hcju/isabelleps huggingface dataset:
preserves_modes_mem_def2: fixes \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set" shows "gen_refine.preserves_modes_mem \<R> = (\<forall>c\<^sub>A mds\<^sub>A mem\<^sub>A c\<^sub>C mds\<^sub>C mem\<^sub>C. (\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> mem\<^sub>A = gen_refine.mem\<^sub>A_of mem\<^sub>C \<and> mds\<^sub>A = gen_refine.mds\<^sub>A_of mds\<^sub>C)"
leq_refl: fixes leq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f"   and x :: "'f" assumes "partial_function_definitions leq lub" shows "leq x x"
\<eta>\<sigma>uni: shows "unifier\<^sub>l\<^sub>s (\<eta> \<cdot> \<sigma>) (L\<^sub>1 \<union> L\<^sub>2\<^sup>C)"
pivot_tableau'pivot_tableau: fixes t :: "(nat \<times> linear_poly) list"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> t"   and "x\<^sub>i \<in> lvars t" shows "pivot_tableau' x\<^sub>i x\<^sub>j t = pivot_tableau x\<^sub>i x\<^sub>j t"
split_option_all: fixes P :: "'f option \<Rightarrow> bool" shows "(\<forall>x. P x) = (P None \<and> (\<forall>x. P (Some x)))"
run_one_step_basic_testop_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x281 :: "t"   and x282 :: "testop"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Testop x281 x282) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
simps_1: fixes n :: "nat" assumes "n \<le> 3" shows "closest_pair_recurrence n = 3 + real n + mergesort_recurrence n + real n * real n"
ltl_lang_equiv_and_false_1: fixes \<phi>\<^sub>1 :: "'a ltln"   and \<phi>\<^sub>2 :: "'a ltln" assumes "\<phi>\<^sub>1 \<sim>\<^sub>L false\<^sub>n" shows "\<phi>\<^sub>1 and\<^sub>n \<phi>\<^sub>2 \<sim>\<^sub>L false\<^sub>n"
REFSTEP: fixes it :: "'S set"   and it' :: "'Sa set"   and \<sigma> :: "'\<sigma>"   and \<sigma>' :: "'\<sigma>a"   and x' :: "'Sa"   and x :: "'S" assumes "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi> it \<sigma>"   and "\<Phi>' it' \<sigma>'"   and "x' = \<alpha> x"   and "it' = \<alpha> ` it"   and "((it, \<sigma>), it', \<sigma>') \<in> R"   and "x \<in> it"   and "x' \<in> it'"   and "c \<sigma>"   and "c' \<sigma>'" shows "f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"
rroots_def: fixes p :: "real poly" shows "rroots p = (if p = 0 \<or> 2 < degree p then [] else if degree p = 0 then [] else if degree p = 1 then [roots1 p] else rroots2 p)"
nonneg_weights: fixes v :: "'V"   and w :: "'W"   and v' :: "'V" assumes "(v, w, v') \<in> gv.E" shows "(0::'W) \<le> w"
coboundedI2: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<le> c" shows "min a b \<le> c"
less_lsum_nchotomy: fixes al :: "'a list"   and f :: "'a \<Rightarrow> nat"   and i :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "i < lsum f al" shows "\<exists>n j. n < length al \<and> j < f (al ! n) \<and> i = lsum f (take n al) + j"
greater_Card: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "\<kappa> < vcard (VPow \<kappa>)"
multiset_order_inclusion_eq: fixes E :: "'b multiset"   and F :: "'b multiset"   and r :: "('b \<times> 'b) set" assumes "E \<subseteq># F"   and "trans r" shows "E = F \<or> (E, F) \<in> mult r"
induct: fixes P :: "nat \<Rightarrow> nat set list \<Rightarrow> nat set list \<Rightarrow> 'b option list set list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat set list"   and a2 :: "nat set list"   and a3 :: "'b option list set list" assumes "\<And>n A_pos A_neg L. \<lbrakk>\<And>x xa x2 xb y xc ya xd yb xe yc xf yba xg yaa xh yab xi yd xj ye xk yf xl yg. \<lbrakk>\<not> length A_pos + length A_neg \<noteq> length L; x = take (length A_pos) L; xa = drop (length A_pos) L; dominate_True A_pos x = None; dominate_False A_pos x A_neg xa = Some x2; (xb, y) = x2; (xc, ya) = xb; (xd, yb) = xc; (xe, yc) = yb; (xf, yba) = ya; (xg, yaa) = yba; (xh, yab) = y; (xi, yd) = xh; (xj, ye) = yd; (xk, yf) = yab; (xl, yg) = yf\<rbrakk> \<Longrightarrow> P n (xd @ xe # yc) (xf @ yaa) ((xi @ bin_join n xe xj False xg xl # ye) @ xk @ yg); \<And>x xa x2 xb y xc ya xd yb xe yaa xf yab xg yac xh yc xi yd xj ye. \<lbrakk>\<not> length A_pos + length A_neg \<noteq> length L; x = take (length A_pos) L; xa = drop (length A_pos) L; dominate_True A_pos x = Some x2; (xb, y) = x2; (xc, ya) = xb; (xd, yb) = ya; (xe, yaa) = yb; (xf, yab) = yaa; (xg, yac) = y; (xh, yc) = yac; (xi, yd) = yc; (xj, ye) = yd\<rbrakk> \<Longrightarrow> P n (xc @ (xd \<union> xf) # xe @ yab) A_neg ((xg @ bin_join n xd xh True xf xj # xi @ ye) @ xa)\<rbrakk> \<Longrightarrow> P n A_pos A_neg L" shows "P a0 a1 a2 a3"
fold_image: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" assumes "inj_on g A" shows "Finite_Set.fold f z (g ` A) = Finite_Set.fold (f \<circ> g) z A"
Gamma_0: shows "Gamma (0::'a) = (0::'a)"
execl_all_def: fixes P :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" shows "P \<Turnstile> cfs\<box> \<equiv> P \<Turnstile> cfs \<and> cfs \<noteq> [] \<and> fst (cfs ! 0) = 0 \<and> fst (cfs ! (length cfs - 1)) \<notin> {0..<size P}"
tailrec_impl_def: shows "local.tailrec_impl \<equiv> gds_init gds \<bind> FOREACH\<^sub>C\<^bsup>\<lambda>it s. gen_rwof s \<and> (\<not> gds_is_break gds s \<longrightarrow> gds_is_empty_stack gds s) \<and> V0 - it \<subseteq> gen_discovered s\<^esup> V0 ((Not \<circ>\<circ> gds_is_break) gds) (\<lambda>v0 s. let s0 = s in if gds_is_discovered gds v0 s then RETURN s else gds_new_root gds v0 s \<bind> WHILE\<^bsup>\<lambda>s. gen_rwof s \<and> insert v0 (gen_discovered s0) \<subseteq> gen_discovered s\<^esup> (\<lambda>s. \<not> gds_is_break gds s \<and> \<not> gds_is_empty_stack gds s) (\<lambda>s. gds_get_pending gds s \<bind> (\<lambda>(u, Vs, s). case Vs of None \<Rightarrow> gds_finish gds u s | Some v \<Rightarrow> if gds_is_discovered gds v s then if gds_is_finished gds v s then gds_cross_edge gds u v s else gds_back_edge gds u v s else gds_discover gds u v s)))"
case_1: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x1 :: "'b" shows "(case Normal x1 of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f1 x1"
tree_sigma_cong: fixes M :: "'b measure"   and M' :: "'b measure" assumes "sets M = sets M'" shows "tree_sigma M = tree_sigma M'"
add_less_imp_less_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less (plus c a) (plus c b)" shows "less a b"
assms: shows "\<S>\<bullet>\<C> \<turnstile> $$* vs : ts' _> ts' @ ts"
hyps: fixes i :: "nat"   and m :: "nat \<Rightarrow> nat" assumes "i < length ls"   and "ls ! i = (p, Read volatile a t # is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "flush_all_until_volatile_write (ls[i := (p, is, \<theta>(t \<mapsto> v), sb @ [Read\<^sub>s\<^sub>b volatile a t v], \<D>', \<O>', \<R>')]) m = flush_all_until_volatile_write ls m"
update_convs_2: fixes edgesL' :: "('b \<times> 'b) list \<Rightarrow> ('b \<times> 'b) list"   and nodesL :: "'b list"   and edgesL :: "('b \<times> 'b) list"   and more :: "'c" shows "edgesL_update edgesL' \<lparr>nodesL = nodesL, edgesL = edgesL, \<dots> = more\<rparr> = \<lparr>nodesL = nodesL, edgesL = edgesL' edgesL, \<dots> = more\<rparr>"
is_pascal_rot_CW: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and E :: "Points"   and F :: "Points" assumes "is_pascal A B C D E F" shows "is_pascal F A B C D E"
iprefix_imp_take_eq: fixes n :: "nat"   and xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" assumes "n \<le> length xs"   and "xs \<sqsubseteq> f" shows "xs \<down> n = f \<Down> n"
lrev_work2_def: shows "lrev_work2 \<equiv> fix\<cdot>lrev_body2"
unwrapExist': fixes j :: "nat"   and A :: "atom fm"   and L :: "real list" shows "eval (ExN (j + 1) A) L = eval (ExN j (ExQ A)) L"
map_bset_eqvt: fixes p :: "perm"   and f :: "'g \<Rightarrow> 'e"   and B :: "'g set['f]" shows "p \<bullet> map_bset f B = map_bset (p \<bullet> f) (p \<bullet> B)"
simps_8: fixes c :: "int"   and e :: "MIR.num" shows "minusinf (Ge (CN 0 c e)) = F"
it_path_SnocI: fixes "is" :: "nat list"   and t :: "'a rose_tree"   and i :: "nat" assumes "is \<in> it_paths t"   and "i < length (children (tree_at t is))" shows "is @ [i] \<in> it_paths t"
subgoal1: shows "\<not> (\<forall>(e1, e2)\<in>edges G. P ((nP(i := \<bottom>)) e1) (nP e2))"
last_snoc: fixes xs :: "'a list"   and x :: "'a" shows "last (xs ## x) = x"
sigma_algebra_disjoint_iff: fixes \<Omega> :: "'a set"   and M :: "'a set set" shows "sigma_algebra \<Omega> M = (algebra \<Omega> M \<and> (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M))"
well_order_on_Field: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" assumes "well_order_on A r" shows "A = Field r"
attach_trace_eq': fixes \<I> :: "('a, 'b) \<I>"   and res1 :: "('a, 'b) resource"   and res2 :: "('a, 'b) resource"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('c, 'd, 'a, 'b) converter" assumes "outs_\<I> \<I> \<turnstile>\<^sub>R res1 \<approx> res2"   and "\<I> \<turnstile>res res1 \<surd>"   and "\<I> \<turnstile>res res2 \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "outs_\<I> \<I>' \<turnstile>\<^sub>R conv \<rhd> res1 \<approx> conv \<rhd> res2"
fBallI: fixes A :: "'a fset"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x |\<in>| A \<Longrightarrow> P x" shows "fBall A P"
NoFalsity: fixes H :: "('a, 'b) fm set" assumes "Hintikka H" shows "\<^bold>\<bottom> \<notin> H"
bprog_vars_def: fixes bp :: "instr array" shows "bprog_vars bp = \<Union> (instr_vars ` set (list_of_array bp))"
finite_dimensional_vector_space_on_axioms_def: fixes U\<^sub>V\<^sub>S :: "'c set"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and basis :: "'c set" shows "finite_dimensional_vector_space_on_axioms U\<^sub>V\<^sub>S scale basis \<equiv> (finite basis \<and> \<not> module_on.dependent scale basis) \<and> module_on.span scale basis = U\<^sub>V\<^sub>S \<and> basis \<subseteq> U\<^sub>V\<^sub>S"
learnLessState_def: shows "learnLessState \<equiv> {(stateA, stateB). getM stateA = getM stateB \<and> getConflictFlag stateA = getConflictFlag stateB \<and> getC stateA = getC stateB \<and> (getF stateA, getF stateB) \<in> learnLess (getC stateA)}"
factors_of_int_poly_4: fixes p :: "int poly"   and qs :: "int poly list" assumes "factors_of_int_poly p = qs" shows "distinct qs"
INV_rule: fixes T :: "('e, 'f) TS_scheme"   and I :: "'e set" assumes "init T \<subseteq> I"   and "{I \<inter> reach T} TS.trans T {> I}" shows "reach T \<subseteq> I"
assms_4: shows "brnL cl (pickFT cl) \<in> II"
parenthesis_impl_tree_path: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (finished s)"   and "w \<in> dom (finished s)"   and "\<delta> s v < \<delta> s w"   and "\<phi> s w < \<phi> s v" shows "(v, w) \<in> (tree_edges s)\<^sup>+"
Syntax_with_Numerals_and_Connectives_axioms: shows "Syntax_with_Numerals_and_Connectives var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi"
vec_simps_12: fixes ag :: "'b"   and bg :: "'b" shows "ag + bg = bg + ag"
relcomp_def: shows "(O) \<equiv> \<lambda>r s. {(xa, x). ((\<lambda>x xa. (x, xa) \<in> r) OO (\<lambda>x xa. (x, xa) \<in> s)) xa x}"
cenvalid_def: fixes A :: "triple set"   and t :: "etriple" shows "A |\<Turnstile>\<^sub>e t \<equiv> \<forall>n. |\<Turnstile>n: A \<longrightarrow> \<Turnstile>n:\<^sub>e t"
stream_smap_nats: fixes s :: "'c stream" shows "s = smap ((!!) s) nats"
right_idem: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.max (signed.max a b) b = signed.max a b"
intros_3: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> boolean_axioms" shows "H \<turnstile> A"
stabiliser_subgroup: fixes x :: "'b" shows "subgroup (stabiliser x) G"
open_UNION_box: fixes M :: "'b set" assumes "open M" shows "M = (\<Union>f\<in>{f \<in> Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) \<subseteq> M}. box (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i))"
bounded_linear_image: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "bounded S"   and "bounded_linear f" shows "bounded (f ` S)"
F_def: shows "F \<equiv> principal (complex_cone' \<alpha> - {0})"
delta': fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'd" assumes "finite S" shows "(\<Sum>k\<in>S. if a = k then b k else (0::'d)) = (if a \<in> S then b a else (0::'d))"
negligible_Un_eq: fixes s :: "'c set"   and t :: "'c set" shows "negligible (s \<union> t) = (negligible s \<and> negligible t)"
simps: fixes a1 :: "binding list"   and a2 :: "POPLmark.type"   and a3 :: "POPLmark.type" shows "(a1 \<turnstile> a2 <: a3) = ((\<exists>\<Gamma> S. a1 = \<Gamma> \<and> a2 = S \<and> a3 = Top \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f S) \<or> (\<exists>\<Gamma> i. a1 = \<Gamma> \<and> a2 = TVar i \<and> a3 = TVar i \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f \<and> \<Gamma> \<turnstile>\<^sub>w\<^sub>f TVar i) \<or> (\<exists>\<Gamma> i U T. a1 = \<Gamma> \<and> a2 = TVar i \<and> a3 = T \<and> \<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB U\<rfloor> \<and> \<Gamma> \<turnstile> \<up>\<^sub>\<tau> (Suc i) 0 U <: T) \<or> (\<exists>\<Gamma> T\<^sub>1 S\<^sub>1 S\<^sub>2 T\<^sub>2. a1 = \<Gamma> \<and> a2 = S\<^sub>1 \<rightarrow> S\<^sub>2 \<and> a3 = T\<^sub>1 \<rightarrow> T\<^sub>2 \<and> \<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1 \<and> \<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2) \<or> (\<exists>\<Gamma> T\<^sub>1 S\<^sub>1 S\<^sub>2 T\<^sub>2. a1 = \<Gamma> \<and> a2 = (\<forall><:S\<^sub>1. S\<^sub>2) \<and> a3 = (\<forall><:T\<^sub>1. T\<^sub>2) \<and> \<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1 \<and> TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2))"
normalize_normalize: fixes v :: "complex vec" shows "vec_normalize (vec_normalize v) = vec_normalize v"
GSubspace_Ker: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'c" assumes "FGModuleEnd G smult V T" shows "RModule (aezfun_setspan G) smult (ker T \<inter> V) \<and> ker T \<inter> V \<subseteq> V"
list_Nil_iff: fixes t :: "'b tree" shows "(list t = []) = (t = \<langle>\<rangle>)"
triv_RepFun: fixes A :: "hf" shows "\<lbrace>x . x \<^bold>\<in> A\<rbrace> = A"
Nml_HcompD_5: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "\<not> is_Prim\<^sub>0 u"
li_trans_l: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset" assumes "locally_irrefl R (A + B)" shows "locally_irrefl R A"
top_ow_def: fixes U :: "'a set"   and top :: "'a" shows "top_ow U top \<equiv> top \<in> U"
orthogonal_matrix_def: fixes Q :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = Finite_Cartesian_Product.mat (1::'a) \<and> Q ** Finite_Cartesian_Product.transpose Q = Finite_Cartesian_Product.mat (1::'a))"
divide_out_primepow_ex: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and n :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "n \<noteq> zero"   and "\<exists>p\<in>#factorial_semiring.prime_factorization zero times one normalize n. P p" obtains p :: "'a" and k :: "nat" and n' :: "'a" where "P p" and "normalization_semidom.prime zero times one normalize p" and "dvd.dvd times p n" and "\<not> dvd.dvd times p n'" and "0 < k" and "n = times (power.power one times p k) n'"
elims: fixes x :: "complex \<times> complex"   and xa :: "complex \<times> complex"   and y :: "complex" assumes "x *\<^sub>v\<^sub>v xa = y" obtains xb :: "complex" and ya :: "complex" and a :: "complex" and b :: "complex" where "x = (xb, ya)" and "xa = (a, b)" and "y = xb * a + ya * b"
bindWT_fmapWT_unitWT: fixes f :: "'h \<rightarrow> 'e"   and m :: "'h\<cdot>('f, 'g) writerT" shows "bindWT\<cdot>(fmapWT\<cdot>f\<cdot>m)\<cdot>unitWT = fmapWT\<cdot>f\<cdot>m"
trancl_trans: fixes R :: "('a \<times> 'a) set"   and P :: "('a \<times> 'a) set" assumes "trans R" shows "(P\<^sup>+ \<subseteq> R) = (P \<subseteq> R)"
subst_subst: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "y \<notin> Fvars \<phi>" shows "subst (subst \<phi> (Var y) x) t y = subst \<phi> t x"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>v. P \<epsilon> v"   and "\<And>va vb. P (va # vb) \<epsilon>"   and "\<And>a u b v. (a = b \<Longrightarrow> P u v) \<Longrightarrow> P (a # u) (b # v)" shows "P a0 a1"
intPI_def: shows "intPI \<equiv> MonotProblem.intPI TE_wtFsym wtPsym TE_arOf TE_resOf parOf TE_\<Phi>"
ide_def: fixes a :: "'c" shows "C.ide a \<equiv> a \<cdot> a \<noteq> C.null \<and> (\<forall>f. (f \<cdot> a \<noteq> C.null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> C.null \<longrightarrow> a \<cdot> f = f))"
Abs_dlist_def: fixes xs :: "'c list" shows "Abs_dlist xs = Abs_dlist' (if equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None then xs else equal_base.list_remdups ceq' xs)"
fmdomE: fixes x :: "'c"   and m :: "('c, 'd) fmap" assumes "x |\<in>| fmdom m" obtains y :: "'d" where "m $$ x = Some y"
simps_1: fixes \<Psi> :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x0 :: "'b"   and t :: "nat \<Rightarrow> 'c" shows "grid_function \<Psi> x0 t 0 = x0"
Fvars_lcnj: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and \<phi>s :: "'c list"   and F :: "'d set" assumes "Syntax_with_Connectives_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls"   and "set \<phi>s \<subseteq> fmla"   and "finite F" shows "Fvars (Syntax_with_Connectives_False.lcnj cnj imp fls \<phi>s) = \<Union> (set (map Fvars \<phi>s))"
induct: fixes P :: "'c\<cdot>maybe \<Rightarrow> bool"   and x :: "'c\<cdot>maybe" assumes "P \<bottom>"   and "P Nothing"   and "\<And>a. P (Just\<cdot>a)" shows "P x"
mpdl: shows "\<pi> m pd\<rightarrow> \<pi> l"
case_2: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Del of Ins \<Rightarrow> f1 | Del \<Rightarrow> f2) = f2"
card_paths_with_new: shows "card paths_with_new = Suc (card paths)"
10: shows "m \<in> {e. f e \<noteq> bot \<and> f e + bot = fe + bot} \<and> (\<forall>y. y \<in> {e. f e \<noteq> bot \<and> f e + bot = fe + bot} \<longrightarrow> enum_lex_less_eq m y)"
noFault_startD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "'e"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Normal t" shows "s \<noteq> Fault f"
greaterThanLessThan_subseteq_greaterThanLessThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.dense_linorder less_eq less" shows "(ord.greaterThanLessThan less a b \<subseteq> ord.greaterThanLessThan less c d) = (less a b \<longrightarrow> less_eq c a \<and> less_eq b d)"
closed_except_def: fixes t :: "'a"   and S :: "name fset" shows "closed_except t S = (frees t |\<subseteq>| S)"
bisimSubstReflexive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<sim>\<^sub>s P"
distinct_7: shows "ConcatOp \<noteq> string_binop.LessOp"
countable_INT'': fixes M :: "'a measure"   and I :: "'b set"   and F :: "'b \<Rightarrow> 'a set" assumes "UNIV \<in> sets M"   and "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M" shows "\<Inter> (F ` I) \<in> sets M"
normal_subgroup_in_kernel_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d"   and L :: "'c set" shows "normal_subgroup_in_kernel \<eta> G composition unit G' composition' unit' L \<equiv> group_homomorphism \<eta> G composition unit G' composition' unit' \<and> normal_subgroup L G composition unit \<and> normal_subgroup_in_kernel_axioms \<eta> G unit' L"
map_simp: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" shows "FF.map f = (F (fst f), F (snd f))"
shortest_path_decr_ek_measure: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "NFlow c s t f"   and "Graph.isShortestPath (residualGraph c f) s p t" shows "ek_analysis_defs.ekMeasure (residualGraph c (NFlow.augment c f (NPreflow.augmentingFlow c f p))) s t < ek_analysis_defs.ekMeasure (residualGraph c f) s t"
mtx_row_length: fixes signs :: "rat list list"   and ls :: "nat list list" shows "list_all (\<lambda>r. length r = length signs) (map (mtx_row signs) ls)"