hcju/isabelleps huggingface dataset:
intro\<^sub>3: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and c\<^sub>1 :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set"   and c\<^sub>2 :: "('a, 'c, 'd) Stmt"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and c :: "('a, 'c, 'd) Stmt"   and \<Gamma>' :: "'a \<Rightarrow> 'd set option"   and \<S>' :: "'a set \<times> 'a set"   and P' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.\<R>\<^sub>3_aux ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type ((c\<^sub>1, mds), mem\<^sub>1) \<Gamma> \<S> P ((c\<^sub>2, mds), mem\<^sub>2)"   and "sifum_types.has_type ev\<^sub>B aexp_vars bexp_vars \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P c \<Gamma>' \<S>' P'" shows "sifum_types.\<R>\<^sub>3_aux ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type ((c\<^sub>1 ;; c, mds), mem\<^sub>1) \<Gamma>' \<S>' P' ((c\<^sub>2 ;; c, mds), mem\<^sub>2)"
permutes_not_in: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "f permutes S"   and "x \<notin> S" shows "f x = x"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) tllist \<Rightarrow> 'a"   and x1 :: "'b" shows "(case TNil x1 of TNil x \<Rightarrow> f1 x | TCons x xa \<Rightarrow> f2 x xa) = f1 x1"
U_seq_st: fixes a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set" shows "((a, A) \<in> R \<cdot> Multirelations.U) = (A = {} \<and> (a, {}) \<in> R \<or> (\<exists>B. B \<noteq> {} \<and> (a, B) \<in> R))"
abel_semigroup_set_axioms: shows "abel_semigroup_set (+) (0::'a)"
obindE: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'e \<Rightarrow> 'd \<Rightarrow> 'c option"   and s :: "'d"   and v :: "'c" assumes "(f |>> g) s = Some v" obtains v' :: "'e" where "f s = Some v'" and "g v' s = Some v"
F_Un_D_card: shows "card (F \<union> D) = nat (int ((nat p - 1) div 2))"
elims_1: fixes x :: "'edge list"   and xa :: "'edge list"   and y :: "bool" assumes "valid_path_rev_aux x xa = y" obtains cs :: "'edge list" where "x = csb__" and "xa = []" and "y"   | cs :: "'edge list" and as :: "'edge list" and a :: "'edge" where "x = cs" and "xa = as @ [a]" and "y = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> valid_path_rev_aux [] as | c' # cs' \<Rightarrow> c' \<in> get_return_edges a \<and> valid_path_rev_aux cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> valid_path_rev_aux (a # cs) as | _ \<Rightarrow> valid_path_rev_aux cs as)"
subring_props_6: fixes h1 :: "'a"   and h2 :: "'a" assumes "h1 \<in> K"   and "h2 \<in> K" shows "h1 \<otimes> h2 \<in> K"
FinFuncD: fixes f :: "'d \<Rightarrow> 'e"   and r :: "('e \<times> 'e) set"   and s :: "('d \<times> 'd) set"   and x :: "'d" assumes "f \<in> FinFunc r s"   and "x \<in> Field s" shows "f x \<in> Field r"
iso_dghm_ObjMap_eqE: fixes a :: "V"   and b :: "V" assumes "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" obtains "a = b"
prems_1: fixes r :: "'a rule" assumes "r \<in> set (r__ # rs)" shows "normalized_nnf_match (get_match r)"
r_def: shows "r \<equiv> run\<^sub>t \<delta>\<^sub>\<R> q\<^sub>\<R> w"
in_gamma_option_iff: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and x :: "'b"   and r :: "'c \<Rightarrow> 'b set"   and u :: "'c option" assumes "Gamma_semilattice num' plus' \<gamma>" shows "(x \<in> \<gamma>_option r u) = (\<exists>u'. u = Some u' \<and> x \<in> r u')"
dbm_entry_val_mono_3: fixes u :: "'c \<Rightarrow> 'd"   and c :: "'c"   and b :: "'d DBMEntry"   and b' :: "'d DBMEntry" assumes "dbm_entry_val u (Some c) None b"   and "b \<preceq> b'" shows "dbm_entry_val u (Some c) None b'"
word_lsb_def: fixes a :: "'a word" shows "lsb a = odd (uint a)"
preserves_reflects_arr: fixes f :: "'a" shows "arr (trg f) = arr f"
sum_singleton: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f {x} = f x"
shift_safe: fixes i :: "nat"   and s :: "'a list"   and t :: "'a list"   and j :: "nat" assumes "\<forall>ii<i. \<not> sublist_at s t ii"   and "t ! (i + j) \<noteq> s ! j"   and "j < length s"   and "\<forall>jj<j. t ! (i + jj) = s ! jj" shows "\<forall>ii<i + (j - \<ff> s j + 1). \<not> sublist_at s t ii"
push_impl_def_opt: fixes x :: "'v"   and s :: "'v list \<times> nat list \<times> ('v \<Rightarrow> node_state option) \<times> (nat \<times> 'v set) list" shows "push_impl x s \<equiv> case s of (S, B, I, P) \<Rightarrow> let _ = stat_newnode (); j = length S; S = S @ [x]; B = B @ [j]; I = I(x \<mapsto> STACK j); P = if E `` {x} = {} then P else P @ [(j, E `` {x})] in (S, B, I, P)"
intro: fixes x :: "'a \<Longrightarrow> 'b" assumes "wb_lens x"   and "mwb_lens x" shows "vwb_lens x"
k: shows "(spr_simAbs (fst k), snd k) \<in> (\<Union>a. SPRsingle.sim_equiv_class a ` SPR.jkbpC) \<times> UNIV"
prems_2: shows "\<forall>i\<le>Suc k. if pcss i = [] then CSS (f i) = css \<and> CS (f i) = pcs i @ cs else CS (f i) = pcs i \<and> CSS (f i) = butlast (pcss i) @ [(fst (last (pcss i)) @ cs, snd (last (pcss i)) @ cs)] @ css"
map_one: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "map f [x] = [f x]"
antisym: fixes a :: "'e"   and b :: "'e" assumes "a \<le> b"   and "b \<le> a" shows "a = b"
assms_1: shows "\<forall>\<^sub>F x in at_top. f x - eval c x = 0"
raw_match_term_dom_res_eq_tvs: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'" shows "dom subs' = tvs t \<union> dom subs"
freshEnv_getEnv1: fixes ys :: "'d"   and y :: "'e"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> ('f, 'g, 'd, 'e, 'h) qTerm set option"   and xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm set" assumes "freshEnv ys y rho"   and "rho xs x = Some X" shows "ys \<noteq> xs \<or> y \<noteq> x"
expand_superset: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "{a. g a \<noteq> (0::'c)} \<subseteq> A" shows "Sum_any g = sum g A"
good_ordSupgrp: shows "ordH \<noteq> (0::'f)"
bounded: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
FunctorM_def: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" shows "FunctorM F \<equiv> PreFunctor F \<and> FunctorM_axioms F"
simps_1: fixes x21 :: "'c"   and x22 :: "'c list"   and y21 :: "'c"   and y22 :: "'c list" shows "(x21 # x22 = y21 # y22) = (x21 = y21 \<and> x22 = y22)"
step_Fault_prop: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Fault f" shows "s' = Fault f"
assms_2: fixes \<pi> :: "perm" assumes "supp (- \<pi>) \<sharp>* fv (Lam [x]. e)"   and "supp \<pi> \<sharp>* Lam [x]. e" shows "F (\<pi> \<bullet> e) (\<pi> \<bullet> x) (Lam [x]. e) = F e x (Lam [x]. e)"
card_of_Plus_empty1: fixes A :: "'a set" shows "|A| =o |A <+> {}|"
inverse_eq_imp_eq: fixes a :: "'a"   and b :: "'a" assumes "inverse a = inverse b" shows "a = b"
inv_f_f: fixes x :: "'a" shows "inv hom (hom x) = x"
content_real_eq_0: fixes a :: "real"   and b :: "real" shows "(content {a..b} = 0) = (b \<le> a)"
\<U>: shows "Ball \<U> (openin Y) \<and> f ` S \<subseteq> \<Union> \<U>"
indicator_simps_1: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "indicator S x = (1::'c)"
assms: shows "x \<succeq> y \<and> y \<succeq> x"
induct: fixes x1 :: "'a OCL_Types.type"   and x2 :: "'a OCL_Types.type"   and P :: "'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "x1 \<sqsubset> x2"   and "\<And>\<tau> \<sigma>. \<tau> \<sqsubset>\<^sub>B \<sigma> \<Longrightarrow> P \<tau>[1] \<sigma>[1]"   and "\<And>\<tau> \<sigma>. \<tau> \<sqsubset>\<^sub>B \<sigma> \<Longrightarrow> P \<tau>[?] \<sigma>[?]"   and "\<And>\<tau>. P \<tau>[1] \<tau>[?]"   and "P OclAny[?] OclSuper"   and "\<And>\<tau> \<sigma>. \<lbrakk>\<tau> \<sqsubset> \<sigma>; P \<tau> \<sigma>\<rbrakk> \<Longrightarrow> P (Collection \<tau>) (Collection \<sigma>)"   and "\<And>\<tau> \<sigma>. \<lbrakk>\<tau> \<sqsubset> \<sigma>; P \<tau> \<sigma>\<rbrakk> \<Longrightarrow> P (Set \<tau>) (Set \<sigma>)"   and "\<And>\<tau> \<sigma>. \<lbrakk>\<tau> \<sqsubset> \<sigma>; P \<tau> \<sigma>\<rbrakk> \<Longrightarrow> P (OrderedSet \<tau>) (OrderedSet \<sigma>)"   and "\<And>\<tau> \<sigma>. \<lbrakk>\<tau> \<sqsubset> \<sigma>; P \<tau> \<sigma>\<rbrakk> \<Longrightarrow> P (Bag \<tau>) (Bag \<sigma>)"   and "\<And>\<tau> \<sigma>. \<lbrakk>\<tau> \<sqsubset> \<sigma>; P \<tau> \<sigma>\<rbrakk> \<Longrightarrow> P (Sequence \<tau>) (Sequence \<sigma>)"   and "\<And>\<tau>. P (Set \<tau>) (Collection \<tau>)"   and "\<And>\<tau>. P (OrderedSet \<tau>) (Collection \<tau>)"   and "\<And>\<tau>. P (Bag \<tau>) (Collection \<tau>)"   and "\<And>\<tau>. P (Sequence \<tau>) (Collection \<tau>)"   and "P (Collection OclSuper) OclSuper"   and "\<And>\<pi> \<xi>. strict_subtuple (\<lambda>\<tau> \<sigma>. \<tau> \<sqsubset> \<sigma> \<and> P \<tau> \<sigma> \<or> \<tau> = \<sigma>) \<pi> \<xi> \<Longrightarrow> P (Tuple \<pi>) (Tuple \<xi>)"   and "\<And>\<pi>. P (Tuple \<pi>) OclSuper" shows "P x1 x2"
injective_iff_kernel_unit: shows "inj_on induced contained.Partition = (induced.Ker = {contained.Class \<one>})"
atLeastAtMost_singleton_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.order less_eq less" shows "(ord.atLeastAtMost less_eq a b = {c}) = (a = b \<and> b = c)"
of_rat_diff: fixes a :: "rat"   and b :: "rat" shows "of_rat (a - b) = of_rat a - of_rat b"
supports_set: fixes X :: "'a set"   and S :: "'b set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "\<forall>x\<in>X. \<forall>a b. a \<notin> S \<and> b \<notin> S \<longrightarrow> [(a, b)] \<bullet> x \<in> X" shows "S supports X"
suminf_eq_lim: fixes f :: "nat \<Rightarrow> 'a" shows "suminf f = lim (\<lambda>n. sum f {..<n})"
basis_subspace_exists: fixes S :: "('b, 'c) vec set" assumes "vec.subspace S" obtains B :: "('b, 'c) vec set" where "finite B" and "B \<subseteq> S" and "vec.independent B" and "vec.span B = S" and "card B = vec.dim S"
many_strongly_connected_iff_1: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>T \<le> x\<^sup>\<star>)"
order_trans_rules_26: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
simps_2: fixes A :: "X5 set" shows "CX5h H2 A = {x \<in> A. x = X5.Xd2'}"
step: fixes k :: "nat"   and Xs :: "event_data option list set list"   and z :: "'a" assumes "progress_regex \<sigma> P r j \<le> k"   and "Suc k \<le> progress_regex \<sigma> P' r j'"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) k \<phi>)) \<phi>s Xs"   and "U k z" shows "U (Suc k) (f Xs (\<tau> \<sigma> k) z)"
mt_class: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and CDef :: "classDef"   and m :: "nat"   and mDef :: "methodDef"   and Bs :: "nat list"   and B :: "nat" assumes "CT C = Some CDef"   and "lookup (cMethods CDef) (\<lambda>md. mName md = m) = Some mDef"   and "varDefs_types (mParams mDef) = Bs"   and "mReturn mDef = B" shows "mtype(CT,m,C) = Bs \<rightarrow> B"
subseq_sum_path_weights_le: fixes xs :: "('c \<times> 'lbl \<times> 'd) list"   and ys :: "('e \<times> 'lbl \<times> 'f) list" assumes "map (\<lambda>(s, l, t). l) xs \<preceq> map (\<lambda>(s, l, t). l) ys" shows "sum_weights (map (\<lambda>(s, l, t). l) xs) \<le> sum_weights (map (\<lambda>(s, l, t). l) ys)"
less_list_iff_less_sets: fixes xs :: "'a list"   and ys :: "'a list" assumes "sorted xs"   and "sorted ys" shows "(xs < ys) = (list.set xs \<lless> list.set ys)"
rep_def: fixes f :: "'a" shows "rep f \<equiv> SOME f'. f' \<in> {f'. f \<cong> f'}"
span': shows "gs.span (set (map of_int_hom.vec_hom fs)) = gs.span (set (map of_int_hom.vec_hom fs'))"
pure_pure: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog" assumes "h \<turnstile> ok f"   and "pure f h" shows "h \<turnstile> f \<rightarrow>\<^sub>h h"
subfieldE_1: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subfield K R" shows "subring K R"
insert_Diff1: fixes x :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "(x::'b) \<in> (B::'b set)" shows "((x::'b) \<triangleright> (A::'b set)) - (B::'b set) = A - B"
ls_lookup_map_of_pre: fixes l :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst l)" shows "ls_lookup k l = map_of l k"
converse_pseudonatural_equivalence_axioms: shows "converse_pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C FG.map FG.cmp I\<^sub>C.map I\<^sub>C.cmp \<epsilon>\<^sub>0 \<epsilon>\<^sub>1"
cod_def: fixes f :: "('o, 'a) arr" shows "cod f = (if codomains f \<noteq> {} then SOME b. b \<in> codomains f else null)"
Split: fixes c :: "('b, 'c) node set \<Rightarrow> ('b, 'c) node set \<Rightarrow> 'a"   and M :: "('b, 'c) node set"   and N :: "('b, 'c) node set" shows "Split c (Scons M N) = c M N"
streamsp_streams_eq: fixes A :: "'c set" shows "streamsp (\<lambda>x. x \<in> A) = (\<lambda>x. x \<in> streams A)"
nat_pow_eq: fixes x :: "nat"   and k :: "nat" shows "x [^]\<^bsub>R\<^esub> k = x ^ k mod p"
countable_UN: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (\<Union> (A ` I))"
e_def: shows "e = \<Prod>\<^sub># (image_mset primitive_part_fract (field_poly.prime_factorization (map_poly to_fract p)))"
read1_def: fixes c :: "'e \<Rightarrow> 'c"   and x :: "('d \<Rightarrow> 'e list) \<times> (('e list \<Rightarrow> 'e list) \<Rightarrow> 'd \<Rightarrow> 'd)"   and S :: "'d \<Rightarrow> 'e set"   and P :: "('c, 'd) action" shows "read1 c x S P \<equiv> iPrefix (\<lambda>A. c (select x A)) (\<lambda>(s, s'). \<exists>a. a \<in> S s \<and> s' = increase x a s) (Denotational_Semantics.Let x) (\<lambda>s. c ` S s) P"
order_iff: fixes b :: "'a"   and a :: "'a" shows "less_eq b a = (a = sup a b)"
ring_simprules_27: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y = \<ominus>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> y)"
lspasl_starr_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" assumes "class.heap_sep_algebra plus zero sep_disj" shows "(heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> sep_algebra.sep_conj plus sep_disj A B h0) = (heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> (A h1 \<or> sep_algebra.sep_conj plus sep_disj A B h0) \<or> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> \<not> (B h2 \<or> sep_algebra.sep_conj plus sep_disj A B h0))"
winning_path_ltl: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and P :: "'c llist" assumes "ParityGame G"   and "ParityGame.winning_path G p P"   and "\<not> lnull P"   and "\<not> lnull (ltl P)" shows "ParityGame.winning_path G p (ltl P)"
nat_add_max_le: fixes n :: "nat"   and i :: "nat"   and j :: "nat"   and m :: "nat" shows "(n + max i j \<le> m) = (n + i \<le> m \<and> n + j \<le> m)"
sdGeZero: fixes ts :: "traffic" shows "\<forall>c. 0 < braking_distance ts c"
t_prod_def: fixes X :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod X Y = {t_fusion u v |u v. u \<in> X \<and> v \<in> Y \<and> Dioid_Models.last u = first v}"
simps_1: fixes f :: "'a" shows "\<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0\<^bold>\<down> = \<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0"
iMODb_0: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m, 0 ] = [r\<dots>,0]"
Ej_def: fixes j :: "nat"   and x :: "nat list" shows "Ej j x = {eij i j - 1 |i. i < length x \<and> dij i j \<le> x ! i}"
adhI: fixes x :: "'a llist"   and A :: "'a llist set" assumes "\<And>U. U \<in> nhds x \<Longrightarrow> U \<inter> A \<noteq> {}" shows "x adh A"
id_quasi_measure_preserving: fixes M :: "'b measure" shows "(\<lambda>x. x) \<in> quasi_measure_preserving M M"
shd_sset: fixes a :: "'b stream" shows "shd a \<in> sset a"
f: fixes C :: "'a set" assumes "C \<in> \<B>" shows "f C \<in> C"
sfilter_strict_scons: fixes p :: "'r \<rightarrow> tr"   and x :: "'r"   and xs :: "[:'r:]" assumes "p\<cdot>\<bottom> = \<bottom>" shows "sfilter\<cdot>p\<cdot>(x :# xs) = If p\<cdot>x then x :# sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs"
strict_prefix_def: fixes xs :: "'a list"   and ys :: "'a list" shows "strict_prefix xs ys = (prefix xs ys \<and> xs \<noteq> ys)"
Gimax: shows "max_normal_subgroup (\<GG> ! i) (G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>)"
s'_def: shows "s' = Restr {((a, x), b, y) |a x b y. (a, b) \<in> {(a\<^sub>1, a\<^sub>2), (a\<^sub>1, a\<^sub>3), (a\<^sub>2, a\<^sub>3)} \<and> (x, y) \<in> s} ({(a\<^sub>1, x) |x. x \<in># X} \<union> {(a\<^sub>2, y) |y. y \<in># Y} \<union> {(a\<^sub>3, z) |z. z \<in># Z})"
funpower_fixespointwise: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat" assumes "fixespointwise f A" shows "fixespointwise (f ^^ n) A"
vdw_lemmaE: fixes k :: "nat"   and m :: "nat"   and l :: "nat"   and n :: "nat"   and a :: "int"   and b :: "int"   and col :: "int \<Rightarrow> nat" assumes "vdw_lemma k m l n"   and "a + int n \<le> b + 1"   and "col \<in> {a..b} \<rightarrow> {..<k}" obtains start :: "int" and steps :: "nat \<Rightarrow> nat" where "\<And>i. i < m \<Longrightarrow> 0 < steps i" and "is_multi_arith_prog_on (l + 1) m start steps a b" and "let f = multi_arith_prog m start steps in \<forall>c\<in>{0..<m} \<rightarrow> {0..l}. \<forall>s<m. (\<forall>j\<le>s. c j < l) \<longrightarrow> col (f c) = col (f (\<lambda>i. if i \<le> s then 0 else c i))"
fimage_eqI: fixes b :: "'m"   and f :: "'n \<Rightarrow> 'm"   and x :: "'n"   and A :: "'n fset" assumes "b = f x"   and "x |\<in>| A" shows "b |\<in>| f |`| A"
preservation: shows "rel_preserves_binary_pred (indRelTEQ TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
all_set_conv_all_nth: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" shows "(\<forall>x\<in>set xs. P x) = (\<forall>i<length xs. P (xs ! i))"
sinvar_all_edges_normal_form_def: fixes sinvar :: "'b graph \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "SecurityInvariant_withOffendingFlows.sinvar_all_edges_normal_form sinvar P \<equiv> \<forall>G nP. sinvar G nP = (\<forall>(e1, e2)\<in>edges G. P (nP e1) (nP e2))"
in_set_conv_nth: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> ran\<^sub>l xs) = (\<exists>i<length xs. xs ! i = x)"
VN_disjoint_def: fixes v :: "'b V_rec" shows "VN_disjoint v \<equiv> V\<^bsub>v\<^esub> \<inter> N\<^bsub>v\<^esub> = {}"
neg_bounds_compare_2: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c \<le>\<^sub>u\<^sub>b b" shows "c >\<^sub>u\<^sub>b b"
ring_one: fixes R :: "('e, 'f) Ring_scheme" assumes "Ring R" shows "1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R"
NewInitDoneReds: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and h :: "heap"   and a :: "nat"   and P :: "(char list list \<times> char list exp) prog"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and h' :: "heap"   and l :: "sfields" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "new_Addr h = \<lfloor>a\<rfloor>"   and "P \<turnstile> C has_fields FDTs"   and "h' = h(a \<mapsto> blank P C)" shows "P \<turnstile> \<langle>new C,(h, l, sh),False\<rangle> \<rightarrow>* \<langle>addr a,(h', l, sh),False\<rangle>"
toList_filter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c odlist" shows "toList (ODList.filter P xs) = List.filter P (toList xs)"
delete_id_if_wbt_notin: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "wbt t"   and "\<not> isin t x" shows "delete x t = t"
matrix_to_iarray_divide_by_norm: fixes A :: "((real, 'c) vec, 'd) vec" shows "matrix_to_iarray (divide_by_norm A) = divide_by_norm_iarray (matrix_to_iarray A)"
valid_Conj_bempty: fixes P :: "'state" shows "P \<Turnstile>\<^sub>S Conj bempty"
refl: fixes t :: "('d, 'e) Term.term" shows "t \<unrhd> t"
has_derivative_at_withinI: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" assumes "(f has_derivative f') (at x)" shows "(f has_derivative f') (at x within s)"
divide_def: fixes p :: "int"   and x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "padic_integers p" shows "padic_integers.divide p x y = (if x = \<zero>\<^bsub>padic_int p\<^esub> then \<zero>\<^bsub>padic_int p\<^esub> else [p] \<cdot>\<^bsub>padic_int p\<^esub> \<one>\<^bsub>padic_int p\<^esub> [^]\<^bsub>padic_int p\<^esub> nat (padic_integers.ord_Zp p x - padic_integers.ord_Zp p y) \<otimes>\<^bsub>padic_int p\<^esub> padic_integers.ac_Zp p x \<otimes>\<^bsub>padic_int p\<^esub> inv\<^bsub>padic_int p\<^esub> padic_integers.ac_Zp p y)"
synch0: shows "\<forall>p. te p 0 = 0"
plus: fixes x :: "real" assumes "x \<in> {0..1}" shows "\<exists>t\<in>kk. x \<in> ball t (ee t / 3)"
piecewise_continuous_onI: fixes I :: "'a set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b"   and l :: "'a \<Rightarrow> 'b"   and u :: "'a \<Rightarrow> 'b" assumes "finite I"   and "continuous_on ({a..b} - I) f"   and "\<And>i. \<lbrakk>i \<in> I; a < i; i \<le> b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> l i) (at_left i)"   and "\<And>i. \<lbrakk>i \<in> I; a \<le> i; i < b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> u i) (at_right i)" shows "piecewise_continuous_on a b I f"
F3_def: shows "F3 = filter (\<lambda>f. \<exists>v\<in>set V3. f \<in> set (facesAt g v)) F2"
simps_1: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "res_seq s 0 = s"
a_b: shows "(a, b) \<in> r\<^sup>*"
hyps_2: assumes "alive (s@@k) s" shows "alive (ref l) s"
less_eqE: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b" obtains c :: "'c multiset" where "b = a + c"
simps_2: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp trans_fun (x # xs) c = f_Exec_Comp trans_fun xs (trans_fun x c)"
compat_def: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" shows "compat r r' f \<equiv> \<forall>a b. (a, b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
eval: shows "G\<turnstile>s0 \<midarrow>Methd D sig-\<succ>v\<rightarrow> s1"