hcju/isabelleps huggingface dataset:
IH: fixes z :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term"   and yb :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" assumes "z \<in> set T"   and "yb \<in> set S"   and "timpls_transformable_to' TI z yb"   and "wf\<^sub>t\<^sub>r\<^sub>m z"   and "\<forall>f\<in>funs_term z. is_Abs f \<longrightarrow> the_Abs f \<in> A" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) z yb"
finished_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "finished r x = (\<nexists>y. r x y)"
SUP_id_eq: fixes Sup :: "'c set \<Rightarrow> 'c"   and A :: "'c set" shows "Sup (id ` A) = Sup A"
eq_on_cyclic_on_iff1: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "cyclic_on f S"   and "x \<in> S" obtains "f x \<in> S" and "(f x = x) = (card S = 1)"
Nonce_Enc: fixes X :: "nonce_t"   and X' :: "msg"   and Y' :: "msg" shows "Nonce X \<noteq> Enc X' Y'"
comp_transfer: fixes B :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "((B ===> C) ===> (A ===> B) ===> A ===> C) (\<circ>) (\<circ>)"
rep_eq: fixes x :: "('c, 'd) fmap" shows "fmdom' x = dom (($$) x)"
filterlim_split_at_real: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter"   and x :: "real" assumes "filterlim f F (at_left x)"   and "filterlim f F (at_right x)" shows "filterlim f F (at x)"
isodefl_cast: fixes t :: "udom defl" shows "isodefl (cast\<cdot>t) t"
assms: shows "fpairs r \<subseteq>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B"
Mp_smult_1: fixes m :: "int"   and a :: "int"   and f :: "int poly" shows "poly_mod.Mp m (smult (poly_mod.M m a) f) = poly_mod.Mp m (smult a f)"
ideal_points_different_5: fixes u :: "complex_homo"   and v :: "complex_homo"   and i1 :: "complex_homo"   and i2 :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v"   and "ideal_points (poincare_line u v) = {i1, i2}" shows "v \<noteq> i2"
f3: fixes a :: "nat"   and b :: "nat"   and c :: "nat" assumes "a < b"   and "b < c" shows "f3 {a, b, c} = f {(a, b), (a, c)}"
IC1_1: shows "callee_invariant_on (callee_of_core core1) IC1 (\<I>E \<oplus>\<^sub>\<I> (\<I>CA \<oplus>\<^sub>\<I> \<I>CU))"
wf_tuple_Suc_fviD: fixes n :: "nat"   and b :: "nat"   and \<phi> :: "Formula.formula"   and v :: "'a option list" assumes "wf_tuple (Suc n) (Formula.fvi b \<phi>) v" shows "wf_tuple n (Formula.fvi (Suc b) \<phi>) (tl v)"
P_new_solves_if_disjoint: assumes "new_last = v1" shows "\<exists>paths'. DisjointPaths G v0 v1 paths' \<and> card paths' = Suc (card paths)"
sepref_relpropI: fixes P :: "'f \<Rightarrow> 'e"   and R :: "'f" shows "P R = CONSTRAINT P R"
choose_minimal: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and v :: "'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "WellOrderedStrategies G S p good r"   and "v \<in> S"   and "(\<sigma>', WellOrderedStrategies.choose good r v) \<in> r - Id" shows "\<sigma>' \<notin> good v"
K3_imp_K2: fixes A :: "agent"   and authTicket :: "msg"   and authK :: "nat"   and T2 :: "nat"   and B :: "agent"   and evs :: "event list" assumes "Says A Tgs \<lbrace>authTicket, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> kerbIV_gets" shows "\<exists>Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"
is_mgu_def: fixes \<sigma> :: "'s"   and AAA :: "'a set set" shows "is_mgu \<sigma> AAA = (is_unifiers \<sigma> AAA \<and> (\<forall>\<tau>. is_unifiers \<tau> AAA \<longrightarrow> (\<exists>\<gamma>. \<tau> = \<sigma> \<odot> \<gamma>)))"
normal_series_snd_to_last: shows "\<HH> ! (length \<HH> - 2) \<lhd> G"
PseudoHA_CompFun_the: fixes S :: "'a"   and A :: "('a, 'b, 'c) seqauto"   and D :: "'c data" assumes "S \<in> States A" shows "the (CompFun (PseudoHA A D) S) = {}"
map_disc_iff: fixes f :: "'h \<Rightarrow> 'g"   and a :: "'h list" shows "(map f a = []) = (a = [])"
that_1: shows "AE x\<in>{a..b} in lebesgue. cmod (f x) \<le> B"
elims_3: fixes x :: "Core.term" assumes "\<not> beta_reducible x" obtains ux_ :: "typ" and t :: "Core.term" where "x = Abs ux_ t" and "\<not> beta_reducible t"   | v :: "String.literal" and va :: "typ" and u :: "Core.term" where "x = Ct v__ va__ $ u__" and "\<not> (beta_reducible (Ct v__ va__) \<or> beta_reducible u__)"   | v :: "variable" and va :: "typ" and u :: "Core.term" where "x = Fv vb__ vaa__ $ ua__" and "\<not> (beta_reducible (Fv vb__ vaa__) \<or> beta_reducible ua__)"   | v :: "nat" and u :: "Core.term" where "x = Bv vc__ $ ub__" and "\<not> (beta_reducible (Bv vc__) \<or> beta_reducible ub__)"   | v :: "Core.term" and va :: "Core.term" and u :: "Core.term" where "x = vd__ $ vab__ $ u" and "\<not> (beta_reducible (vd__ $ vab__) \<or> beta_reducible u)"   | v :: "String.literal" and va :: "typ" where "x = Ct ve__ vac__"   | v :: "variable" and va :: "typ" where "x = Fv vf__ va"   | v :: "nat" where "x = Bv v"
map_cong: fixes x :: "'c + 'd"   and ya :: "'c + 'd"   and f1 :: "'c \<Rightarrow> 'e"   and g1 :: "'c \<Rightarrow> 'e"   and f2 :: "'d \<Rightarrow> 'f"   and g2 :: "'d \<Rightarrow> 'f" assumes "x = ya"   and "\<And>z1. z1 \<in> Basic_BNFs.setl ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> Basic_BNFs.setr ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_sum f1 f2 x = map_sum g1 g2 ya"
proj_arcs_H_in_arcs_G: fixes a :: "'b" assumes "a \<in> arcs H" shows "proj_arcs_H a \<in> arcs G"
induct: fixes P :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list \<Rightarrow> ('sets set \<times> 'sets set) list \<Rightarrow> (('fun, 'b, 'sets) prot_fun, 'c) Term.term \<Rightarrow> bool"   and a0 :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term list"   and a1 :: "('sets set \<times> 'sets set) list"   and a2 :: "(('fun, 'b, 'sets) prot_fun, 'c) Term.term" assumes "\<And>M TI x. P M TI (Var x)"   and "\<And>M TI f T. (\<And>z. z \<in> set T \<Longrightarrow> P M TI z) \<Longrightarrow> P M TI (Fun f T)" shows "P a0 a1 a2"
leftmost_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and i :: "nat"   and s :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.leftmost \<NN> \<TT> i s = (i < length s \<and> CFG.is_word \<TT> (take i s) \<and> CFG.is_nonterminal \<NN> (s ! i))"
coboundedI1: fixes a :: "'a Word.word"   and c :: "'a Word.word"   and b :: "'a Word.word" assumes "a \<le>s c" shows "signed.min a b \<le>s c"
wtE_o: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm" assumes "wtE \<xi>"   and "wtSB \<rho>" shows "wtE (int \<xi> \<circ> \<rho>)"
coincide_on_cl: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and C :: "'b literal set" assumes "coincide_on \<sigma> \<eta> (vars_of_cl C)" shows "subst_cl C \<sigma> = subst_cl C \<eta>"
lift_Suc_antimono_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f (Suc n)) (f n)"   and "n \<le> n'" shows "less_eq (f n') (f n)"
mdecl_visible: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "String.literal"   and S :: "String.literal"   and fs :: "fdecl list"   and ms :: "'a option mdecl list"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a option" assumes "wf_prog wf_md P"   and "class P C = \<lfloor>(S, fs, ms)\<rfloor>"   and "(M, Ts, T, m) \<in> set ms" shows "P \<turnstile> C sees M: Ts\<rightarrow>T = m in C"
galois_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (inf x (uminus y)) z = less_eq x (sup y z)"
analz_synth_Un: fixes G :: "msg set"   and H :: "msg set" shows "analz (synth G \<union> H) = analz (G \<union> H) \<union> synth G"
option_rel_Some1: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e"   and y :: "'f option" shows "rel_option A (Some x) y = (\<exists>y'. y = Some y' \<and> A x y')"
inverse: fixes \<eta> :: "nat" shows "wiring (\<I>_ideal \<eta>) (\<I>_ideal \<eta>) (cnv \<eta> \<odot> sim \<eta>) (id, id)"
l_coset_swap: fixes G :: "('c, 'd) monoid_scheme"   and y :: "'c"   and x :: "'c"   and H :: "'c set" assumes "Group.group G"   and "y \<in> x <#\<^bsub>G\<^esub> H"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> y <#\<^bsub>G\<^esub> H"
thm_noncont_propos_1: fixes v :: "i" shows "[Necessary p\<^sub>0 in v]"
f_def: shows "f \<equiv> \<lambda>x. (x, One /\<^sub>R setdist {x} (- U))"
simps_7: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas_inner (\<phi> M\<^sub>n \<psi>) = subformulas\<^sub>\<nu> (\<phi> M\<^sub>n \<psi>) \<union> subformulas\<^sub>\<mu> (\<phi> M\<^sub>n \<psi>)"
mod_h_bot_indep: fixes P :: "assn"   and h :: "heap" shows "((h, {}) \<Turnstile> P) = (h\<^sub>\<bottom> \<Turnstile> P)"
simps_39: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'b"   and f3 :: "('c, 'd) fm \<Rightarrow> 'b"   and f4 :: "('c, 'd) fm \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and f5 :: "('c, 'd) fm \<Rightarrow> 'b"   and f6 :: "'d \<Rightarrow> ('c, 'd) fm \<Rightarrow> 'b"   and x3 :: "('c, 'd) fm" shows "(case \<^bold>\<not> x3 of Pro x \<Rightarrow> f1 x | Nom x \<Rightarrow> f2 x | \<^bold>\<not> x \<Rightarrow> f3 x | x \<^bold>\<or> xa \<Rightarrow> f4 x xa | \<^bold>\<diamond> x \<Rightarrow> f5 x | \<^bold>@ x xa \<Rightarrow> f6 x xa) = f3 x3"
intro: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L"   and "weak_total_order_axioms L" shows "weak_total_order L"
ring_iso_imp_img_field: fixes R :: "('c, 'd) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and S :: "('e, 'f) ring_scheme" assumes "field R"   and "h \<in> ring_iso R S" shows "field (S\<lparr>zero := h \<zero>\<^bsub>R\<^esub>\<rparr>)"
neighbourhood_base_at_unlocalized: fixes P :: "'c set \<Rightarrow> bool"   and X :: "'c topology"   and x :: "'c" assumes "\<And>S T. \<lbrakk>P S; openin X T; x \<in> T; T \<subseteq> S\<rbrakk> \<Longrightarrow> P T" shows "neighbourhood_base_at x P X = (x \<in> topspace X \<longrightarrow> (\<exists>U V. openin X U \<and> P V \<and> x \<in> U \<and> U \<subseteq> V \<and> V \<subseteq> topspace X))"
coerce_Just: fixes x :: "'d" shows "COERCE('d\<cdot>maybe, 'c\<cdot>maybe)\<cdot>(Just\<cdot>x) = Just\<cdot>(COERCE('d, 'c)\<cdot>x)"
nlists_Suc: fixes A :: "'c set"   and n :: "nat" shows "nlists A (Suc n) = (\<Union>x\<in>A. (#) x ` nlists A n)"
inf0: fixes a :: "'h" shows "a \<equiv> a \<sqinter> \<top>"
RM_1_b: fixes \<chi> :: "\<o>"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<chi> in v] \<Longrightarrow> [\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<^bold>\<box>\<chi> in v]" shows "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
star_of_int_def: fixes z :: "int" shows "of_int z = star_of (of_int z)"
UN_I: fixes a :: "'d"   and A :: "'d set"   and b :: "'e"   and B :: "'d \<Rightarrow> 'e set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
distinct_182: fixes x11 :: "Syntax.term"   and x71 :: "Syntax.term"   and x72 :: "Syntax.term" shows "Prj1 x11 \<noteq> Syntax.Pair x71 x72"
identity_\<Pi>\<^sub>3_def: shows "(\<^bold>=) \<equiv> (\<^bold>=\<^sub>3)"
SeqSubstForm_def: fixes v :: "hf"   and u :: "hf"   and x :: "hf"   and x' :: "hf"   and s :: "hf"   and k :: "hf" shows "SeqSubstForm v u x x' s k \<equiv> BuildSeq2 (SubstAtomic v u) SubstMakeForm s k x x'"
exhaust_sel: fixes option :: "'g option" obtains "option = None"   | "option = Some (the option)"
tendsto_sandwich: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and net :: "'b filter"   and h :: "'b \<Rightarrow> 'a"   and c :: "'a" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in net. less_eq (f n) (g n)"   and "\<forall>\<^sub>F n in net. less_eq (g n) (h n)"   and "filterlim f (topological_space.nhds open c) net"   and "filterlim h (topological_space.nhds open c) net" shows "filterlim g (topological_space.nhds open c) net"
Conj: fixes xset :: "('b, 'pred, 'act, 'effect) formula set['b]"   and F :: "'effect fs_set" assumes "finite (supp xset)"   and "\<And>x. x \<in> set_bset xset \<Longrightarrow> x \<in> \<A>[F]" shows "Conj xset \<in> \<A>[F]"
simps_2: fixes l :: "'b llist"   and i :: "nat" shows "l !! Suc i = (case l of LNil \<Rightarrow> None | x ## xs \<Rightarrow> xs !! i)"
seq1: fixes c1 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and m' :: "'id \<Rightarrow> 'val"   and c2 :: "('exp, 'id) MWLsCom" assumes "\<langle>c1,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>None,m'\<rangle>" shows "\<langle>c1;c2,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>Some c2,m'\<rangle>"
minus_real_alg_def: fixes x :: "real_alg"   and y :: "real_alg" shows "x - y = x + - y"
row_add_iarray_def: fixes A :: "'b iarray iarray"   and a :: "nat"   and b :: "nat"   and q :: "'b" shows "row_add_iarray A a b q = IArray.of_fun (\<lambda>n. if n = a then A !! a + mult_iarray (A !! b) q else A !! n) (IArray.length A)"
sum_over_product_right: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "CEM"   and "O y z" shows "P ((x \<oplus> y) \<otimes> (x \<oplus> z)) (x \<oplus> y \<otimes> z)"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 is-x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n ns n'' n'. \<lbrakk>n is-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' V n'. \<lbrakk>n is-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'; n'' \<noteq> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'"   and "\<And>n ns n'' p n'. \<lbrakk>n is-ns\<rightarrow>\<^sub>d* n''; P n ns n''; n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'\<rbrakk> \<Longrightarrow> P n (ns @ [n'']) n'" shows "P x1 x2 x3"
simps_6: fixes v :: "'b formula" shows "is_nnf (\<^bold>\<not> v) = is_lit_plus (\<^bold>\<not> v)"
divide_divide_eq_left': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (divide a b) c = divide a (times c b)"
lossless_bind_spmf: fixes p :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf" shows "lossless_spmf (p \<bind> f) = (lossless_spmf p \<and> (\<forall>x\<in>set_spmf p. lossless_spmf (f x)))"
\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c_matching: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<alpha> :: "action \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" shows "Matching_Ternary.matches (\<beta>\<^sub>m\<^sub>a\<^sub>g\<^sub>i\<^sub>c \<gamma>, \<alpha>) m a p = Semantics.matches \<gamma> m p"
comparable: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Y"   and "y \<in> Y" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
simps_5: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "negation_free (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = (negation_free \<phi>\<^sub>1 \<and> negation_free \<phi>\<^sub>2)"
L_B_subset_A: fixes l :: "'lbl" assumes "l \<in> set (L B)" shows "l \<in> set (L A)"
concat_map_all_distinct: fixes f :: "'y \<Rightarrow> 'x list"   and xs :: "'y list"   and x :: "'y" assumes "distinct (concat (map f xs))"   and "x \<in> set xs" shows "distinct (f x)"
dg_Hom_vifunion_in_Vset: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "B \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom \<BB> a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>"
pochhammer_eq_0_imp_nonpos_Int: fixes x :: "'a"   and n :: "nat" assumes "pochhammer x n = (0::'a)" shows "x \<in> \<int>\<^sub>\<le>\<^sub>0"
simps_5: fixes e1 :: "expr"   and e2 :: "expr" shows "randomfree (LET e1 IN e2) = (randomfree e1 \<and> randomfree e2)"
le_less_trans: fixes D :: "('d, 'e) Order_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<preceq>\<^bsub>D\<^esub> b"   and "b \<prec>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
accD_pos: fixes x :: "'s"   and y :: "'s" assumes "(x, y) \<in> acc" shows "\<exists>n. 0 < p x y n"
simps_1: fixes r :: "real"   and x :: "complex" shows "Re (r *\<^sub>R x) = r * Re x"
backward_finite_path_root_iff: fixes x :: "'a" shows "(\<exists>r. backward_finite_path_root r x) = backward_finite_path x"
total_v_num: fixes k :: "nat" assumes "\<And>v u. \<lbrakk>v \<in> V; u \<in> V; v \<noteq> u\<rbrakk> \<Longrightarrow> \<exists>!n. adjacent v n \<and> adjacent u n"   and "finite E"   and "finite V"   and "V \<noteq> {}"   and "\<forall>v\<in>V. degree v G = k"   and "0 < k" shows "card V = k * k - k + 1"
max_rank_of_def: fixes \<Sigma> :: "'a set set"   and \<psi> :: "'a ltl" shows "max_rank_of \<Sigma> \<psi> \<equiv> semi_mojmir_def.max_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M (theG \<psi>))"
equivclp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "equivclp r = (symclp r)\<^sup>*\<^sup>*"
mircfr: shows "\<forall>bs p. qfree p \<longrightarrow> qfree (mircfr p) \<and> Ifm bs (mircfr p) = Ifm bs (E p)"
Quo3: fixes n :: "nat"   and m :: "nat" shows "n * Quo n m = m - m mod n"
intros_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n\<^sub>0 :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n\<^sub>1 :: "'b SDG_node"   and p :: "'f"   and n\<^sub>2 :: "'b SDG_node"   and V :: "'c"   and ns' :: "'b SDG_node list"   and n\<^sub>3 :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.realizable sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>0 ns n\<^sub>1"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>1 None (Some (p, True)) n\<^sub>2 \<or> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>1 (Some V) (Some (p, True)) n\<^sub>2"   and "SDG.matched sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>2 ns' n\<^sub>3" shows "SDG.realizable sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n\<^sub>0 (ns @ n\<^sub>1 # ns') n\<^sub>3"
trancl_trans_induct: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "(x, y) \<in> r\<^sup>+"   and "\<And>x y. (x, y) \<in> r \<Longrightarrow> P x y"   and "\<And>x y z. \<lbrakk>(x, y) \<in> r\<^sup>+; P x y; (y, z) \<in> r\<^sup>+; P y z\<rbrakk> \<Longrightarrow> P x z" shows "P x y"
CHAOS_def: fixes A :: "'a set" shows "CHAOS A \<equiv> \<mu> x. STOP |-| Mprefix A (\<lambda>xa. x)"
as_needed_asses_submap_exec_i: fixes v :: "'d"   and a :: "('d, 'e) fmap \<times> 'f"   and s :: "('d, 'e) fmap" assumes "v \<in> fmdom' (action_needed_asses a s)" shows "fmlookup (action_needed_asses a s) v = fmlookup s v \<and> fmlookup (action_needed_asses a s) v = fmlookup (fst a) v"
less_trans: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b"   and "b \<prec>\<^bsub>D\<^esub> c" shows "a \<prec>\<^bsub>D\<^esub> c"
zero_hom_0_axioms: shows "zero_hom_0 to_fract"
set_acc_lengths: fixes ls :: "'a list list"   and acc :: "nat" assumes "ls \<in> lists (- {[]})" shows "list.set (acc_lengths acc ls) \<subseteq> {acc<..}"
trg_vcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "seq \<nu> \<mu>" shows "trg (\<nu> \<cdot> \<mu>) = trg \<nu>"
UM_Struct_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and parOf :: "'b \<Rightarrow> 'c list"   and D :: "univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" shows "UM_Struct wtFsym wtPsym arOf parOf D intF intP \<equiv> M_Signature TYPE('c) wtFsym wtPsym \<and> CU.Struct wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) D intF intP"
sum_UNIV_type_def: fixes f :: "'c \<Rightarrow> 'b"   and uu :: "'c itself" shows "sum_UNIV_type f uu = sum f UNIV"
exhaust: fixes y :: "('a, 'b) graph_node" obtains x1 :: "'a" where "y = Assumption x1"   | x2 :: "'a" where "y = Conclusion x2"   | x3 :: "'b" where "y = Rule x3"   | "y = Helper"
weight_0: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n" shows "(\<Sum>(f, \<pi>)\<in>Cauchy_Binet.Z_not_inj n m. Cauchy_Binet.weight n A B f \<pi>) = (0::'b)"
rep_eq: fixes x :: "natural"   and xa :: "natural" shows "(x < xa) = (nat_of_natural x < nat_of_natural xa)"
bind_left_identity: fixes a :: "'e"   and f :: "'e \<Rightarrow> ('c, 'd) state" shows "pure a \<bind> f = f a"
simps: fixes X :: "'d \<Rightarrow> bool"   and I :: "'d \<Rightarrow> intv"   and r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d \<Rightarrow> real" shows "regionp X I r a = (\<exists>u X\<^sub>0. a = u \<and> (\<forall>x\<in>{x. X x}. 0 \<le> u x) \<and> (\<forall>x\<in>{x. X x}. intv_elem x u (I x)) \<and> X\<^sub>0 = {x. X x \<and> (\<exists>d. I x = Intv d)} \<and> (\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. r x y = (frac (u x) \<le> frac (u y))))"
arc_lp: shows "arc (linepath (a - complex_of_real d) (a + complex_of_real e))"
access_alloc': fixes l :: "Location"   and s :: "Store"   and t :: "New" assumes "\<not> isArrLenLoc l" shows "s\<langle>t\<rangle>@@l = s@@l"