formal/mizar/abcmiz_a.miz

:: A Model of Mizar Concepts -- Unification
::  by Grzegorz Bancerek

environ

 vocabularies TARSKI, QC_LANG3, PBOOLE, MSUALG_1, CATALG_1, FINSEQ_1, XBOOLE_0,
      ZFMISC_1, ARYTM_3, CARD_1, NAT_1, NUMBERS, XXREAL_0, ZF_LANG1, ORDINAL1,
      TREES_A, ABIAN, CARD_3, MEMBER_1, FINSET_1, FUNCOP_1, FUNCT_1, TREES_4,
      TREES_2, MSATERM, RELAT_1, MCART_1, MSAFREE, ZF_MODEL, AOFA_000,
      FINSEQ_2, PARTFUN1, QC_LANG1, FUNCT_2, ORDINAL4, CAT_3, TREES_3,
      ABCMIZ_0, ABCMIZ_1, ABCMIZ_A, STRUCT_0, FACIRC_1, INSTALG1, MSUALG_2,
      COMPUT_1, BINTREE1, TREES_9, ARYTM_1, FUNCT_6, SUBSET_1, MARGREL1,
      SETLIM_2;
 notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, XFAMILY, SUBSET_1, DOMAIN_1,
      RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FACIRC_1, ENUMSET1, FUNCOP_1,
      XCMPLX_0, XXREAL_0, ORDINAL1, NAT_D, MCART_1, FINSET_1, CARD_1, NUMBERS,
      CARD_3, FINSEQ_1, FINSEQ_2, FINSEQ_4, FUNCT_6, TREES_1, TREES_2, TREES_3,
      TREES_4, TREES_9, PBOOLE, STRUCT_0, MSUALG_1, MSUALG_2, MSAFREE,
      EQUATION, MSATERM, INSTALG1, CATALG_1, MSAFREE3, AOFA_000, ABCMIZ_1;
 constructors RELSET_1, DOMAIN_1, WELLORD2, TREES_9, EQUATION, NAT_D, FINSEQ_4,
      CATALG_1, FACIRC_1, ABCMIZ_1, PRE_POLY, XTUPLE_0, XFAMILY;
 registrations XBOOLE_0, SUBSET_1, XREAL_0, ORDINAL1, FUNCT_1, FINSET_1,
      STRUCT_0, PBOOLE, MSUALG_2, FINSEQ_1, NAT_1, CARD_1, TREES_3, TREES_2,
      FUNCOP_1, RELAT_1, INDEX_1, INSTALG1, MSAFREE3, WAYBEL26, FACIRC_1,
      ABCMIZ_1, MSATERM, RELSET_1, XTUPLE_0;
 requirements BOOLE, SUBSET, NUMERALS, ARITHM, REAL;
 definitions TARSKI, XBOOLE_0, RELAT_1, FUNCT_1, PBOOLE, ABCMIZ_1;
 equalities TARSKI, SUBSET_1, FINSEQ_1, MSAFREE, MSAFREE3, MSUALG_1, ABCMIZ_1,
      ORDINAL1;
 expansions TARSKI, XBOOLE_0, FUNCT_1, PBOOLE, ABCMIZ_1, ORDINAL1;
 theorems TARSKI, XBOOLE_0, XBOOLE_1, TREES_1, XXREAL_0, XREAL_1, ZFMISC_1,
      FUNCT_1, FUNCT_2, FINSEQ_1, FINSEQ_2, ENUMSET1, FUNCT_6, INSTALG1, NAT_1,
      MCART_1, PBOOLE, RELAT_1, RELSET_1, CARD_1, CARD_5, ORDINAL1, MSUALG_2,
      TREES_3, TREES_4, FINSEQ_3, FUNCOP_1, MSAFREE, MSATERM, MSAFREE3,
      YELLOW11, PARTFUN1, WELLORD2, ABCMIZ_1, TREES_9, XTUPLE_0, XREGULAR;
 schemes FUNCT_1, NAT_1, RECDEF_1, CLASSES1, FINSEQ_1;

begin :: Preliminary

reserve i,j for Nat;

::$CT

scheme MinimalElement{X() -> finite non empty set, R[set,set]}:
  ex x being set st x in X() &
  for y being set st y in X() holds not R[y,x]
  provided
A1: for x,y being set st x in X() & y in X() & R[x,y] holds not R[y,x] and
A2: for x,y,z being set st x in X() & y in X() & z in X() & R[x,y] & R[y,z]
    holds R[x,z]
  proof
    assume
A3: for x being set st x in X() ex y being set st y in X() & R[y,x];
    set n = card X();
    set x0 = the Element of X();
    defpred P[Nat,set,set] means $2 in X() implies $3 in X() & R[ $3,$2];
A4: for m being Nat st 1 <= m & m < n+1
    for x being set ex y being set st P[m,x,y]
    proof
      let m be Nat; assume 1 <= m & m < n+1;
      let x be set;
      per cases;
      suppose
A5:     x nin X();
        set y = the set;
        take y;
        thus P[m,x,y] by A5;
      end;
      suppose
        x in X(); then
        consider y being set such that
A6:     y in X() & R[y,x] by A3;
        take y; thus thesis by A6;
      end;
    end;
    consider p being FinSequence such that
A7: len p = n+1 and
A8: p.1 = x0 or n+1 = 0 and
A9: for i being Nat st 1 <= i & i < n+1 holds P[i, p.i, p.(i+1)]
    from RECDEF_1:sch 3(A4);
    defpred Q[Nat] means $1 in dom p implies p.$1 in X();
A10: Q[ 0] by FINSEQ_3:25;
A11: now let i be Nat; assume
A12:   Q[i];
      thus Q[i+1] proof assume
        i+1 in dom p; then i+1 <= n+1 by A7,FINSEQ_3:25; then
A13:     i < n+1 by NAT_1:13;
        per cases;
        suppose i = 0;
          hence thesis by A8;
        end;
        suppose i > 0; then
          i >= 0+1 & i is Element of NAT by NAT_1:13,ORDINAL1:def 12;
          hence thesis by A12,A7,A9,A13,FINSEQ_3:25;
        end;
      end;
    end;
A14: for i being Nat holds Q[i] from NAT_1:sch 2(A10,A11);
A15: rng p c= X()
    proof
      let x be object; assume x in rng p; then
      ex i being object st i in dom p & x = p.i by FUNCT_1:def 3;
      hence thesis by A14;
    end;
A16: for i,j being Nat st 1 <= i & i < j & j <= n+1 holds R[p.j, p.i]
    proof
      let i,j be Nat; assume
A17:   1 <= i;
      assume
A18:   i < j; then
      i+1 <= j by NAT_1:13; then
      consider k being Nat such that
A19:   j = i+1+k by NAT_1:10;
      assume
A20:   j <= n+1; then i <= n+1 by A18,XXREAL_0:2; then
A21:   i in dom p by A17,A7,FINSEQ_3:25;
      defpred S[Nat] means i+1+$1 <= n+1 implies R[p.(i+1+$1), p.i];
A22:   S[ 0] proof assume i+1+0 <= n+1; then
A23:     i < n+1 by NAT_1:13;
        p.i in X() & i is Element of NAT by A14,A21;
        hence R[p.(i+1+0), p.i] by A9,A17,A23;
      end;
A24:   now let k be Nat; assume
A25:     S[k];
        thus S[k+1] proof assume
A26:       i+1+(k+1) <= n+1;
A27:       i+1+(k+1) = i+1+k+1; then
A28:       i+1+k < n+1 by A26,NAT_1:13;
A29:       p.i in X() by A14,A21;
          i+1+k = 1+(i+k); then
A30:       1 <= i+1+k by NAT_1:11; then
          i+1+k in dom p by A7,A28,FINSEQ_3:25; then
A31:       p.(i+1+k) in X() & i+1+k is Element of NAT by A14; then
          p.(i+1+(k+1)) in X() & R[p.(i+1+(k+1)), p.(i+1+k)] by A9,A28,A27,A30;
          hence R[p.(i+1+(k+1)), p.i] by A2,A28,A25,A31,A29;
        end;
      end;
      for k being Nat holds S[k] from NAT_1:sch 2(A22,A24);
      hence R[p.j, p.i] by A19,A20;
    end;
A32: dom p = Seg(n+1) & card Seg(n+1) = n+1 by A7,FINSEQ_1:57,def 3;
    Segm card rng p c= Segm card X() by A15,CARD_1:11; then
    card rng p <= n & n < n+1 by NAT_1:19,39; then
    not dom p, rng p are_equipotent by A32,CARD_1:5; then
    p is not one-to-one by WELLORD2:def 4; then
    consider i,j being object such that
A33: i in dom p & j in dom p & p.i = p.j & i <> j;
    reconsider i,j as Nat by A33;
A34: 1 <= i & 1 <= j & i <= n+1 & j <= n+1 by A7,A33,FINSEQ_3:25;
    p.i in rng p by A33,FUNCT_1:def 3; then
A35: p.i in X() by A15;
    i < j or j < i by A33,XXREAL_0:1; then
    R[p.i,p.i] by A16,A33,A34;
    hence contradiction by A1,A35;
  end;

scheme FiniteC{X() -> finite set, P[set]}:
  P[X()]
  provided
A1
: for A being Subset of X() st for B being set st B c< A holds P[B] holds P[A]
  proof
    defpred Q[Nat] means for A being Subset of X() st card A = $1 holds P[A];
A2: for n being Nat st for i being Nat st i < n holds Q[i] holds Q[n]
    proof
      let n be Nat such that
A3:   for i being Nat st i < n holds Q[i];
      let A be Subset of X() such that
A4:   card A = n;
      now
        let B be set such that
A5:     B c< A;
        B c= A by A5; then
        reconsider B9 = B as Subset of X() by XBOOLE_1:1;
        card B9 < n by A4,A5,TREES_1:6;
        hence P[B] by A3;
      end;
      hence thesis by A1;
    end;
    for n being Nat holds Q[n] from NAT_1:sch 4(A2); then
    Q[card X()] & [#]X() = X();
    hence thesis;
  end;

scheme Numeration{X() -> finite set, R[set, set]}:
  ex s being one-to-one FinSequence st rng s = X() &
  for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j
  provided
A1: for x,y being set st x in X() & y in X() & R[x,y] holds not R[y,x] and
A2: for x,y,z being set st x in X() & y in X() & z in X() & R[x,y] & R[y,z]
    holds R[x,z]
  proof
    defpred P[set] means
    ex s being one-to-one FinSequence st rng s = $1 &
    for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j;
A3: P[{}]
    proof
      reconsider s = {} as one-to-one FinSequence;
      take s; thus thesis;
    end;
A4: for A being Subset of X() st for B being set st B c< A holds P[B]
    holds P[A]
    proof
      let A be Subset of X() such that
A5:   for B being set st B c< A holds P[B];
      per cases;
      suppose A is empty;
        hence P[A] by A3;
      end;
      suppose A is non empty; then
        reconsider A9 = A as non empty finite set;
A6:     for x,y being set st x in A9 & y in A9 & R[x,y] holds not R[y,x] by A1;
A7:     for x,y,z being set st x in A9 & y in A9 & z in A9 & R[x,y] & R[y,z]
        holds R[x,z] by A2;
        consider x being set such that
A8:     x in A9 & for y being set st y in A9 holds not R[y,x]
        from MinimalElement(A6,A7);
        set B = A\{x};
A9:     x nin B & B c= A by ZFMISC_1:56; then
        B c< A by A8; then
        consider s being one-to-one FinSequence such that
A10:     rng s = B and
A11:     for i,j st i in dom s & j in dom s & R[s.i, s.j] holds i < j by A5;
        <*x*> is one-to-one & rng <*x*> = {x} & {x} misses B
        by FINSEQ_1:39,FINSEQ_3:93,XBOOLE_1:79; then
        reconsider s9 = <*x*>^s as one-to-one FinSequence by A10,FINSEQ_3:91;
A12:     {x} c= A by A8,ZFMISC_1:31;
A13:     len <*x*> = 1 by FINSEQ_1:40;
        thus P[A]
        proof
          take s9;
          thus
          rng s9 = (rng <*x*>)\/ rng s by FINSEQ_1:31
          .= {x} \/ B by A10,FINSEQ_1:38 .= A by A12,XBOOLE_1:45;
          let i,j such that
A14:       i in dom s9 & j in dom s9 & R[s9.i, s9.j];
A15:       dom <*x*> = Seg 1 by FINSEQ_1:38;
          per cases by A13,A14,FINSEQ_1:25;
          suppose i in dom <*x*> & j in dom <*x*>; then
            i = 1 & j = 1 by A15,FINSEQ_1:2,TARSKI:def 1; then
            s9.i = x & s9.j = x by FINSEQ_1:41;
            hence i < j by A8,A14;
          end;
          suppose
A16:         i in dom <*x*> & ex n being Nat st n in dom s & j = 1 + n; then
A17:         i = 1 by A15,FINSEQ_1:2,TARSKI:def 1;
            consider n being Nat such that
A18:         n in dom s & j = 1+n by A16;
            1 <= n by A18,FINSEQ_3:25;
            hence i < j by A17,A18,NAT_1:13;
          end;
          suppose
A19:         j in dom <*x*> & ex n being Nat st n in dom s & i = 1 + n; then
            j = 1 by A15,FINSEQ_1:2,TARSKI:def 1; then
A20:         s9.j = x by FINSEQ_1:41;
            consider n being Nat such that
A21:         n in dom s & i = 1+n by A19;
            s9.i = s.n by A13,A21,FINSEQ_1:def 7; then
            s9.i in rng s by A21,FUNCT_1:def 3;
            hence i < j by A8,A14,A20,A9,A10;
          end;
          suppose (ex n being Nat st n in dom s & i = 1 + n) &
            ex n being Nat st n in dom s & j = 1 + n; then
            consider ni,nj being Nat such that
A22:         ni in dom s & i = 1+ni & nj in dom s & j = 1+nj;
            s9.i = s.ni & s9.j = s.nj by A13,A22,FINSEQ_1:def 7; then
            ni < nj by A11,A14,A22;
            hence i < j by A22,XREAL_1:6;
          end;
        end;
      end;
    end;
    thus P[X()] from FiniteC(A4);
  end;

theorem Th2:
  for x being variable holds varcl vars x = vars x
  proof
    let x be variable; x in Vars; then
    consider A being Subset of Vars, j being Element of NAT such that
A1: x = [varcl A,j] & A is finite by ABCMIZ_1:18;
    vars x = varcl A by A1;
    hence thesis;
  end;

theorem Th3:
 for C being initialized ConstructorSignature for e being expression of C holds
  e is compound iff not ex x being Element of Vars st e = x-term C
  proof let C be initialized ConstructorSignature;
    let e be expression of C;
    (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)) by ABCMIZ_1:53;
    hence thesis;
  end;

begin :: Standardized Constructor Signature

registration
  cluster empty for quasi-loci;
  existence by ABCMIZ_1:29;
end;

definition
 let C be ConstructorSignature;
 attr C is standardized means:
Def1:
  for o being OperSymbol of C st o is constructor holds o in Constructors &
    o`1 = the_result_sort_of o &
    card o`2`1 = len the_arity_of o;
end;

theorem Th4:
  for C being ConstructorSignature st C is standardized
  for o being OperSymbol of C
  holds o is constructor iff o in Constructors
  proof
    let C be ConstructorSignature such that
A1: C is standardized;
    let o be OperSymbol of C;
    thus o is constructor implies o in Constructors by A1;
    assume o in Constructors; then
    not o in {*, non_op} by ABCMIZ_1:39,XBOOLE_0:3;
    hence o <> * & o <> non_op by TARSKI:def 2;
  end;

registration
 cluster MaxConstrSign -> standardized;
 coherence
  proof let o be OperSymbol of MaxConstrSign;
A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors
     by ABCMIZ_1:def 24;
   assume
A2: o is constructor; then
A3: (the ResultSort of MaxConstrSign).o = o`1 &
    card ((the Arity of MaxConstrSign).o) = card o`2`1
     by ABCMIZ_1:def 24;
    o <> * & o <> non_op by A2; then
    not o in {*, non_op} by TARSKI:def 2;
   hence thesis by A3,A1,XBOOLE_0:def 3;
  end;
end;

registration
 cluster initialized standardized strict for ConstructorSignature;
 existence
  proof take MaxConstrSign;
   thus thesis;
  end;
end;

definition
  let C be initialized standardized ConstructorSignature;
  let c be constructor OperSymbol of C;
  func loci_of c -> quasi-loci equals c`2`1;
  coherence
  proof
    reconsider c as Element of Constructors by Th4;
    loci_of c is quasi-loci;
    hence thesis;
  end;
end;

registration
 let C be ConstructorSignature;
 cluster constructor for Subsignature of C;
 existence
  proof reconsider S = C as Subsignature of C by INSTALG1:15;
    take S; thus thesis;
  end;
end;

registration
 let C be initialized ConstructorSignature;
 cluster initialized for constructor Subsignature of C;
 existence
  proof reconsider S = C as constructor Subsignature of C by INSTALG1:15;
    take S; thus thesis;
  end;
end;

registration
 let C be standardized ConstructorSignature;
 cluster -> standardized for constructor Subsignature of C;
 coherence
  proof let S be constructor Subsignature of C;
   let o be OperSymbol of S such that
A1: o <> * & o <> non_op;
A2: the carrier' of S c= the carrier' of C by INSTALG1:10;
   reconsider c = o as OperSymbol of C by A2;
A3: c is constructor by A1;
    the Arity of S = (the Arity of C)|the carrier' of S &
    the ResultSort of S = (the ResultSort of C)|the carrier' of S
     by INSTALG1:12; then
    the_result_sort_of c = the_result_sort_of o &
    the_arity_of c = the_arity_of o by FUNCT_1:49;
   hence thesis by A3,Def1;
  end;
end;

theorem
 for S1,S2 being standardized ConstructorSignature
  st the carrier' of S1 = the carrier' of S2
  holds the ManySortedSign of S1 = the ManySortedSign of S2
  proof let S1,S2 be standardized ConstructorSignature such that
A1: the carrier' of S1 = the carrier' of S2;
A2: the carrier of S1 = 3 & the carrier of S2 = 3 by ABCMIZ_1:def 9,YELLOW11:1;
    now let o be OperSymbol of S1;
     reconsider o2 = o as OperSymbol of S2 by A1;
     per cases;
     suppose o = * or o = non_op; then
       (the Arity of S1).o = <*an_Adj*> & (the Arity of S2).o = <*an_Adj*> or
       (the Arity of S1).o = <*an_Adj,a_Type*> &
       (the Arity of S2).o = <*an_Adj,a_Type*> by ABCMIZ_1:def 9;
      hence (the Arity of S1).o = (the Arity of S2).o;
     end;
     suppose o is constructor & o2 is constructor; then
      card o`2`1 = len the_arity_of o & card o`2`1 = len the_arity_of o2 &
      the_arity_of o = (len the_arity_of o) |-> a_Term &
      the_arity_of o2 = (len the_arity_of o2) |-> a_Term
       by Def1,ABCMIZ_1:37;
      hence (the Arity of S1).o = the_arity_of o2 .= (the Arity of S2).o;
     end;
    end; then
A3: the Arity of S1 = the Arity of S2 by A1,A2,FUNCT_2:63;
    now let o be OperSymbol of S1;
     reconsider o2 = o as OperSymbol of S2 by A1;
     per cases;
     suppose o = * or o = non_op; then
       (the ResultSort of S1).o = a_Type & (the ResultSort of S2).o = a_Type or
       (the ResultSort of S1).o = an_Adj &
       (the ResultSort of S2).o = an_Adj by ABCMIZ_1:def 9;
      hence (the ResultSort of S1).o = (the ResultSort of S2).o;
     end;
     suppose o is constructor & o2 is constructor; then
       the_result_sort_of o = o`1 & the_result_sort_of o2 = o`1
       by Def1;
      hence (the ResultSort of S1).o = the_result_sort_of o2
         .= (the ResultSort of S2).o;
     end;
    end;
   hence thesis by A1,A2,A3,FUNCT_2:63;
  end;

theorem
 for C being ConstructorSignature holds
   C is standardized iff C is Subsignature of MaxConstrSign
  proof let C be ConstructorSignature;
A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors
     by ABCMIZ_1:def 24;
A2: dom the Arity of MaxConstrSign = the carrier' of MaxConstrSign
     by FUNCT_2:def 1;
A3: dom the ResultSort of MaxConstrSign = the carrier' of MaxConstrSign
     by FUNCT_2:def 1;
   thus C is standardized implies C is Subsignature of MaxConstrSign
     proof assume
A4:    for o being OperSymbol of C st o is constructor
        holds o in Constructors &
         o`1 = the_result_sort_of o &
         card o`2`1 = len the_arity_of o;
A5:    the carrier of C = 3 & the carrier of MaxConstrSign = 3
        by ABCMIZ_1:def 9,YELLOW11:1;
A6:    the Arity of C c= the Arity of MaxConstrSign
        proof let x,y be object; assume
A7:       [x,y] in the Arity of C; then
         reconsider x as OperSymbol of C by ZFMISC_1:87;
          x = * or x = non_op or x is constructor; then
          x in {*, non_op} or x in Constructors by A4,TARSKI:def 2; then
         reconsider c = x as OperSymbol of MaxConstrSign by A1,XBOOLE_0:def 3;
A8:       y = (the Arity of C).x by A7,FUNCT_1:1;
         per cases;
         suppose x = * or x = non_op; then
           c = * & y = <*an_Adj,a_Type*> or c = non_op & y = <*an_Adj*>
            by A8,ABCMIZ_1:def 9; then
           y = (the Arity of MaxConstrSign).c by ABCMIZ_1:def 9;
          hence thesis by A2,FUNCT_1:def 2;
         end;
         suppose
A9:        x is constructor; then
A10:        x <> * & x <> non_op; then
A11:        c is constructor;
           reconsider y as set by TARSKI:1;
           card x`2`1 = len the_arity_of x by A4,A9
                     .= card y by A7,FUNCT_1:1; then
A12:        card y = card ((the Arity of MaxConstrSign).c)
            by A11,ABCMIZ_1:def 24;
           y in {a_Term}* & (the Arity of MaxConstrSign).c in {a_Term}*
            by A8,A10,ABCMIZ_1:def 9; then
           y = (the Arity of MaxConstrSign).c by A12,ABCMIZ_1:6;
          hence thesis by A2,FUNCT_1:def 2;
         end;
        end;
       the ResultSort of C c= the ResultSort of MaxConstrSign
        proof let x,y be object; assume
A13:       [x,y] in the ResultSort of C; then
         reconsider x as OperSymbol of C by ZFMISC_1:87;
          x is constructor or x = * or x = non_op; then
          x in {*, non_op} or x in Constructors by A4,TARSKI:def 2; then
         reconsider c = x as OperSymbol of MaxConstrSign by A1,XBOOLE_0:def 3;
A14:       y = (the ResultSort of C).x by A13,FUNCT_1:1;
         per cases;
         suppose x = * or x = non_op; then
           c = * & y = a_Type or c = non_op & y = an_Adj
            by A14,ABCMIZ_1:def 9; then
           y = (the ResultSort of MaxConstrSign).c by ABCMIZ_1:def 9;
          hence thesis by A3,FUNCT_1:def 2;
         end;
         suppose
A15:        x is constructor & c is constructor; then
           x`1 = the_result_sort_of x by A4
              .= y by A13,FUNCT_1:1; then
           y = the_result_sort_of c by A15,Def1
            .= (the ResultSort of MaxConstrSign).c;
          hence thesis by A3,FUNCT_1:def 2;
         end;
        end;
      hence thesis by A5,A6,INSTALG1:13;
     end;
   assume
A16: C is Subsignature of MaxConstrSign;
   let o be OperSymbol of C such that
A17: o <> * & o <> non_op;
    the carrier' of C c= the carrier' of MaxConstrSign &
    o in the carrier' of C by A16,INSTALG1:10; then
   reconsider c = o as OperSymbol of MaxConstrSign;
A18: c is constructor by A17;
    not c in {*, non_op} by A17,TARSKI:def 2;
   hence o in Constructors by A1,XBOOLE_0:def 3;
   thus o`1 = (the ResultSort of MaxConstrSign).c by A18,ABCMIZ_1:def 24
           .= ((the ResultSort of MaxConstrSign)|the carrier' of C).o
             by FUNCT_1:49
           .= (the ResultSort of C).o by A16,INSTALG1:12
           .= the_result_sort_of o;
   thus card o`2`1 = card ((the Arity of MaxConstrSign).c)
         by A18,ABCMIZ_1:def 24
           .= card (((the Arity of MaxConstrSign)|the carrier' of C).o)
             by FUNCT_1:49
           .= card ((the Arity of C).o) by A16,INSTALG1:12
           .= len the_arity_of o;
  end;

registration
 let C be initialized ConstructorSignature;
 cluster non compound for quasi-term of C;
 existence
  proof set x = the Element of Vars;
   take x-term C; thus thesis;
  end;
end;

registration
 cluster -> pair for Element of Vars;
 coherence
  proof let x be Element of Vars;
    Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT:
    A is finite} & x in Vars by ABCMIZ_1:18; then
    ex A being Subset of Vars, j being Element of NAT st
    x = [varcl A, j] & A is finite;
   hence thesis;
  end;
end;

theorem Th7:
 for x being Element of Vars st vars x is natural holds vars x = 0
  proof let x be Element of Vars;
   assume x`1 is natural; then
   reconsider n = x`1 as Element of NAT;
    Vars = {[varcl A, j] where A is Subset of Vars, j is Element of NAT:
    A is finite} & x in Vars by ABCMIZ_1:18; then
   consider A being Subset of Vars, j being Element of NAT such that
A1: x = [varcl A, j] & A is finite;
   set i = the Element of n;
   assume A2: x`1 <> 0; then
A3: i in n;
   reconsider i as Element of NAT by A2,ORDINAL1:10;
    n = varcl A & vars x c= Vars by A1; then
    i in Vars by A3;
   hence thesis;
  end;

theorem Th8:
 Vars misses Constructors proof assume Vars meets Constructors; then
   consider x being object such that
A1: x in Vars & x in Constructors by XBOOLE_0:3;
   reconsider x as Element of Vars by A1;
   consider A being Subset of Vars, j being Element of NAT such that
A2: x = [varcl A, j] & A is finite by A1,ABCMIZ_1:18;
    x in Modes \/ Attrs or x in Funcs by A1,XBOOLE_0:def 3; then
    x in Modes or x in Attrs or x in Funcs by XBOOLE_0:def 3; then
    x`2 in [:QuasiLoci,NAT:] & x`2 = j by A2,MCART_1:10; then
    ex a,b being object st a in QuasiLoci & b in NAT & [a,b] = j
     by ZFMISC_1:def 2;
   hence thesis;
  end;

theorem
 for x being Element of Vars holds x <> * & x <> non_op;

theorem Th10:
 for C being standardized ConstructorSignature holds
   Vars misses the carrier' of C
  proof let C be standardized ConstructorSignature;
   assume Vars meets the carrier' of C; then
   consider x being object such that
A1: x in Vars & x in the carrier' of C by XBOOLE_0:3;
   reconsider x as Element of Vars by A1;
   reconsider c = x as OperSymbol of C by A1;
   x = * or x = non_op or c is constructor; then
    x = * or x = non_op or x in Constructors & Vars misses Constructors
     by Th8,Def1;
   hence thesis by XBOOLE_0:3;
  end;

theorem Th11: :: see ABCMIZ_1:51
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  holds
   (ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]) or
   (ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
    ( o in Constructors or o = * or o = non_op ))
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C;
   set X = MSVars C;
   set Y = X (\/) ((the carrier of C) --> {0});
   reconsider q = e as Term of C,Y by MSAFREE3:8;
   per cases by MSATERM:2;
   suppose
     ex s being SortSymbol of C, v being Element of Y.s st q.{} = [v,s]; then
    consider s being SortSymbol of C, v being Element of Y.s such that
A1:  q.{} = [v,s];
    consider z being object such that
A2:  z in dom the Sorts of Free(C,X) & e in (the Sorts of Free(C, X)).z
      by CARD_5:2;
    reconsider z as SortSymbol of C by A2;
     the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then
A3:  z = a_Type or z = an_Adj or z = a_Term by ENUMSET1:def 1;
A4:  q = root-tree [v,s] by A1,MSATERM:5; then
A5:  the_sort_of q = s by MSATERM:14;
A6:  the Sorts of Free(C, X) = C-Terms(X,Y) by MSAFREE3:24; then
     the Sorts of Free(C, X) c= the Sorts of FreeMSA Y by PBOOLE:def 18; then
     (the Sorts of Free(C, X)).z c= (the Sorts of FreeMSA Y).z &
     FreeMSA Y = MSAlgebra(#FreeSort Y, FreeOper Y#); then
     q in (the Sorts of FreeMSA Y).z &
     (the Sorts of FreeMSA Y).z = FreeSort(Y, z) by A2,MSAFREE:def 11; then
A7:  s = z by A5,MSATERM:def 5; then
     v in (MSVars C).z by A4,A2,A6,MSAFREE3:18; then
A8:  v in Vars & z = a_Term by A3,ABCMIZ_1:def 25; then
    reconsider x = v as Element of Vars;
     e = x-term C by A1,A7,A8,MSATERM:5;
    hence thesis by A1,A7,A8;
   end;
   suppose
     q.{} in [:the carrier' of C,{the carrier of C}:]; then
    consider o,s being object such that
A9:  o in the carrier' of C & s in {the carrier of C} & q.{} = [o,s]
      by ZFMISC_1:def 2;
    reconsider o as OperSymbol of C by A9;
     o is constructor iff o <> * & o <> non_op; then
     s = the carrier of C & (o in Constructors or o = * or o = non_op)
      by A9,Def1,TARSKI:def 1;
    hence thesis by A9;
   end;
  end;

registration
 let C be initialized standardized ConstructorSignature;
 let e be expression of C;
 cluster e.{} -> pair;
 coherence
  proof
    (ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]) or
    (ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op )) by Th11;
   hence thesis;
  end;
end;

theorem Th12:
 for C being initialized ConstructorSignature for e being expression of C
 for o being OperSymbol of C st e.{} = [o, the carrier of C]
  holds
    e is expression of C, the_result_sort_of o
  proof let C be initialized ConstructorSignature;
   let e be expression of C;
   let o be OperSymbol of C such that
A1: e.{} = [o, the carrier of C];
   set X = MSVars C, Y = X (\/) ((the carrier of C) --> {0});
   reconsider t = e as Term of C, Y by MSAFREE3:8;
    variables_in t c= X by MSAFREE3:27; then
    e in {t1 where t1 is Term of C, Y:
          the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then
    e in C-Terms(X,Y).the_sort_of t by MSAFREE3:def 5; then
A2: e in (the Sorts of Free(C, X)).the_sort_of t by MSAFREE3:24;
    the_sort_of t = the_result_sort_of o by A1,MSATERM:17;
   hence thesis by A2,ABCMIZ_1:def 28;
  end;

theorem Th13:
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  holds
    ( (e.{})`1 = * implies e is expression of C, a_Type C ) &
    ( (e.{})`1 = non_op implies e is expression of C, an_Adj C )
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C;
   per cases by Th11;
   suppose
     ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then
    consider x being Element of Vars such that
A1:  e = x-term C & e.{} = [x, a_Term];
    thus thesis by A1;
   end;
   suppose
     (ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op )); then
    consider o being OperSymbol of C such that
A2:  e.{} = [o, the carrier of C];
    set X = MSVars C, Y = X (\/) ((the carrier of C) --> {0});
    reconsider t = e as Term of C, Y by MSAFREE3:8;
     variables_in t c= X by MSAFREE3:27; then
     e in {t1 where t1 is Term of C, Y:
           the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then
     e in C-Terms(X,Y).the_sort_of t by MSAFREE3:def 5; then
A3:  e in (the Sorts of Free(C, X)).the_sort_of t by MSAFREE3:24;
A4:  the_result_sort_of non_op C = an_Adj C &
     the_result_sort_of ast C = a_Type C by ABCMIZ_1:38;
A5:  (e.{})`1 = o & non_op C = non_op & ast C = * by A2;
     the_sort_of t = the_result_sort_of o by A2,MSATERM:17;
    hence thesis by A3,A4,A5,ABCMIZ_1:def 28;
   end;
  end;

theorem Th14:
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  holds
   (e.{})`1 in Vars & (e.{})`2 = a_Term & e is quasi-term of C or
   (e.{})`2 = the carrier of C &
    ( (e.{})`1 in Constructors & (e.{})`1 in the carrier' of C or
      (e.{})`1 = * or (e.{})`1 = non_op )
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C;
   per cases by Th11;
   suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term];
    then consider x being Element of Vars such that
A1:  e = x-term C & e.{} = [x, a_Term];
    thus thesis by A1;
   end;
   suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op ); then
    consider o being OperSymbol of C such that
A2:  e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op );
    thus thesis by A2;
   end;
  end;

theorem
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  st (e.{})`1 in Constructors
  holds e in (the Sorts of Free(C, MSVars C)).(e.{})`1`1
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C;
   assume
A1: (e.{})`1 in Constructors;
   per cases by Th11;
   suppose
     ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]; then
    consider x being Element of Vars such that
A2:  e = x-term C & e.{} = [x, a_Term];
     (e.{})`1 = x by A2;
    hence thesis by A1,Th8,XBOOLE_0:3;
   end;
   suppose
     ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op ); then
    consider o being OperSymbol of C such that
A3:  e.{} = [o, the carrier of C];
A4:  (e.{})`1 = o by A3;
     * in {*, non_op} & non_op in {*, non_op} by TARSKI:def 2; then
     o <> * & o <> non_op by A1,A4,ABCMIZ_1:39,XBOOLE_0:3; then
A5:  o is constructor;
    set X = MSVars C;
    reconsider t = e as Term of C, X (\/) ((the carrier of C) --> {0})
      by MSAFREE3:8;
A6:  the_sort_of t = the_result_sort_of o by A3,MSATERM:17
         .= o`1 by A5,Def1;
     variables_in t c= X by MSAFREE3:27; then
     e in {t1 where t1 is Term of C, X (\/) ((the carrier of C) --> {0}):
           the_sort_of t1 = the_sort_of t & variables_in t1 c= X}; then
     e in C-Terms(X, X (\/) ((the carrier of C)-->{0})).the_sort_of t
       by MSAFREE3:def 5;
    hence e in (the Sorts of Free(C, MSVars C)).(e.{})`1`1
      by A4,A6,MSAFREE3:23;
   end;
  end;

theorem
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  holds not (e.{})`1 in Vars iff (e.{})`1 is OperSymbol of C
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C;
A1:  (e.{})`1 in Vars or (e.{})`1 in the carrier' of C or
     (e.{})`1 = ast C or (e.{})`1 = non_op C by Th14;
     Vars misses the carrier' of C by Th10;
    hence not (e.{})`1 in Vars iff (e.{})`1 is OperSymbol of C
     by A1,XBOOLE_0:3;
  end;

theorem Th17:
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  st (e.{})`1 in Vars
 ex x being Element of Vars st x = (e.{})`1 & e = x-term C
  proof let C be initialized standardized ConstructorSignature;
   let t be expression of C such that
A1: (t.{})`1 in Vars;
   set X = MSVars C;
   set V = X (\/) ((the carrier of C) --> {0});
   reconsider q = t as Term of C, V by MSAFREE3:8;
   per cases by MSATERM:2;
   suppose q.{} in [:the carrier' of C, {the carrier of C}:]; then
     (q.{})`1 in the carrier' of C &
     the carrier' of C misses Vars by Th10,MCART_1:10;
    hence thesis by A1,XBOOLE_0:3;
   end;
   suppose
     ex s being SortSymbol of C, v being Element of V.s st q.{} = [v,s]; then
    consider s being SortSymbol of C, v being Element of V.s such that
A2:  t.{} = [v,s];
A3:  q = root-tree [v,s] by A2,MSATERM:5;
    reconsider x = v as Element of Vars by A1,A2;
    take x;
     the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9; then
A4:  s = a_Term or s = a_Type or s = an_Adj by ENUMSET1:def 1;
     ((the carrier of C) --> {0}).s = {0}; then
     V.s = X.s \/ {0} by PBOOLE:def 4; then
A5:  s = a_Term or V.s = {} \/ {0} by A4,ABCMIZ_1:def 25;
     v in V.s & x <> 0;
    hence thesis by A2,A3,A5;
   end;
  end;

theorem Th18:
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  st (e.{})`1 = *
 ex a being expression of C, an_Adj C, q being expression of C, a_Type C st
  e = [*,3]-tree(a,q)
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C such that
A1: (e.{})`1 = *;
    not ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]
       by A1;
   then consider o being OperSymbol of C such that
A2: e.{} = [o, the carrier of C] and
    o in Constructors or o = * or o = non_op by Th11;
   set Y = (MSVars C) (\/) ((the carrier of C) --> {0});
   reconsider t = e as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0})
     by MSAFREE3:8;
   consider aa being ArgumentSeq of
      Sym(o, (MSVars C) (\/) ((the carrier of C) --> {0})) such that
A3: t = [o, the carrier of C]-tree aa by A2,MSATERM:10;
A4: * = [o, the carrier of C]`1 by A1,A3,TREES_4:def 4 .= o;
A5: the_arity_of ast C = <*an_Adj C, a_Type C*> by ABCMIZ_1:38;
A6: len aa = len the_arity_of o by MSATERM:22
          .= 2 by A4,A5,FINSEQ_1:44; then
    dom aa = Seg 2 by FINSEQ_1:def 3; then
A7: 1 in dom aa & 2 in dom aa; then
   reconsider t1 = aa.1, t2 = aa.2 as Term of C,
      (MSVars C) (\/) ((the carrier of C) --> {0}) by MSATERM:22;
A8: len doms aa = len aa by TREES_3:38;
    (doms aa).1 = dom t1 & (doms aa).2 = dom t2 by A7,FUNCT_6:22; then
A9: 0 < 2 & 0+1 = 1 & 1 < 2 & 1+1 = 2 & {} in (doms aa).1 & {} in (doms aa).2 &
    <* 0*>^<*>NAT = <* 0*> & <* 1*>^<*>NAT = <* 1*>
     by FINSEQ_1:34,TREES_1:22;
    dom t = tree doms aa by A3,TREES_4:10; then
   reconsider 00 = <* 0*>, 01 = <* 1*> as Element of dom t
     by A6,A8,A9,TREES_3:def 15;
    0 < 2 & 1 = 0+1 & 1 < 2 & 2 = 1+1 & aa is DTree-yielding; then
    t1 = t|00 & t2 = t|01 by A3,A6,TREES_4:def 4; then
A10: t1 is expression of C & t2 is expression of C &
    variables_in t1 c= variables_in t & variables_in t2 c= variables_in t
     by MSAFREE3:32,33; then
A11: variables_in t1 c= MSVars C & variables_in t2 c= MSVars C by MSAFREE3:27;
    the_sort_of t1 = (the_arity_of o).1 by A7,MSATERM:23
                  .= an_Adj C by A4,A5,FINSEQ_1:44; then
    t1 in {s where s is Term of C,Y: the_sort_of s = an_Adj C &
           variables_in s c= MSVars C} by A11; then
    t1 in (C-Terms(MSVars C, Y)).an_Adj C by MSAFREE3:def 5; then
    t1 in (the Sorts of Free(C, MSVars C)).an_Adj C by MSAFREE3:24; then
   reconsider a = t1 as expression of C, an_Adj C by A10,ABCMIZ_1:def 28;
    the_sort_of t2 = (the_arity_of o).2 by A7,MSATERM:23
                  .= a_Type C by A4,A5,FINSEQ_1:44; then
    t2 in {s where s is Term of C,Y: the_sort_of s = a_Type C &
           variables_in s c= MSVars C} by A11; then
    t2 in (C-Terms(MSVars C, Y)).a_Type C by MSAFREE3:def 5; then
    t2 in (the Sorts of Free(C, MSVars C)).a_Type C by MSAFREE3:24; then
   reconsider q = t2 as expression of C, a_Type C by A10,ABCMIZ_1:def 28;
   take a,q;
A12: the carrier of C = 3 by ABCMIZ_1:def 9,YELLOW11:1;
    aa = <*a,q*> by A6,FINSEQ_1:44;
   hence thesis by A3,A4,A12,TREES_4:def 6;
  end;

theorem Th19:
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  st (e.{})`1 = non_op
 ex a being expression of C, an_Adj C st e = [non_op,3]-tree a
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C such that
A1: (e.{})`1 = non_op;
   not ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term]
    by A1;
   then consider o being OperSymbol of C such that
A2: e.{} = [o, the carrier of C] and
    o in Constructors or o = * or o = non_op by Th11;
   set Y = (MSVars C) (\/) ((the carrier of C) --> {0});
   reconsider t = e as Term of C, (MSVars C) (\/) ((the carrier of C) --> {0})
     by MSAFREE3:8;
   consider aa being ArgumentSeq of
      Sym(o, (MSVars C) (\/) ((the carrier of C) --> {0})) such that
A3: t = [o, the carrier of C]-tree aa by A2,MSATERM:10;
A4: non_op = [o, the carrier of C]`1 by A1,A3,TREES_4:def 4 .= o;
A5: the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38;
A6: len aa = len the_arity_of o by MSATERM:22
          .= 1 by A4,A5,FINSEQ_1:40; then
    dom aa = Seg 1 by FINSEQ_1:def 3; then
A7: 1 in dom aa; then
   reconsider t1 = aa.1 as Term of C,
      (MSVars C) (\/) ((the carrier of C) --> {0}) by MSATERM:22;
A8: len doms aa = len aa by TREES_3:38;
    (doms aa).1 = dom t1 by A7,FUNCT_6:22; then
A9: 0 < 1 & 0+1 = 1 & {} in (doms aa).1 & <* 0*>^<*>NAT = <* 0*>
     by FINSEQ_1:34,TREES_1:22;
    dom t = tree doms aa by A3,TREES_4:10; then
   reconsider 00 = <* 0*> as Element of dom t by A6,A8,A9,TREES_3:def 15;
    t1 = t|00 by A3,A6,A9,TREES_4:def 4; then
A10: t1 is expression of C & variables_in t1 c= variables_in t
     by MSAFREE3:32,33; then
A11: variables_in t1 c= MSVars C by MSAFREE3:27;
    the_sort_of t1 = (the_arity_of o).1 by A7,MSATERM:23
                  .= an_Adj C by A4,A5,FINSEQ_1:40; then
    t1 in {s where s is Term of C,Y: the_sort_of s = an_Adj C &
           variables_in s c= MSVars C} by A11; then
    t1 in (C-Terms(MSVars C, Y)).an_Adj C by MSAFREE3:def 5; then
    t1 in (the Sorts of Free(C, MSVars C)).an_Adj C by MSAFREE3:24; then
   reconsider a = t1 as expression of C, an_Adj C by A10,ABCMIZ_1:def 28;
   take a;
A12: the carrier of C = 3 by ABCMIZ_1:def 9,YELLOW11:1;
    aa = <*a*> by A6,FINSEQ_1:40;
   hence thesis by A3,A4,A12,TREES_4:def 5;
  end;

theorem Th20:
 for C being initialized standardized ConstructorSignature
 for e being expression of C
  st (e.{})`1 in Constructors
 ex o being OperSymbol of C st o = (e.{})`1 & the_result_sort_of o = o`1 &
   e is expression of C, the_result_sort_of o
  proof let C be initialized standardized ConstructorSignature;
   let e be expression of C such that
A1: (e.{})`1 in Constructors;
   per cases by Th11;
   suppose ex x being Element of Vars st e = x-term C & e.{} = [x, a_Term];
    then consider x being Element of Vars such that
A2:  e = x-term C & e.{} = [x, a_Term];
     (e.{})`1 = x & x in Vars by A2;
    hence thesis by A1,Th8,XBOOLE_0:3;
   end;
   suppose ex o being OperSymbol of C st e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op ); then
    consider o being OperSymbol of C such that
A3:  e.{} = [o, the carrier of C] &
     ( o in Constructors or o = * or o = non_op );
    take o;
A4:  (e.{})`1 = o & (e.{})`2 = the carrier of C by A3;
     * in {*, non_op} & non_op in {*, non_op} by TARSKI:def 2; then
     o <> * & o <> non_op by A1,A4,ABCMIZ_1:39,XBOOLE_0:3; then
     o is constructor;
    hence o = (e.{})`1 & the_result_sort_of o = o`1
      by A3,Def1;
    set X = MSVars C;
    set V = X (\/) ((the carrier of C) --> {0});
    reconsider q = e as Term of C,V by MSAFREE3:8;
A5:  variables_in q c= X by MSAFREE3:27;
A6:  the_sort_of q = the_result_sort_of o by A3,MSATERM:17;
     (the Sorts of Free(C, MSVars C)).the_result_sort_of o
      = C-Terms(X,V).the_result_sort_of o by MSAFREE3:24
     .= {a where a is Term of C,V: the_sort_of a = the_result_sort_of o &
         variables_in a c= X} by MSAFREE3:def 5;
    hence
     e in (the Sorts of Free(C, MSVars C)).the_result_sort_of o by A5,A6;
   end;
  end;

theorem Th21:
 for C being initialized standardized ConstructorSignature
 for t being quasi-term of C holds
   t is compound iff (t.{})`1 in Constructors & (t.{})`1`1 = a_Term
  proof let C be initialized standardized ConstructorSignature;
   set X = MSVars C;
   set V = X (\/) ((the carrier of C) --> {0});
   let t be quasi-term of C;
    C-Terms(X, V) c= the Sorts of FreeMSA V &
    the Sorts of Free(C, X) = C-Terms(X, V) by MSAFREE3:24,PBOOLE:def 18; then
A1: FreeMSA V = MSAlgebra(#FreeSort V, FreeOper V#) &
    (C-Terms(X, V)).a_Term C c= (the Sorts of FreeMSA V).a_Term C &
     t in C-Terms(X,V).a_Term C
     by ABCMIZ_1:def 28; then
    t in (FreeSort V).a_Term C; then
A2: t in FreeSort(V,a_Term C) by MSAFREE:def 11;
A3: (MSVars C).a_Term = Vars & a_Term C = a_Term & a_Term = 2
     by ABCMIZ_1:def 25;
   reconsider q = t as Term of C, V by MSAFREE3:8;
   per cases by MSATERM:2;
   suppose
     ex s being SortSymbol of C, v being Element of V.s st q.{} = [v,s]; then
    consider s being SortSymbol of C, v being Element of V.s such that
A4:  t.{} = [v,s];
A5:  q = root-tree [v,s] & the_sort_of q = a_Term C
      by A2,A4,MSATERM:5,def 5; then
A6:  a_Term C = s & (t.{})`1 = v by A4,MSATERM:14; then
    reconsider x = v as Element of Vars by A3,A5,A1,MSAFREE3:18;
     q = x-term C & vars x <> 2 by A5,A6,Th7;
    hence thesis by A6;
   end;
   suppose
     q.{} in [:the carrier' of C,{the carrier of C}:]; then
    consider o, k being object such that
A7:  o in the carrier' of C & k in {the carrier of C} & q.{} = [o,k]
      by ZFMISC_1:def 2;
    reconsider o as OperSymbol of C by A7;
     k = the carrier of C by A7,TARSKI:def 1; then
A8:  the_result_sort_of o = the_sort_of q by A7,MSATERM:17
       .= a_Term C by A1,MSAFREE3:17; then
     o <> ast C & o <> non_op C by ABCMIZ_1:38; then
A9:  o is constructor; then
A10:  a_Term C = o`1 by A8,Def1 .= (q.{})`1`1 by A7;
A11:  (q.{})`1 = o by A7;
     now given x being Element of Vars such that
A12:    q = x-term C;
       q.{} = [x,a_Term] by A12,TREES_4:3; then
       k = a_Term by A7,XTUPLE_0:1; then
       2 = the carrier of C by A7,TARSKI:def 1;
      hence contradiction by ABCMIZ_1:def 9,YELLOW11:1;
     end;
    hence thesis by A9,A10,A11,Th3,Def1;
   end;
  end;

theorem Th22:
 for C being initialized standardized ConstructorSignature
 for t being expression of C holds
   t is non compound quasi-term of C iff (t.{})`1 in Vars
  proof let C be initialized standardized ConstructorSignature;
   let t be expression of C;
   thus
    now assume t is non compound quasi-term of C; then
     consider x being Element of Vars such that
A1:   t = x-term C by Th3;
      t.{} = [x,a_Term] & x in Vars by A1,TREES_4:3;
     hence (t.{})`1 in Vars;
    end;
   assume (t.{})`1 in Vars; then
    ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17;
   hence thesis;
  end;

theorem
 for C being initialized standardized ConstructorSignature
 for t being expression of C holds
   t is quasi-term of C iff
      (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars
  proof let C be initialized standardized ConstructorSignature;
   let t be expression of C;
   hereby assume t is quasi-term of C; then
     reconsider tr = t as quasi-term of C;
     tr is compound or tr is non compound;
     hence (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or
           (t.{})`1 in Vars by Th21,Th22;
    end;
   assume that
A1: (t.{})`1 in Constructors & (t.{})`1`1 = a_Term or (t.{})`1 in Vars and
A2: t is not quasi-term of C;
A3: (t.{})`1 in Vars implies
      ex x being Element of Vars st x = (t.{})`1 & t = x-term C by Th17; then
    (t.{})`1 in Constructors & (t.{})`1`1 = a_Term by A1,A2; then
    ex o being OperSymbol of C st o = (t.{})`1 & the_result_sort_of o = o`1 &
     t is expression of C, the_result_sort_of o by Th20;
   hence thesis by A2,A3,A1;
  end;

theorem Th24:
 for C being initialized standardized ConstructorSignature
 for a being expression of C holds
   a is positive quasi-adjective of C iff
    (a .{})`1 in Constructors & (a .{})`1`1 = an_Adj
  proof let C be initialized standardized ConstructorSignature;
   set X = MSVars C;
   set V = X (\/) ((the carrier of C) --> {0});
   let t be expression of C;
   consider A being MSSubset of FreeMSA V such that
A1: Free(C, X) = GenMSAlg A & A = (Reverse V)""X by MSAFREE3:def 1;
    the Sorts of Free(C, X) is MSSubset of FreeMSA V
     by A1,MSUALG_2:def 9; then
    the Sorts of Free(C, X) c= the Sorts of FreeMSA V by PBOOLE:def 18; then
A2: (the Sorts of Free(C, MSVars C)).an_Adj C c=
    (the Sorts of FreeMSA V).an_Adj C;
   per cases by Th14;
   suppose (t.{})`1 in Vars & (t.{})`2 = a_Term & t is quasi-term of C;
    hence thesis by Th8,ABCMIZ_1:77,XBOOLE_0:3;
   end;
   suppose that
A3:  (t.{})`2 = the carrier of C and
A4:  (t.{})`1 in Constructors and
A5:  (t.{})`1 in the carrier' of C;
    reconsider o = (t.{})`1 as OperSymbol of C by A5;
    reconsider tt = t as Term of C, V by MSAFREE3:8;
     not o in {*, non_op} by A4,ABCMIZ_1:39,XBOOLE_0:3; then
     o <> * & o <> non_op by TARSKI:def 2; then
     o is constructor; then
A6:  o`1 = the_result_sort_of o by Def1;
A7:  t.{} = [o, (t.{})`2]; then
A8:  the_sort_of tt = the_result_sort_of o by A3,MSATERM:17;
    hereby assume t is positive quasi-adjective of C; then
A9:    t in (the Sorts of Free(C, MSVars C)).an_Adj C by ABCMIZ_1:def 28;
      thus (t.{})`1 in Constructors by A4;
      assume (t.{})`1`1 <> an_Adj;
      hence contradiction by A2,A9,A6,A8,MSAFREE3:7;
     end;
    assume (t.{})`1 in Constructors;
    assume (t.{})`1`1 = an_Adj; then
    reconsider t as expression of C, an_Adj C by A3,A6,A7,Th12;
     t is positive proof given a being expression of C, an_Adj C such that
A10:     t = (non_op C)term a;
        t = [non_op, the carrier of C]-tree <*a*> by A10,ABCMIZ_1:43; then
        t.{} = [non_op, the carrier of C] by TREES_4:def 4; then
        (t.{})`1 = non_op; then
        (t.{})`1 in {*, non_op} by TARSKI:def 2;
       hence thesis by A4,ABCMIZ_1:39,XBOOLE_0:3;
      end;
    hence thesis;
   end;
   suppose
A11:  (t.{})`1 = *; then
     (t.{})`1 in {*, non_op} by TARSKI:def 2; then
A12:  not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
    consider a being expression of C, an_Adj C,
    q being expression of C, a_Type C such that
A13:  t = [*,3]-tree(a,q) by A11,Th18;
     t = [*,3]-tree<*a,q*> by A13,TREES_4:def 6; then
     t.{} = [*, 3] by TREES_4:def 4; then
     (t.{})`1 = *; then
     t is expression of C, a_Type C & a_Type C = a_Type & a_Type = 0 &
     an_Adj C = an_Adj & an_Adj = 1 by Th13;
    hence thesis by A12,ABCMIZ_1:48;
   end;
   suppose
A14:  (t.{})`1 = non_op; then
     (t.{})`1 in {*, non_op} by TARSKI:def 2; then
A15:  not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
    consider a being expression of C, an_Adj C such that
A16:  t = [non_op,3]-tree a by A14,Th19;
     t = [non_op,3]-tree <*a*> by A16,TREES_4:def 5
      .= [non_op, the carrier of C]-tree <*a*> by ABCMIZ_1:def 9,YELLOW11:1
      .= (non_op C)term a by ABCMIZ_1:43;
    hence thesis by A15,ABCMIZ_1:def 37;
   end;
  end;

theorem
 for C being initialized standardized ConstructorSignature
 for a being quasi-adjective of C holds
   a is negative iff (a .{})`1 = non_op
  proof let C be initialized standardized ConstructorSignature;
   let t be quasi-adjective of C;
   per cases;
   suppose
A1:  t is positive expression of C, an_Adj C; then
     (t.{})`1 in Constructors & non_op in {*, non_op} by Th24,TARSKI:def 2;
    hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3;
   end;
   suppose
A2:  t is negative expression of C, an_Adj C; then
    consider a being expression of C, an_Adj C such that
A3:  a is positive & t = (non_op C)term a by ABCMIZ_1:def 38;
     t = [non_op, the carrier of C]-tree <*a*> by A3,ABCMIZ_1:43; then
     t.{} = [non_op, the carrier of C] by TREES_4:def 4;
    hence thesis by A2;
   end;
  end;

theorem
 for C being initialized standardized ConstructorSignature
 for t being expression of C holds
   t is pure expression of C, a_Type C iff
    (t.{})`1 in Constructors & (t.{})`1`1 = a_Type
  proof let C be initialized standardized ConstructorSignature;
   set X = MSVars C;
   set V = X (\/) ((the carrier of C) --> {0});
   let t be expression of C;
   consider A being MSSubset of FreeMSA V such that
A1: Free(C, X) = GenMSAlg A & A = (Reverse V)""X by MSAFREE3:def 1;
    the Sorts of Free(C, X) is MSSubset of FreeMSA V
     by A1,MSUALG_2:def 9; then
    the Sorts of Free(C, X) c= the Sorts of FreeMSA V by PBOOLE:def 18; then
A2: (the Sorts of Free(C, MSVars C)).a_Type C c=
    (the Sorts of FreeMSA V).a_Type C;
   per cases by Th14;
   suppose (t.{})`1 in Vars & (t.{})`2 = a_Term & t is quasi-term of C;
    hence thesis by Th8,ABCMIZ_1:48,XBOOLE_0:3;
   end;
   suppose that
A3:  (t.{})`2 = the carrier of C and
A4:  (t.{})`1 in Constructors and
A5:  (t.{})`1 in the carrier' of C;
    reconsider o = (t.{})`1 as OperSymbol of C by A5;
    reconsider tt = t as Term of C, V by MSAFREE3:8;
     not o in {*, non_op} by A4,ABCMIZ_1:39,XBOOLE_0:3; then
     o <> * & o <> non_op by TARSKI:def 2; then
     o is constructor; then
A6:  o`1 = the_result_sort_of o by Def1;
A7:  t.{} = [o, (t.{})`2]; then
A8:  the_sort_of tt = the_result_sort_of o by A3,MSATERM:17;
    thus
     now assume t is pure expression of C, a_Type C; then
A9:    t in (the Sorts of Free(C, MSVars C)).a_Type C by ABCMIZ_1:def 28;
      thus (t.{})`1 in Constructors by A4;
      assume (t.{})`1`1 <> a_Type;
      hence contradiction by A2,A9,A6,A8,MSAFREE3:7;
     end;
    assume (t.{})`1 in Constructors;
    assume (t.{})`1`1 = a_Type; then
    reconsider t as expression of C, a_Type C by A3,A7,Th12,A6;
     t is pure proof given a being expression of C, an_Adj C,
                  q being expression of C, a_Type C such that
A10:     t = (ast C)term(a,q);
        t = [*, the carrier of C]-tree <*a,q*> by A10,ABCMIZ_1:46; then
        t.{} = [*, the carrier of C] by TREES_4:def 4; then
        (t.{})`1 = *; then
        (t.{})`1 in {*, non_op} by TARSKI:def 2;
       hence thesis by A4,ABCMIZ_1:39,XBOOLE_0:3;
      end;
    hence thesis;
   end;
   suppose
A11:  (t.{})`1 = *; then
     (t.{})`1 in {*, non_op} by TARSKI:def 2; then
A12:  not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
    consider a being expression of C, an_Adj C,
    q being expression of C, a_Type C
    such that
A13:  t = [*,3]-tree(a,q) by A11,Th18;
     t = [*,3]-tree<*a,q*> by A13,TREES_4:def 6
      .= [*, the carrier of C]-tree <*a,q*> by ABCMIZ_1:def 9,YELLOW11:1
      .= (ast C)term(a,q) by ABCMIZ_1:46;
    hence thesis by A12,ABCMIZ_1:def 41;
   end;
   suppose
A14:  (t.{})`1 = non_op; then
     (t.{})`1 in {*, non_op} by TARSKI:def 2; then
A15:  not (t.{})`1 in Constructors by ABCMIZ_1:39,XBOOLE_0:3;
    consider a being expression of C, an_Adj C such that
A16:  t = [non_op,3]-tree a by A14,Th19;
     t = [non_op,3]-tree <*a*> by A16,TREES_4:def 5; then
     t.{} = [non_op,3] by TREES_4:def 4; then
     (t.{})`1 = non_op; then
     t is expression of C, an_Adj C by Th13;
    hence thesis by A15,ABCMIZ_1:48;
   end;
  end;

begin :: Expressions

reserve i,j for Nat,
  x for variable,
  l for quasi-loci;

set MC = MaxConstrSign;

definition
  mode expression is expression of MaxConstrSign;
  mode valuation is valuation of MaxConstrSign;
  mode quasi-adjective is quasi-adjective of MaxConstrSign;
  func QuasiAdjs -> Subset of Free(MaxConstrSign, MSVars MaxConstrSign) equals
  QuasiAdjs MaxConstrSign; coherence;
  mode quasi-term is quasi-term of MaxConstrSign;
  func QuasiTerms -> Subset of Free(MaxConstrSign, MSVars MaxConstrSign) equals
  QuasiTerms MaxConstrSign; coherence;
  mode quasi-type is quasi-type of MaxConstrSign;
  func QuasiTypes -> set equals QuasiTypes MaxConstrSign; coherence;
end;

registration
  cluster QuasiAdjs -> non empty; coherence;
  cluster QuasiTerms -> non empty; coherence;
  cluster QuasiTypes -> non empty; coherence;
end;

definition
  redefine func Modes -> non empty Subset of Constructors;
  coherence
  proof
    Modes c= Modes \/ Attrs & Modes \/ Attrs c= Constructors by XBOOLE_1:7;
    hence thesis by XBOOLE_1:1;
  end;
  redefine func Attrs -> non empty Subset of Constructors;
  coherence
  proof
    Attrs c= Modes \/ Attrs & Modes \/ Attrs c= Constructors by XBOOLE_1:7;
    hence thesis by XBOOLE_1:1;
  end;
  redefine func Funcs -> non empty Subset of Constructors;
  coherence by XBOOLE_1:7;
end;

reserve C for initialized ConstructorSignature,
  c for constructor OperSymbol of C;

definition
  func set-constr -> Element of Modes equals [a_Type,[{},0]];
  coherence
  proof
    a_Type in {a_Type} & [{},0] in [:QuasiLoci,NAT:]
    by ABCMIZ_1:29,TARSKI:def 1,ZFMISC_1:def 2;
    hence thesis by ZFMISC_1:def 2;
  end;
end;

theorem
  kind_of set-constr = a_Type & loci_of set-constr = {} &
  index_of set-constr = 0;

theorem Th28:
 Constructors = [:{a_Type, an_Adj, a_Term}, [:QuasiLoci, NAT:]:]
  proof
   thus Constructors = [:{a_Type},[:QuasiLoci,NAT:]:] \/
         [:{an_Adj},[:QuasiLoci,NAT:]:] \/ Funcs
      .= [:{a_Type} \/ {an_Adj}, [:QuasiLoci,NAT:]:] \/ Funcs by ZFMISC_1:97
      .= [:{a_Type, an_Adj}, [:QuasiLoci,NAT:]:] \/ Funcs by ENUMSET1:1
      .= [:{a_Type, an_Adj} \/ {a_Term}, [:QuasiLoci,NAT:]:] by ZFMISC_1:97
      .= [:{a_Type, an_Adj, a_Term}, [:QuasiLoci, NAT:]:] by ENUMSET1:3;
  end;

theorem Th29:
  [rng l, i] in Vars & l^<*[rng l,i]*> is quasi-loci
  proof
    varcl rng l = rng l by ABCMIZ_1:33;
    hence [rng l, i] in Vars by ABCMIZ_1:17; then
    reconsider x = [rng l, i] as variable;
    rng l in {rng l, i} & {rng l, i} in x by TARSKI:def 2; then
    vars x = rng l & x nin rng l by XREGULAR:7;
    hence thesis by ABCMIZ_1:31;
  end;

theorem Th30:
  ex l st len l = i
  proof
    defpred P[Nat] means ex l st len l = $1;
    <*>Vars is quasi-loci & len <*>Vars = 0 by ABCMIZ_1:29; then
A1: P[ 0 ];
A2: P[j] implies P[j+1] proof given l such that
A3:   len l = j;
      reconsider l1 = l^<*[rng l, 1]*> as quasi-loci by Th29;
      take l1; thus thesis by A3,FINSEQ_2:16;
    end;
    P[j] from NAT_1:sch 2(A1,A2);
    hence thesis;
  end;

theorem Th31:
  for X being finite Subset of Vars ex l st rng l = varcl X
  proof
    let X be finite Subset of Vars;
    reconsider Y = varcl X as finite Subset of Vars by ABCMIZ_1:24;
    defpred R[set, set] means $1 in $2`1;
A1: for x,y being set st x in Y & y in Y & R[x,y] holds not R[y,x]
    proof
      let x,y be set such that
A2:   x in Y & y in Y & R[x,y] & R[y,x];
      x in Vars by A2; then
      consider A being Subset of Vars, j being Element of NAT such that
A3:   x = [varcl A, j] & A is finite by ABCMIZ_1:18;
      y in Vars by A2; then
      consider B being Subset of Vars, k being Element of NAT such that
A4:   y = [varcl B, k] & B is finite by ABCMIZ_1:18;
A5:   y in varcl A & x in varcl B by A2,A3,A4;
A6:   varcl A in {varcl A} & varcl B in {varcl B} by TARSKI:def 1;
      {varcl A} in x & {varcl B} in y by A4,A3,TARSKI:def 2;
      hence thesis by A5,A6,XREGULAR:10;
    end;
A7: for x,y,z being set st x in Y & y in Y & z in Y & R[x,y] & R[y,z]
    holds R[x,z]
    proof
      let x,y,z be set such that
A8:   x in Y & y in Y & z in Y & R[x,y] & R[y,z];
      y in Vars by A8; then
      consider B being Subset of Vars, k being Element of NAT such that
A9:   y = [varcl B, k] & B is finite by ABCMIZ_1:18;
      z in Vars by A8; then
      consider C being Subset of Vars, j being Element of NAT such that
A10:   z = [varcl C, j] & C is finite by ABCMIZ_1:18;
A11:   z`1 = varcl C & y`1 = varcl B by A10,A9; then
      varcl B c= varcl C by A8,A9,ABCMIZ_1:def 1;
      hence R[x,z] by A11,A8;
    end;
    consider l being one-to-one FinSequence such that
A12: rng l = Y and
A13: for i,j st i in dom l & j in dom l & R[l.i, l.j] holds i < j
    from Numeration(A1,A7);
    reconsider l as one-to-one FinSequence of Vars by A12,FINSEQ_1:def 4;
    now let i be Nat, x be variable; assume
A14:   i in dom l & x = l.i;
      let y be variable; assume
A15:   y in vars x;
      x in Vars; then
      consider A being Subset of Vars, j being Element of NAT such that
A16:   x = [varcl A, j] & A is finite by ABCMIZ_1:18;
      x in rng l & vars x = varcl A by A14,A16,FUNCT_1:def 3; then
      vars x c= Y by A12,A16,ABCMIZ_1:def 1; then
      consider a being object such that
A17:   a in dom l & y = l.a by A12,A15,FUNCT_1:def 3;
      reconsider a as Nat by A17;
      take a;
      thus a in dom l & a < i & y = l.a by A13,A14,A15,A17;
    end; then
    reconsider l as quasi-loci by ABCMIZ_1:30;
    take l;
    thus rng l = varcl X by A12;
  end;

theorem Th32:  :: to mozna uogolnic na X zamkniety na poddrzewa
               :: (troche dodatkowych pojec i twierdzen)
  for S being non empty non void ManySortedSign
  for Y being non empty-yielding ManySortedSet of the carrier of S
  for X,o being set, p being DTree-yielding FinSequence
  st ex C st X = Union the Sorts of Free(S, Y)
  holds o-tree p in X implies p is FinSequence of X
  proof
    let S be non empty non void ManySortedSign;
    let Y be non empty-yielding ManySortedSet of the carrier of S;
    let X,o be set;
    let p be DTree-yielding FinSequence;
    given C such that
A1: X = Union the Sorts of Free(S, Y);
    assume o-tree p in X; then
    reconsider e = o-tree p as Element of Free(S, Y) by A1;
    rng p c= X
    proof
      let z be object; assume z in rng p; then
      consider i being object such that
A2:   i in dom p & z = p.i by FUNCT_1:def 3;
      reconsider i as Nat by A2;
      reconsider ppi = p.i as DecoratedTree by A2,TREES_3:24;
A3:   1 <= i & i <= len p by A2,FINSEQ_3:25; then
A4:   (i-'1)+1 = i by XREAL_1:235;
then A5:   i-'1 < len p by A3,NAT_1:13;
A6:   len doms p = len p by TREES_3:38;
A7:   (doms p).i = dom ppi by A2,FUNCT_6:22;
A8:   dom e = tree doms p by TREES_4:10;
      <*i-'1*>^<*>NAT = <*i-'1*> & <*>NAT in dom ppi
      by FINSEQ_1:34,TREES_1:22; then
      reconsider q = <*i-'1*> as Element of dom e
      by A4,A5,A6,A7,A8,TREES_3:def 15;
      e|q = z by A2,A4,A5,TREES_4:def 4; then
      z is Element of Free(S, Y) by MSAFREE3:33;
      hence thesis by A1;
    end;
    hence p is FinSequence of X by FINSEQ_1:def 4;
  end;

definition
  let C;
  let e be expression of C;
  mode subexpression of e -> expression of C means
  it in Subtrees e;
  existence by TREES_9:11;
  func constrs e -> set equals
  (proj1 rng e)/\the set of all o where o is constructor OperSymbol of C;
  coherence;
end;

definition
  let S be non empty non void ManySortedSign;
  let X be non empty-yielding ManySortedSet of the carrier of S;
  let e be Element of Free(S,X);
  func main-constr e -> set
equals:   :: dobre dla zestandaryzowanych (nie ma def)
Def9:
  (e.{})`1 if e is compound otherwise {};

:: x-term C = [x, a_Term]-tree {}
:: (ast C)term(a,t) = [*, the carrier of C]-tree<*a,t*>
:: (non_op C)term a = [non_op, the carrier of C]-tree<*a*>
:: c-trm p = [c, the carrier of C]-tree p
::   problem gdy '{}' moze byc 'c'
  correctness;
  func args e -> DTree-yielding FinSequence means: ARGS:
  e = (e.{})-tree it;
  existence
  proof
    consider v being set, p being DTree-yielding FinSequence such that
A1: e = v-tree p by TREES_9:8;
A2: v = e.{} by A1,TREES_4:def 4;
    thus thesis by A1,A2;
  end;
  uniqueness by TREES_4:15;
end;

definition
  let S be non empty non void ManySortedSign;
  let X be non empty-yielding ManySortedSet of the carrier of S;
  let e be Element of Free(S,X);
  redefine func args e -> FinSequence of Free(S, X);
  coherence
  proof
A1: e = (e.{})-tree args e by ARGS;
    args e is FinSequence of Free(S, X) by A1,Th32;
    hence thesis;
  end;
end;

theorem Th33:
  for C for e being expression of C holds e is subexpression of e
  proof
    let C be initialized ConstructorSignature;
    let e be expression of C;
    thus e in Subtrees e by TREES_9:11;
  end;

theorem
  main-constr (x -term C) = {} by Def9;

theorem Th35:
  for c being constructor OperSymbol of C
  for p being FinSequence of QuasiTerms C st len p = len the_arity_of c
  holds main-constr (c-trm p) = c
  proof
    let c be constructor OperSymbol of C;
    let p be FinSequence of QuasiTerms C;
    assume len p = len the_arity_of c; then
    c-trm p = [c, the carrier of C]-tree p by ABCMIZ_1:def 35; then
    (c-trm p).{} = [c, the carrier of C] by TREES_4:def 4;
    hence main-constr (c-trm p) = [c, the carrier of C]`1 by Def9
    .= c;
  end;

definition
  let C;
  let e be expression of C;
  attr e is constructor means:
Def11:
  e is compound & main-constr e is constructor OperSymbol of C;
end;

registration
  let C;
  cluster constructor -> compound for expression of C;
  coherence;
end;

registration
  let C;
  cluster constructor for expression of C;
  existence
  proof
    consider m, a being OperSymbol of C such that
A1: the_result_sort_of m = a_Type & the_arity_of m = {} &
    the_result_sort_of a = an_Adj & the_arity_of a = {} by ABCMIZ_1:def 12;
    the_arity_of ast C = <*an_Adj C, a_Type C*> &
    the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38; then
    reconsider m as constructor OperSymbol of C by A1,ABCMIZ_1:def 11;
    set p = <*>QuasiTerms C;
    take e = m-trm p; thus e is compound;
    len p = len the_arity_of m by A1;
    hence thesis by Th35;
  end;
end;

registration
  let C;
  let e be constructor expression of C;
  cluster constructor for subexpression of e;
  existence
  proof
    e is subexpression of e by Th33;
    hence thesis;
  end;
end;

registration
  let S be non void Signature;
  let X be non empty-yielding ManySortedSet of S;
  let t be Element of Free(S,X);
  cluster rng t -> Relation-like;
  coherence
  proof
    set Z = (the carrier of S)-->{0};
    set Y = X (\/) Z;
    t is Term of S,Y by MSAFREE3:8; then
    rng t c= the carrier of DTConMSA Y by RELAT_1:def 19;
    hence thesis;
  end;
end;

theorem
  for e being constructor expression of C holds main-constr e in constrs e
  proof
    let e be constructor expression of C;
A1: main-constr e = (e.{})`1 by Def9;
    {} in dom e by TREES_1:22; then
    e.{} in rng e by FUNCT_1:def 3; then
A2: main-constr e in proj1 rng e by A1,MCART_1:86;
    main-constr e is constructor OperSymbol of C by Def11; then
    main-constr e in the set of all c;
    hence main-constr e in constrs e by A2,XBOOLE_0:def 4;
  end;

begin :: Arity

reserve a,a9 for quasi-adjective,
  t,t1,t2 for quasi-term,
  T for quasi-type,

  c for Element of Constructors;

definition
  let C be non void Signature;
  attr C is arity-rich means: Def12:
  for n being Nat, s being SortSymbol of C holds
  {o where o is OperSymbol of C: the_result_sort_of o = s &
        len the_arity_of o = n} is infinite;
  let o be OperSymbol of C;
  attr o is nullary means: Def13: the_arity_of o = {};
  attr o is unary means: Def14: len the_arity_of o = 1;
  attr o is binary means: Def15: len the_arity_of o = 2;
end;

theorem Th37:
  for C being non void Signature for o being OperSymbol of C holds
  (o is nullary implies o is not unary) &
  (o is nullary implies o is not binary) &
  (o is unary implies o is not binary);

registration
  let C be ConstructorSignature;
  cluster non_op C -> unary;
  coherence
  proof
    the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38;
    hence len the_arity_of non_op C = 1 by FINSEQ_1:39;
  end;
  cluster ast C -> binary;
  coherence
  proof
    the_arity_of ast C = <*an_Adj C, a_Type C*> by ABCMIZ_1:38;
    hence len the_arity_of ast C = 2 by FINSEQ_1:44;
  end;
end;

registration
  let C be ConstructorSignature;
  cluster nullary -> constructor for OperSymbol of C;
  coherence
  proof
    let o be OperSymbol of C such that
A1: the_arity_of o = {};
    the_arity_of ast C = <*an_Adj C, a_Type C*> &
    the_arity_of non_op C = <*an_Adj C*> by ABCMIZ_1:38;
    hence o <> * & o <> non_op by A1;
  end;
end;

theorem Th38:
  for C being ConstructorSignature holds C is initialized iff
  ex m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C st
  m is nullary & a is nullary
  proof
    let C be ConstructorSignature;
    hereby assume C is initialized; then
      consider m, a being OperSymbol of C such that
A1:   the_result_sort_of m = a_Type & the_arity_of m = {} &
      the_result_sort_of a = an_Adj & the_arity_of a = {};
      reconsider m as OperSymbol of a_Type C by A1,ABCMIZ_1:def 32;
      reconsider a as OperSymbol of an_Adj C by A1,ABCMIZ_1:def 32;
      take m, a;
      thus m is nullary by A1;
      thus a is nullary by A1;
    end;
    given m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C
    such that
A2: m is nullary & a is nullary;
    take m,a;
    the_result_sort_of non_op C = an_Adj C &
    the_result_sort_of ast C = a_Type C by ABCMIZ_1:38;
    hence thesis by A2,ABCMIZ_1:def 32;
  end;

registration
  let C be initialized ConstructorSignature;
  cluster nullary constructor for OperSymbol of a_Type C;
  existence
  proof
    consider m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C
    such that
A1: m is nullary & a is nullary by Th38;
    take m; thus thesis by A1;
  end;
  cluster nullary constructor for OperSymbol of an_Adj C;
  existence
  proof
    consider m being OperSymbol of a_Type C, a being OperSymbol of an_Adj C
    such that
A2: m is nullary & a is nullary by Th38;
    take a; thus thesis by A2;
  end;
end;

registration
  let C be initialized ConstructorSignature;
  cluster nullary constructor for OperSymbol of C;
  existence
  proof
    set o = the nullary constructor OperSymbol of a_Type C;
    take o; thus thesis;
  end;
end;

registration
  cluster arity-rich -> with_an_operation_for_each_sort for
non void Signature;
  coherence
  proof
    let S be non void Signature such that
A1: for n being Nat, s being SortSymbol of S holds
    {o where o is OperSymbol of S: the_result_sort_of o = s &
        len the_arity_of o = n} is infinite;
    let v be object;
    assume v in the carrier of S; then
    reconsider v as SortSymbol of S;
    set A = {o where o is OperSymbol of S: the_result_sort_of o = v &
    len the_arity_of o = 0};
    reconsider A as infinite set by A1;
    set a = the Element of A;
    a in A; then
    consider o being OperSymbol of S such that
A2: a = o & the_result_sort_of o = v & len the_arity_of o = 0;
    thus thesis by A2,FUNCT_2:4;
  end;
  cluster arity-rich -> initialized for ConstructorSignature;
  coherence
  proof
    let C be ConstructorSignature such that
A3: C is arity-rich;
    set Xt = {o where o is OperSymbol of C: the_result_sort_of o = a_Type C &
    len the_arity_of o = 0};
    set x = the Element of Xt;
    Xt is infinite set by A3; then
    x in Xt; then
    consider m being OperSymbol of C such that
A4: x = m & the_result_sort_of m = a_Type C & len the_arity_of m = 0;
    set Xa = {o where o is OperSymbol of C: the_result_sort_of o = an_Adj C &
    len the_arity_of o = 0};
    set x = the Element of Xa;
    Xa is infinite set by A3; then
    x in Xa; then
    consider a being OperSymbol of C such that
A5: x = a & the_result_sort_of a = an_Adj C & len the_arity_of a = 0;
    take m, a; thus thesis by A4,A5;
  end;
end;

registration
  cluster MaxConstrSign -> arity-rich;
  coherence
  proof set C = MaxConstrSign;
    let n be Nat, s be SortSymbol of C;
A1: the carrier of C = {0,1,2} by ABCMIZ_1:def 9;
    set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
    len the_arity_of o = n};
    consider l being quasi-loci such that
A2: len l = n by Th30;
    deffunc F(object) = [s,[l,$1]];
    consider f being Function such that
A3: dom f = NAT & for i being object st i in NAT holds f.i = F(i)
    from FUNCT_1:sch 3;
    f is one-to-one
    proof
      let i,j be object; assume i in dom f & j in dom f; then
      reconsider i,j as Element of NAT by A3;
      f.i = F(i) & f.j = F(j) by A3; then
      f.i = f.j implies [l,i] = [l,j] by XTUPLE_0:1;
      hence thesis by XTUPLE_0:1;
    end; then
A4: rng f is infinite by A3,CARD_1:59;
    rng f c= X
    proof
      let y be object; assume y in rng f; then
      consider x being object such that
A5:   x in dom f & y = f.x by FUNCT_1:def 3;
      reconsider x as Element of NAT by A3,A5;
A6:   [l,x] in [:QuasiLoci, NAT:] & y = F(x) by A3,A5; then
      y in Constructors by A1,Th28,ZFMISC_1:def 2; then
      y in {*,non_op}\/Constructors by XBOOLE_0:def 3; then
      reconsider y as OperSymbol of C by ABCMIZ_1:def 24;
A7:   y is constructor by A6; then
A8:   the_result_sort_of y = y`1 by ABCMIZ_1:def 24 .= s by A6;
      len the_arity_of y = card y`2`1 by A7,ABCMIZ_1:def 24
      .= card [l,x]`1 by A6
      .= len l;
      hence thesis by A2,A8;
    end;
    hence X is infinite by A4;
  end;
end;

registration
  cluster arity-rich initialized for ConstructorSignature;
  existence
  proof
    take MaxConstrSign; thus thesis;
  end;
end;

registration
  let C be arity-rich ConstructorSignature;
  let s be SortSymbol of C;
  cluster nullary constructor for OperSymbol of s;
  existence
  proof
    set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
    len the_arity_of o = 0};
    X is infinite by Def12; then
    consider m1,m2 being object such that
A1: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10;
    consider o1 being OperSymbol of C such that
A2: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 0 by A1;
    reconsider m1 as OperSymbol of s by A2,ABCMIZ_1:def 32;
    the_arity_of m1 = {} by A2; then
 m1 is nullary;
    hence thesis;
  end;
  cluster unary constructor for OperSymbol of s;
  existence
  proof
    set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
    len the_arity_of o = 1};
    X is infinite by Def12; then
    consider m1,m2 being object such that
A3: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10;
    consider o1 being OperSymbol of C such that
A4: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 1 by A3;
    consider o2 being OperSymbol of C such that
A5: m2 = o2 & the_result_sort_of o2 = s & len the_arity_of o2 = 1 by A3;
    reconsider m1,m2 as OperSymbol of s by A4,A5,ABCMIZ_1:def 32;
A6: m1 is unary & m2 is unary by A4,A5; then
A7: m1 <> ast C & m2 <> ast C by Th37;
    m1 <> non_op or m2 <> non_op by A3; then
    m1 is constructor or m2 is constructor by A7;
    hence thesis by A6;
  end;
  cluster binary constructor for OperSymbol of s;
  existence
  proof
    set X = {o where o is OperSymbol of C: the_result_sort_of o = s &
    len the_arity_of o = 2};
    X is infinite by Def12; then
    consider m1,m2 being object such that
A8: m1 in X & m2 in X & m1 <> m2 by ZFMISC_1:def 10;
    consider o1 being OperSymbol of C such that
A9: m1 = o1 & the_result_sort_of o1 = s & len the_arity_of o1 = 2 by A8;
    consider o2 being OperSymbol of C such that
A10: m2 = o2 & the_result_sort_of o2 = s & len the_arity_of o2 = 2 by A8;
    reconsider m1,m2 as OperSymbol of s by A9,A10,ABCMIZ_1:def 32;
A11: m1 is binary & m2 is binary by A9,A10; then
A12: m1 <> non_op C & m2 <> non_op C by Th37;
    m1 <> * or m2 <> * by A8; then
    m1 is constructor or m2 is constructor by A12;
    hence thesis by A11;
  end;
end;

registration
  let C be arity-rich ConstructorSignature;
  cluster unary constructor for OperSymbol of C;
  existence
  proof
    set o = the unary constructor OperSymbol of a_Type C;
    take o; thus thesis;
  end;
  cluster binary constructor for OperSymbol of C;
  existence
  proof
    set o = the binary constructor OperSymbol of a_Type C;
    take o; thus thesis;
  end;
end;

theorem Th39:
 for o being nullary OperSymbol of C holds
   [o, the carrier of C]-tree {} is expression of C, the_result_sort_of o
  proof
   let o be nullary OperSymbol of C;
   set X = MSVars C;
   set Z = (the carrier of C)-->{0};
   set Y = X (\/) Z;
A1: the_arity_of o = {} by Def13;
A2: the Sorts of Free(C, X) = C-Terms(X, Y) by MSAFREE3:24;
    for i being Nat st i in dom {} ex t being Term of C,Y st t = {}.i &
    the_sort_of t = (the_arity_of o).i; then
   reconsider p = {} as ArgumentSeq of Sym(o, Y) by A1,MSATERM:24;
A3: 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
       ex t being DecoratedTree st t in rng p & x in (C variables_in t).s9
       by MSAFREE3:11;
      hence thesis;
     end;
   set s9 = the_result_sort_of o;
A4: 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 A2,MSAFREE3:def 5; then
    [o, the carrier of C]-tree {} in (the Sorts of Free(C, X)).s9 by A3,A4;
   hence thesis by ABCMIZ_1:41;
  end;

definition
  let C be initialized ConstructorSignature;
  let m be nullary constructor OperSymbol of a_Type C;
  redefine func m term -> pure expression of C, a_Type C;
  coherence
  proof
    set T = m term;
    the_arity_of m = 0 by Def13; then
    len the_arity_of m = 0; then
A1: T = [m, the carrier of C]-tree {} by ABCMIZ_1:def 29;
    ex m, a being OperSymbol of C st
    the_result_sort_of m = a_Type & the_arity_of m = {} &
    the_result_sort_of a = an_Adj & the_arity_of a = {}
    by ABCMIZ_1:def 12; then
    the_result_sort_of m = a_Type C by ABCMIZ_1:def 32; then
    reconsider T as expression of C, a_Type C by A1,Th39;
    T is pure
    proof
      given a being expression of C, an_Adj C,
      t being expression of C, a_Type C such that
A2:   T = (ast C)term(a,t);
      T = [ *, the carrier of C]-tree <*a,t*> by A2,ABCMIZ_1:46;
      hence thesis by A1,TREES_4:15;
    end;
    hence thesis;
  end;
end;

definition
  let c be Element of Constructors;
  func @c -> constructor OperSymbol of MaxConstrSign equals c;
  coherence
  proof
    * in {*, non_op} & non_op in {*, non_op} &
    the carrier' of MC = {*, non_op} \/ Constructors
    by ABCMIZ_1:def 24,TARSKI:def 2; then
    c <> * & c <> non_op & c in the carrier' of MC
    by ABCMIZ_1:39,XBOOLE_0:3,def 3;
    hence thesis by ABCMIZ_1:def 11;
  end;
end;

definition
  let m be Element of Modes;
  redefine func @m -> constructor OperSymbol of a_Type MaxConstrSign;
  coherence
  proof
A1: m`1 in {a_Type} by MCART_1:10;
    the_result_sort_of @m = m`1 by ABCMIZ_1:def 24
    .= a_Type by A1,TARSKI:def 1;
    hence thesis by ABCMIZ_1:def 32;
  end;
end;

registration
  cluster @set-constr -> nullary;
  coherence
  proof
    len the_arity_of @set-constr = card set-constr`2`1 by ABCMIZ_1:def 24
    .= card [{},0]`1 .= card 0;
    hence the_arity_of @set-constr = {};
  end;
end;

theorem
  the_arity_of @set-constr = {} by Def13;

definition
  func set-type -> quasi-type equals
  ({}QuasiAdjs MaxConstrSign) ast ((@set-constr) term);
  coherence;
end;

theorem
  adjs set-type = {} & the_base_of set-type = (@set-constr) term;

definition
  let l be FinSequence of Vars;
  func args l -> FinSequence of QuasiTerms MaxConstrSign means:
Def18:
  len it = len l & for i st i in dom l holds it.i = (l/.i)-term MaxConstrSign;
  existence
  proof
    deffunc F(Nat) = (l/.$1)-term MaxConstrSign;
    consider g being FinSequence such that
A1: len g = len l and
A2: for i st i in dom g holds g.i = F(i) from FINSEQ_1:sch 2;
A3: dom g = dom l by A1,FINSEQ_3:29;
    rng g c= QuasiTerms MaxConstrSign
    proof
      let j be object; assume j in rng g; then
      consider z being object such that
A4:   z in dom g & j = g.z by FUNCT_1:def 3;
      reconsider z as Nat by A4;
      j = F(z) by A2,A4;
      hence thesis by ABCMIZ_1:49;
    end; then
    reconsider g as FinSequence of QuasiTerms MaxConstrSign by FINSEQ_1:def 4;
    take g;
    thus thesis by A1,A2,A3;
  end;
  uniqueness
  proof
    let a1,a2 be FinSequence of QuasiTerms MaxConstrSign such that
A5: len a1 = len l and
A6: for i st i in dom l holds a1.i = (l/.i)-term MaxConstrSign and
A7: len a2 = len l and
A8: for i st i in dom l holds a2.i = (l/.i)-term MaxConstrSign;
A9: dom a1 = dom l & dom a2 = dom l by A5,A7,FINSEQ_3:29;
    now let i;
      assume i in dom a1; then
      a1.i = (l/.i)-term MaxConstrSign & a2.i = (l/.i)-term MaxConstrSign
      by A6,A8,A9;
      hence a1.i = a2.i;
    end;
    hence thesis by A9;
  end;
end;

definition
  let c;
  func base_exp_of c -> expression equals (@c)-trm args loci_of c;
  coherence;
end;

theorem Th42:
  for o being OperSymbol of MaxConstrSign holds
  o is constructor iff o in Constructors
  proof
    let o be OperSymbol of MaxConstrSign;
A1: the carrier' of MaxConstrSign = {*, non_op} \/ Constructors
    by ABCMIZ_1:def 24;
    o is constructor iff o nin {*,non_op} by TARSKI:def 2;
    hence thesis by A1,ABCMIZ_1:39,XBOOLE_0:3,def 3;
  end;

theorem
  for m being nullary OperSymbol of MaxConstrSign holds
  main-constr (m term) = m
  proof set C = MaxConstrSign;
    let m be nullary OperSymbol of C;
    the_arity_of m = 0 by Def13; then
    len the_arity_of m = 0 & len {} = 0; then
A1: m term = [m, the carrier of C]-tree {} &
    m-trm(<*>QuasiTerms C) = [m, the carrier of C]-tree {}
    by ABCMIZ_1:def 29,def 35;
    hence main-constr (m term) = ((m term).{})`1 by Def9
    .= [m, the carrier of C]`1 by A1,TREES_4:def 4
    .= m;
  end;

theorem
  for m being unary constructor OperSymbol of MaxConstrSign
  for t holds main-constr (m term t) = m
  proof set C = MaxConstrSign;
    let m be unary constructor OperSymbol of C;
    let t;
    reconsider w = t as Element of QuasiTerms C by ABCMIZ_1:49;
    reconsider p = <*w*> as FinSequence of QuasiTerms C;
A1: len the_arity_of m = 1 by Def14; then
    the_arity_of m = 1 |-> a_Term by ABCMIZ_1:37
    .= <*a_Term*> by FINSEQ_2:59; then
    len p = 1 & (the_arity_of m).1 = a_Term C by FINSEQ_1:40; then
A2: m term t = [m, the carrier of C]-tree <*t*> &
    m-trm p = [m, the carrier of C]-tree <*t*> by A1,ABCMIZ_1:def 30,def 35;
    hence main-constr (m term t) = ((m term t).{})`1 by Def9
    .= [m, the carrier of C]`1 by A2,TREES_4:def 4
    .= m;
  end;

theorem
  for a holds main-constr ((non_op MaxConstrSign)term a) = non_op
  proof set C = MaxConstrSign;
    set m = non_op C;
    let a;
A1: len the_arity_of m = 1 by Def14;
    the_arity_of m = <*an_Adj*> by ABCMIZ_1:38; then
    (the_arity_of m).1 = an_Adj C by FINSEQ_1:40; then
A2: m term a = [m, the carrier of C]-tree <*a*> by A1,ABCMIZ_1:def 30;
    thus main-constr (m term a) = ((m term a).{})`1 by Def9
    .= [m, the carrier of C]`1 by A2,TREES_4:def 4
    .= non_op;
  end;

theorem
  for m being binary constructor OperSymbol of MaxConstrSign
  for t1,t2 holds main-constr (m term(t1,t2)) = m
  proof set C = MaxConstrSign;
    let m be binary constructor OperSymbol of C;
    let t1,t2;
    reconsider w1 = t1, w2 = t2 as Element of QuasiTerms C by ABCMIZ_1:49;
    reconsider p = <*w1,w2*> as FinSequence of QuasiTerms C;
A1: len the_arity_of m = 2 by Def15; then
    the_arity_of m = 2 |-> a_Term by ABCMIZ_1:37
    .= <*a_Term,a_Term*> by FINSEQ_2:61; then
    (the_arity_of m).1 = a_Term C & (the_arity_of m).2 = a_Term C & len p = 2
    by FINSEQ_1:44; then
A2: m term(t1,t2) = [m, the carrier of C]-tree <*t1,t2*> &
    m-trm p = [m, the carrier of C]-tree p by A1,ABCMIZ_1:def 31,def 35;
    hence main-constr (m term(t1,t2)) = ((m term(t1,t2)).{})`1 by Def9
    .= [m, the carrier of C]`1 by A2,TREES_4:def 4
    .= m;
  end;

theorem
  for q being expression of MaxConstrSign, a_Type MaxConstrSign
  for a holds main-constr ((ast MaxConstrSign)term(a,q)) = *
  proof set C = MaxConstrSign;
    set m = ast C;
    let q be expression of MaxConstrSign, a_Type MaxConstrSign;
    let a;
A1: len the_arity_of m = 2 by Def15;
    the_arity_of m = <*an_Adj C,a_Type C*> by ABCMIZ_1:38; then
    (the_arity_of m).1 = an_Adj C & (the_arity_of m).2 = a_Type C
    by FINSEQ_1:44; then
A2: m term(a,q) = [m, the carrier of C]-tree <*a,q*> by A1,ABCMIZ_1:def 31;
    thus main-constr (m term(a,q)) = ((m term(a,q)).{})`1 by Def9
    .= [m, the carrier of C]`1 by A2,TREES_4:def 4
    .= *;
  end;

definition
  let T be quasi-type;
  func constrs T -> set equals
  (constrs the_base_of T) \/ union {constrs a: a in adjs T};
  coherence;
end;

theorem
  for q being pure expression of MaxConstrSign, a_Type MaxConstrSign
  for A being finite Subset of QuasiAdjs MaxConstrSign holds
  constrs(A ast q) = (constrs q) \/ union {constrs a: a in A};

theorem
  constrs(a ast T) = (constrs a) \/ (constrs T)
  proof
    set A = {constrs a9: a9 in {a} \/ adjs T};
    set B = {constrs a9: a9 in adjs T};
A1: A = B \/{constrs a}
    proof
      thus A c= B \/{constrs a} proof let z be object;
        assume z in A; then
        consider a9 such that
A2:     z = constrs a9 & a9 in {a} \/ adjs T;
        a9 in {a} or a9 in adjs T by A2,XBOOLE_0:def 3; then
        a9 = a or a9 in adjs T by TARSKI:def 1; then
        z in {constrs a} or z in B by A2,TARSKI:def 1;
        hence thesis by XBOOLE_0:def 3;
      end;
      let z be object; assume
A3:   z in B\/{constrs a};
A4:   a in {a} by TARSKI:def 1;
      per cases by A3,XBOOLE_0:def 3;
      suppose z in B; then
        consider a9 such that
A5:     z = constrs a9 & a9 in adjs T;
        a9 in {a} \/ adjs T by A5,XBOOLE_0:def 3;
        hence thesis by A5;
      end;
      suppose z in {constrs a}; then
        z = constrs a & a in {a} \/ adjs T by A4,TARSKI:def 1,XBOOLE_0:def 3;
        hence thesis;
      end;
    end;
    thus constrs(a ast T) = (constrs the_base_of (a ast T)) \/ union A
    .= (constrs the_base_of T) \/ union A
    .= (constrs the_base_of T) \/ ((union {constrs a}) \/ union B)
    by A1,ZFMISC_1:78
    .= (constrs the_base_of T) \/ ((constrs a) \/ union B) by ZFMISC_1:25
    .= (constrs the_base_of T) \/ (constrs a) \/ union B by XBOOLE_1:4
    .= (constrs a) \/ ((constrs the_base_of T) \/ union B) by XBOOLE_1:4
    .= (constrs a) \/ (constrs T);
  end;

begin :: Unification

definition
  let C be initialized ConstructorSignature;
  let t,p be expression of C;
  pred t matches_with p means
  ex f being valuation of C st t = p at f;
  reflexivity
  proof let t be expression of C;
    take f = the empty valuation of C;
    thus t at f = t by ABCMIZ_1:139;
  end;
end;

theorem
 for t1,t2,t3 being expression of C st t1 matches_with t2 & t2 matches_with t3
  holds t1 matches_with t3
  proof
    let t1,t2,t3 be expression of C;
    given f1 being valuation of C such that
A1: t1 = t2 at f1;
    given f2 being valuation of C such that
A2: t2 = t3 at f2;
    take f2 at f1;
    thus thesis by A1,A2,ABCMIZ_1:149;
  end;

definition
  let C be initialized ConstructorSignature;
  let A,B be Subset of QuasiAdjs C;
  pred A matches_with B means
  ex f being valuation of C st B at f c= A;
  reflexivity
  proof let t be Subset of QuasiAdjs C;
    take f = the empty valuation of C;
    let x be object; assume x in t at f; then
    ex a being quasi-adjective of C st x = a at f & a in t;
    hence x in t by ABCMIZ_1:139;
  end;
end;

theorem
 for A1,A2,A3 being Subset of QuasiAdjs C
  st A1 matches_with A2 & A2 matches_with A3
  holds A1 matches_with A3
  proof
    let t1,t2,t3 be Subset of QuasiAdjs C;
    given f1 being valuation of C such that
A1: t2 at f1 c= t1;
    given f2 being valuation of C such that
A2: t3 at f2 c= t2;
    take f2 at f1;
    (t3 at f2) at f1 c= t2 at f1 by A2,ABCMIZ_1:146; then
    (t3 at f2) at f1 c= t1 by A1;
    hence thesis by ABCMIZ_1:150;
  end;

definition
  let C be initialized ConstructorSignature;
  let T,P be quasi-type of C;
  pred T matches_with P means
  ex f being valuation of C st
   (adjs P) at f c= adjs T & (the_base_of P) at f = the_base_of T;
  reflexivity
  proof let t be quasi-type of C;
    take f = the empty valuation of C;
    thus (adjs t) at f c= adjs t
    proof let x be object; assume x in (adjs t) at f; then
      ex a being quasi-adjective of C st x = a at f & a in adjs t;
      hence x in adjs t by ABCMIZ_1:139;
    end;
    thus thesis by ABCMIZ_1:139;
  end;
end;

theorem
 for T1,T2,T3 being quasi-type of C st T1 matches_with T2 & T2 matches_with T3
  holds T1 matches_with T3
  proof
    let t1,t2,t3 be quasi-type of C;
    given f1 being valuation of C such that
A1: (adjs t2) at f1 c= adjs t1 & (the_base_of t2) at f1 = the_base_of t1;
    given f2 being valuation of C such that
A2: (adjs t3) at f2 c= adjs t2 & (the_base_of t3) at f2 = the_base_of t2;
    take f2 at f1;
    ((adjs t3) at f2) at f1 c= (adjs t2) at f1 by A2,ABCMIZ_1:146; then
    ((adjs t3) at f2) at f1 c= adjs t1 by A1;
    hence thesis by A1,A2,ABCMIZ_1:149,150;
  end;

definition
  let C be initialized ConstructorSignature;
  let t1,t2 be expression of C;
  let f be valuation of C;
::$N Unification of Mizar terms
  pred f unifies t1,t2 means
  t1 at f = t2 at f;
end;

theorem
 for t1,t2 being expression of C for f being valuation of C st f unifies t1,t2
  holds f unifies t2,t1;

definition
  let C be initialized ConstructorSignature;
  let t1,t2 be expression of C;
  pred t1,t2 are_unifiable means
  ex f being valuation of C st f unifies t1,t2;
  reflexivity
  proof
    let t be expression of C;
    set f = the valuation of C;
    take f; thus t at f = t at f;
  end;
  symmetry
  proof
    let t1,t2 be expression of C;
    given f being valuation of C such that
A1: f unifies t1,t2;
    take f;
    thus t2 at f = t1 at f by A1;
  end;
end;

definition
  let C be initialized ConstructorSignature;
  let t1,t2 be expression of C;
  pred t1,t2 are_weakly-unifiable means
  ex g being irrelevant one-to-one valuation of C
  st variables_in t2 c= dom g & t1,t2 at g are_unifiable;
  reflexivity
  proof
    let t be expression of C;
    take C idval variables_in t;
    thus thesis by ABCMIZ_1:131,137;
  end;

::  symmetry;
end;

:: theorem
::  for t1,t2 being expression of C st t1 matches_with t2
::   holds t1,t2 are_weakly-unifiable;

theorem
 for t1,t2 being expression of C st t1,t2 are_unifiable
  holds t1,t2 are_weakly-unifiable
  proof
   let t1,t2 be expression of C;
   given f being valuation of C such that
A1: f unifies t1,t2;
   take g = C idval variables_in t2;
   thus variables_in t2 c= dom g by ABCMIZ_1:131;
   take f; thus f unifies t1,t2 at g by A1,ABCMIZ_1:137;
  end;

definition
  let C be initialized ConstructorSignature;
  let t,t1,t2 be expression of C;
  pred t is_a_unification_of t1,t2 means
  ex f being valuation of C st f unifies t1,t2 & t = t1 at f;
end;

theorem
 for t1,t2,t being expression of C st t is_a_unification_of t1,t2
  holds t is_a_unification_of t2,t1
  proof
   let t1,t2,t be expression of C;
   given f being valuation of C such that
A1: f unifies t1,t2 & t = t1 at f;
   take f;
   thus f unifies t2,t1 & t = t2 at f by A1;
  end;

theorem
 for t1,t2,t being expression of C st t is_a_unification_of t1,t2
  holds t matches_with t1 & t matches_with t2
  proof
   let t1,t2,t be expression of C;
   given f being valuation of C such that
A1: f unifies t1,t2 & t = t1 at f;
   thus ex f being valuation of C st t = t1 at f by A1;
   take f; thus t = t2 at f by A1;
  end;

definition
  let C be initialized ConstructorSignature;
  let t,t1,t2 be expression of C;
  pred t is_a_general-unification_of t1,t2 means
  t is_a_unification_of t1,t2 &
  for u being expression of C st u is_a_unification_of t1,t2
  holds u matches_with t;
end;

:: theorem
::  for t1,t2 being expression of C st t1,t2 are_unifiable
::  ex t being expression of C st t is_a_general-unification_of t1,t2;

begin :: Type distribution

theorem Th57:
  for n being Nat for s being SortSymbol of MaxConstrSign
  ex m being constructor OperSymbol of s
  st len the_arity_of m = n
  proof set C = MaxConstrSign;
    let n be Nat;
    let s be SortSymbol of C;
    deffunc F(Nat) = [{},$1];
    consider l being FinSequence such that
A1: len l = n and
A2: for i st i in dom l holds l.i = F(i) from FINSEQ_1:sch 2;
A3: l is one-to-one
    proof
      let i,j be object such that
A4:   i in dom l & j in dom l & l.i = l.j;
      reconsider i,j as Nat by A4;
      l.i = F(i) & l.i = F(j) by A2,A4; then
      i = F(j)`2;
      hence thesis;
    end;
    rng l c= Vars
    proof
      let z be object; assume z in rng l; then
      consider a being object such that
A5:   a in dom l & z = l.a by FUNCT_1:def 3;
      reconsider a as Nat by A5;
      z = F(a) by A2,A5;
      hence thesis by ABCMIZ_1:25;
    end; then
    reconsider l as one-to-one FinSequence of Vars by A3,FINSEQ_1:def 4;
    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 i,x; assume i in dom l & x = l.i; then
      x = F(i) by A2;
      hence thesis;
    end; then
    reconsider l as quasi-loci by ABCMIZ_1:30;
    set m = [s,[l,0]];
 the carrier of C = {a_Type, an_Adj, a_Term} by ABCMIZ_1:def 9;
then A6: m in Constructors by Th28; then
    m in {*,non_op}\/Constructors by XBOOLE_0:def 3; then
    reconsider m as constructor OperSymbol of C by A6,Th42,ABCMIZ_1:def 24;
    the_result_sort_of m = m`1 by ABCMIZ_1:def 24 .= s; then
    reconsider m as constructor OperSymbol of s by ABCMIZ_1:def 32;
    take m;
    thus len the_arity_of m = card m`2`1 by ABCMIZ_1:def 24
      .= card [l,0]`1
      .= card l
      .= n by A1;
  end;

theorem Th58:
  for l for s being SortSymbol of MaxConstrSign
  for m being constructor OperSymbol of s st len the_arity_of m = len l
  holds variables_in (m-trm args l) = rng l
  proof
    let l; set X = rng l;
    set n = len l;
    set C = MaxConstrSign;
    let s be SortSymbol of C;
    let m be constructor OperSymbol of s such that
A1: len the_arity_of m = n;
    set p = args l;
    set Y = {variables_in t where t is quasi-term of C: t in rng p};
A2: len p = len the_arity_of m by A1,Def18; then
A3: variables_in (m-trm p) = union Y by ABCMIZ_1:90;
A4: dom p = dom l by A1,A2,FINSEQ_3:29;
    thus variables_in (m-trm p) c= X
    proof
      let s be object; assume s in variables_in (m-trm p); then
      consider A being set such that
A5:   s in A & A in Y by A3,TARSKI:def 4;
      consider t being quasi-term of C such that
A6:   A = variables_in t & t in rng p by A5;
      consider z being object such that
A7:   z in dom p & t = p.z by A6,FUNCT_1:def 3;
      reconsider z as Element of NAT by A7;
      l.z = l/.z by A4,A7,PARTFUN1:def 6; then
A8:   l/.z in X by A4,A7,FUNCT_1:def 3;
      p.z = (l/.z)-term C by A4,A7,Def18; then
      A = {l/.z} by A6,A7,ABCMIZ_1:86;
      hence thesis by A5,A8,TARSKI:def 1;
    end;
    let s be object; assume s in X; then
    consider z being object such that
A9: z in dom l & s = l.z by FUNCT_1:def 3;
    reconsider z as Element of NAT by A9;
    set t = (l/.z)-term C;
    p.z = t & l.z = l/.z by A9,Def18,PARTFUN1:def 6; then
    variables_in t = {s} & t in rng p by A4,A9,ABCMIZ_1:86,FUNCT_1:def 3; then
    s in {s} & {s} in Y by TARSKI:def 1;
    hence thesis by A3,TARSKI:def 4;
  end;

theorem Th59:
  for X being finite Subset of Vars st varcl X = X
  for s being SortSymbol of MaxConstrSign
  ex m being constructor OperSymbol of s st ::a_Type MaxConstrSign
  ex p being FinSequence of QuasiTerms MaxConstrSign
  st len p = len the_arity_of m & vars (m-trm p) = X
  proof
    let X be finite Subset of Vars;
    assume
A1: varcl X = X; then
    consider l such that
A2: rng l = X by Th31;
    set n = len l;
    set C = MaxConstrSign;
    let s be SortSymbol of C;
    consider m being constructor OperSymbol of s such that
A3: len the_arity_of m = n by Th57;
    take m;
    set p = args l;
    take p;
    thus len p = len the_arity_of m by A3,Def18;
    thus thesis by A1,A2,A3,Th58;
  end;

definition
  let d be PartFunc of Vars, QuasiTypes;
  attr d is even means
  for x,T st x in dom d & T = d.x holds vars T = vars x;
end;

definition
  let l be quasi-loci;
  mode type-distribution of l -> PartFunc of Vars, QuasiTypes means:
Def30:
  dom it = rng l & it is even;
  existence
  proof
    defpred P[object,object] means
ex x,T st $1 = x & $2 = T & vars T = vars x;
A1: for z being object st z in rng l ex y being object st P[z,y]
    proof
      set C = MaxConstrSign;
      let z be object; assume z in rng l; then
      reconsider x = z as variable;
      varcl vars x = vars x by Th2; then
      consider m being constructor OperSymbol of a_Type C,
      p being FinSequence of QuasiTerms C such that
A2:   len p = len the_arity_of m & vars (m-trm p) = vars x by Th59;
      a_Type C in the carrier of C &
      the carrier of C c= rng the ResultSort of C by ABCMIZ_1:def 54; then
      consider o being object such that
A3:   o in dom the ResultSort of C & a_Type C = (the ResultSort of C).o
      by FUNCT_1:def 3;
      reconsider o as OperSymbol of C by A3;
      the_result_sort_of o = a_Type C by A3; then
      the_result_sort_of m = a_Type C by ABCMIZ_1:def 32; then
      reconsider q = m-trm p as pure expression of C, a_Type C
      by A2,ABCMIZ_1:75;
      set B = {} QuasiAdjs C;
      reconsider T = B ast q as quasi-type;
      take T, x, T;
      thus thesis by A2,ABCMIZ_1:106;
    end;
    consider f being Function such that
A4: dom f = rng l and
A5: for z being object st z in rng l holds P[z,f.z] from CLASSES1:sch 1(A1);
    rng f c= QuasiTypes
    proof
      let y be object; assume y in rng f; then
      consider z being object such that
A6:   z in dom f & y = f.z by FUNCT_1:def 3;
      P[z,y] by A4,A5,A6;
      hence thesis by ABCMIZ_1:def 43;
    end; then
    reconsider f as PartFunc of Vars, QuasiTypes by A4,RELSET_1:4;
    take f; thus dom f = rng l by A4;
    let x,T; assume x in dom f & T = f.x; then
    P[x,T] by A4,A5;
    hence thesis;
  end;
end;

theorem
  for l being empty quasi-loci holds {} is type-distribution of l
  proof
    let l be empty quasi-loci;
    reconsider d = {} as PartFunc of Vars, QuasiTypes by RELSET_1:12;
    dom d = rng l & d is even;
    hence thesis by Def30;
  end;