object; assume x in (variables_in (Sym(o, Y)-tree p)).s; then consider t being DecoratedTree such that A17: t in rng p and A18: x in (C variables_in t).s9 by MSAFREE3:11; A19: C variables_in a c= X by MSAFREE3:27; A20: C variables_in b c= X by MSAFREE3:27; A21: rng p = {a,b} by FINSEQ_2:127; A22: (C variables_in a).s9 c= X.s9 by A19; A23: (C variables_in b).s9 c= X.s9 by A20; t = a or t = b by A17,A21,TARSKI:def 2; hence thesis by A18,A22,A23; end; set s9 = the_result_sort_of o; A24: the_sort_of (Sym(o, Y)-tree p) = the_result_sort_of o by MSATERM:20; (the Sorts of Free(C, X)).s9 = {t where t is Term of C,Y: the_sort_of t = s9 & variables_in t c= X} by A9,MSAFREE3:def 5; then [o, the carrier of C]-tree <*a,b*> in (the Sorts of Free(C, X)).s9 by A16,A24; hence thesis by Th41; end; definition let C,o such that A1: len the_arity_of o = 2; let e1,e2 be expression of C such that A2: ex s1,s2 being SortSymbol of C st s1 = (the_arity_of o).1 & s2 = (the_arity_of o).2 & e1 is expression of C, s1 & e2 is expression of C, s2; func o term(e1,e2) -> expression of C equals : Def31: [o, the carrier of C]-tree<*e1,e2*>; coherence by A1,A2,Th45; end; reserve t, t1,t2 for expression of C, a_Type C; theorem Th46: (ast C)term(a,t) is expression of C, a_Type C & (ast C)term(a,t) = [ *, the carrier of C]-tree <*a,t*> proof A1: the_result_sort_of ast C = a_Type C by Def9; A2: the_arity_of ast C = <*an_Adj C, a_Type C*> by Def9; then A3: len the_arity_of ast C = 2 by FINSEQ_1:44; A4: (the_arity_of ast C).1 = an_Adj C by A2,FINSEQ_1:44; A5: (the_arity_of ast C).2 = a_Type C by A2,FINSEQ_1:44; then (ast C)term(a,t) = [ *, the carrier of C]-tree <*a,t*> by A3,A4,Def31; hence thesis by A1,A3,A4,A5,Th45; end; theorem (ast C)term(a,t1) = (ast C)term(b,t2) implies a = b & t1 = t2 proof assume (ast C)term(a,t1) = (ast C)term(b,t2); then [ *, the carrier of C]-tree<*a,t1*> = (ast C)term(b,t2) by Th46 .= [ *, the carrier of C]-tree<*b,t2*> by Th46; then <*a,t1*> = <*b,t2*> by TREES_4:15; hence thesis by FINSEQ_1:77; end; registration let C,a,t; cluster (ast C)term(a,t) -> compound; coherence proof (ast C)term(a,t) = [ *, the carrier of C]-tree <*a,t*> by Th46; then ((ast C)term(a,t)).{} = [ast C, the carrier of C] by TREES_4:def 4; hence ((ast C)term(a,t)).{} in [:the carrier' of C, {the carrier of C}:] by ZFMISC_1:106; end; end; definition let S be non void Signature; let s be SortSymbol of S such that A1: ex o being OperSymbol of S st the_result_sort_of o = s; mode OperSymbol of s -> OperSymbol of S means the_result_sort_of it = s; existence by A1; end; definition let C be ConstructorSignature; redefine func non_op C -> OperSymbol of an_Adj C; coherence proof the_result_sort_of non_op C = an_Adj C by Def9; hence ex o being OperSymbol of C st the_result_sort_of o = an_Adj C; thus thesis by Def9; end; redefine func ast C -> OperSymbol of a_Type C; coherence proof the_result_sort_of ast C = a_Type C by Def9; hence ex o being OperSymbol of C st the_result_sort_of o = a_Type C; thus thesis by Def9; end; end; theorem Th48: for s1,s2 being SortSymbol of C st s1 <> s2 for t1 being expression of C, s1 for t2 being expression of C, s2 holds t1 <> t2 proof set X = MSVars C; set Y = X (\/) ((the carrier of C) --> {0}); A1: ex A being MSSubset of FreeMSA Y st ( Free(C, X) = GenMSAlg A)&( A = (Reverse Y)""X) by MSAFREE3:def 1; let s1,s2 be SortSymbol of C; the Sorts of Free(C, X) is MSSubset of FreeMSA Y by A1,MSUALG_2:def 9; then A2: the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by PBOOLE:def 18; then A3: (the Sorts of Free(C,X)).s1 c= (the Sorts of FreeMSA Y).s1; A4: (the Sorts of Free(C,X)).s2 c= (the Sorts of FreeMSA Y).s2 by A2; assume s1 <> s2; then A5: (the Sorts of FreeMSA Y).s1 misses (the Sorts of FreeMSA Y).s2 by PROB_2:def 2; let t1 be expression of C, s1; let t2 be expression of C, s2; A6: t1 in (the Sorts of Free(C,X)).s1 by Def28; t2 in (the Sorts of Free(C,X)).s2 by Def28; hence thesis by A3,A4,A5,A6,XBOOLE_0:3; end; begin :: Quasi-terms definition let C; A1: (the Sorts of Free(C, MSVars C)).a_Term C c= Union the Sorts of Free(C, MSVars C) proof let x be object; dom the Sorts of Free(C, MSVars C) = the carrier of C by PARTFUN1:def 2; hence thesis by CARD_5:2; end; func QuasiTerms C -> Subset of Free(C, MSVars C) equals (the Sorts of Free(C, MSVars C)).a_Term C; coherence by A1; end; registration let C; cluster QuasiTerms C -> non empty constituted-DTrees; coherence; end; definition let C; mode quasi-term of C is expression of C, a_Term C; end; theorem z is quasi-term of C iff z in QuasiTerms C by Th41; definition let x be variable; let C; func x-term C -> quasi-term of C equals root-tree [x, a_Term]; coherence proof (MSVars C).a_Term = Vars by Def25; then root-tree [x, a_Term] in QuasiTerms C by MSAFREE3:4; hence thesis by Th41; end; end; theorem Th50: for x1,x2 being variable for C1,C2 being initialized ConstructorSignature st x1-term C1 = x2-term C2 holds x1 = x2 proof let x1,x2 be variable; let C1,C2 be initialized ConstructorSignature; assume x1-term C1 = x2-term C2; then [x1, a_Term] = [x2, a_Term] by TREES_4:4; hence thesis by XTUPLE_0:1; end; registration let x be variable; let C; cluster x-term C -> non compound; coherence proof a_Term C in the carrier of C; then A1: a_Term C <> the carrier of C; A2: (x-term C).{} = [x, a_Term C] by TREES_4:3; a_Term C nin {the carrier of C} by A1,TARSKI:def 1; hence (x-term C).{} nin [:the carrier' of C, {the carrier of C}:] by A2,ZFMISC_1:87; end; end; theorem Th51: for p being DTree-yielding FinSequence holds [c, the carrier of C]-tree p is expression of C iff len p = len the_arity_of c & p in (QuasiTerms C)* proof set o = c; A1: o <> * by Def11; A2: o <> non_op by Def11; let p be DTree-yielding FinSequence; set V = (MSVars C) (\/) ((the carrier of C) --> {0}); A3: the Sorts of Free(C, MSVars C) = C-Terms(MSVars C, V) by MSAFREE3:24; thus now assume A4: [o, the carrier of C]-tree p is expression of C; then A5: [o, the carrier of C]-tree p is Term of C, V by MSAFREE3:8; then A6: p is ArgumentSeq of Sym(o,V) by MSATERM:1; hence len p = len the_arity_of o by MSATERM:22; reconsider q = p as ArgumentSeq of Sym(o,V) by A5,MSATERM:1; A7: the_sort_of ((Sym(o,V))-tree q) = the_result_sort_of o by MSATERM:20; A8: variables_in ((Sym(o,V))-tree q) c= MSVars C by A4,MSAFREE3:27; (C-Terms(MSVars C,V)).the_result_sort_of o = {t where t is Term of C,V: the_sort_of t = the_result_sort_of o & variables_in t c= MSVars C} by MSAFREE3:def 5; then Sym(o,V)-tree p in (C-Terms(MSVars C,V)).the_result_sort_of o by A7,A8; then A9: rng p c= Union (C-Terms(MSVars C,V)) by A6,MSAFREE3:19; rng p c= QuasiTerms C proof let a be object; assume A10: a in rng p; then reconsider ta = a as expression of C by A9,MSAFREE3:24; consider i being object such that A11: i in dom p and A12: a = p.i by A10,FUNCT_1:def 3; reconsider i as Nat by A11; reconsider t = p.i as Term of C, V by A6,A11,MSATERM:22; A13: (the Arity of C).o in {a_Term}* by A1,A2,Def9; A14: dom p = dom the_arity_of o by A6,MSATERM:22; A15: the_arity_of o is FinSequence of {a_Term} by A13,FINSEQ_1:def 11; A16: (the_arity_of o).i in rng the_arity_of o by A11,A14,FUNCT_1:def 3; rng the_arity_of o c= {a_Term C} by A15,FINSEQ_1:def 4; then (the_arity_of o).i = a_Term C by A16,TARSKI:def 1; then A17: the_sort_of t = a_Term C by A6,A11,MSATERM:23; t = ta by A12; then variables_in t c= MSVars C by MSAFREE3:27; then t in {T where T is Term of C,V: the_sort_of T = a_Term C & variables_in T c= MSVars C} by A17; then t in (C-Terms(MSVars C,V)).a_Term C by MSAFREE3:def 5; hence thesis by A12,MSAFREE3:23; end; then p is FinSequence of QuasiTerms C by FINSEQ_1:def 4; hence p in (QuasiTerms C)* by FINSEQ_1:def 11; end; assume A18: len p = len the_arity_of o; assume A19: p in (QuasiTerms C)*; Free(C, MSVars C) = (FreeMSA V)|(C-Terms(MSVars C, V)) by MSAFREE3:25; then the Sorts of Free(C, MSVars C) is ManySortedSubset of the Sorts of FreeMSA V by MSUALG_2:def 9; then the Sorts of Free(C, MSVars C) c= the Sorts of FreeMSA V by PBOOLE:def 18; then A20: QuasiTerms C c= (the Sorts of FreeMSA V).a_Term C; A21: p is FinSequence of QuasiTerms C by A19,FINSEQ_1:def 11; then A22: rng p c= QuasiTerms C by FINSEQ_1:def 4; now let i be Nat; assume A23: i in dom p; then p.i in rng p by FUNCT_1:def 3; then A24: p.i in QuasiTerms C by A22; then reconsider t = p.i as expression of C; A25: (the Arity of C).o in {a_Term}* by A1,A2,Def9; A26: dom p = dom the_arity_of o by A18,FINSEQ_3:29; A27: the_arity_of o is FinSequence of {a_Term} by A25,FINSEQ_1:def 11; A28: (the_arity_of o).i in rng the_arity_of o by A23,A26,FUNCT_1:def 3; rng the_arity_of o c= {a_Term C} by A27,FINSEQ_1:def 4; then A29: (the_arity_of o).i = a_Term C by A28,TARSKI:def 1; reconsider T = t as Term of C,V by MSAFREE3:8; take T; thus T = p.i; T in (the Sorts of FreeMSA V).a_Term C by A20,A24; then T in FreeSort(V, a_Term C) by MSAFREE:def 11; hence the_sort_of T = (the_arity_of o).i by A29,MSATERM:def 5; end; then A30: p is ArgumentSeq of Sym(o,V) by A18,MSATERM:24; A31: dom the Sorts of Free(C, MSVars C) = the carrier of C by PARTFUN1:def 2; rng p c= Union (C-Terms(MSVars C, V)) by A3,A21,FINSEQ_1:def 4; then Sym(o,V)-tree p in (C-Terms(MSVars C, V)).the_result_sort_of o by A30,MSAFREE3:19; hence thesis by A3,A31,CARD_5:2; end; reserve p for FinSequence of QuasiTerms C; definition let C,c; let p such that A1: len p = len the_arity_of c; A2: p in (QuasiTerms C)* by FINSEQ_1:def 11; func c-trm p -> compound expression of C equals : Def35: [c, the carrier of C]-tree p; coherence proof reconsider t = [c, the carrier of C]-tree p as expression of C by A1,A2 ,Th51; t.{} = [c, the carrier of C] by TREES_4:def 4; then t.{} in [:the carrier' of C, {the carrier of C}:] by ZFMISC_1:106; hence thesis by Def27; end; end; theorem Th52: len p = len the_arity_of c implies c-trm p is expression of C, the_result_sort_of c proof set X = MSVars C; set V = X(\/)((the carrier of C)-->{0}); assume len p = len the_arity_of c; then A1: Sym(c,V)-tree p = c-trm p by Def35; A2: the Sorts of Free(C,X) = C-Terms(X,V) by MSAFREE3:24; c-trm p is Term of C,V by MSAFREE3:8; then reconsider q = p as ArgumentSeq of Sym(c,V) by A1,MSATERM:1; rng q c= Union the Sorts of Free(C,X) by FINSEQ_1:def 4; then c-trm p in (C-Terms(X,V)).the_result_sort_of c by A1,A2,MSAFREE3:19; hence thesis by A2,Def28; end; theorem Th53: for e being expression of C holds (ex x being variable st e = x-term C) or (ex c being constructor OperSymbol of C st ex p being FinSequence of QuasiTerms C st len p = len the_arity_of c & e = c-trm p) or (ex a being expression of C, an_Adj C st e = (non_op C)term a) or ex a being expression of C, an_Adj C st ex t being expression of C, a_Type C st e = (ast C)term(a,t) proof let t be expression of C; set X = MSVars C; set V = X(\/)((the carrier of C)-->{0}); per cases by Th7; suppose ex s being SortSymbol of C, v being set st t = root-tree [v,s] & v in X.s; then consider s being SortSymbol of C, v being set such that A1: t = root-tree [v,s] and A2: v in X.s; the carrier of C = {a_Type, an_Adj, a_Term} by Def9; then A3: s = a_Term or s = an_Adj or s = a_Type by ENUMSET1:def 1; then reconsider v as variable by A2,Def25; t = v-term C by A1,A2,A3,Def25; hence thesis; end; suppose ex o being OperSymbol of C, p being FinSequence of Free(C,X) st t = [o,the carrier of C]-tree p & len p = len the_arity_of o & p is DTree-yielding & p is ArgumentSeq of Sym(o,V); then consider o being OperSymbol of C, p being FinSequence of Free(C,X) such that A4: t = [o, the carrier of C]-tree p and A5: len p = len the_arity_of o and p is DTree-yielding and A6: p is ArgumentSeq of Sym(o,V); per cases; suppose A7: o = *; then A8: the_arity_of o = <*an_Adj,a_Type*> by Def9; A9: dom p = dom the_arity_of o by A6,MSATERM:22; A10: dom the_arity_of o = Seg 2 by A8,FINSEQ_1:89; A11: len the_arity_of o = 2 by A8,FINSEQ_1:44; A12: 1 in Seg 2; A13: 2 in Seg 2; A14: p.1 in rng p by A9,A10,A12,FUNCT_1:3; p.2 in rng p by A9,A10,A13,FUNCT_1:3; then reconsider p1 = p.1, p2 = p.2 as expression of C by A14; reconsider t1 = p1, t2 = p2 as Term of C,V by MSAFREE3:8; A15: C variables_in p1 c= X by MSAFREE3:27; A16: variables_in t1 = C variables_in t1; A17: C variables_in p2 c= X by MSAFREE3:27; A18: variables_in t2 = C variables_in t2; A19: <*an_Adj,a_Type*>.2 = a_Type C by FINSEQ_1:44; A20: <*an_Adj,a_Type*>.1 = an_Adj C by FINSEQ_1:44; the_sort_of t1 = (the_arity_of o).1 by A6,A9,A10,A12,MSATERM:23; then t1 in {q where q is Term of C,V: the_sort_of q = an_Adj C & variables_in q c= X} by A8,A15,A16,A20; then p1 in C-Terms(X,V).an_Adj C by MSAFREE3:def 5; then p1 in (the Sorts of Free(C,X)).an_Adj C by MSAFREE3:24; then reconsider a = p1 as expression of C, an_Adj C by Def28; the_sort_of t2 = (the_arity_of o).2 by A6,A9,A10,A13,MSATERM:23; then t2 in {q where q is Term of C,V: the_sort_of q = a_Type C & variables_in q c= X} by A8,A17,A18,A19; then p2 in C-Terms(X,V).a_Type C by MSAFREE3:def 5; then p2 in (the Sorts of Free(C,X)).a_Type C by MSAFREE3:24; then reconsider q = p2 as expression of C, a_Type C by Def28; p = <*a,q*> by A5,A11,FINSEQ_1:44; then t = (ast C)term(a,q) by A4,A7,A8,A11,A19,A20,Def31; hence thesis; end; suppose A21: o = non_op; then A22: the_arity_of o = <*an_Adj*> by Def9; A23: dom p = dom the_arity_of o by A6,MSATERM:22; A24: dom the_arity_of o = Seg 1 by A22,FINSEQ_1:38; A25: len the_arity_of o = 1 by A22,FINSEQ_1:39; A26: 1 in Seg 1; then p.1 in rng p by A23,A24,FUNCT_1:3; then reconsider p1 = p.1 as expression of C; reconsider t1 = p1 as Term of C,V by MSAFREE3:8; A27: C variables_in p1 c= X by MSAFREE3:27; A28: variables_in t1 = C variables_in t1; A29: <*an_Adj*>.1 = an_Adj C by FINSEQ_1:40; the_sort_of t1 = (the_arity_of o).1 by A6,A23,A24,A26,MSATERM:23; then t1 in {q where q is Term of C,V: the_sort_of q = an_Adj C & variables_in q c= X} by A22,A27,A28,A29; then p1 in C-Terms(X,V).an_Adj C by MSAFREE3:def 5; then p1 in (the Sorts of Free(C,X)).an_Adj C by MSAFREE3:24; then reconsider a = p1 as expression of C, an_Adj C by Def28; p = <*a*> by A5,A25,FINSEQ_1:40; then t = (non_op C)term(a) by A4,A21,A22,A25,A29,Def30; hence thesis; end; suppose o is constructor; then reconsider o as constructor OperSymbol of C; t = [o, the carrier of C]-tree p by A4; then p in (QuasiTerms C)* by Th51; then reconsider p as FinSequence of QuasiTerms C by FINSEQ_1:def 11; t = o-trm p by A4,A5,Def35; hence thesis by A5; end; end; end; theorem Th54: len p = len the_arity_of c implies c-trm p <> (non_op C)term a proof assume len p = len the_arity_of c; then c-trm p = [c, the carrier of C]-tree p by Def35; then A1: (c-trm p).{} = [c, the carrier of C] by TREES_4:def 4; assume c-trm p = (non_op C)term a; then c-trm p = [non_op, the carrier of C]-tree<*a*> by Th43; then [c, the carrier of C] = [non_op, the carrier of C] by A1,TREES_4:def 4; then c = non_op by XTUPLE_0:1; hence thesis by Def11; end; theorem Th55: len p = len the_arity_of c implies c-trm p <> (ast C)term(a,t) proof assume len p = len the_arity_of c; then c-trm p = [c, the carrier of C]-tree p by Def35; then A1: (c-trm p).{} = [c, the carrier of C] by TREES_4:def 4; assume c-trm p = (ast C)term(a,t); then c-trm p = [ *, the carrier of C]-tree<*a,t*> by Th46; then [c, the carrier of C] = [ *, the carrier of C] by A1,TREES_4:def 4; then c = * by XTUPLE_0:1; hence thesis by Def11; end; theorem (non_op C)term a <> (ast C)term(b,t) proof assume (non_op C)term a = (ast C)term(b,t); then (non_op C)term a = [ *, the carrier of C]-tree<*b,t*> by Th46; then ((non_op C)term a).{} = [ *, the carrier of C] by TREES_4:def 4; then ([non_op,the carrier of C]-tree<*a*>).{} = [ *, the carrier of C] by Th43; then [non_op, the carrier of C] = [ *, the carrier of C] by TREES_4:def 4; hence thesis by XTUPLE_0:1; end; reserve e for expression of C; theorem Th57: e.{} = [non_op, the carrier of C] implies ex a st e = (non_op C)term a proof assume A1: e.{} = [non_op, the carrier of C]; non_op C in the carrier' of C; then A2: e.{} in [:the carrier' of C, {the carrier of C}:] by A1,ZFMISC_1:106; per cases by Th53; suppose ex x being variable st e = x-term C; hence thesis by A2,Def27; end; suppose ex c,p st len p = len the_arity_of c & e = c-trm p; then consider c being constructor OperSymbol of C, p being FinSequence of QuasiTerms C such that A3: len p = len the_arity_of c and A4: e = c-trm p; e = [c, the carrier of C]-tree p by A3,A4,Def35; then e.{} = [c, the carrier of C] by TREES_4:def 4; then non_op = c by A1,XTUPLE_0:1; hence thesis by Def11; end; suppose ex a st e = (non_op C)term a; hence thesis; end; suppose ex a,t st e = (ast C)term(a,t); then consider a,t such that A5: e = (ast C)term(a,t); e = [ *, the carrier of C]-tree <*a,t*> by A5,Th46; then e.{} = [ *, the carrier of C] by TREES_4:def 4; hence thesis by A1,XTUPLE_0:1; end; end; theorem Th58: e.{} = [ *, the carrier of C] implies ex a, t st e = (ast C)term(a,t) proof assume A1: e.{} = [ *, the carrier of C]; ast C in the carrier' of C; then A2: e.{} in [:the carrier' of C, {the carrier of C}:] by A1,ZFMISC_1:106; per cases by Th53; suppose ex x being variable st e = x-term C; hence thesis by A2,Def27; end; suppose ex c,p st len p = len the_arity_of c & e = c-trm p; then consider c being constructor OperSymbol of C, p being FinSequence of QuasiTerms C such that A3: len p = len the_arity_of c and A4: e = c-trm p; e = [c, the carrier of C]-tree p by A3,A4,Def35; then e.{} = [c, the carrier of C] by TREES_4:def 4; then * = c by A1,XTUPLE_0:1; hence thesis by Def11; end; suppose ex a being expression of C, an_Adj C st e = (non_op C)term a; then consider a being expression of C, an_Adj C such that A5: e = (non_op C)term a; e = [non_op, the carrier of C]-tree <*a*> by A5,Th43; then e.{} = [non_op, the carrier of C] by TREES_4:def 4; hence thesis by A1,XTUPLE_0:1; end; suppose ex a,t st e = (ast C)term(a,t); hence thesis; end; end; begin :: Quasi-adjectives reserve a,a9 for expression of C, an_Adj C; definition let C,a; func Non a -> expression of C, an_Adj C equals : Def36: a|<* 0 *> if ex a9 st a = (non_op C)term a9 otherwise (non_op C)term a; coherence proof thus now given a9 being expression of C, an_Adj C such that A1: a = (non_op C)term a9; A2: a = [non_op, the carrier of C]-tree <*a9*> by A1,Th43; len <*a9*> = 1 by FINSEQ_1:40; then a|<* 0*> = <*a9*>.(0+1) by A2,TREES_4:def 4; hence a|<* 0*> is expression of C, an_Adj C by FINSEQ_1:40; end; thus thesis by Th43; end; consistency; end; definition let C,a; attr a is positive means : Def37: not ex a9 st a = (non_op C)term a9; end; registration let C; cluster positive for expression of C, an_Adj C; existence proof consider m, a being OperSymbol of C such that the_result_sort_of m = a_Type and the_arity_of m = {} and A1: the_result_sort_of a = an_Adj and A2: the_arity_of a = {} by Def12; set X = MSVars C; root-tree [a, the carrier of C] in (the Sorts of Free(C, X)).an_Adj by A1 ,A2,MSAFREE3:5; then reconsider v = root-tree [a, the carrier of C] as expression of C, an_Adj C by Th41; take v; given a9 being expression of C, an_Adj C such that A3: v = (non_op C)term a9; v = [non_op, the carrier of C]-tree<*a9*> by A3,Th43; then [non_op, the carrier of C] = v.{} by TREES_4:def 4 .= [a, the carrier of C] by TREES_4:3; then a = non_op C by XTUPLE_0:1; hence contradiction by A2,Def9; end; end; theorem Th59: for a being positive expression of C, an_Adj C holds Non a = (non_op C)term a proof let a be positive expression of C, an_Adj C; not ex a9 being expression of C, an_Adj C st a = (non_op C)term a9 by Def37; hence thesis by Def36; end; definition let C,a; attr a is negative means : Def38: ex a9 st a9 is positive & a = (non_op C)term a9; end; registration let C; let a be positive expression of C, an_Adj C; cluster Non a -> negative non positive; coherence proof thus Non a is negative proof take a; thus thesis by Th59; end; take a; thus thesis by Th59; end; end; registration let C; cluster negative non positive for expression of C, an_Adj C; existence proof set a = the positive expression of C, an_Adj C; take Non a; thus thesis; end; end; theorem Th60: for a being non positive expression of C, an_Adj C ex a9 being expression of C, an_Adj C st a = (non_op C)term a9 & Non a = a9 proof let a be non positive expression of C, an_Adj C; consider a9 being expression of C, an_Adj C such that A1: a = (non_op C)term a9 by Def37; A2: a = [non_op, the carrier of C]-tree<*a9*> by A1,Th43; take a9; len <*a9*> = 1 by FINSEQ_1:40; then a|<* 0*> = <*a9*>.(0+1) by A2,TREES_4:def 4 .= a9 by FINSEQ_1:40; hence thesis by A1,Def36; end; theorem Th61: for a being negative expression of C, an_Adj C ex a9 being positive expression of C, an_Adj C st a = (non_op C)term a9 & Non a = a9 proof let a be negative expression of C, an_Adj C; consider a9 being expression of C, an_Adj C such that A1: a9 is positive and A2: a = (non_op C)term a9 by Def38; A3: a = [non_op, the carrier of C]-tree<*a9*> by A2,Th43; reconsider a9 as positive expression of C, an_Adj C by A1; take a9; len <*a9*> = 1 by FINSEQ_1:40; then a|<* 0*> = <*a9*>.(0+1) by A3,TREES_4:def 4 .= a9 by FINSEQ_1:40; hence thesis by A2,Def36; end; theorem Th62: for a being non positive expression of C, an_Adj C holds (non_op C)term (Non a) = a proof let a be non positive expression of C, an_Adj C; ex a9 being expression of C, an_Adj C st ( a = (non_op C) term a9)&( Non a = a9) by Th60; hence thesis; end; registration let C; let a be negative expression of C, an_Adj C; cluster Non a -> positive; coherence proof ex a9 being positive expression of C, an_Adj C st a = (non_op C)term a9 & Non a = a9 by Th61; hence thesis; end; end; definition let C,a; attr a is regular means : Def39: a is positive or a is negative; end; registration let C; cluster positive -> regular non negative for expression of C, an_Adj C; coherence; cluster negative -> regular non positive for expression of C, an_Adj C; coherence; end; registration let C; cluster regular for expression of C, an_Adj C; existence proof set a = the positive expression of C, an_Adj C; take a; thus thesis; end; end; definition let C; set X = {a: a is regular}; A1: X c= Union the Sorts of Free(C, MSVars C) proof let x be object; assume x in X; then ex a st x = a & a is regular; hence thesis; end; func QuasiAdjs C -> Subset of Free(C, MSVars C) equals {a: a is regular}; coherence by A1; end; registration let C; cluster QuasiAdjs C -> non empty constituted-DTrees; coherence proof set v = the positive expression of C, an_Adj C; v in {a: a is regular}; hence QuasiAdjs C is non empty; let x be object; assume x in QuasiAdjs C; hence thesis; end; end; definition let C; mode quasi-adjective of C is regular expression of C, an_Adj C; end; theorem Th63: z is quasi-adjective of C iff z in QuasiAdjs C proof z in QuasiAdjs C iff ex a st z = a & a is regular; hence thesis; end; theorem z is quasi-adjective of C iff z is positive expression of C, an_Adj C or z is negative expression of C, an_Adj C by Def39; registration let C; cluster non positive -> negative for quasi-adjective of C; coherence by Def39; cluster non negative -> positive for quasi-adjective of C; coherence; end; registration let C; cluster positive for quasi-adjective of C; existence proof set a = the positive expression of C, an_Adj C; a is quasi-adjective of C; hence thesis; end; cluster negative for quasi-adjective of C; existence proof set a = the negative expression of C, an_Adj C; a is quasi-adjective of C; hence thesis; end; end; theorem Th65: for a being positive quasi-adjective of C ex v being constructor OperSymbol of C st the_result_sort_of v = an_Adj C & ex p st len p = len the_arity_of v & a = v-trm p proof let e be positive quasi-adjective of C; per cases by Th53; suppose ex x being variable st e = x-term C; hence thesis by Th48; end; suppose ex c being constructor OperSymbol of C st ex p being FinSequence of QuasiTerms C st len p = len the_arity_of c & e = c-trm p; then consider c being constructor OperSymbol of C, p being FinSequence of QuasiTerms C such that A1: len p = len the_arity_of c and A2: e = c-trm p; take c; e is expression of C, the_result_sort_of c by A1,A2,Th52; hence the_result_sort_of c = an_Adj C by Th48; take p; thus thesis by A1,A2; end; suppose ex a st e = (non_op C)term a; hence thesis by Def37; end; suppose ex a,t st e = (ast C)term(a,t); then e is expression of C, a_Type C by Th46; hence thesis by Th48; end; end; theorem Th66: for v being constructor OperSymbol of C st the_result_sort_of v = an_Adj C & len p = len the_arity_of v holds v-trm p is positive quasi-adjective of C proof let v be constructor OperSymbol of C such that A1: the_result_sort_of v = an_Adj C; assume A2: len p = len the_arity_of v; then reconsider a = v-trm p as expression of C, an_Adj C by A1,Th52; a is positive by A2,Th54; hence thesis; end; registration let C; let a be quasi-adjective of C; cluster Non a -> regular; coherence proof per cases; suppose a is positive; then reconsider a9 = a as positive expression of C, an_Adj C; Non a9 is negative; hence thesis; end; suppose a is negative; then reconsider a9 = a as negative expression of C, an_Adj C; Non a9 is positive; hence thesis; end; end; end; theorem Th67: for a being quasi-adjective of C holds Non Non a = a proof let a be quasi-adjective of C; per cases; suppose a is positive; then reconsider a9 = a as positive expression of C, an_Adj C; A1: ex b being positive expression of C, an_Adj C st ( Non a9 = (non_op C)term b)&( Non Non a9 = b) by Th61; Non a9 = (non_op C)term a by Th59; hence thesis by A1,Th44; end; suppose a is negative; then reconsider a9 = a as negative expression of C, an_Adj C; ex b being positive expression of C, an_Adj C st a9 = (non_op C)term b & Non a9 = b by Th61; hence thesis by Th59; end; end; theorem for a1,a2 being quasi-adjective of C st Non a1 = Non a2 holds a1 = a2 proof let a1,a2 be quasi-adjective of C; Non Non a1 = a1 by Th67; hence thesis by Th67; end; theorem for a being quasi-adjective of C holds Non a <> a proof let a be quasi-adjective of C; per cases; suppose a is positive; then reconsider a9 = a as positive quasi-adjective of C; Non a9 is negative quasi-adjective of C; hence thesis; end; suppose a is negative; then reconsider a9 = a as negative quasi-adjective of C; Non a9 is positive quasi-adjective of C; hence thesis; end; end; begin :: Quasi-types definition let C; let q be expression of C, a_Type C; attr q is pure means : Def41: not ex a, t st q = (ast C)term(a,t); end; theorem Th70: for m being OperSymbol of C st the_result_sort_of m = a_Type & the_arity_of m = {} ex t st t = root-tree [m, the carrier of C] & t is pure proof let m be OperSymbol of C such that A1: the_result_sort_of m = a_Type and A2: the_arity_of m = {}; set X = MSVars C; root-tree [m, the carrier of C] in (the Sorts of Free(C, X)).a_Type by A1,A2, MSAFREE3:5; then reconsider T = root-tree [m, the carrier of C] as expression of C, a_Type C by Th41; take T; thus T = root-tree [m, the carrier of C]; given a,t such that A3: T = (ast C)term(a,t); T = [ *, the carrier of C]-tree<*a,t*> by A3,Th46; then [ *, the carrier of C] = T.{} by TREES_4:def 4 .= [m, the carrier of C] by TREES_4:3; then m = ast C by XTUPLE_0:1; hence contradiction by A2,Def9; end; theorem Th71: for v being OperSymbol of C st the_result_sort_of v = an_Adj & the_arity_of v = {} ex a st a = root-tree [v, the carrier of C] & a is positive proof let m be OperSymbol of C such that A1: the_result_sort_of m = an_Adj and A2: the_arity_of m = {}; set X = MSVars C; root-tree [m, the carrier of C] in (the Sorts of Free(C, X)).an_Adj by A1,A2,MSAFREE3:5; then reconsider T = root-tree [m, the carrier of C] as expression of C, an_Adj C by Th41; take T; thus T = root-tree [m, the carrier of C]; given a being expression of C, an_Adj C such that A3: T = (non_op C)term a; T = [non_op, the carrier of C]-tree<*a*> by A3,Th43; then [non_op, the carrier of C] = T.{} by TREES_4:def 4 .= [m, the carrier of C] by TREES_4:3; then m = non_op by XTUPLE_0:1; hence contradiction by A2,Def9; end; registration let C; cluster pure for expression of C, a_Type C; existence proof consider m, a being OperSymbol of C such that A1: the_result_sort_of m = a_Type and A2: the_arity_of m = {} and the_result_sort_of a = an_Adj and the_arity_of a = {} by Def12; ex t being expression of C, a_Type C st t = root-tree [m, the carrier of C] & t is pure by A1,A2,Th70; hence thesis; end; end; reserve q for pure expression of C, a_Type C, A for finite Subset of QuasiAdjs C; definition let C; func QuasiTypes C -> set equals {[A,t]: t is pure}; coherence; end; registration let C; cluster QuasiTypes C -> non empty; coherence proof set q = the pure expression of C, a_Type C; {} is finite Subset of QuasiAdjs C by XBOOLE_1:2; then [{},q] in {[A,t]: t is pure}; hence thesis; end; end; definition let C; mode quasi-type of C -> set means : Def43: it in QuasiTypes C; existence proof set T = the Element of QuasiTypes C; take T; thus thesis; end; end; theorem Th72: z is quasi-type of C iff ex A,q st z = [A,q] proof z in QuasiTypes C iff ex t,A st z = [A,t] & t is pure; hence thesis by Def43; end; theorem Th73: [x,y] is quasi-type of C iff x is finite Subset of QuasiAdjs C & y is pure expression of C, a_Type C proof thus now assume [x,y] is quasi-type of C; then ex A,q st ( [x,y] = [A,q]) by Th72; hence x is finite Subset of QuasiAdjs C & y is pure expression of C, a_Type C by XTUPLE_0:1; end; thus thesis by Th72; end; reserve T for quasi-type of C; registration let C; cluster -> pair for quasi-type of C; coherence proof let x be quasi-type of C; ex A,q st x = [A,q] by Th72; hence thesis; end; end; theorem Th74: ex m being constructor OperSymbol of C st the_result_sort_of m = a_Type C & ex p st len p = len the_arity_of m & q = m-trm p proof set e = q; per cases by Th53; suppose ex x being variable st e = x-term C; hence thesis by Th48; end; suppose ex c being constructor OperSymbol of C st ex p being FinSequence of QuasiTerms C st len p = len the_arity_of c & e = c-trm p; then consider c being constructor OperSymbol of C, p being FinSequence of QuasiTerms C such that A1: len p = len the_arity_of c and A2: e = c-trm p; take c; e is expression of C, the_result_sort_of c by A1,A2,Th52; hence the_result_sort_of c = a_Type C by Th48; take p; thus thesis by A1,A2; end; suppose ex a st e = (non_op C)term a; then e is expression of C, an_Adj C by Th43; hence thesis by Th48; end; suppose ex a st ex q being expression of C, a_Type C st e = (ast C)term(a,q); hence thesis by Def41; end; end; theorem Th75: for m being constructor OperSymbol of C st the_result_sort_of m = a_Type C & len p = len the_arity_of m holds m-trm p is pure expression of C, a_Type C proof let v be constructor OperSymbol of C such that A1: the_result_sort_of v = a_Type C; assume A2: len p = len the_arity_of v; then reconsider a = v-trm p as expression of C, a_Type C by A1,Th52; a is pure by A2,Th55; hence thesis; end; theorem QuasiTerms C misses QuasiAdjs C & QuasiTerms C misses QuasiTypes C & QuasiTypes C misses QuasiAdjs C proof set X = MSVars C; set Y = X (\/) ((the carrier of C) --> {0}); ex A being MSSubset of FreeMSA Y st ( Free(C, X) = GenMSAlg A)&( A = (Reverse Y)""X) by MSAFREE3:def 1; then the Sorts of Free(C, X) is MSSubset of FreeMSA Y by MSUALG_2:def 9; then A1: the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by PBOOLE:def 18; then A2: QuasiTerms C c= (the Sorts of FreeMSA Y).a_Term C; A3: (the Sorts of Free(C,X)).an_Adj C c= (the Sorts of FreeMSA Y).an_Adj C by A1; QuasiAdjs C c= (the Sorts of Free(C,X)).an_Adj C proof let x be object; assume x in QuasiAdjs C; then ex a st x = a & a is regular; hence thesis by Def28; end; then A4: QuasiAdjs C c= (the Sorts of FreeMSA Y).an_Adj C by A3; (the Sorts of FreeMSA Y).a_Term C misses (the Sorts of FreeMSA Y).an_Adj C by PROB_2:def 2; hence QuasiTerms C misses QuasiAdjs C by A2,A4,XBOOLE_1:64; now let x be object; assume that A5: x in QuasiTerms C and A6: x in QuasiTypes C; x is quasi-type of C by A6,Def43; hence contradiction by A5; end; hence QuasiTerms C misses QuasiTypes C by XBOOLE_0:3; now let x be object; assume that A7: x in QuasiAdjs C and A8: x in QuasiTypes C; x is quasi-type of C by A8,Def43; hence contradiction by A7; end; hence thesis by XBOOLE_0:3; end; theorem for e being set holds (e is quasi-term of C implies e is not quasi-adjective of C) & (e is quasi-term of C implies e is not quasi-type of C) & (e is quasi-type of C implies e is not quasi-adjective of C) by Th48; notation let C,A,q; synonym A ast q for [A,q]; end; definition let C,A,q; redefine func A ast q -> quasi-type of C; coherence by Th73; end; registration let C,T; cluster T`1 -> finite for set; coherence proof ex A,q st T = [A,q] by Th72; hence thesis; end; end; notation let C,T; synonym adjs T for T`1; synonym the_base_of T for T`2; end; definition let C,T; redefine func adjs T -> Subset of QuasiAdjs C; coherence proof ex A,q st T = [A,q] by Th72; hence thesis; end; redefine func the_base_of T -> pure expression of C, a_Type C; coherence proof ex A,q st T = [A,q] by Th72; hence thesis; end; end; theorem adjs (A ast q) = A & the_base_of (A ast q) = q; theorem for A1,A2 being finite Subset of QuasiAdjs C for q1,q2 being pure expression of C, a_Type C st A1 ast q1 = A2 ast q2 holds A1 = A2 & q1 = q2 by XTUPLE_0:1; theorem Th80: T = (adjs T) ast the_base_of T; theorem for T1,T2 being quasi-type of C st adjs T1 = adjs T2 & the_base_of T1 = the_base_of T2 holds T1 = T2 proof let T1,T2 be quasi-type of C; T1 = (adjs T1) ast the_base_of T1; hence thesis by Th80; end; definition let C,T; let a be quasi-adjective of C; func a ast T -> quasi-type of C equals [{a} \/ adjs T, the_base_of T]; coherence proof a in QuasiAdjs C; then {a} c= QuasiAdjs C by ZFMISC_1:31; then {a} \/ adjs T is Subset of QuasiAdjs C by XBOOLE_1:8; hence thesis by Th73; end; end; theorem for a being quasi-adjective of C holds adjs (a ast T) = {a} \/ adjs T & the_base_of (a ast T) = the_base_of T; theorem for a being quasi-adjective of C holds a ast (a ast T) = a ast T proof let a be quasi-adjective of C; thus a ast (a ast T) = [{a} \/ ({a} \/ adjs T), the_base_of (a ast T)] .= [{a} \/ {a} \/ adjs T, the_base_of (a ast T)] by XBOOLE_1:4 .= a ast T; end; theorem for a,b being quasi-adjective of C holds a ast (b ast T) = b ast (a ast T) by XBOOLE_1:4; begin :: Variables in quasi-types registration let S be non void Signature; let s be SortSymbol of S; let X be non-empty ManySortedSet of the carrier of S; let t be Term of S,X; cluster (variables_in t).s -> finite; coherence proof defpred P[non empty Relation] means for s being SortSymbol of S holds (S variables_in $1).s is finite; A1: for z being SortSymbol of S, v being Element of X.z holds P[root-tree[v,z]] proof let z be SortSymbol of S, v be Element of X.z; let s be SortSymbol of S; s = z or s <> z; hence thesis by MSAFREE3:10; end; A2: for o being OperSymbol of S, p being ArgumentSeq of Sym(o,X) st for t being Term of S,X st t in rng p holds P[t] holds P[[o,the carrier of S]-tree p] proof let o be OperSymbol of S, p be ArgumentSeq of Sym(o,X) such that A3: for t being Term of S,X st t in rng p for s being SortSymbol of S holds (S variables_in t).s is finite; let s be SortSymbol of S; deffunc F(Term of S,X) = (S variables_in $1).s; set A = {F(q) where q is Term of S,X: q in rng p}; A4: rng p is finite; A5: A is finite from FRAENKEL:sch 21(A4); now let B be set; assume B in A; then ex q being Term of S,X st B = (S variables_in q).s & q in rng p; hence B is finite by A3; end; then A6: union A is finite by A5,FINSET_1:7; (S variables_in ([o,the carrier of S]-tree p)).s c= union A proof let x be object; assume x in (S variables_in ([o,the carrier of S]-tree p)).s; then consider t being DecoratedTree such that A7: t in rng p and A8: x in (S variables_in t).s by MSAFREE3:11; consider i being object such that A9: i in dom p and A10: t = p.i by A7,FUNCT_1:def 3; reconsider i as Nat by A9; reconsider t = p.i as Term of S,X by A9,MSATERM:22; (S variables_in t).s in A by A7,A10; hence thesis by A8,A10,TARSKI:def 4; end; hence thesis by A6; end; for t being Term of S,X holds P[t] from MSATERM:sch 1(A1,A2); hence thesis; end; end; registration let S be non void Signature; let s be SortSymbol of S; let X be non empty-yielding ManySortedSet of the carrier of S; let t be Element of Free(S,X); cluster (S variables_in t).s -> finite; coherence proof reconsider t as Term of S, X (\/) ((the carrier of S) --> {0}) by MSAFREE3:8; (S variables_in t).s = (variables_in t).s; hence thesis; end; end; definition let S be non void Signature; let X be non empty-yielding ManySortedSet of the carrier of S; let s be SortSymbol of S; func (X,s) variables_in -> Function of Union the Sorts of Free(S,X), bool (X.s) means : Def45: for t being Element of Free(S,X) holds it.t = (S variables_in t).s; uniqueness proof let f1,f2 be Function of Union the Sorts of Free(S,X), bool (X.s) such that A1: for t being Element of Free(S,X) holds f1.t = (S variables_in t).s and A2: for t being Element of Free(S,X) holds f2.t = (S variables_in t).s; now let x be Element of Union the Sorts of Free(S,X); reconsider t = x as Element of Free(S,X); thus f1.x = (S variables_in t).s by A1 .= f2.x by A2; end; hence thesis by FUNCT_2:63; end; existence proof defpred P[object,object] means ex t being Element of Free(S,X) st t = $1 & $2 = (S variables_in t).s; A3: now let x be object; assume x in Union the Sorts of Free(S,X); then reconsider t = x as Element of Free(S,X); S variables_in t c= X by MSAFREE3:27; then (S variables_in t).s c= X.s; hence ex y being object st y in bool (X.s) & P[x,y]; end; consider f being Function such that A4: dom f = Union the Sorts of Free(S,X) & rng f c= bool (X.s) and A5: for x being object st x in Union the Sorts of Free(S,X) holds P[x, f.x] from FUNCT_1:sch 6(A3); reconsider f as Function of Union the Sorts of Free(S,X), bool (X.s) by A4,FUNCT_2:2; take f; let x be Element of Free(S,X); ex t being Element of Free(S,X) st t = x & f.x = (S variables_in t).s by A5; hence thesis; end; end; definition let C be initialized ConstructorSignature; let e be expression of C; func variables_in e -> Subset of Vars equals (C variables_in e).a_Term C; coherence proof A1: (MSVars C).a_Term C = Vars by Def25; C variables_in e c= MSVars C by MSAFREE3:27; hence thesis by A1; end; end; registration let C,e; cluster variables_in e -> finite; coherence; end; definition let C,e; func vars e -> finite Subset of Vars equals varcl variables_in e; coherence by Th24; end; theorem varcl vars e = vars e; theorem for x being variable holds variables_in (x-term C) = {x} by MSAFREE3:10; theorem for x being variable holds vars (x-term C) = {x} \/ vars x proof let x be variable; thus vars (x-term C) = varcl {x} by MSAFREE3:10 .= {x} \/ vars x by Th27; end; theorem Th88: for p being DTree-yielding FinSequence st e = [c, the carrier of C]-tree p holds variables_in e = union {variables_in t where t is quasi-term of C: t in rng p} proof let p be DTree-yielding FinSequence; set X = {variables_in t where t is quasi-term of C: t in rng p}; assume A1: e = [c, the carrier of C]-tree p; then p in (QuasiTerms C)* by Th51; then p is FinSequence of QuasiTerms C by FINSEQ_1:def 11; then A2: rng p c= QuasiTerms C by FINSEQ_1:def 4; thus variables_in e c= union X proof let a be object; assume a in variables_in e; then consider t being DecoratedTree such that A3: t in rng p and A4: a in (C variables_in t).a_Term C by A1,MSAFREE3:11; reconsider t as quasi-term of C by A2,A3,Th41; variables_in t in X by A3; hence thesis by A4,TARSKI:def 4; end; let a be object; assume a in union X; then consider Y being set such that A5: a in Y and A6: Y in X by TARSKI:def 4; ex t being quasi-term of C st Y = variables_in t & t in rng p by A6; hence thesis by A1,A5,MSAFREE3:11; end; theorem Th89: for p being DTree-yielding FinSequence st e = [c, the carrier of C]-tree p holds vars e = union {vars t where t is quasi-term of C: t in rng p} proof let p be DTree-yielding FinSequence; assume A1: e = [c, the carrier of C]-tree p; set A = {variables_in t where t is quasi-term of C: t in rng p}; set B = {vars t where t is quasi-term of C: t in rng p}; per cases; suppose A2: A = {}; set b = the Element of B; now assume B <> {}; then b in B; then consider t being quasi-term of C such that b = vars t and A3: t in rng p; variables_in t in A by A3; hence contradiction by A2; end; hence thesis by A1,A2,Th8,Th88,ZFMISC_1:2; end; suppose A <> {}; then reconsider A as non empty set; set D = the set of all varcl s where s is Element of A; A4: B c= D proof let a be object; assume a in B; then consider t being quasi-term of C such that A5: a = vars t and A6: t in rng p; variables_in t in A by A6; then reconsider s = variables_in t as Element of A; a = varcl s by A5; hence thesis; end; A7: D c= B proof let a be object; assume a in D; then consider s being Element of A such that A8: a = varcl s; s in A; then consider t being quasi-term of C such that A9: s = variables_in t and A10: t in rng p; vars t = a by A8,A9; hence thesis by A10; end; thus vars e = varcl union A by A1,Th88 .= union D by Th10 .= union B by A4,A7,XBOOLE_0:def 10; end; end; theorem len p = len the_arity_of c implies variables_in (c-trm p) = union {variables_in t where t is quasi-term of C: t in rng p} proof assume len p = len the_arity_of c; then c-trm p = [c, the carrier of C]-tree p by Def35; hence thesis by Th88; end; theorem len p = len the_arity_of c implies vars (c-trm p) = union {vars t where t is quasi-term of C: t in rng p} proof assume len p = len the_arity_of c; then c-trm p = [c, the carrier of C]-tree p by Def35; hence thesis by Th89; end; theorem for S being ManySortedSign, o being set holds S variables_in ([o, the carrier of S]-tree {}) = EmptyMS the carrier of S proof let S be ManySortedSign, o be set; now let s be object; assume A1: s in the carrier of S; now let x be object; rng {} = {}; then x in (S variables_in ([o, the carrier of S]-tree {})).s iff ex q being DecoratedTree st q in {} & x in (S variables_in q).s by A1,MSAFREE3:11; hence x in (S variables_in ([o, the carrier of S]-tree {})).s iff x in (EmptyMS the carrier of S).s; end; hence (S variables_in ([o, the carrier of S]-tree {})).s = (EmptyMS the carrier of S).s by TARSKI:2; end; hence thesis; end; theorem Th93: for S being ManySortedSign, o being set, t being DecoratedTree holds S variables_in ([o, the carrier of S]-tree <*t*>) = S variables_in t proof let S be ManySortedSign, o be set, t be DecoratedTree; now let s be object; assume A1: s in the carrier of S; A2: t in {t} by TARSKI:def 1; now let x be object; rng <*t*> = {t} by FINSEQ_1:39; then x in (S variables_in ([o, the carrier of S]-tree <*t*>)).s iff ex q being DecoratedTree st q in {t} & x in (S variables_in q).s by A1,MSAFREE3:11; hence x in (S variables_in ([o, the carrier of S]-tree <*t*>)).s iff x in (S variables_in t).s by A2,TARSKI:def 1; end; hence (S variables_in ([o, the carrier of S]-tree <*t*>)).s = (S variables_in t).s by TARSKI:2; end; hence thesis; end; theorem Th94: variables_in ((non_op C)term a) = variables_in a proof (non_op C)term a = [non_op, the carrier of C]-tree <*a*> by Th43; hence thesis by Th93; end; theorem vars ((non_op C)term a) = vars a by Th94; theorem Th96: for S being ManySortedSign, o being set, t1,t2 being DecoratedTree holds S variables_in ([o, the carrier of S]-tree <*t1,t2*>) = (S variables_in t1) (\/) (S variables_in t2) proof let S be ManySortedSign, o be set, t1,t2 be DecoratedTree; now let s be object; assume A1: s in the carrier of S; A2: t1 in {t1,t2} by TARSKI:def 2; A3: t2 in {t1,t2} by TARSKI:def 2; now let x be object; rng <*t1,t2*> = {t1,t2} by FINSEQ_2:127; then x in (S variables_in ([o, the carrier of S]-tree <*t1,t2*>)).s iff ex q being DecoratedTree st q in {t1,t2} & x in (S variables_in q).s by A1,MSAFREE3:11; then x in (S variables_in ([o, the carrier of S]-tree <*t1,t2*>)).s iff x in (S variables_in t1).s or x in (S variables_in t2).s by A2,A3,TARSKI:def 2; then x in (S variables_in ([o, the carrier of S]-tree <*t1,t2*>)).s iff x in (S variables_in t1).s \/ (S variables_in t2).s by XBOOLE_0:def 3; hence x in (S variables_in ([o, the carrier of S]-tree <*t1,t2*>)).s iff x in ((S variables_in t1) (\/) (S variables_in t2)).s by A1,PBOOLE:def 4; end; hence (S variables_in ([o, the carrier of S]-tree <*t1,t2*>)).s = ((S variables_in t1) (\/) (S variables_in t2)).s by TARSKI:2; end; hence thesis; end; theorem Th97: variables_in ((ast C)term(a,t)) = (variables_in a)\/(variables_in t) proof (ast C)term(a,t) = [ *, the carrier of C]-tree <*a,t*> by Th46; then variables_in ((ast C)term(a,t)) = ((C variables_in a)(\/)(C variables_in t)).a_Term by Th96; hence thesis by PBOOLE:def 4; end; theorem vars ((ast C)term(a,t)) = (vars a)\/(vars t) proof thus vars ((ast C)term(a,t)) = varcl((variables_in a)\/(variables_in t)) by Th97 .= (vars a)\/(vars t) by Th11; end; theorem Th99: variables_in Non a = variables_in a proof per cases; suppose a is non positive; then consider a9 being expression of C, an_Adj C such that A1: a = (non_op C)term a9 and A2: Non a = a9 by Th60; [non_op C, the carrier of C]-tree<*a9*> = a by A1,Th43; hence thesis by A2,Th93; end; suppose a is positive; then Non a = (non_op C)term a by Th59 .= [non_op, the carrier of C]-tree <*a*> by Th43; hence thesis by Th93; end; end; theorem vars Non a = vars a by Th99; definition let C; let T be quasi-type of C; func variables_in T -> Subset of Vars equals (union (((MSVars C, a_Term C) variables_in).:adjs T)) \/ variables_in the_base_of T; coherence proof A1: ((MSVars C, a_Term C) variables_in).:adjs T is Subset of bool Vars by Def25 ; union bool Vars = Vars by ZFMISC_1:81; then (union (((MSVars C, a_Term C) variables_in).:adjs T)) c= Vars by A1,ZFMISC_1:77; hence thesis by XBOOLE_1:8; end; end; registration let C; let T be quasi-type of C; cluster variables_in T -> finite; coherence proof now let A be set; assume A in ((MSVars C, a_Term C) variables_in).:adjs T; then consider x being object such that A1: x in Union the Sorts of Free(C, MSVars C) and x in adjs T and A2: A = ((MSVars C, a_Term C) variables_in).x by FUNCT_2:64; reconsider x as expression of C by A1; A = (C variables_in x).a_Term C by A2,Def45; hence A is finite; end; then union (((MSVars C, a_Term C) variables_in).:adjs T) is finite by FINSET_1:7; hence thesis; end; end; definition let C; let T be quasi-type of C; func vars T -> finite Subset of Vars equals varcl variables_in T; coherence by Th24; end; theorem for T being quasi-type of C holds varcl vars T = vars T; theorem Th102: for T being quasi-type of C for a being quasi-adjective of C holds variables_in (a ast T) = (variables_in a) \/ (variables_in T) proof let T be quasi-type of C; let a be quasi-adjective of C; A1: dom ((MSVars C, a_Term C) variables_in) = Union the Sorts of Free(C, MSVars C) by FUNCT_2:def 1; thus variables_in (a ast T) = (union (((MSVars C, a_Term C) variables_in).:adjs(a ast T))) \/ variables_in the_base_of T .= (union (((MSVars C, a_Term C) variables_in).:({a} \/ adjs T))) \/ variables_in the_base_of T .= (union ((((MSVars C, a_Term C) variables_in).:{a}) \/ (((MSVars C, a_Term C) variables_in).:adjs T))) \/ variables_in the_base_of T by RELAT_1:120 .= (union (((MSVars C, a_Term C) variables_in).:{a})) \/ (union (((MSVars C, a_Term C) variables_in).:adjs T)) \/ variables_in the_base_of T by ZFMISC_1:78 .= (union (Im((MSVars C, a_Term C) variables_in,a))) \/ variables_in T by XBOOLE_1:4 .= (union {((MSVars C, a_Term C) variables_in).a}) \/ variables_in T by A1,FUNCT_1:59 .= (((MSVars C, a_Term C) variables_in).a) \/ variables_in T by ZFMISC_1:25 .= (variables_in a) \/ variables_in T by Def45; end; theorem for T being quasi-type of C for a being quasi-adjective of C holds vars (a ast T) = (vars a) \/ (vars T) proof let T be quasi-type of C; let a be quasi-adjective of C; thus vars (a ast T) = varcl((variables_in a)\/variables_in T) by Th102 .= (vars a) \/ vars T by Th11; end; theorem Th104: variables_in (A ast q) = (union {variables_in a where a is quasi-adjective of C: a in A}) \/ (variables_in q) proof set X = ((MSVars C, a_Term C) variables_in).:A; set Y = {variables_in a where a is quasi-adjective of C: a in A}; A1: X c= Y proof let z be object; assume z in X; then consider a being object such that a in dom ((MSVars C, a_Term C) variables_in) and A2: a in A and A3: z = ((MSVars C, a_Term C) variables_in).a by FUNCT_1:def 6; reconsider a as quasi-adjective of C by A2,Th63; z = variables_in a by A3,Def45; hence thesis by A2; end; A4: Y c= X proof let z be object; assume z in Y; then consider a being quasi-adjective of C such that A5: z = variables_in a and A6: a in A; A7: z = ((MSVars C, a_Term C) variables_in).a by A5,Def45; dom ((MSVars C, a_Term C) variables_in) = Union the Sorts of Free(C, MSVars C) by FUNCT_2:def 1; hence thesis by A6,A7,FUNCT_1:def 6; end; thus variables_in (A ast q) = (union (((MSVars C, a_Term C) variables_in).:adjs(A ast q))) \/ variables_in q .= (union (((MSVars C, a_Term C) variables_in).:A)) \/ variables_in q .= (union {variables_in a where a is quasi-adjective of C: a in A}) \/ (variables_in q) by A1,A4,XBOOLE_0:def 10; end; theorem vars (A ast q) = (union {vars a where a is quasi-adjective of C: a in A}) \/ (vars q) proof set X = {variables_in a where a is quasi-adjective of C: a in A}; set Y = {vars a where a is quasi-adjective of C: a in A}; A1: union X c= union Y proof let x be object; assume x in union X; then consider Z being set such that A2: x in Z and A3: Z in X by TARSKI:def 4; consider a being quasi-adjective of C such that A4: Z = variables_in a and A5: a in A by A3; A6: Z c= vars a by A4,Def1; vars a in Y by A5; hence thesis by A2,A6,TARSKI:def 4; end; for x,y st [x,y] in union Y holds x c= union Y proof let x,y; assume [x,y] in union Y; then consider Z being set such that A7: [x,y] in Z and A8: Z in Y by TARSKI:def 4; ex a being quasi-adjective of C st ( Z = vars a)&( a in A) by A8; then A9: x c= Z by A7,Def1; Z c= union Y by A8,ZFMISC_1:74; hence thesis by A9; end; then A10: varcl union X c= union Y by A1,Def1; A11: union Y c= varcl union X proof let x be object; assume x in union Y; then consider Z being set such that A12: x in Z and A13: Z in Y by TARSKI:def 4; consider a being quasi-adjective of C such that A14: Z = vars a and A15: a in A by A13; variables_in a in X by A15; then vars a c= varcl union X by Th9,ZFMISC_1:74; hence thesis by A12,A14; end; thus vars (A ast q) = varcl((union X) \/ (variables_in q)) by Th104 .= (varcl union X) \/ (vars q) by Th11 .= (union Y) \/ (vars q) by A10,A11,XBOOLE_0:def 10; end; theorem Th106: variables_in (({}QuasiAdjs C) ast q) = variables_in q proof set A = {}QuasiAdjs C; set AA = {variables_in a where a is quasi-adjective of C: a in A}; AA c= {} proof let x be object; assume x in AA; then ex a being quasi-adjective of C st x = variables_in a & a in A; hence thesis; end; then A1: AA = {}; variables_in (A ast q) = (union AA) \/ (variables_in q) by Th104; hence thesis by A1,ZFMISC_1:2; end; theorem Th107: e is ground iff variables_in e = {} proof thus e is ground implies variables_in e = {} by Th1,XBOOLE_1:3; assume that A1: variables_in e = {} and A2: Union (C variables_in e) <> {}; set x = the Element of Union (C variables_in e); A3: ex y being object st ( y in dom (C variables_in e))&( x in (C variables_in e).y) by A2,CARD_5:2; A4: dom (C variables_in e) = the carrier of C by PARTFUN1:def 2 .= {a_Type, an_Adj, a_Term} by Def9; A5: C variables_in e c= MSVars C by MSAFREE3:27; A6: (MSVars C).an_Adj = {} by Def25; A7: (MSVars C).a_Type = {} by Def25; A8: (C variables_in e).an_Adj C c= {} by A5,A6; (C variables_in e).a_Type C c= {} by A5,A7; hence thesis by A1,A3,A4,A8,ENUMSET1:def 1; end; definition let C; let T be quasi-type of C; attr T is ground means : Def50: variables_in T = {}; end; registration let C; cluster ground pure for expression of C, a_Type C; existence proof consider m, a being OperSymbol of C such that A1: the_result_sort_of m = a_Type and A2: the_arity_of m = {} and the_result_sort_of a = an_Adj and the_arity_of a = {} by Def12; root-tree [m, the carrier of C] in (the Sorts of Free(C,MSVars C)).a_Type C by A1,A2,MSAFREE3:5; then reconsider mm = root-tree [m, the carrier of C] as expression of C, a_Type C by Th41; take mm; set p = <*>Union the Sorts of Free(C, MSVars C); A3: mm = [m, the carrier of C]-tree p by TREES_4:20; A4: m <> * by A2,Def9; m <> non_op by A1,Def9; then A5: m is constructor by A4; variables_in mm c= {} proof let x be object; assume x in variables_in mm; then x in union {variables_in t where t is quasi-term of C: t in rng p} by A3,A5,Th88; then consider Y such that x in Y and A6: Y in {variables_in t where t is quasi-term of C: t in rng p} by TARSKI:def 4; ex t being quasi-term of C st Y = variables_in t & t in rng p by A6; hence thesis; end; then variables_in mm = {}; hence mm is ground by Th107; ex t being expression of C, a_Type C st t = root-tree [m, the carrier of C] & t is pure by A1,A2,Th70; hence thesis; end; cluster ground for quasi-adjective of C; existence proof consider m, a being OperSymbol of C such that the_result_sort_of m = a_Type and the_arity_of m = {} and A7: the_result_sort_of a = an_Adj and A8: the_arity_of a = {} by Def12; consider mm being expression of C, an_Adj C such that A9: mm = root-tree [a, the carrier of C] and A10: mm is positive by A7,A8,Th71; reconsider mm as quasi-adjective of C by A10; take mm; set p = <*>Union the Sorts of Free(C, MSVars C); A11: mm = [a, the carrier of C]-tree p by A9,TREES_4:20; A12: a <> * by A7,Def9; a <> non_op by A8,Def9; then A13: a is constructor by A12; variables_in mm c= {} proof let x be object; assume x in variables_in mm; then x in union {variables_in t where t is quasi-term of C: t in rng p} by A11,A13,Th88; then consider Y such that x in Y and A14: Y in {variables_in t where t is quasi-term of C: t in rng p} by TARSKI:def 4; ex t being quasi-term of C st Y = variables_in t & t in rng p by A14; hence thesis; end; then variables_in mm = {}; hence thesis by Th107; end; end; theorem Th108: for t being ground pure expression of C, a_Type C holds ({} QuasiAdjs C) ast t is ground proof let t be ground pure expression of C, a_Type C; set T = ({} QuasiAdjs C) ast t; thus variables_in T = variables_in t by Th106 .= {} by Th107; end; registration let C; let t be ground pure expression of C, a_Type C; cluster ({} QuasiAdjs C) ast t -> ground for quasi-type of C; coherence by Th108; end; registration let C; cluster ground for quasi-type of C; existence proof set t = the ground pure expression of C, a_Type C; take ({} QuasiAdjs C) ast t; thus thesis; end; end; registration let C; let T be ground quasi-type of C; let a be ground quasi-adjective of C; cluster a ast T -> ground; coherence proof thus variables_in(a ast T) = (variables_in a)\/variables_in T by Th102 .= {}\/variables_in T by Th107 .= {} by Def50; end; end; begin :: Smooth Type Widening :: Type widening is smooth iff :: vars-function is sup-semilattice homomorphism from widening sup-semilattice :: into VarPoset definition func VarPoset -> strict non empty Poset equals (InclPoset the set of all varcl A where A is finite Subset of Vars)opp; coherence proof set A0 = the finite Subset of Vars; set V = the set of all varcl A where A is finite Subset of Vars; varcl A0 in V; then reconsider V as non empty set; reconsider P = InclPoset V as non empty Poset; P opp is non empty; hence thesis; end; end; theorem Th109: for x, y being Element of VarPoset holds x <= y iff y c= x proof let x, y be Element of VarPoset; set V = the set of all varcl A where A is finite Subset of Vars; set A0 = the finite Subset of Vars; varcl A0 in V; then reconsider V as non empty set; reconsider a = x, b = y as Element of (InclPoset V) opp; x <= y iff ~a >= ~b by YELLOW_7:1; hence thesis by YELLOW_1:3; end; :: registration :: let V1,V2 be Element of VarPoset; :: identify V1 <= V2 with V2 c= V1; :: compatibility by Th22; :: end; theorem Th110: for x holds x is Element of VarPoset iff x is finite Subset of Vars & varcl x = x proof let x; set V = the set of all varcl A where A is finite Subset of Vars; set A0 = the finite Subset of Vars; varcl A0 in V; then reconsider V as non empty set; the carrier of InclPoset V = V by YELLOW_1:1; then x is Element of VarPoset iff x in V; then x is Element of VarPoset iff ex A being finite Subset of Vars st x = varcl A; hence thesis by Th24; end; registration cluster VarPoset -> with_infima with_suprema; coherence proof set V = the set of all varcl A where A is finite Subset of Vars; set A0 = the finite Subset of Vars; varcl A0 in V; then reconsider V as non empty set; now let x,y; assume x in V; then consider A1 being finite Subset of Vars such that A1: x = varcl A1; assume y in V; then consider A2 being finite Subset of Vars such that A2: y = varcl A2; x \/ y = varcl (A1 \/ A2) by A1,A2,Th11; hence x \/ y in V; end; then InclPoset V is with_suprema by YELLOW_1:11; hence VarPoset is with_infima by LATTICE3:10; now let x,y; assume x in V; then consider A1 being finite Subset of Vars such that A3: x = varcl A1; assume y in V; then consider A2 being finite Subset of Vars such that A4: y = varcl A2; reconsider V1 = varcl A1, V2 = varcl A2 as finite Subset of Vars by Th24; x /\ y = varcl (V1 /\ V2) by A3,A4,Th13; hence x /\ y in V; end; then InclPoset V is with_infima by YELLOW_1:12; hence thesis by YELLOW_7:16; end; end; theorem Th111: for V1, V2 being Element of VarPoset holds V1 "\/" V2 = V1 /\ V2 & V1 "/\" V2 = V1 \/ V2 proof let V1, V2 be Element of VarPoset; set V = the set of all varcl A where A is finite Subset of Vars; set A0 = the finite Subset of Vars; varcl A0 in V; then reconsider V as non empty set; A1: VarPoset = (InclPoset V) opp; A2: the carrier of InclPoset V = V by YELLOW_1:1; reconsider v1 = V1, v2 = V2 as Element of (InclPoset V) opp; reconsider a1 = V1, a2 = V2 as Element of InclPoset V; V1 in V by A2; then consider A1 being finite Subset of Vars such that A3: V1 = varcl A1; V2 in V by A2; then consider A2 being finite Subset of Vars such that A4: V2 = varcl A2; A5: a1~ = v1; A6: a2~ = v2; A7: InclPoset V is with_infima with_suprema by A1,LATTICE3:10,YELLOW_7:16; reconsider x1 = V1, x2 = V2 as finite Subset of Vars by A3,A4,Th24; V1 /\ V2 = varcl (x1 /\ x2) by A3,A4,Th13; then V1 /\ V2 in V; then a1 "/\" a2 = V1 /\ V2 by YELLOW_1:9; hence V1 "\/" V2 = V1 /\ V2 by A5,A6,A7,YELLOW_7:21; V1 \/ V2 = varcl (A1 \/ A2) by A3,A4,Th11; then a1 \/ a2 in V; then a1 "\/" a2 = V1 \/ V2 by YELLOW_1:8; hence thesis by A5,A6,A7,YELLOW_7:23; end; registration let V1,V2 be Element of VarPoset; identify V1 "\/" V2 with V1 /\ V2; compatibility by Th111; identify V1 "/\" V2 with V1 \/ V2; compatibility by Th111; end; theorem Th112: for X being non empty Subset of VarPoset holds ex_sup_of X, VarPoset & sup X = meet X proof let X be non empty Subset of VarPoset; set a = the Element of X; A1: meet X c= a by SETFAM_1:3; A2: a is finite Subset of Vars by Th110; then A3: meet X c= Vars by A1,XBOOLE_1:1; for a being Element of X holds varcl a = a by Th110; then varcl meet X = meet X by Th12; then reconsider m = meet X as Element of VarPoset by A1,A2,A3,Th110; A4: now thus X is_<=_than m by SETFAM_1:3,Th109; let b be Element of VarPoset; assume A5: X is_<=_than b; for Y st Y in X holds b c= Y by Th109,A5; then b c= m by SETFAM_1:5; hence m <= b by Th109; end; hence ex_sup_of X, VarPoset by YELLOW_0:15; hence thesis by A4,YELLOW_0:def 9; end; registration cluster VarPoset -> up-complete; coherence proof for X being non empty directed Subset of VarPoset holds ex_sup_of X, VarPoset by Th112; hence thesis by WAYBEL_0:75; end; end; theorem Top VarPoset = {} proof set V = the set of all varcl A where A is finite Subset of Vars; A1: {} Vars in V by Th8; A2: VarPoset opp is lower-bounded by YELLOW_7:31; (Bottom InclPoset V)~ = {} by A1,YELLOW_1:13; hence thesis by A2,YELLOW_7:33; end; definition let C; func vars-function C -> Function of QuasiTypes C, the carrier of VarPoset means for T being quasi-type of C holds it.T = vars T; uniqueness proof let f1,f2 be Function of QuasiTypes C, the carrier of VarPoset such that A1: for T being quasi-type of C holds f1.T = vars T and A2: for T being quasi-type of C holds f2.T = vars T; now let T be Element of QuasiTypes C; reconsider t = T as quasi-type of C by Def43; thus f1.T = vars t by A1 .= f2.T by A2; end; hence thesis by FUNCT_2:63; end; existence proof defpred P[object,object] means ex T being quasi-type of C st $1 = T & $2 = vars T; A3: for x being object st x in QuasiTypes C ex y being object st P[x,y] proof let x be object; assume x in QuasiTypes C; then reconsider T = x as quasi-type of C by Def43; take vars T, T; thus thesis; end; consider f being Function such that A4: dom f = QuasiTypes C and A5: for x being object st x in QuasiTypes C holds P[x,f.x] from CLASSES1:sch 1(A3); rng f c= the carrier of VarPoset proof let y be object; assume y in rng f; then consider x being object such that A6: x in dom f and A7: y = f.x by FUNCT_1:def 3; consider T being quasi-type of C such that x = T and A8: y = vars T by A4,A5,A6,A7; varcl vars T = vars T; then y is Element of VarPoset by A8,Th110; hence thesis; end; then reconsider f as Function of QuasiTypes C, the carrier of VarPoset by A4,FUNCT_2:2; take f; let x be quasi-type of C; x in QuasiTypes C by Def43; then ex T being quasi-type of C st x = T & f.x = vars T by A5; hence thesis; end; end; definition let L be non empty Poset; attr L is smooth means ex C being initialized ConstructorSignature, f being Function of L, VarPoset st the carrier of L c= QuasiTypes C & f = (vars-function C)|the carrier of L & for x,y being Element of L holds f preserves_sup_of {x,y}; end; registration let C be initialized ConstructorSignature; let T be ground quasi-type of C; cluster RelStr(#{T}, id {T}#) -> smooth; coherence proof set L = RelStr(#{T}, id {T}#); A1: T in QuasiTypes C by Def43; then {T} c= QuasiTypes C by ZFMISC_1:31; then reconsider f = (vars-function C)|{T} as Function of L, VarPoset by FUNCT_2:32; take C, f; thus the carrier of L c= QuasiTypes C by A1,ZFMISC_1:31; thus f = (vars-function C)|the carrier of L; let x,y be Element of L; set F = {x,y}; assume ex_sup_of F, L; A2: x = T by TARSKI:def 1; y = T by TARSKI:def 1; then A3: F = {T} by A2,ENUMSET1:29; dom f = {T} by FUNCT_2:def 1; then A4: Im(f,T) = {f.x} by A2,FUNCT_1:59; hence ex_sup_of f.:F, VarPoset by A3,YELLOW_0:38; thus sup (f.:F) = f.x by A3,A4,YELLOW_0:39 .= f.sup F by A2,TARSKI:def 1; end; end; begin :: Structural induction scheme StructInd {C() -> initialized ConstructorSignature, P[set], t() -> expression of C()}: P[t()] provided A1: for x being variable holds P[x-term C()] and A2: for c being constructor OperSymbol of C() for p being FinSequence of QuasiTerms C() st len p = len the_arity_of c & for t being quasi-term of C() st t in rng p holds P[t] holds P[c-trm p] and A3: for a being expression of C(), an_Adj C() st P[a] holds P[(non_op C())term a] and A4: for a being expression of C(), an_Adj C() st P[a] for t being expression of C(), a_Type C() st P[t] holds P[(ast C())term(a,t)] proof defpred Q[set] means $1 is expression of C() implies P[ $1 ]; set X = MSVars C(); set V = X (\/) ((the carrier of C())-->{0}); set S = C(), C = C(); A5: t() is Term of S,V by MSAFREE3:8; A6: for s being SortSymbol of S, v being Element of V.s holds Q[root-tree [v,s]] proof let s be SortSymbol of S; let v be Element of V.s; set t = root-tree [v,s]; assume A7: t is expression of S; A8: t.{} = [v,s] by TREES_4:3; A9: s in the carrier of C; A10: (t.{})`2 = s by A8; A11: s <> the carrier of C by A9; per cases by A7,Th53; suppose ex x being variable st t = x-term C; hence thesis by A1; end; suppose ex c being constructor OperSymbol of C st ex p being FinSequence of QuasiTerms C st len p = len the_arity_of c & t = c-trm p; then consider c being constructor OperSymbol of C, p being FinSequence of QuasiTerms C such that A12: len p = len the_arity_of c and A13: t = c-trm p; t = [c, the carrier of C]-tree p by A12,A13,Def35; then t.{} = [c, the carrier of C] by TREES_4:def 4; hence thesis by A10,A11; end; suppose ex a being expression of C(), an_Adj C() st t = (non_op C)term a; then consider a being expression of C(), an_Adj C() such that A14: t = (non_op C)term a; A15: the_arity_of non_op C = <*an_Adj C*> by Def9; A16: <*an_Adj C*>.1 = an_Adj C by FINSEQ_1:40; len <*an_Adj C*> = 1 by FINSEQ_1:40; then t = [non_op C, the carrier of C]-tree<*a*> by A14,A15,A16,Def30; then t.{} = [non_op C, the carrier of C] by TREES_4:def 4; hence thesis by A10,A11; end; suppose ex a being expression of C(), an_Adj C() st ex q being expression of C, a_Type C st t = (ast C)term(a,q); then consider a being expression of C, an_Adj C, q being expression of C, a_Type C such that A17: t = (ast C)term(a,q); A18: the_arity_of ast C = <*an_Adj C,a_Type C*> by Def9; A19: <*an_Adj C,a_Type C*>.1 = an_Adj C by FINSEQ_1:44; A20: <*an_Adj C,a_Type C*>.2 = a_Type C by FINSEQ_1:44; len <*an_Adj C,a_Type C*> = 2 by FINSEQ_1:44; then t = [ast C, the carrier of C]-tree<*a,q*> by A17,A18,A19,A20,Def31; then t.{} = [ast C, the carrier of C] by TREES_4:def 4; hence thesis by A10,A11; end; end; A21: for o being OperSymbol of S, p being ArgumentSeq of Sym(o,V) st for t being Term of S,V st t in rng p holds Q[t] holds Q[[o,the carrier of S]-tree p] proof let o be OperSymbol of S; let p be ArgumentSeq of Sym(o,V) such that A22: for t being Term of S,V st t in rng p holds Q[t]; set t = [o,the carrier of S]-tree p; assume A23: t is expression of S; per cases by A23,Th53; suppose ex x being variable st t = x-term C; hence thesis by A1; end; suppose ex c being constructor OperSymbol of C st ex p being FinSequence of QuasiTerms C st len p = len the_arity_of c & t = c-trm p; then consider c being constructor OperSymbol of C, q being FinSequence of QuasiTerms C such that A24: len q = len the_arity_of c and A25: t = c-trm q; t = [c, the carrier of C]-tree q by A24,A25,Def35; then A26: p = q by TREES_4:15; now let t be quasi-term of C; t is Term of S,V by MSAFREE3:8; hence t in rng q implies P[t] by A22,A26; end; hence thesis by A2,A24,A25; end; suppose ex a being expression of C(), an_Adj C() st t = (non_op C)term a; then consider a being expression of C(), an_Adj C() such that A27: t = (non_op C)term a; A28: the_arity_of non_op C = <*an_Adj C*> by Def9; A29: <*an_Adj C*>.1 = an_Adj C by FINSEQ_1:40; len <*an_Adj C*> = 1 by FINSEQ_1:40; then t = [non_op C, the carrier of C]-tree<*a*> by A27,A28,A29,Def30; then A30: p = <*a*> by TREES_4:15; A31: rng <*a*> = {a} by FINSEQ_1:39; A32: a in {a} by TARSKI:def 1; a is Term of S,V by MSAFREE3:8; hence thesis by A3,A22,A27,A30,A31,A32; end; suppose ex a being expression of C(), an_Adj C() st ex q being expression of C, a_Type C st t = (ast C)term(a,q); then consider a being expression of C, an_Adj C, q being expression of C, a_Type C such that A33: t = (ast C)term(a,q); A34: the_arity_of ast C = <*an_Adj C,a_Type C*> by Def9; A35: <*an_Adj C,a_Type C*>.1 = an_Adj C by FINSEQ_1:44; A36: <*an_Adj C,a_Type C*>.2 = a_Type C by FINSEQ_1:44; len <*an_Adj C,a_Type C*> = 2 by FINSEQ_1:44; then t = [ast C, the carrier of C]-tree<*a,q*> by A33,A34,A35,A36,Def31; then A37: p = <*a,q*> by TREES_4:15; A38: rng <*a,q*> = {a,q} by FINSEQ_2:127; A39: a in {a,q} by TARSKI:def 2; A40: q in {a,q} by TARSKI:def 2; A41: a is Term of S,V by MSAFREE3:8; A42: q is Term of S,V by MSAFREE3:8; P[a] by A22,A37,A38,A39,A41; hence thesis by A4,A22,A33,A37,A38,A40,A42; end; end; for t being Term of S,V holds Q[t] from MSATERM:sch 1(A6,A21); hence thesis by A5; end; definition let S be ManySortedSign; attr S is with_an_operation_for_each_sort means : Def54: the carrier of S c= rng the ResultSort of S; let X be ManySortedSet of the carrier of S; attr X is with_missing_variables means X"{{}} c= rng the ResultSort of S; end; theorem Th114: for S being non void Signature for X being ManySortedSet of the carrier of S holds X is with_missing_variables iff for s being SortSymbol of S st X.s = {} ex o being OperSymbol of S st the_result_sort_of o = s proof let S be non void Signature; let X be ManySortedSet of the carrier of S; A1: dom X = the carrier of S by PARTFUN1:def 2; hereby assume X is with_missing_variables; then A2: X"{{}} c= rng the ResultSort of S; let s be SortSymbol of S; assume X.s = {}; then X.s in {{}} by TARSKI:def 1; then s in X"{{}} by A1,FUNCT_1:def 7; then consider o being object such that A3: o in the carrier' of S and A4: (the ResultSort of S).o = s by A2,FUNCT_2:11; reconsider o as OperSymbol of S by A3; take o; thus the_result_sort_of o = s by A4; end; assume A5: for s being SortSymbol of S st X.s = {} ex o being OperSymbol of S st the_result_sort_of o = s; let x be object; assume A6: x in X"{{}}; then A7: X.x in {{}} by FUNCT_1:def 7; reconsider x as SortSymbol of S by A1,A6,FUNCT_1:def 7; X.x = {} by A7,TARSKI:def 1; then ex o being OperSymbol of S st the_result_sort_of o = x by A5; hence thesis by FUNCT_2:4; end; registration cluster MaxConstrSign -> with_an_operation_for_each_sort; coherence proof set C = MaxConstrSign; set m = [a_Type, [{}, 0]], a = [an_Adj, [{}, 0]], f = [a_Term, [{}, 0]]; A1: a_Type in {a_Type} by TARSKI:def 1; A2: an_Adj in {an_Adj} by TARSKI:def 1; A3: a_Term in {a_Term} by TARSKI:def 1; A4: [<*> Vars, 0] in [:QuasiLoci, NAT:] by Th29,ZFMISC_1:def 2; then A5: m in Modes by A1,ZFMISC_1:def 2; A6: a in Attrs by A2,A4,ZFMISC_1:def 2; A7: f in Funcs by A3,A4,ZFMISC_1:def 2; A8: m in Modes \/ Attrs by A5,XBOOLE_0:def 3; A9: a in Modes \/ Attrs by A6,XBOOLE_0:def 3; A10: m in Constructors by A8,XBOOLE_0:def 3; A11: a in Constructors by A9,XBOOLE_0:def 3; A12: f in Constructors by A7,XBOOLE_0:def 3; the carrier' of MaxConstrSign = {*, non_op} \/ Constructors by Def24; then reconsider m,a,f as OperSymbol of MaxConstrSign by A10,A11,A12, XBOOLE_0:def 3; A13: m is constructor; A14: a is constructor; A15: f is constructor; A16: (the ResultSort of C).m = m`1 by A13,Def24; A17: (the ResultSort of C).a = a`1 by A14,Def24; A18: (the ResultSort of C).f = f`1 by A15,Def24; A19: (the ResultSort of C).m = a_Type by A16; A20: (the ResultSort of C).a = an_Adj by A17; A21: (the ResultSort of C).f = a_Term by A18; A22: the carrier of C = {a_Type, an_Adj, a_Term} by Def9; let x be object; assume x in the carrier of C; then x = a_Type or x = an_Adj or x = a_Term by A22,ENUMSET1:def 1; hence thesis by A19,A20,A21,FUNCT_2:4; end; let C be ConstructorSignature; cluster MSVars C -> with_missing_variables; coherence proof set X = MSVars C; let x be object; assume A23: x in X"{{}}; then A24: x in dom X by FUNCT_1:def 7; A25: X.x in {{}} by A23,FUNCT_1:def 7; x in the carrier of C by A24; then x in {a_Type, an_Adj, a_Term} by Def9; then A26: x = a_Type or x = an_Adj or x = a_Term by ENUMSET1:def 1; A27: X.x = {} by A25,TARSKI:def 1; A28: (the ResultSort of C).(ast C) = a_Type by Def9; (the ResultSort of C).(non_op C) = an_Adj by Def9; hence thesis by A26,A27,A28,Def25,FUNCT_2:4; end; end; registration let S be ManySortedSign; cluster non-empty -> with_missing_variables for ManySortedSet of the carrier of S; coherence proof let X be ManySortedSet of the carrier of S such that A1: X is non-empty; let x be object; assume A2: x in X"{{}}; then A3: x in dom X by FUNCT_1:def 7; A4: X.x in {{}} by A2,FUNCT_1:def 7; A5: X.x in rng X by A3,FUNCT_1:def 3; X.x = {} by A4,TARSKI:def 1; hence thesis by A1,A5; end; end; registration let S be ManySortedSign; cluster with_missing_variables for ManySortedSet of the carrier of S; existence proof set A = the non-empty ManySortedSet of the carrier of S; take A; thus thesis; end; end; registration cluster initialized with_an_operation_for_each_sort strict for ConstructorSignature; existence proof take MaxConstrSign; thus thesis; end; end; registration let C be with_an_operation_for_each_sort ManySortedSign; cluster -> with_missing_variables for ManySortedSet of the carrier of C; coherence proof let X be ManySortedSet of the carrier of C; A1: X"{{}} c= dom X by RELAT_1:132; A2: dom X = the carrier of C by PARTFUN1:def 2; the carrier of C c= rng the ResultSort of C by Def54; hence X"{{}} c= rng the ResultSort of C by A1,A2; end; end; definition let G be non empty DTConstrStr; redefine func Terminals G -> Subset of G; coherence proof the carrier of G = Terminals G \/NonTerminals G by LANG1:1; hence thesis by XBOOLE_1:7; end; redefine func NonTerminals G -> Subset of G; coherence proof the carrier of G = Terminals G \/NonTerminals G by LANG1:1; hence thesis by XBOOLE_1:7; end; end; theorem Th115: for D1,D2 being non empty DTConstrStr st the Rules of D1 c= the Rules of D2 holds NonTerminals D1 c= NonTerminals D2 & (the carrier of D1) /\ Terminals D2 c= Terminals D1 & (Terminals D1 c= Terminals D2 implies the carrier of D1 c= the carrier of D2) proof let D1,D2 be non empty DTConstrStr such that A1: the Rules of D1 c= the Rules of D2; thus A2: NonTerminals D1 c= NonTerminals D2 proof let x be object; assume x in NonTerminals D1; then ex s being Symbol of D1 st x = s & ex n being FinSequence st s ==> n; then consider s being Symbol of D1, n being FinSequence such that A3: x = s and A4: s ==> n; A5: [s,n] in the Rules of D1 by A4; then [s,n] in the Rules of D2 by A1; then reconsider s9 = s as Symbol of D2 by ZFMISC_1:87; s9 ==> n by A1,A5; hence thesis by A3; end; hereby let x be object; assume A6: x in (the carrier of D1) /\ Terminals D2; then A7: x in Terminals D2 by XBOOLE_0:def 4; reconsider s9 = x as Symbol of D1 by A6,XBOOLE_0:def 4; reconsider s = x as Symbol of D2 by A6; assume not x in Terminals D1; then consider n being FinSequence such that A8: s9 ==> n; [s9,n] in the Rules of D1 by A8; then s ==> n by A1; then not ex s being Symbol of D2 st x = s & not ex n being FinSequence st s ==> n; hence contradiction by A7; end; assume Terminals D1 c= Terminals D2; then Terminals D1 \/ NonTerminals D1 c= Terminals D2 \/ NonTerminals D2 by A2,XBOOLE_1:13; then Terminals D1 \/ NonTerminals D1 c= the carrier of D2 by LANG1:1; hence thesis by LANG1:1; end; theorem Th116: for D1,D2 being non empty DTConstrStr st Terminals D1 c= Terminals D2 & the Rules of D1 c= the Rules of D2 holds TS D1 c= TS D2 proof let G,G9 be non empty DTConstrStr such that A1: Terminals G c= Terminals G9 and A2: the Rules of G c= the Rules of G9; A3: the carrier of G9 = (Terminals G9) \/ NonTerminals G9 by LANG1:1; A4: the carrier of G c= the carrier of G9 by A1,A2,Th115; defpred P[set] means $1 in TS G9; A5: for s being Symbol of G st s in Terminals G holds P[root-tree s] proof let s be Symbol of G; assume A6: s in Terminals G; then reconsider s9 = s as Symbol of G9 by A1,A3,XBOOLE_0:def 3; root-tree s = root-tree s9; hence thesis by A1,A6,DTCONSTR:def 1; end; A7: for nt being Symbol of G, ts being FinSequence of TS(G) st nt ==> roots ts & for t being DecoratedTree of the carrier of G st t in rng ts holds P[t] holds P[nt-tree ts] proof let n be Symbol of G; let s be FinSequence of TS(G) such that A8: [n, roots s] in the Rules of G and A9: for t being DecoratedTree of the carrier of G st t in rng s holds P[t]; rng s c= TS G9 by A9; then reconsider s9 = s as FinSequence of TS G9 by FINSEQ_1:def 4; reconsider n9 = n as Symbol of G9 by A4; n9 ==> roots s9 by A2,A8; hence thesis by DTCONSTR:def 1; end; A10: for t being DecoratedTree of the carrier of G st t in TS(G) holds P[t] from DTCONSTR:sch 7(A5,A7); let x be object; assume A11: x in TS G; then reconsider t = x as Element of FinTrees(the carrier of G); P[t] by A10,A11; hence thesis; end; theorem Th117: for S being ManySortedSign for X,Y being ManySortedSet of the carrier of S st X c= Y holds X is with_missing_variables implies Y is with_missing_variables proof let S be ManySortedSign; let X,Y be ManySortedSet of the carrier of S such that A1: X c= Y and A2: X"{{}} c= rng the ResultSort of S; let x be object; assume A3: x in Y"{{}}; then A4: x in dom Y by FUNCT_1:def 7; A5: Y.x in {{}} by A3,FUNCT_1:def 7; A6: dom X = the carrier of S by PARTFUN1:def 2; A7: Y.x = {} by A5,TARSKI:def 1; X.x c= Y.x by A1,A4; then X.x = {} by A7; then X.x in {{}} by TARSKI:def 1; then x in X"{{}} by A4,A6,FUNCT_1:def 7; hence thesis by A2; end; theorem Th118: for S being set for X,Y being ManySortedSet of S st X c= Y holds Union coprod X c= Union coprod Y proof let S be set; let X,Y be ManySortedSet of S such that A1: X c= Y; A2: dom Y = S by PARTFUN1:def 2; let x be object; assume A3: x in Union coprod X; then A4: x`2 in dom X by CARD_3:22; A5: x`1 in X.x`2 by A3,CARD_3:22; A6: x = [x`1,x`2] by A3,CARD_3:22; X.x`2 c= Y.x`2 by A1,A4; hence thesis by A2,A4,A5,A6,CARD_3:22; end; theorem for S being non void Signature for X,Y being ManySortedSet of the carrier of S st X c= Y holds the carrier of DTConMSA X c= the carrier of DTConMSA Y by Th118,XBOOLE_1:9; theorem Th120: for S being non void Signature for X being ManySortedSet of the carrier of S st X is with_missing_variables holds NonTerminals DTConMSA X = [:the carrier' of S,{the carrier of S}:] & Terminals DTConMSA X = Union coprod X proof let S be non void Signature; let X be ManySortedSet of the carrier of S such that A1: X is with_missing_variables; set D = DTConMSA X, A = [:the carrier' of S,{the carrier of S}:] \/ Union (coprod (X qua ManySortedSet of the carrier of S)); A2: Union(coprod X) misses [:the carrier' of S,{the carrier of S}:] by MSAFREE:4; A3: (Terminals D) misses (NonTerminals D) by DTCONSTR:8; thus NonTerminals DTConMSA X c= [:the carrier' of S,{the carrier of S}:] by MSAFREE:6; thus A4: [:the carrier' of S,{the carrier of S}:] c= NonTerminals D proof let o,x2 be object; assume A5: [o,x2] in [:the carrier' of S,{the carrier of S}:]; then A6: x2 in {the carrier of S} by ZFMISC_1:87; reconsider o as OperSymbol of S by A5,ZFMISC_1:87; A7: the carrier of S = x2 by A6,TARSKI:def 1; then reconsider xa = [o,the carrier of S] as Element of (the carrier of D) by A5,XBOOLE_0:def 3; set O = the_arity_of o; defpred P[object,object] means $2 in A & (X.(O.$1) <> {} implies $2 in coprod(O.$1,X)) & (X.(O.$1) = {} implies ex o being OperSymbol of S st $2 = [o,the carrier of S] & the_result_sort_of o = O.$1); A8: for a be object st a in Seg len O ex b be object st P[a,b] proof let a be object; assume a in Seg len O; then A9: a in dom O by FINSEQ_1:def 3; then A10: O.a in rng O by FUNCT_1:def 3; then reconsider s = O.a as SortSymbol of S; per cases; suppose X.(O.a) is non empty; then consider x be object such that A11: x in X.(O.a) by XBOOLE_0:def 1; take y = [x,O.a]; A12: y in coprod(O.a,X) by A10,A11,MSAFREE:def 2; A13: O.a in rng O by A9,FUNCT_1:def 3; dom coprod(X) = the carrier of S by PARTFUN1:def 2; then (coprod(X)).(O.a) in rng coprod(X) by A13,FUNCT_1:def 3; then coprod(O.a,X) in rng coprod(X) by A13,MSAFREE:def 3; then y in Union coprod(X) by A12,TARSKI:def 4; hence thesis by A10,A11,MSAFREE:def 2,XBOOLE_0:def 3; end; suppose A14: X.(O.a) = {}; then consider o being OperSymbol of S such that A15: the_result_sort_of o = s by A1,Th114; take y = [o,the carrier of S]; the carrier of S in {the carrier of S} by TARSKI:def 1; then y in [:the carrier' of S,{the carrier of S}:] by ZFMISC_1:87; hence thesis by A14,A15,XBOOLE_0:def 3; end; end; consider b be Function such that A16: dom b = Seg len O & for a be object st a in Seg len O holds P[a,b.a] from CLASSES1:sch 1(A8); reconsider b as FinSequence by A16,FINSEQ_1:def 2; rng b c= A proof let a be object; assume a in rng b; then ex c being object st c in dom b & b.c = a by FUNCT_1:def 3; hence thesis by A16; end; then reconsider b as FinSequence of A by FINSEQ_1:def 4; reconsider b as Element of A* by FINSEQ_1:def 11; A17: len b = len O by A16,FINSEQ_1:def 3; now let c be set; assume A18: c in dom b; then A19: P[c,b.c] by A16; dom O = Seg len O by FINSEQ_1:def 3; then A20: O.c in rng O by A16,A18,FUNCT_1:def 3; dom coprod(X) = the carrier of S by PARTFUN1:def 2; then (coprod(X)).(O.c) in rng coprod(X) by A20,FUNCT_1:def 3; then coprod(O.c,X) in rng coprod(X) by A20,MSAFREE:def 3; then X.(O.c) <> {} implies b.c in Union coprod(X) by A19,TARSKI:def 4; hence b.c in [:the carrier' of S,{the carrier of S}:] implies for o1 being OperSymbol of S st [o1,the carrier of S] = b.c holds the_result_sort_of o1 = O.c by A2,A19,XBOOLE_0:3,XTUPLE_0:1; assume A21: b.c in Union (coprod X); now assume X.(O.c) = {}; then A22: ex o being OperSymbol of S st ( b.c = [o,the carrier of S]) &( the_result_sort_of o = O.c) by A16,A18; the carrier of S in {the carrier of S} by TARSKI:def 1; then b.c in [:the carrier' of S,{the carrier of S}:] by A22,ZFMISC_1:87; hence contradiction by A2,A21,XBOOLE_0:3; end; hence b.c in coprod(O.c,X) by A16,A18; end; then [xa,b] in REL(X) by A17,MSAFREE:5; then xa ==> b; hence thesis by A7; end; thus Terminals D c= Union coprod X proof let x be object; assume A23: x in Terminals D; then not x in [:the carrier' of S,{the carrier of S}:] by A3,A4,XBOOLE_0:3; hence thesis by A23,XBOOLE_0:def 3; end; thus thesis by MSAFREE:6; end; theorem for S being non void Signature for X,Y being ManySortedSet of the carrier of S st X c= Y & X is with_missing_variables holds Terminals DTConMSA X c= Terminals DTConMSA Y & the Rules of DTConMSA X c= the Rules of DTConMSA Y & TS DTConMSA X c= TS DTConMSA Y proof let S be non void Signature; let X,Y be ManySortedSet of the carrier of S such that A1: X c= Y and A2: X is with_missing_variables; A3: Y is with_missing_variables by A1,A2,Th117; set G = DTConMSA X, G9 = DTConMSA Y; A4: the carrier of G c= the carrier of G9 by A1,Th118,XBOOLE_1:9; A5: Terminals G = Union coprod X by A2,Th120; A6: Terminals G9 = Union coprod Y by A3,Th120; hence Terminals G c= Terminals G9 by A1,A5,Th118; A7: (the carrier of G)* c= (the carrier of G9)* by A4,FINSEQ_1:62; thus the Rules of G c= the Rules of G9 proof let a,b be object; assume A8: [a,b] in the Rules of G; then A9: a in [:the carrier' of S,{the carrier of S}:] by MSAFREE1:2; reconsider a as Element of [:the carrier' of S,{the carrier of S}:] \/ Union coprod X by A9,XBOOLE_0:def 3; reconsider a9 = a as Element of [:the carrier' of S,{the carrier of S}:] \/ Union coprod Y by A9,XBOOLE_0:def 3; reconsider b as Element of ([:the carrier' of S,{the carrier of S}:] \/ Union coprod X)* by A8, MSAFREE1:2; reconsider b9 = b as Element of ([:the carrier' of S,{the carrier of S}:] \/ Union coprod Y)* by A7; now let o be OperSymbol of S; assume A10: [o,the carrier of S] = a9; hence A11: len b9 = len (the_arity_of o) by A8,MSAFREE:def 7; let x be set; assume A12: x in dom b9; hence b9.x in [:the carrier' of S,{the carrier of S}:] implies for o1 be OperSymbol of S st [o1,the carrier of S] = b.x holds the_result_sort_of o1 = (the_arity_of o).x by A8,A10,MSAFREE:def 7; A13: Union coprod Y misses [:the carrier' of S,{the carrier of S}:] by MSAFREE:4; A14: b.x in [:the carrier' of S,{the carrier of S}:] \/ Union coprod X by A12,DTCONSTR:2; A15: dom b9 = Seg len b9 by FINSEQ_1:def 3; dom the_arity_of o = Seg len b9 by A11,FINSEQ_1:def 3; then A16: (the_arity_of o).x in the carrier of S by A12,A15,DTCONSTR:2; assume A17: b9.x in Union coprod Y; b.x in [:the carrier' of S,{the carrier of S}:] or b.x in Union coprod X by A14,XBOOLE_0:def 3; then b.x in coprod((the_arity_of o).x,X) by A8,A10,A12,A13,A17, MSAFREE:def 7,XBOOLE_0:3; then A18: ex a being set st ( a in X.((the_arity_of o).x))&( b.x = [a , (the_arity_of o).x]) by A16,MSAFREE:def 2; X.((the_arity_of o).x) c= Y.((the_arity_of o).x) by A1,A16; hence b9.x in coprod((the_arity_of o).x,Y) by A16,A18,MSAFREE:def 2; end; hence thesis by A9,MSAFREE:def 7; end; hence thesis by A1,A5,A6,Th116,Th118; end; theorem Th122: for t being set holds t in Terminals DTConMSA MSVars C iff ex x being variable st t = [x,a_Term C] proof let t be set; set X = MSVars C; A1: Terminals DTConMSA X = Union coprod X by Th120; A2: dom X = the carrier of C by PARTFUN1:def 2; A3: the carrier of C = {a_Type, an_Adj, a_Term} by Def9; A4: X.a_Type = {} by Def25; A5: X.an_Adj = {} by Def25; A6: X.a_Term = Vars by Def25; hereby assume A7: t in Terminals DTConMSA X; then A8: t`2 in dom X by A1,CARD_3:22; A9: t`1 in X.t`2 by A1,A7,CARD_3:22; A10: t`2 = a_Type or t`2 = an_Adj or t`2 = a_Term by A3,A8,ENUMSET1:def 1; reconsider x = t`1 as variable by A3,A4,A5,A6,A8,A9,ENUMSET1:def 1; take x; thus t = [x,a_Term C] by A1,A4,A5,A7,A10,CARD_3:22; end; given x being variable such that A11: t = [x,a_Term C]; A12: t`1 = x by A11; t`2 = a_Term by A11; hence thesis by A1,A2,A6,A11,A12,CARD_3:22; end; theorem Th123: for t being set holds t in NonTerminals DTConMSA MSVars C iff t = [ast C, the carrier of C] or t = [non_op C, the carrier of C] or ex c being constructor OperSymbol of C st t = [c, the carrier of C] proof let t be set; set X = MSVars C; A1: NonTerminals DTConMSA X = [:the carrier' of C,{the carrier of C}:] by Th120; hereby assume t in NonTerminals DTConMSA MSVars C; then consider a,b being object such that A2: a in the carrier' of C and A3: b in {the carrier of C} and A4: t = [a,b] by A1,ZFMISC_1:def 2; reconsider a as OperSymbol of C by A2; A5: b = the carrier of C by A3,TARSKI:def 1; a is constructor or a is not constructor; hence t = [ast C, the carrier of C] or t = [non_op C, the carrier of C] or ex c being constructor OperSymbol of C st t = [c, the carrier of C] by A4,A5; end; the carrier of C in {the carrier of C} by TARSKI:def 1; hence thesis by A1,ZFMISC_1:87; end; theorem Th124: for S being non void Signature for X being with_missing_variables ManySortedSet of the carrier of S for t being set st t in Union the Sorts of Free(S,X) holds t is Term of S, X (\/) ((the carrier of S)-->{0}) proof let S be non void Signature; let X be with_missing_variables ManySortedSet of the carrier of S; set V = X (\/) ((the carrier of S)-->{0}); set A = Free(S, X); set U = the Sorts of A; A1: U = S-Terms(X, V) by MSAFREE3:24; let t be set; assume t in Union U; then consider s being object such that A2: s in dom U and A3: t in U.s by CARD_5:2; reconsider s as SortSymbol of S by A2; U.s = {r where r is Term of S,V: the_sort_of r = s & variables_in r c= X} by A1,MSAFREE3:def 5; then ex r being Term of S,V st t = r & the_sort_of r = s & variables_in r c= X by A3; hence thesis; end; theorem for S being non void Signature for X being with_missing_variables ManySortedSet of the carrier of S for t being Term of S, X (\/) ((the carrier of S)-->{0}) st t in Union the Sorts of Free(S,X) holds t in (the Sorts of Free(S,X)).the_sort_of t proof let S be non void Signature; let X be with_missing_variables ManySortedSet of the carrier of S; set V = X (\/) ((the carrier of S)-->{0}); set A = Free(S, X); set U = the Sorts of A; A1: U = S-Terms(X, V) by MSAFREE3:24; let t be Term of S, X (\/) ((the carrier of S)-->{0}); assume t in Union U; then consider s being object such that A2: s in dom U and A3: t in U.s by CARD_5:2; reconsider s as SortSymbol of S by A2; U.s = {r where r is Term of S,V: the_sort_of r = s & variables_in r c= X} by A1,MSAFREE3:def 5; then ex r being Term of S,V st t = r & the_sort_of r = s & variables_in r c= X by A3; hence thesis by A3; end; theorem for G being non empty DTConstrStr for s being Element of G for p being FinSequence st s ==> p holds p is FinSequence of the carrier of G proof let G be non empty DTConstrStr; let s be Element of G; let p be FinSequence; assume s ==> p; then [s,p] in the Rules of G; then p in (the carrier of G)* by ZFMISC_1:87; hence thesis by FINSEQ_1:def 11; end; theorem Th127: for S being non void Signature for X,Y being ManySortedSet of the carrier of S for g1 being Symbol of DTConMSA X for g2 being Symbol of DTConMSA Y for p1 being FinSequence of the carrier of DTConMSA X for p2 being FinSequence of the carrier of DTConMSA Y st g1 = g2 & p1 = p2 & g1 ==> p1 holds g2 ==> p2 proof let S be non void Signature; let X,Y be ManySortedSet of the carrier of S; A1: dom Y = the carrier of S by PARTFUN1:def 2; set G1 = DTConMSA X; set G2 = DTConMSA Y; let g1 be Symbol of G1; let g2 be Symbol of G2; let p1 be FinSequence of the carrier of G1; let p2 be FinSequence of the carrier of G2; assume that A2: g1 = g2 and A3: p1 = p2 and A4: g1 ==> p1; A5: [g1, p1] in REL X by A4; then A6: p1 in ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X))* by ZFMISC_1:87; then A7: g1 in [:the carrier' of S,{the carrier of S}:] by A5,MSAFREE:def 7; A8: p2 in ([:the carrier' of S, {the carrier of S}:] \/ Union (coprod Y))* by FINSEQ_1:def 11; now let o9 be OperSymbol of S; assume A9: [o9,the carrier of S] = g2; hence A10: len p2 = len the_arity_of o9 by A2,A3,A5,A6,MSAFREE:def 7; let x be set; assume A11: x in dom p2; hence p2.x in [:the carrier' of S,{the carrier of S}:] implies for o1 be OperSymbol of S st [o1,the carrier of S] = p2.x holds the_result_sort_of o1 = (the_arity_of o9).x by A2,A3,A5,A6,A9,MSAFREE:def 7; x in dom the_arity_of o9 by A10,A11,FINSEQ_3:29; then (the_arity_of o9).x in rng the_arity_of o9 by FUNCT_1:def 3; then reconsider i = (the_arity_of o9).x as SortSymbol of S; assume A12: p2.x in Union coprod Y; then A13: (p2.x)`2 in dom Y by CARD_3:22; A14: (p2.x)`1 in Y.(p2.x)`2 by A12,CARD_3:22; A15: p2.x = [(p2.x)`1,(p2.x)`2] by A12,CARD_3:22; reconsider nn = the carrier of S as set; A: not nn in nn; p2.x in rng p1 by A3,A11,FUNCT_1:def 3; then the carrier of S nin the carrier of S & p2.x in [:the carrier' of S,{the carrier of S}:] or p2.x in Union coprod X by XBOOLE_0:def 3,A; then p2.x in coprod(i,X) by A1,A2,A3,A5,A6,A9,A11,A13,A15,MSAFREE:def 7,ZFMISC_1:106; then ex a being set st ( a in X.i)&( p2.x = [a,i]) by MSAFREE:def 2; then i = (p2.x)`2; hence p2.x in coprod((the_arity_of o9).x,Y) by A14,A15,MSAFREE:def 2; end; then [g2, p2] in REL Y by A2,A7,A8,MSAFREE:def 7; hence thesis; end; theorem Th128: for S being non void Signature for X being with_missing_variables ManySortedSet of the carrier of S holds Union the Sorts of Free(S, X) = TS DTConMSA X proof let S be non void Signature; let X be with_missing_variables ManySortedSet of the carrier of S; set V = X (\/) ((the carrier of S)-->{0}); set A = Free(S, X); set U = the Sorts of A; set G = DTConMSA X; A1: U = S-Terms(X, V) by MSAFREE3:24; A2: dom U = the carrier of S by PARTFUN1:def 2; defpred P[set] means $1 in Union U implies $1 in TS G; A3: for s being SortSymbol of S, v being Element of V.s holds P[root-tree [v,s]] proof let s be SortSymbol of S; let v be Element of V.s; assume root-tree [v,s] in Union U; then consider s1 being object such that A4: s1 in dom U and A5: root-tree [v,s] in U.s1 by CARD_5:2; reconsider s1 as SortSymbol of S by A4; U.s1={t where t is Term of S,V: the_sort_of t = s1 & variables_in t c= X} by A1,MSAFREE3:def 5; then consider t being Term of S,V such that A6: root-tree [v,s] = t and the_sort_of t = s1 and A7: variables_in t c= X by A5; (variables_in t).s = {v} by A6,MSAFREE3:10; then {v} c= X.s by A7; then v in X.s by ZFMISC_1:31; then [v,s] in Terminals G by MSAFREE:7; hence thesis by DTCONSTR:def 1; end; A8: for o being OperSymbol of S, p being ArgumentSeq of Sym(o,V) st for t being Term of S,V st t in rng p holds P[t] holds P[[o,the carrier of S]-tree p] proof let o be OperSymbol of S; let p be ArgumentSeq of Sym(o,V) such that A9: for t being Term of S,V st t in rng p holds P[t] and A10: [o,the carrier of S]-tree p in Union U; consider s being object such that A11: s in dom U and A12: [o,the carrier of S]-tree p in U.s by A10,CARD_5:2; reconsider s as SortSymbol of S by A11; U.s={t where t is Term of S,V: the_sort_of t = s & variables_in t c= X} by A1,MSAFREE3:def 5; then consider t being Term of S,V such that A13: [o,the carrier of S]-tree p = t and A14: the_sort_of t = s and variables_in t c= X by A12; t.{} = [o,the carrier of S] by A13,TREES_4:def 4; then the_result_sort_of o = s by A14,MSATERM:17; then A15: rng p c= Union U by A1,A12,MSAFREE3:19; rng p c= TS G proof let x be object; assume A16: x in rng p; then x is Term of S,V by A15,Th124; hence thesis by A9,A15,A16; end; then reconsider q = p as FinSequence of TS G by FINSEQ_1:def 4; NonTerminals G = [:the carrier' of S,{the carrier of S}:] by Th120; then [o,the carrier of S] in NonTerminals G by ZFMISC_1:106; then reconsider oo = [o,the carrier of S] as Symbol of G; Sym(o,V) ==> roots p by MSATERM:21; then oo ==> roots q by Th127; hence thesis by DTCONSTR:def 1; end; A17: for t being Term of S,V holds P[t] from MSATERM:sch 1(A3,A8); A18: NonTerminals DTConMSA X = [:the carrier' of S,{the carrier of S}:] by Th120; A19: Terminals DTConMSA X = Union coprod X by Th120; defpred Q[set] means $1 in Union U; A20: for s being Symbol of G st s in Terminals G holds Q[root-tree s] proof let s be Symbol of G; assume A21: s in Terminals G; then A22: s`2 in dom X by A19,CARD_3:22; A23: s`1 in X.s`2 by A19,A21,CARD_3:22; A24: s = [s`1,s`2] by A19,A21,CARD_3:22; A25: dom U = the carrier of S by PARTFUN1:def 2; root-tree s in (the Sorts of Free(S,X)).s`2 by A22,A23,A24,MSAFREE3:4; hence thesis by A22,A25,CARD_5:2; end; A26: for nt being Symbol of G, ts being FinSequence of TS G st nt ==> roots ts & for t being DecoratedTree of the carrier of G st t in rng ts holds Q[t] holds Q[nt-tree ts] proof let nt be Symbol of G; let ts be FinSequence of TS G such that A27: nt ==> roots ts and A28: for t being DecoratedTree of the carrier of G st t in rng ts holds Q[t]; nt in NonTerminals G by A27; then consider o,z being object such that A29: o in the carrier' of S and A30: z in {the carrier of S} and A31: nt = [o,z] by A18,ZFMISC_1:def 2; reconsider o as OperSymbol of S by A29; A32: rng ts c= Union U by A28; rng ts c= TS DTConMSA V proof let a be object; assume a in rng ts; then A33: a is Element of S-TermsV by A32,Th124; S-TermsV = TS DTConMSA V by MSATERM:def 1; hence thesis by A33; end; then reconsider p = ts as FinSequence of TS DTConMSA V by FINSEQ_1:def 4; reconsider q = p as FinSequence of S-TermsV by MSATERM:def 1; A34: z = the carrier of S by A30,TARSKI:def 1; then Sym(o, V) ==> roots p by A27,A31,Th127; then reconsider q as ArgumentSeq of Sym(o, V) by MSATERM:21; set t = Sym(o, V)-tree q; t in U.the_result_sort_of o by A1,A32,MSAFREE3:19; hence thesis by A2,A31,A34,CARD_5:2; end; A35: for t being DecoratedTree of the carrier of G st t in TS G holds Q[t] from DTCONSTR:sch 7(A20,A26); thus Union U c= TS DTConMSA X proof let x be object; assume A36: x in Union U; then consider s being object such that A37: s in dom U and A38: x in U.s by CARD_5:2; reconsider s as SortSymbol of S by A37; x in U.s by A38; then x is Term of S,V by A1,MSAFREE3:16; hence thesis by A17,A36; end; let x be object; assume A39: x in TS G; then reconsider TG = TS G as non empty Subset of FinTrees(the carrier of G); x is Element of TG by A39; hence thesis by A35; end; definition let S be non void Signature; let X be ManySortedSet of the carrier of S; mode term-transformation of S,X -> UnOp of Union the Sorts of Free(S,X) means :Def56: for s being SortSymbol of S holds it.:((the Sorts of Free(S,X)).s) c= (the Sorts of Free(S,X)).s; existence proof set f = id Union the Sorts of Free(S,X); A1: dom f = Union the Sorts of Free(S,X); rng f = Union the Sorts of Free(S,X); then reconsider f as UnOp of Union the Sorts of Free(S,X) by A1,FUNCT_2:2; take f; thus thesis by Th4; end; end; theorem Th129: for S being non void Signature for X being non empty ManySortedSet of the carrier of S for f being UnOp of Union the Sorts of Free(S,X) holds f is term-transformation of S,X iff for s being SortSymbol of S for a being set st a in (the Sorts of Free(S,X)).s holds f.a in (the Sorts of Free(S,X)).s proof let S be non void Signature; let X be non empty ManySortedSet of the carrier of S; A1: dom the Sorts of Free(S,X) = the carrier of S by PARTFUN1:def 2; let f be UnOp of Union the Sorts of Free(S,X); A2: dom f = Union the Sorts of Free(S,X) by FUNCT_2:52; hereby assume A3: f is term-transformation of S,X; let s be SortSymbol of S; A4: f.:((the Sorts of Free(S,X)).s) c= (the Sorts of Free(S,X)).s by A3,Def56; (the Sorts of Free(S,X)).s in rng the Sorts of Free(S,X) by A1,FUNCT_1:def 3; then A5: (the Sorts of Free(S,X)).s c= Union the Sorts of Free(S,X) by ZFMISC_1:74; let a be set; assume a in (the Sorts of Free(S,X)).s; then f.a in f.:((the Sorts of Free(S,X)).s) by A2,A5,FUNCT_1:def 6; hence f.a in (the Sorts of Free(S,X)).s by A4; end; assume A6: for s being SortSymbol of S for a being set st a in (the Sorts of Free(S,X)).s holds f.a in (the Sorts of Free(S,X)).s; let s be SortSymbol of S; let x be object; assume x in f.:((the Sorts of Free(S,X)).s); then ex a being object st a in dom f & a in (the Sorts of Free(S,X)).s & x = f.a by FUNCT_1:def 6; hence thesis by A6; end; theorem Th130: for S being non void Signature for X being non empty ManySortedSet of the carrier of S for f being term-transformation of S,X for s being SortSymbol of S for p being FinSequence of (the Sorts of Free(S,X)).s holds f*p is FinSequence of (the Sorts of Free(S,X)).s & card (f*p) = len p proof let S be non void Signature; let X be non empty ManySortedSet of the carrier of S; set A = Free(S,X); let f be term-transformation of S,X; let s be SortSymbol of S; let p be FinSequence of (the Sorts of A).s; A1: Union the Sorts of A = {} or Union the Sorts of A <> {}; A2: dom the Sorts of A = the carrier of S by PARTFUN1:def 2; A3: dom f = Union the Sorts of A by A1,FUNCT_2:def 1; (the Sorts of A).s in rng the Sorts of A by A2,FUNCT_1:def 3; then (the Sorts of A).s c= Union the Sorts of A by ZFMISC_1:74; then rng p c= dom f by A3; then A4: dom (f*p) = dom p by RELAT_1:27; dom p = Seg len p by FINSEQ_1:def 3; then A5: f*p is FinSequence by A4,FINSEQ_1:def 2; A6: rng(f*p) c= (the Sorts of A).s proof let z be object; assume z in rng(f*p); then consider i being object such that A7: i in dom(f*p) and A8: z = (f*p).i by FUNCT_1:def 3; p.i in rng p by A4,A7,FUNCT_1:def 3; then f.(p.i) in (the Sorts of A).s by Th129; hence thesis by A7,A8,FUNCT_1:12; end; hence f*p is FinSequence of (the Sorts of Free(S,X)).s by A5,FINSEQ_1:def 4; reconsider q = f*p as FinSequence of (the Sorts of A).s by A5,A6, FINSEQ_1:def 4; thus card(f*p) = len q .= len p by A4,FINSEQ_3:29; end; definition let S be non void Signature; let X be ManySortedSet of the carrier of S; let t be term-transformation of S,X; attr t is substitution means for o being OperSymbol of S for p,q being FinSequence of Free(S,X) st [o, the carrier of S]-tree p in Union the Sorts of Free(S,X) & q = t*p holds t.([o, the carrier of S]-tree p) = [o, the carrier of S]-tree q; end; scheme StructDef {C() -> initialized ConstructorSignature, V,N(set) -> (expression of C()), F,A(set,set) -> (expression of C())}: ex f being term-transformation of C(), MSVars C() st (for x being variable holds f.(x-term C()) = V(x)) & (for c being constructor OperSymbol of C() for p,q being FinSequence of QuasiTerms C() st len p = len the_arity_of c & q = f*p holds f.(c-trm p) = F(c, q)) & (for a being expression of C(), an_Adj C() holds f.((non_op C())term a) = N(f.a)) & for a being expression of C(), an_Adj C() for t being expression of C(), a_Type C() holds f.((ast C())term(a,t)) = A(f.a, f.t) provided A1: for x being variable holds V(x) is quasi-term of C() and A2: for c being constructor OperSymbol of C() for p being FinSequence of QuasiTerms C() st len p = len the_arity_of c holds F(c, p) is expression of C(), the_result_sort_of c and A3: for a being expression of C(), an_Adj C() holds N(a) is expression of C(), an_Adj C() and A4: for a being expression of C(), an_Adj C() for t being expression of C(), a_Type C() holds A(a,t) is expression of C(), a_Type C() proof set V = MSVars C(); set X = V(\/)((the carrier of C())-->{0}); set A = Free(C(), V); set U = the Sorts of A; set D = Union U; set G = DTConMSA V; deffunc TermVal(Symbol of G) = V($1`1); deffunc NTermVal(Symbol of G, FinSequence, Function) = IFEQ($1`1,*, A($3.1,$3.2), IFEQ($1`1,non_op, N($3.1), F($1`1,$3))); consider f being Function of TS G, D such that A5: for t being Symbol of G st t in Terminals G holds f.(root-tree t) = TermVal(t) and A6: for nt being Symbol of G, ts being FinSequence of TS G st nt ==> roots ts holds f.(nt-tree ts) = NTermVal(nt, roots ts, f * ts) from DTCONSTR:sch 8; D = TS G by Th128; then reconsider f as Function of D,D; f is term-transformation of C(), V proof let s be SortSymbol of C(); let x be object; assume x in f.:((the Sorts of A).s); then consider a being Element of D such that A7: a in (the Sorts of A).s and A8: x = f.a by FUNCT_2:65; defpred P[expression of C()] means for s being SortSymbol of C() st $1 in (the Sorts of A).s holds f.$1 in (the Sorts of A).s; A9: for x being variable holds P[x-term C()] proof let y be variable; set a = y-term C(); let s be SortSymbol of C(); assume A10: a in (the Sorts of A).s; A11: [y,a_Term C()] in Terminals G by Th122; then reconsider t = [y,a_Term C()] as Symbol of G; f.a = TermVal(t) by A5,A11 .= V(y); then A12: f.a is quasi-term of C() by A1; a is expression of C(), s by A10,Def28; then s = a_Term C() by Th48; hence thesis by A12,Def28; end; A13: for c being constructor OperSymbol of C() for p being FinSequence of QuasiTerms C() st len p = len the_arity_of c & for t being quasi-term of C() st t in rng p holds P[t] holds P[c-trm p] proof let c be constructor OperSymbol of C(); let p be FinSequence of QuasiTerms C(); assume that A14: len p = len the_arity_of c and A15: for t being quasi-term of C() st t in rng p holds P[t]; set a = c-trm p; set nt = [c, the carrier of C()]; let s be SortSymbol of C() such that A16: a in (the Sorts of A).s; nt in NonTerminals G by Th123; then reconsider nt as Symbol of G; reconsider ts = p as FinSequence of TS G by Th128; A17: a = nt-tree ts by A14,Def35; reconsider aa = a as Term of C(), X by MSAFREE3:8; the Sorts of A = C()-Terms(V,X) by MSAFREE3:24; then the Sorts of A c= the Sorts of FreeMSA X by PBOOLE:def 18; then (the Sorts of A).s c= (the Sorts of FreeMSA X).s; then aa in (FreeSort X).s by A16; then aa in FreeSort(X,s) by MSAFREE:def 11; then A18: the_sort_of aa = s by MSATERM:def 5; A19: c <> * by Def11; A20: c <> non_op by Def11; A21: rng p c= QuasiTerms C() by FINSEQ_1:def 4; dom f = D by FUNCT_2:def 1; then A22: rng p c= dom f; rng(f*p) c= QuasiTerms C() proof let z be object; assume z in rng(f*p); then consider i being object such that A23: i in dom(f*p) and A24: z = (f*p).i by FUNCT_1:def 3; i in dom p by A22,A23,RELAT_1:27; then A25: p.i in rng p by FUNCT_1:def 3; then reconsider pi1 = p.i as quasi-term of C() by A21,Th41; pi1 in (the Sorts of A).a_Term C() by Th41; then f.pi1 in (the Sorts of A).a_Term C() by A15,A25; hence thesis by A23,A24,FUNCT_1:12; end; then reconsider q = f*p as FinSequence of QuasiTerms C() by FINSEQ_1:def 4; rng p c= C()-Terms X proof let z be object; assume z in rng p; then z is Element of C()-TermsX by MSAFREE3:8; hence thesis; end; then reconsider r = p as FinSequence of C()-Terms X by FINSEQ_1:def 4; A26: len q = len p by A22,FINSEQ_2:29; a is Term of C(), X by MSAFREE3:8; then A27: r is ArgumentSeq of Sym(c, X) by A17,MSATERM:1; then A28: the_result_sort_of c = s by A17,A18,MSATERM:20; Sym(c, X) ==> roots r by A27,MSATERM:21; then nt ==> roots ts by Th127; then f.a = NTermVal(nt, roots ts, f * ts) by A6,A17 .= IFEQ(c,non_op, N((f * ts).1), F(c, f * ts)) by A19, FUNCOP_1:def 8 .= F(c, f * ts) by A20,FUNCOP_1:def 8; then f.a is expression of C(), the_result_sort_of c by A2,A14,A26; hence thesis by A28,Def28; end; A29: for a being expression of C(), an_Adj C() st P[a] holds P[(non_op C())term a] proof let v be expression of C(), an_Adj C() such that A30: P[v]; A31: v in U.an_Adj C() by Def28; then f.v in U.an_Adj C() by A30; then reconsider fv = f.v as expression of C(), an_Adj C() by Def28; let s be SortSymbol of C(); assume A32: (non_op C())term v in U.s; A33: (non_op C())term v is expression of C(), an_Adj C() by Th43; (non_op C())term v is expression of C(), s by A32,Def28; then A34: s = an_Adj C() by A33,Th48; set QA = U.an_Adj C(); rng <*v*> = {v} by FINSEQ_1:38; then rng <*v*> c= QA by A31,ZFMISC_1:31; then reconsider p = <*v*> as FinSequence of QA by FINSEQ_1:def 4; set c = non_op C(); set a = (non_op C())term v; set nt = [c, the carrier of C()]; nt in NonTerminals G by Th123; then reconsider nt as Symbol of G; reconsider ts = p as FinSequence of TS G by Th128; A35: a = nt-tree ts by Th43; dom f = D by FUNCT_2:def 1; then A36: f*p = <*fv*> by FINSEQ_2:34; rng p c= C()-Terms X proof let z be object; assume z in rng p; then z is expression of C(), an_Adj C() by Th41; then z is Element of C()-TermsX by MSAFREE3:8; hence thesis; end; then reconsider r = p as FinSequence of C()-Terms X by FINSEQ_1:def 4; a is Term of C(), X by MSAFREE3:8; then r is ArgumentSeq of Sym(c, X) by A35,MSATERM:1; then Sym(c, X) ==> roots r by MSATERM:21; then nt ==> roots ts by Th127; then f.a = NTermVal(nt, roots ts, f * ts) by A6,A35 .= IFEQ(c,non_op, N((f * ts).1), F(c, f * ts)) by FUNCOP_1:def 8 .= N((f*ts).1) by FUNCOP_1:def 8 .= N(fv) by A36,FINSEQ_1:40; then f.a is expression of C(), an_Adj C() by A3; hence thesis by A34,Def28; end; A37: for a being expression of C(), an_Adj C() st P[a] for t being expression of C(), a_Type C() st P[t] holds P[(ast C())term(a,t)] proof let v be expression of C(), an_Adj C() such that A38: P[v]; let t be expression of C(), a_Type C() such that A39: P[t]; A40: v in U.an_Adj C() by Def28; A41: t in U.a_Type C() by Def28; A42: f.v in U.an_Adj C() by A38,A40; A43: f.t in U.a_Type C() by A39,A41; reconsider fv = f.v as expression of C(), an_Adj C() by A42,Def28; reconsider ft = f.t as expression of C(), a_Type C() by A43,Def28; let s be SortSymbol of C(); assume A44: (ast C())term(v,t) in U.s; A45: (ast C())term(v,t) is expression of C(), a_Type C() by Th46; (ast C())term(v,t) is expression of C(), s by A44,Def28; then A46: s = a_Type C() by A45,Th48; reconsider p = <*v,t*> as FinSequence of D; set c = ast C(); set a = (ast C())term(v,t); set nt = [c, the carrier of C()]; nt in NonTerminals G by Th123; then reconsider nt as Symbol of G; reconsider ts = p as FinSequence of TS G by Th128; A47: a = nt-tree ts by Th46; A48: f*p = <*fv,ft*> by FINSEQ_2:36; rng p c= C()-Terms X proof let z be object; assume z in rng p; then z is Element of C()-TermsX by MSAFREE3:8; hence thesis; end; then reconsider r = p as FinSequence of C()-Terms X by FINSEQ_1:def 4; a is Term of C(), X by MSAFREE3:8; then r is ArgumentSeq of Sym(c, X) by A47,MSATERM:1; then Sym(c, X) ==> roots r by MSATERM:21; then nt ==> roots ts by Th127; then f.a = NTermVal(nt, roots ts, f * ts) by A6,A47 .= A((f*ts).1,(f*ts).2) by FUNCOP_1:def 8 .= A(fv,(f*ts).2) by A48,FINSEQ_1:44 .= A(fv,ft) by A48,FINSEQ_1:44; then f.a is expression of C(), a_Type C() by A4; hence thesis by A46,Def28; end; P[a] from StructInd(A9,A13,A29,A37); hence thesis by A7,A8; end; then reconsider f as term-transformation of C(), MSVars C(); take f; hereby let x be variable; x in Vars; then A49: x in V.a_Term C() by Def25; reconsider x9 = x as Element of V.a_Term C() by Def25; reconsider xx = [x9,a_Term C()] as Symbol of G by A49,MSAFREE3:2; xx in Terminals G by A49,MSAFREE:7; hence f.(x-term C()) = V(xx`1) by A5 .= V(x); end; hereby let c be constructor OperSymbol of C(); let p,q be FinSequence of QuasiTerms C(); assume that A50: len p = len the_arity_of c and A51: q = f*p; set a = c-trm p; set nt = [c, the carrier of C()]; nt in NonTerminals G by Th123; then reconsider nt as Symbol of G; reconsider ts = p as FinSequence of TS G by Th128; A52: a = nt-tree ts by A50,Def35; A53: c <> * by Def11; A54: c <> non_op by Def11; rng p c= C()-Terms X proof let z be object; assume z in rng p; then z is Element of C()-TermsX by MSAFREE3:8; hence thesis; end; then reconsider r = p as FinSequence of C()-Terms X by FINSEQ_1:def 4; a is Term of C(), X by MSAFREE3:8; then r is ArgumentSeq of Sym(c, X) by A52,MSATERM:1; then Sym(c, X) ==> roots r by MSATERM:21; then nt ==> roots ts by Th127; then f.a = NTermVal(nt, roots ts, f * ts) by A6,A52 .= IFEQ(c,non_op, N((f * ts).1), F(c, f * ts)) by A53,FUNCOP_1:def 8 .= F(c, f * ts) by A54,FUNCOP_1:def 8; hence f.(c-trm p) = F(c, q) by A51; end; hereby let v be expression of C(), an_Adj C(); A55: v in U.an_Adj C() by Def28; then f.v in U.an_Adj C() by Th129; then reconsider fv = f.v as expression of C(), an_Adj C() by Def28; set QA = U.an_Adj C(); rng <*v*> = {v} by FINSEQ_1:38; then rng <*v*> c= QA by A55,ZFMISC_1:31; then reconsider p = <*v*> as FinSequence of QA by FINSEQ_1:def 4; set c = non_op C(); set a = (non_op C())term v; set nt = [c, the carrier of C()]; nt in NonTerminals G by Th123; then reconsider nt as Symbol of G; reconsider ts = p as FinSequence of TS G by Th128; A56: a = nt-tree ts by Th43; dom f = D by FUNCT_2:def 1; then A57: f*p = <*fv*> by FINSEQ_2:34; rng p c= C()-Terms X proof let z be object; assume z in rng p; then z is expression of C(), an_Adj C() by Th41; then z is Element of C()-TermsX by MSAFREE3:8; hence thesis; end; then reconsider r = p as FinSequence of C()-Terms X by FINSEQ_1:def 4; a is Term of C(), X by MSAFREE3:8; then r is ArgumentSeq of Sym(c, X) by A56,MSATERM:1; then Sym(c, X) ==> roots r by MSATERM:21; then nt ==> roots ts by Th127; then f.a = NTermVal(nt, roots ts, f * ts) by A6,A56 .= IFEQ(c,non_op, N((f * ts).1), F(c, f * ts)) by FUNCOP_1:def 8 .= N((f*ts).1) by FUNCOP_1:def 8; hence f.((non_op C())term v) = N(f.v) by A57,FINSEQ_1:40; end; let v be expression of C(), an_Adj C(); let t be expression of C(), a_Type C(); A58: v in U.an_Adj C() by Def28; A59: t in U.a_Type C() by Def28; A60: f.v in U.an_Adj C() by A58,Th129; A61: f.t in U.a_Type C() by A59,Th129; reconsider fv = f.v as expression of C(), an_Adj C() by A60,Def28; reconsider ft = f.t as expression of C(), a_Type C() by A61,Def28; reconsider p = <*v,t*> as FinSequence of D; set c = ast C(); set a = (ast C())term(v,t); set nt = [c, the carrier of C()]; nt in NonTerminals G by Th123; then reconsider nt as Symbol of G; reconsider ts = p as FinSequence of TS G by Th128; A62: a = nt-tree ts by Th46; A63: f*p = <*fv,ft*> by FINSEQ_2:36; rng p c= C()-Terms X proof let z be object; assume z in rng p; then z is Element of C()-TermsX by MSAFREE3:8; hence thesis; end; then reconsider r = p as FinSequence of C()-Terms X by FINSEQ_1:def 4; a is Term of C(), X by MSAFREE3:8; then r is ArgumentSeq of Sym(c, X) by A62,MSATERM:1; then Sym(c, X) ==> roots r by MSATERM:21; then nt ==> roots ts by Th127; then f.a = NTermVal(nt, roots ts, f * ts) by A6,A62 .= A((f*ts).1,(f*ts).2) by FUNCOP_1:def 8 .= A(fv,(f*ts).2) by A63,FINSEQ_1:44; hence thesis by A63,FINSEQ_1:44; end; begin :: Substitution definition let A be set; let x,y be set; let a,b be Element of A; redefine func IFIN(x,y,a,b) -> Element of A; coherence by MATRIX_7:def 1; end; definition let C be initialized ConstructorSignature; mode valuation of C is PartFunc of Vars, QuasiTerms C; end; definition let C be initialized ConstructorSignature; let f be valuation of C; attr f is irrelevant means : Def58: for x being variable st x in dom f ex y being variable st f.x = y-term C; end; notation let C be initialized ConstructorSignature; let f be valuation of C; antonym f is relevant for f is irrelevant; end; registration let C be initialized ConstructorSignature; cluster empty -> irrelevant for valuation of C; coherence; end; registration let C be initialized ConstructorSignature; cluster empty for valuation of C; existence proof take {}(Vars, QuasiTerms C); thus thesis; end; end; definition let C be initialized ConstructorSignature; let X be Subset of Vars; func C idval X -> valuation of C equals {[x, x-term C] where x is variable: x in X}; coherence proof set f = {[x, x-term C] where x is variable: x in X}; defpred P[variable,set] means $2 = $1-term C; A1: now let x be variable; reconsider t = x-term C as Element of QuasiTerms C by Def28; take t; thus P[x,t]; end; consider g being Function of Vars, QuasiTerms C such that A2: for x being variable holds P[x,g.x] from FUNCT_2:sch 3(A1); f c= g proof let a be object; assume a in f; then consider x being variable such that A3: a = [x, x-term C] and x in X; A4: g.x = x-term C by A2; dom g = Vars by FUNCT_2:def 1; hence thesis by A3,A4,FUNCT_1:1; end; hence thesis by RELSET_1:1; end; end; theorem Th131: for X being Subset of Vars holds dom (C idval X) = X & for x being variable st x in X holds (C idval X).x = x-term C proof let X be Subset of Vars; set f = C idval X; thus dom f c= X proof let a being object; assume a in dom f; then [a,f.a] in f by FUNCT_1:def 2; then ex x being variable st [a,f.a] = [x,x-term C] & x in X; hence thesis by XTUPLE_0:1; end; hereby let x be object; assume A1: x in X; then reconsider a = x as variable; [a,a-term C] in f by A1; hence x in dom f by FUNCT_1:1; end; let x be variable; assume x in X; then [x,x-term C] in C idval X; hence thesis by FUNCT_1:1; end; registration let C be initialized ConstructorSignature; let X be Subset of Vars; cluster C idval X -> irrelevant one-to-one; coherence proof set f = C idval X; A1: dom f = X by Th131; hereby let x be variable; assume A2: x in dom f; take y = x; thus f.x = y-term C by A1,A2,Th131; end; let x,y be object; assume that A3: x in dom f and A4: y in dom f; reconsider x,y as variable by A3,A4; A5: f.x = x-term C by A1,A3,Th131; f.y = y-term C by A1,A4,Th131; hence thesis by A5,Th50; end; end; registration let C be initialized ConstructorSignature; let X be empty Subset of Vars; cluster C idval X -> empty; coherence proof dom (C idval X) = X by Th131; hence thesis; end; end; definition let C; let f be valuation of C; func f# -> term-transformation of C, MSVars C means : Def60: (for x being variable holds (x in dom f implies it.(x-term C) = f.x) & (not x in dom f implies it.(x-term C) = x-term C)) & (for c being constructor OperSymbol of C for p,q being FinSequence of QuasiTerms C st len p = len the_arity_of c & q = it*p holds it.(c-trm p) = c-trm q) & (for a being expression of C, an_Adj C holds it.((non_op C)term a) = (non_op C)term (it.a)) & for a being expression of C, an_Adj C for t being expression of C, a_Type C holds it.((ast C)term(a,t)) = (ast C)term(it.a, it.t); existence proof deffunc V(variable) = IFIN($1, dom f, (f/.$1 qua Element of (QuasiTerms C) qua non empty Subset of Free(C, MSVars C)) qua (expression of C), $1-term C); deffunc F(constructor OperSymbol of C,FinSequence of QuasiTerms C) = $1-trm $2; deffunc N(expression of C) = (non_op C)term $1; deffunc A((expression of C), expression of C) = (ast C)term($1,$2); A1: for x being variable holds V(x) is quasi-term of C proof let x be variable; f/.x is quasi-term of C by Th41; hence thesis by MATRIX_7:def 1; end; A2: for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len the_arity_of c holds F(c, p) is expression of C, the_result_sort_of c by Th52; A3: for a holds N(a) is expression of C, an_Adj C by Th43; A4: for a,t holds A(a,t) is expression of C, a_Type C by Th46; consider f9 being term-transformation of C, MSVars C such that A5: (for x being variable holds f9.(x-term C) = V(x)) & (for c being constructor OperSymbol of C for p,q being FinSequence of QuasiTerms C st len p = len the_arity_of c & q = f9*p holds f9.(c-trm p) = F(c, q)) & (for a holds f9.((non_op C)term a) = N(f9.a)) & for a,t holds f9.((ast C)term(a,t)) = A(f9.a, f9.t) from StructDef(A1,A2,A3,A4); take f9; hereby let x be variable; A6: f9.(x-term C) = V(x) by A5; x in dom f implies f/.x = f.x by PARTFUN1:def 6; hence x in dom f implies f9.(x-term C) = f.x by A6,MATRIX_7:def 1; thus not x in dom f implies f9.(x-term C) = x-term C by A6,MATRIX_7:def 1 ; end; thus thesis by A5; end; correctness proof let f1,f2 be term-transformation of C, MSVars C such that A7: for x being variable holds (x in dom f implies f1.(x-term C) = f.x) & (not x in dom f implies f1.(x-term C) = x-term C) and A8: for c being constructor OperSymbol of C for p,q being FinSequence of QuasiTerms C st len p = len the_arity_of c & q = f1*p holds f1.(c-trm p) = c -trm q and A9: for a being expression of C, an_Adj C holds f1.((non_op C)term a) = (non_op C)term (f1.a) and A10: for a being expression of C, an_Adj C for t being expression of C, a_Type C holds f1.((ast C)term(a,t)) = (ast C)term(f1.a, f1.t) and A11: for x being variable holds (x in dom f implies f2.(x-term C) = f.x ) & (not x in dom f implies f2.(x-term C) = x-term C) and A12: for c being constructor OperSymbol of C for p,q being FinSequence of QuasiTerms C st len p = len the_arity_of c & q = f2*p holds f2.(c-trm p) = c -trm q and A13: for a being expression of C, an_Adj C holds f2.((non_op C)term a) = (non_op C)term (f2.a) and A14: for a being expression of C, an_Adj C for t being expression of C, a_Type C holds f2.((ast C)term(a,t)) = (ast C)term(f2.a, f2.t); set D = Union the Sorts of Free(C, MSVars C); A15: dom f1 = D by FUNCT_2:def 1; A16: dom f2 = D by FUNCT_2:def 1; defpred P[expression of C] means f1.$1 = f2.$1; A17: for x being variable holds P[x-term C] proof let x be variable; x in dom f & f1.(x-term C) = f.x or x nin dom f & f1.(x-term C) = x-term C by A7; hence thesis by A11; end; A18: for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len the_arity_of c & for t being quasi-term of C st t in rng p holds P[t] holds P[c-trm p] proof let c be constructor OperSymbol of C; let p be FinSequence of QuasiTerms C; assume that A19: len p = len the_arity_of c and A20: for t being quasi-term of C st t in rng p holds P[t]; A21: rng p c= QuasiTerms C by FINSEQ_1:def 4; A22: rng(f1*p) = f1.:rng p by RELAT_1:127; A23: rng(f2*p) = f2.:rng p by RELAT_1:127; A24: rng(f1*p) c= f1.:QuasiTerms C by A21,A22,RELAT_1:123; A25: rng(f2*p) c= f2.:QuasiTerms C by A21,A23,RELAT_1:123; A26: f1.:QuasiTerms C c= QuasiTerms C by Def56; A27: f2.:QuasiTerms C c= QuasiTerms C by Def56; A28: rng(f1*p) c= QuasiTerms C by A24,A26; rng(f2*p) c= QuasiTerms C by A25,A27; then reconsider q1 = f1*p, q2 = f2*p as FinSequence of QuasiTerms C by A28,FINSEQ_1:def 4; A29: rng p c= D; then A30: dom q1 = dom p by A15,RELAT_1:27; A31: dom q2 = dom p by A16,A29,RELAT_1:27; now let i be Nat; assume A32: i in dom p; then A33: q1.i = f1.(p.i) by FUNCT_1:13; A34: q2.i = f2.(p.i) by A32,FUNCT_1:13; A35: p.i in rng p by A32,FUNCT_1:def 3; then p.i is quasi-term of C by A21,Th41; hence q1.i = q2.i by A20,A33,A34,A35; end; then f1.(c-trm p) = c-trm q2 by A8,A19,A30,A31,FINSEQ_1:13; hence thesis by A12,A19; end; A36: for a being expression of C, an_Adj C st P[a] holds P[(non_op C)term a] proof let a be expression of C, an_Adj C; assume P[a]; then f1.((non_op C)term a) = (non_op C)term (f2.a) by A9; hence thesis by A13; end; A37: for a being expression of C, an_Adj C st P[a] for t being expression of C, a_Type C st P[t] holds P[(ast C)term(a,t)] proof let a be expression of C, an_Adj C such that A38: P[a]; let t be expression of C, a_Type C; assume P[t]; then f1.((ast C)term(a,t)) = (ast C)term(f2.a,f2.t) by A10,A38; hence thesis by A14; end; now let t be expression of C; thus P[t] from StructInd(A17,A18,A36,A37); end; hence thesis by FUNCT_2:63; end; end; registration let C; let f be valuation of C; cluster f# -> substitution; coherence proof let o be OperSymbol of C; let p,q be FinSequence of Free(C, MSVars C) such that A1: [o, the carrier of C]-tree p in Union the Sorts of Free(C, MSVars C) and A2: q = f# *p; A3: dom (f# ) = Union the Sorts of Free(C, MSVars C) by FUNCT_2:def 1; reconsider t = [o, the carrier of C]-tree p as expression of C by A1; A4: t.{} = [o, the carrier of C] by TREES_4:def 4; per cases; suppose o is constructor; then reconsider c = o as constructor OperSymbol of C; A5: t = [c, the carrier of C]-tree p; then A6: len p = len the_arity_of c by Th51; p in (QuasiTerms C)* by A5,Th51; then reconsider p9 = p as FinSequence of QuasiTerms C by FINSEQ_1:def 11; reconsider q9 = f# *p9 as FinSequence of QuasiTerms C by Th130; A7: len q9 = len p by Th130; thus f# .([o, the carrier of C]-tree p) = f# .(c-trm p9) by A6,Def35 .= c-trm q9 by A6,Def60 .= [o, the carrier of C]-tree q by A2,A6,A7,Def35; end; suppose A8: o = *; then consider a being expression of C, an_Adj C, s being expression of C, a_Type C such that A9: t = (ast C)term(a,s) by A4,Th58; a in (the Sorts of Free(C, MSVars C)).an_Adj C by Def28; then f#.a in (the Sorts of Free(C, MSVars C)).an_Adj C by Th129; then reconsider fa = f#.a as expression of C, an_Adj C by Th41; s in (the Sorts of Free(C, MSVars C)).a_Type C by Def28; then f#.s in (the Sorts of Free(C, MSVars C)).a_Type C by Th129; then reconsider fs = f#.s as expression of C, a_Type C by Th41; t = [ast C, the carrier of C]-tree <*a,s*> by A9,Th46; then p = <*a,s*> by TREES_4:15; then q = <*fa, fs*> by A2,A3,FINSEQ_2:125; then [o, the carrier of C]-tree q = (ast C)term(fa, fs) by A8,Th46; hence thesis by A9,Def60; end; suppose A10: o = non_op; then consider a such that A11: t = (non_op C)term a by A4,Th57; a in (the Sorts of Free(C, MSVars C)).an_Adj C by Def28; then f#.a in (the Sorts of Free(C, MSVars C)).an_Adj C by Th129; then reconsider fa = f#.a as expression of C, an_Adj C by Th41; t = [non_op C, the carrier of C]-tree <*a*> by A11,Th43; then p = <*a*> by TREES_4:15; then q = <*fa*> by A2,A3,FINSEQ_2:34; then [o, the carrier of C]-tree q = (non_op C)term fa by A10,Th43; hence thesis by A11,Def60; end; end; end; reserve f for valuation of C; definition let C,f,e; func e at f -> expression of C equals f#.e; coherence; end; definition let C,f; let p be FinSequence such that A1: rng p c= Union the Sorts of Free(C, MSVars C); func p at f -> FinSequence equals : Def62: f# * p; coherence proof set A = Free(C, MSVars C); dom (f# ) = Union the Sorts of A by FUNCT_2:def 1; then A2: dom (f# *p) = dom p by A1,RELAT_1:27; dom p = Seg len p by FINSEQ_1:def 3; hence thesis by A2,FINSEQ_1:def 2; end; end; definition let C,f; let p be FinSequence of QuasiTerms C; redefine func p at f -> FinSequence of QuasiTerms C equals f# * p; coherence proof A1: f# *p is FinSequence of QuasiTerms C by Th130; rng p c= Union the Sorts of Free(C, MSVars C); hence thesis by A1,Def62; end; compatibility proof rng p c= Union the Sorts of Free(C, MSVars C); hence thesis by Def62; end; end; reserve x for variable; theorem not x in dom f implies (x-term C)at f = x-term C by Def60; theorem x in dom f implies (x-term C)at f = f.x by Def60; theorem len p = len the_arity_of c implies (c-trm p)at f = c-trm p at f by Def60; theorem ((non_op C)term a)at f = (non_op C)term(a at f) by Def60; theorem ((ast C)term(a,t))at f = (ast C)term(a at f,t at f) by Def60; theorem Th137: for X being Subset of Vars holds e at (C idval X) = e proof set t = e; let X be Subset of Vars; set f = C idval X; defpred P[expression of C] means $1 at f = $1; A1: for x being variable holds P[x-term C] proof let x be variable; A2: x in X or x nin X; A3: dom f = X by Th131; x in X implies f.x = x-term C by Th131; hence thesis by A2,A3,Def60; end; A4: for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len the_arity_of c & for t being quasi-term of C st t in rng p holds P[t] holds P[c-trm p] proof let c be constructor OperSymbol of C; let p be FinSequence of QuasiTerms C such that A5: len p = len the_arity_of c and A6: for t being quasi-term of C st t in rng p holds P[t]; len (p at f) = len p by Th130; then A7: dom (p at f) = dom p by FINSEQ_3:29; now let i be Nat; assume A8: i in dom p; then A9: p.i in rng p by FUNCT_1:def 3; rng p c= QuasiTerms C by FINSEQ_1:def 4; then reconsider pi1 = p.i as quasi-term of C by A9,Th41; (p at f).i = pi1 at f by A8,FUNCT_1:13; hence (p at f).i = p.i by A6,A9; end; then p at f = p by A7; hence thesis by A5,Def60; end; A10: for a being expression of C, an_Adj C st P[a] holds P[(non_op C)term a] by Def60; A11: for a being expression of C, an_Adj C st P[a] for t being expression of C, a_Type C st P[t] holds P[(ast C)term(a,t)] by Def60; thus P[t] from StructInd(A1,A4,A10,A11); end; theorem for X being Subset of Vars holds (C idval X)# = id Union the Sorts of Free(C, MSVars C) proof let X be Subset of Vars; set f = C idval X; A1: dom (f# ) = Union the Sorts of Free(C, MSVars C) by FUNCT_2:def 1; now let x be object; assume x in Union the Sorts of Free(C, MSVars C); then reconsider t = x as expression of C; thus (f# ).x = t at f .= x by Th137; end; hence thesis by A1,FUNCT_1:17; end; theorem Th139: for f being empty valuation of C holds e at f = e proof let f be empty valuation of C; f = C idval {}Vars; hence thesis by Th137; end; theorem for f being empty valuation of C holds f# = id Union the Sorts of Free(C, MSVars C) proof let f be empty valuation of C; A1: dom (f# ) = Union the Sorts of Free(C, MSVars C) by FUNCT_2:def 1; now let x be object; assume x in Union the Sorts of Free(C, MSVars C); then reconsider t = x as expression of C; thus (f# ).x = t at f .= x by Th139; end; hence thesis by A1,FUNCT_1:17; end; definition let C,f; let t be quasi-term of C; redefine func t at f -> quasi-term of C; coherence proof t in QuasiTerms C by Def28; then t at f in QuasiTerms C by Th129; hence thesis by Th41; end; end; definition let C,f; let a be expression of C, an_Adj C; redefine func a at f -> expression of C, an_Adj C; coherence proof a in (the Sorts of Free(C, MSVars C)).an_Adj C by Def28; then a at f in (the Sorts of Free(C, MSVars C)).an_Adj C by Th129; hence thesis by Th41; end; end; registration let C,f; let a be positive expression of C, an_Adj C; cluster a at f -> positive for expression of C, an_Adj C; coherence proof consider v being constructor OperSymbol of C such that A1: the_result_sort_of v = an_Adj C and A2: ex p being FinSequence of QuasiTerms C st len p = len the_arity_of v & a = v-trm p by Th65; consider p being FinSequence of QuasiTerms C such that A3: len p = len the_arity_of v and A4: a = v-trm p by A2; A5: len (p at f) = len p by Th130; a at f = v-trm(p at f) by A3,A4,Def60; hence thesis by A1,A3,A5,Th66; end; end; registration let C,f; let a be negative expression of C, an_Adj C; cluster a at f -> negative for expression of C, an_Adj C; coherence proof (non_op C)term (Non a) = a by Th62; then a at f = (non_op C)term((Non a)at f) by Def60 .= Non ((Non a)at f) by Th59; hence thesis; end; end; definition let C,f; let a be quasi-adjective of C; redefine func a at f -> quasi-adjective of C; coherence proof per cases; suppose a is positive; then reconsider a as positive quasi-adjective of C; a at f is positive; hence thesis; end; suppose a is negative; then reconsider a as negative quasi-adjective of C; a at f is negative; hence thesis; end; end; end; theorem (Non a) at f = Non (a at f) proof per cases; suppose a is positive; then reconsider b = a as positive expression of C, an_Adj C; reconsider af = b at f as positive expression of C, an_Adj C; thus (Non a) at f = ((non_op C)term b) at f by Th59 .= (non_op C)term af by Def60 .= Non (a at f) by Th59; end; suppose a is non positive; then consider b being expression of C, an_Adj C such that A1: a = (non_op C)term b and A2: Non a = b by Th60; A3: a at f = (non_op C)term(b at f) by A1,Def60; then a at f is non positive; then ex k being expression of C, an_Adj C st a at f = (non_op C)term k & Non(a at f) = k by Th60; hence thesis by A2,A3,Th44; end; end; definition let C,f; let t be expression of C, a_Type C; redefine func t at f -> expression of C, a_Type C; coherence proof t in (the Sorts of Free(C, MSVars C)).a_Type C by Def28; then t at f in (the Sorts of Free(C, MSVars C)).a_Type C by Th129; hence thesis by Th41; end; end; registration let C,f; let t be pure expression of C, a_Type C; cluster t at f -> pure for expression of C, a_Type C; coherence proof consider m being constructor OperSymbol of C such that A1: the_result_sort_of m = a_Type C and A2: ex p being FinSequence of QuasiTerms C st len p = len the_arity_of m & t = m-trm p by Th74; consider p being FinSequence of QuasiTerms C such that A3: len p = len the_arity_of m and A4: t = m-trm p by A2; A5: len (p at f) = len p by Th130; t at f = m-trm(p at f) by A3,A4,Def60; hence thesis by A1,A3,A5,Th75; end; end; theorem for f being irrelevant one-to-one valuation of C ex g being irrelevant one-to-one valuation of C st for x,y being variable holds x in dom f & f.x = y-term C iff y in dom g & g.y = x-term C proof let f be irrelevant one-to-one valuation of C; set Y = {x where x is variable: x-term C in rng f}; defpred P[object,object] means ex x being set st x in dom f & f.x = root-tree [ $1, a_Term] & $2 = root-tree [x, a_Term]; A1: for x being object st x in Y ex y being object st P[x,y] proof let x be object; assume x in Y; then A2: ex z being variable st x = z & z-term C in rng f; then reconsider z = x as variable; consider y being object such that A3: y in dom f and A4: z-term C = f.y by A2,FUNCT_1:def 3; reconsider y as variable by A3; take y-term C; thus thesis by A3,A4; end; consider g being Function such that A5: dom g = Y and A6: for y being object st y in Y holds P[y,g.y] from CLASSES1:sch 1(A1); A7: Y c= Vars proof let x be object; assume x in Y; then ex z being variable st x = z & z-term C in rng f; hence thesis; end; rng g c= QuasiTerms C proof let y be object; assume y in rng g; then consider x being object such that A8: x in dom g and A9: y = g.x by FUNCT_1:def 3; reconsider x as variable by A5,A7,A8; consider z being set such that A10: z in dom f and f.z = x-term C and A11: g.x = root-tree [z,a_Term] by A5,A6,A8; reconsider z as variable by A10; y = z-term C by A9,A11; hence thesis by Def28; end; then reconsider g as valuation of C by A5,A7,RELSET_1:4; A12: g is irrelevant proof let x be variable; assume x in dom g; then consider y being set such that A13: y in dom f and f.y = x-term C and A14: g.x = root-tree [y,a_Term] by A5,A6; reconsider y as variable by A13; take y; thus thesis by A14; end; g is one-to-one proof let z1,z2 be object; assume that A15: z1 in dom g and A16: z2 in dom g and A17: g.z1 = g.z2; reconsider z1,z2 as variable by A15,A16; consider x1 being set such that A18: x1 in dom f and A19: f.x1 = z1-term C and A20: g.z1 = root-tree[x1,a_Term] by A5,A6,A15; consider x2 being set such that A21: x2 in dom f and A22: f.x2 = z2-term C and A23: g.z1 = root-tree[x2,a_Term] by A5,A6,A16,A17; reconsider x1,x2 as variable by A18,A21; x1-term C = x2-term C by A20,A23; then x1 = x2 by Th50; hence thesis by A19,A22,Th50; end; then reconsider g as irrelevant one-to-one valuation of C by A12; take g; let x,y be variable; hereby assume that A24: x in dom f and A25: f.x = y-term C; f.x in rng f by A24,FUNCT_1:def 3; hence y in dom g by A5,A25; then P[y,g.y] by A5,A6; hence g.y = x-term C by A24,A25,FUNCT_1:def 4; end; assume that A26: y in dom g and A27: g.y = x-term C; consider z being set such that A28: z in dom f and A29: f.z = root-tree [y, a_Term] and A30: x-term C = root-tree [z, a_Term] by A5,A6,A26,A27; reconsider z as variable by A28; x-term C = z-term C by A30; hence thesis by A28,A29,Th50; end; theorem for f,g being irrelevant one-to-one valuation of C st for x,y being variable holds x in dom f & f.x = y-term C implies y in dom g & g.y = x-term C for e st variables_in e c= dom f holds e at f at g = e proof let f,g be irrelevant one-to-one valuation of C such that A1: for x,y being variable holds x in dom f & f.x = y-term C implies y in dom g & g.y = x-term C; let t be expression of C; defpred P[expression of C] means variables_in $1 c= dom f implies $1 at f at g = $1; A2: for x being variable holds P[x-term C] proof let x be variable; assume variables_in (x-term C) c= dom f; then {x} c= dom f by MSAFREE3:10; then A3: x in dom f by ZFMISC_1:31; then consider y being variable such that A4: f.x = y-term C by Def58; A5: y in dom g by A1,A3,A4; A6: g.y = x-term C by A1,A3,A4; (x-term C) at f = y-term C by A3,A4,Def60; hence thesis by A5,A6,Def60; end; A7: for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len the_arity_of c & for t being quasi-term of C st t in rng p holds P[t] holds P[c-trm p] proof let c be constructor OperSymbol of C; let p be FinSequence of QuasiTerms C such that A8: len p = len the_arity_of c and A9: for t being quasi-term of C st t in rng p holds P[t] and A10: variables_in (c-trm p) c= dom f; c-trm p = [c, the carrier of C]-tree p by A8,Def35; then A11: variables_in (c-trm p) = union {variables_in s where s is quasi-term of C: s in rng p} by Th88; A12: len (p at f) = len p by Th130; A13: len (p at f at g) = len (p at f) by Th130; A14: dom (p at f) = dom p by A12,FINSEQ_3:29; A15: dom (p at f at g) = dom (p at f) by A13,FINSEQ_3:29; now let i be Nat; assume A16: i in dom p; then A17: (p at f).i = f# .(p.i) by FUNCT_1:13; A18: p.i in rng p by A16,FUNCT_1:def 3; rng p c= QuasiTerms C by FINSEQ_1:def 4; then reconsider pi1 = p.i as quasi-term of C by A18,Th41; variables_in pi1 in {variables_in s where s is quasi-term of C: s in rng p} by A18; then A19: variables_in pi1 c= variables_in (c-trm p) by A11,ZFMISC_1:74; (p at f at g).i = pi1 at f at g by A14,A16,A17,FUNCT_1:13; hence (p at f at g).i = p.i by A9,A10,A18,A19,XBOOLE_1:1; end; then A20: p at f at g = p by A14,A15; (c-trm p) at f = c-trm (p at f) by A8,Def60; hence thesis by A8,A12,A20,Def60; end; A21: for a being expression of C, an_Adj C st P[a] holds P[(non_op C)term a] proof let a be expression of C, an_Adj C such that A22: P[a] and A23: variables_in ((non_op C)term a) c= dom f; A24: (non_op C)term a = [non_op, the carrier of C]-tree <*a*> by Th43; thus ((non_op C)term a) at f at g = ((non_op C)term (a at f)) at g by Def60 .= (non_op C)term a by A22,A23,A24,Def60,Th93; end; A25: for a being expression of C, an_Adj C st P[a] for t being expression of C, a_Type C st P[t] holds P[(ast C)term(a,t)] proof let a be expression of C, an_Adj C such that A26: P[a]; let t be expression of C, a_Type C such that A27: P[t] and A28: variables_in ((ast C)term(a,t)) c= dom f; (ast C)term(a,t) = [ *, the carrier of C]-tree <*a,t*> by Th46; then A29: variables_in ((ast C)term(a,t)) = ((C variables_in a)(\/)(C variables_in t)).a_Term by Th96 .= (variables_in a)\/variables_in t by PBOOLE:def 4; thus ((ast C)term(a,t)) at f at g = ((ast C)term (a at f, t at f)) at g by Def60 .= (ast C)term(a,t) by A26,A27,A28,A29,Def60,XBOOLE_1:11; end; thus P[t] from StructInd(A2,A7,A21,A25); end; definition let C,f; let A be Subset of QuasiAdjs C; func A at f -> Subset of QuasiAdjs C equals {a at f where a is quasi-adjective of C: a in A}; coherence proof set X = {a at f where a is quasi-adjective of C: a in A}; X c= QuasiAdjs C proof let x be object; assume x in X; then ex a being quasi-adjective of C st x = a at f & a in A; hence thesis; end; hence thesis; end; end; theorem Th144: for A being Subset of QuasiAdjs C for a being quasi-adjective of C st A = {a} holds A at f = {a at f} proof let A be Subset of QuasiAdjs C; let a be quasi-adjective of C such that A1: A = {a}; thus A at f c= {a at f} proof let x be object; assume x in A at f; then ex b being quasi-adjective of C st x = b at f & b in A; then x = a at f by A1,TARSKI:def 1; hence thesis by TARSKI:def 1; end; let x be object; assume x in {a at f}; then A2: x = a at f by TARSKI:def 1; a in A by A1,TARSKI:def 1; hence thesis by A2; end; theorem Th145: for A,B being Subset of QuasiAdjs C holds (A \/ B) at f = (A at f) \/ (B at f) proof let A,B be Subset of QuasiAdjs C; thus (A \/ B) at f c= (A at f) \/ (B at f) proof let x be object; assume x in (A \/ B) at f; then consider a being quasi-adjective of C such that A1: x = a at f and A2: a in A \/ B; a in A or a in B by A2,XBOOLE_0:def 3; then x in A at f or x in B at f by A1; hence thesis by XBOOLE_0:def 3; end; let x be object; assume x in (A at f) \/ (B at f); then x in (A at f) or x in (B at f) by XBOOLE_0:def 3; then A c= A\/B & (ex a being quasi-adjective of C st x = a at f & a in A) or B c= A\/B & ex a being quasi-adjective of C st x = a at f & a in B by XBOOLE_1:7; hence thesis; end; theorem for A,B being Subset of QuasiAdjs C st A c= B holds A at f c= B at f proof let A,B be Subset of QuasiAdjs C; assume A c= B; then A\/B = B by XBOOLE_1:12; then B at f = (A at f)\/(B at f) by Th145; hence thesis by XBOOLE_1:7; end; registration let C be initialized ConstructorSignature; let f be valuation of C; let A be finite Subset of QuasiAdjs C; cluster A at f -> finite; coherence proof A1: A is finite; deffunc F(expression of C) = $1 at f; A2: { F(w) where w is expression of C: w in A } is finite from FRAENKEL:sch 21(A1); A at f c= { F(w) where w is expression of C: w in A } proof let x be object; assume x in A at f; then ex a being quasi-adjective of C st x = a at f & a in A; hence thesis; end; hence thesis by A2; end; end; definition let C be initialized ConstructorSignature; let f be valuation of C; let T be quasi-type of C; func T at f -> quasi-type of C equals ((adjs T) at f)ast((the_base_of T) at f); coherence; end; theorem for T being quasi-type of C holds adjs(T at f) = (adjs T) at f & the_base_of (T at f) = (the_base_of T) at f; theorem for T being quasi-type of C for a being quasi-adjective of C holds (a ast T) at f = (a at f) ast (T at f) proof let T be quasi-type of C; let a be quasi-adjective of C; a in QuasiAdjs C; then reconsider A = {a} as Subset of QuasiAdjs C by ZFMISC_1:31; thus (a ast T) at f = [(adjs (a ast T)) at f,((the_base_of T) at f)] .= [(A\/(adjs T)) at f,((the_base_of T) at f)] .= [(A at f)\/((adjs T) at f),(the_base_of T) at f] by Th145 .= [{a at f}\/((adjs T) at f),(the_base_of T) at f] by Th144 .= [{a at f}\/(adjs (T at f)),(the_base_of T) at f] .= (a at f) ast (T at f); end; definition let C be initialized ConstructorSignature; let f,g be valuation of C; func f at g -> valuation of C means : Def66: dom it = (dom f) \/ dom g & for x being variable st x in dom it holds it.x = ((x-term C) at f) at g; existence proof deffunc h(object) = ((In($1,Vars)-term C) at f) at g; consider h being Function such that A1: dom h = (dom f) \/ dom g and A2: for x being object st x in (dom f) \/ dom g holds h.x = h(x) from FUNCT_1:sch 3; rng h c= QuasiTerms C proof let y be object; assume y in rng h; then consider x being object such that A3: x in dom h and A4: y = h.x by FUNCT_1:def 3; y = h(x) by A1,A2,A3,A4; hence thesis by Def28; end; then reconsider h as valuation of C by A1,RELSET_1:4; take h; thus dom h = (dom f) \/ dom g by A1; let x be variable; assume x in dom h; then h.x = h(x) by A1,A2; hence thesis; end; uniqueness proof let h1,h2 be valuation of C such that A5: dom h1 = (dom f) \/ dom g and A6: for x being variable st x in dom h1 holds h1.x = ((x-term C) at f) at g and A7: dom h2 = (dom f) \/ dom g and A8: for x being variable st x in dom h2 holds h2.x = ((x-term C) at f) at g; now let x be variable; assume A9: x in dom h1; then h1.x = ((x-term C) at f) at g by A6; hence h1.x = h2.x by A5,A7,A8,A9; end; hence thesis by A5,A7; end; end; registration let C be initialized ConstructorSignature; let f,g be irrelevant valuation of C; cluster f at g -> irrelevant; coherence proof let x be variable; assume A1: x in dom (f at g); then A2: (f at g).x = ((x-term C) at f) at g by Def66; A3: dom (f at g) = dom f \/ dom g by Def66; per cases; suppose A4: x in dom f; then consider y being variable such that A5: f.x = y-term C by Def58; A6: (x-term C) at f = y-term C by A4,A5,Def60; then A7: y in dom g implies (f at g).x = g.y by A2,Def60; y nin dom g implies (f at g).x = y-term C by A2,A6,Def60; hence thesis by A7,Def58; end; suppose A8: x nin dom f; then A9: (x-term C) at f = x-term C by Def60; A10: x in dom g by A1,A3,A8,XBOOLE_0:def 3; then (f at g).x = g.x by A2,A9,Def60; hence thesis by A10,Def58; end; end; end; theorem Th149: for f1,f2 being valuation of C holds (e at f1) at f2 = e at (f1 at f2) proof set t = e; let f1,f2 be valuation of C; A1: dom (f1 at f2) = (dom f1) \/ dom f2 by Def66; defpred P[expression of C] means ($1 at f1) at f2 = $1 at (f1 at f2); A2: for x being variable holds P[x-term C] proof let x be variable; per cases; suppose A3: x in dom (f1 at f2); then (x-term C) at (f1 at f2) = (f1 at f2).x by Def60; hence thesis by A3,Def66; end; suppose A4: x nin dom (f1 at f2); then A5: x nin dom f1 by A1,XBOOLE_0:def 3; A6: x nin dom f2 by A1,A4,XBOOLE_0:def 3; A7: (x-term C) at f1 = x-term C by A5,Def60; (x-term C) at f2 = x-term C by A6,Def60; hence thesis by A4,A7,Def60; end; end; A8: for c being constructor OperSymbol of C for p being FinSequence of QuasiTerms C st len p = len the_arity_of c & for t being quasi-term of C st t in rng p holds P[t] holds P[c-trm p] proof let c be constructor OperSymbol of C; let p be FinSequence of QuasiTerms C such that A9: len p = len the_arity_of c and A10: for t being quasi-term of C st t in rng p holds P[t]; A11: len (p at f1) = len p by Th130; A12: len (p at (f1 at f2)) = len p by Th130; A13: len ((p at f1) at f2) = len (p at f1) by Th130; A14: dom (p at f1) = dom p by A11,FINSEQ_3:29; A15: dom (p at (f1 at f2)) = dom p by A12,FINSEQ_3:29; A16: dom ((p at f1) at f2) = dom p by A11,A13,FINSEQ_3:29; now let i be Nat; assume A17: i in dom p; then A18: ((p at f1) at f2).i = f2# .((p at f1).i) by A14,FUNCT_1:13; A19: p.i in rng p by A17,FUNCT_1:def 3; rng p c= QuasiTerms C by FINSEQ_1:def 4; then reconsider pi1 = p.i as quasi-term of C by A19,Th41; thus (p at f1 at f2).i = (pi1 at f1) at f2 by A17,A18,FUNCT_1:13 .= pi1 at (f1 at f2) by A10,A19 .= (p at (f1 at f2)).i by A17,FUNCT_1:13; end; then A20: p at f1 at f2 = p at (f1 at f2) by A15,A16; thus (c-trm p) at f1 at f2 = (c-trm(p at f1)) at f2 by A9,Def60 .= c-trm (p at (f1 at f2)) by A9,A11,A20,Def60 .= (c-trm p) at (f1 at f2) by A9,Def60; end; A21: for a being expression of C, an_Adj C st P[a] holds P[(non_op C)term a] proof let a be expression of C, an_Adj C; assume P[a]; then ((non_op C)term (a at f1)) at f2 = (non_op C)term (a at (f1 at f2)) by Def60 .= ((non_op C)term a) at (f1 at f2) by Def60; hence thesis by Def60; end; A22: for a being expression of C, an_Adj C st P[a] for t being expression of C, a_Type C st P[t] holds P[(ast C)term(a,t)] proof let a be expression of C, an_Adj C such that A23: P[a]; let t be expression of C, a_Type C; assume P[t]; then ((ast C)term (a at f1,t at f1)) at f2 = (ast C)term (a at (f1 at f2),t at (f1 at f2)) by A23,Def60 .= ((ast C)term(a,t)) at (f1 at f2) by Def60; hence thesis by Def60; end; thus P[t] from StructInd(A2,A8,A21,A22); end; theorem Th150: for A being Subset of QuasiAdjs C for f1,f2 being valuation of C holds (A at f1) at f2 = A at (f1 at f2) proof let A be Subset of QuasiAdjs C; let f1,f2 be valuation of C; thus (A at f1) at f2 c= A at (f1 at f2) proof let x be object; assume x in (A at f1) at f2; then consider a being quasi-adjective of C such that A1: x = a at f2 and A2: a in A at f1; consider b being quasi-adjective of C such that A3: a = b at f1 and A4: b in A by A2; x = b at (f1 at f2) by A1,A3,Th149; hence thesis by A4; end; let x be object; assume x in A at (f1 at f2); then consider a being quasi-adjective of C such that A5: x = a at (f1 at f2) and A6: a in A; A7: x = a at f1 at f2 by A5,Th149; a at f1 in A at f1 by A6; hence thesis by A7; end; theorem for T being quasi-type of C for f1,f2 being valuation of C holds (T at f1) at f2 = T at (f1 at f2) proof let T be quasi-type of C; let f1,f2 be valuation of C; thus (T at f1) at f2 = (((adjs T) at f1) at f2)ast((the_base_of (T at f1))at f2) .= ((adjs T) at (f1 at f2))ast((the_base_of (T at f1))at f2) by Th150 .= ((adjs T) at (f1 at f2))ast(((the_base_of T) at f1)at f2) .= T at (f1 at f2) by Th149; end;