hcju/isabelleps huggingface dataset:
ba_def: fixes \<phi> :: "'a \<Rightarrow> bool" shows "dynamic_component.ba \<phi> \<equiv> \<lambda>t n. \<phi> (t n)"
inject_1: fixes x2 :: "xname"   and y2 :: "xname" shows "(SXcpt' x2 = SXcpt' y2) = (x2 = y2)"
simps_1: fixes \<alpha> :: "'a" shows "bn (S_action.Act \<alpha>) = bn \<alpha>"
ap_def: fixes f :: "('d, 'f \<Rightarrow> 'e) state"   and x :: "('d, 'f) state" shows "ap f x = State (\<lambda>s. case run_state f s of (g, s') \<Rightarrow> case run_state x s' of (y, s'') \<Rightarrow> (g y, s''))"
DiffD2: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set"   and P :: "bool" assumes "c \<in> A - B"   and "c \<in> B" shows "P"
r_tuple_nth_def: shows "r_tuple_nth \<equiv> Cn 3 r_ifle [recf.Id 3 2, recf.Id 3 0, r_nth_inbounds, r_constn 2 0]"
arr_char: fixes f :: "'a" shows "L.arr f = left a f"
Out2_def: fixes y :: "nat"   and x :: "nat"   and s2 :: "nat" shows "Out2 y x s2 = (let s1 = (x * y + (q - s2)) mod q in return_spmf (s1, s2))"
advantage_nonneg: fixes \<A> :: "('y, 'z) distinguisher"   and res1 :: "('y, 'z) resource"   and res2 :: "('y, 'z) resource" shows "0 \<le> advantage \<A> res1 res2"
image_ident: fixes Y :: "'c set" shows "(\<lambda>x. x) ` Y = Y"
minus_bcontfun_def: shows "(-) \<equiv> map_fun apply_bcontfun (map_fun apply_bcontfun Bcontfun) (\<lambda>f g x. f x - g x)"
subgroup_inter: fixes A :: "'a set"   and B :: "'a set" assumes "subgroup A"   and "subgroup B" shows "subgroup (A \<inter> B)"
PO_m1_refines_ir_a0i: shows "refines (R_a0im1_ir \<inter> reach a0i \<times> reach m1) med_a0im1_ir a0i m1"
unitary_exexH_k: fixes k :: "nat" assumes "k < n" shows "unitary (exexH_k k)"
solvable_imp_trivial_derived_seq: fixes H :: "'e set" assumes "solvable_seq K H" shows "\<exists>n. (derived K ^^ n) H = {\<one>\<^bsub>K\<^esub>}"
rHom_mem: fixes f :: "'e \<Rightarrow> 'f"   and A :: "('e, 'g) Ring_scheme"   and R :: "('f, 'h) Ring_scheme"   and a :: "'e" assumes "f \<in> rHom A R"   and "a \<in> carrier A" shows "f a \<in> carrier R"
cases_scheme: fixes r :: "('a, 'b, 'c) rec_scheme" obtains rec1 :: "'a" and rec2 :: "'b" and more :: "'c" where "r = \<lparr>rec1 = rec1, rec2 = rec2, \<dots> = more\<rparr>"
field_differentiable_mult: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "f field_differentiable at a within S"   and "g field_differentiable at a within S" shows "(\<lambda>z. f z * g z) field_differentiable at a within S"
mono_Chains: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r \<subseteq> s" shows "Chains r \<subseteq> Chains s"
isAbr_simps_3: fixes f :: "'b" shows "isAbr (Fault f) = False"
conversep_inject: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(r\<inverse>\<inverse> = s\<inverse>\<inverse>) = (r = s)"
exhaust_sel: fixes simple_rule :: "'a simple_rule" obtains "simple_rule = SimpleRule (match_sel simple_rule) (action_sel simple_rule)"
RPDs_trans: fixes r :: "mregex"   and s :: "mregex"   and t :: "mregex" assumes "r \<in> RPD s"   and "s \<in> RPDs t" shows "r \<in> RPDs t"
Interp_as_UNION: fixes C :: "'a literal multiset" shows "Interp C = \<Union> (production ` {D. D \<le> C})"
le1: shows "adjoint M0 * M0 + adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"
source_code_axioms_def: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option" shows "source_code_axioms M b fi X enc dec \<equiv> (simple_distributed M X fi \<and> b = 2) \<and> (\<forall>x. dec (enc x) = Some x) \<and> (\<forall>w. (enc w = []) = (w = [])) \<and> (\<forall>x. x \<noteq> [] \<longrightarrow> enc x = enc [hd x] @ enc (tl x))"
rem: fixes Cn :: "'b set \<Rightarrow> 'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn" shows "Tarskian_logic.remainder Cn A \<phi> \<equiv> {B. B \<subseteq> A \<and> \<phi> \<notin> Cn B \<and> (\<forall>B'\<subseteq>A. B \<subset> B' \<longrightarrow> \<phi> \<in> Cn B')}"
Bex_set_list_ex: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
I_inj_on_S: fixes l :: "nat"   and u :: "nat" assumes "l < length s \<and> u < length s \<and> I (s ! l) = I (s ! u)" shows "l = u"
mult: fixes f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "real_polynomial_function f"   and "real_polynomial_function g" shows "real_polynomial_function (\<lambda>x. f x * g x)"
s_def: fixes t :: "real" shows "ego.s t = (if t \<le> 0 then s\<^sub>e else if t \<le> ego2.t_stop then ego.p t else ego.p_max)"
suffix_length_le: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>s ys" shows "\<^bold>|xs\<^bold>| \<le> \<^bold>|ys\<^bold>|"
vors_3: shows "filter (\<lambda>v. v = u \<or> v \<in> set us) xs = u # us"
ipresIGFreshAbs_def: fixes hA :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'k, 'c, 'l) model_scheme"   and MOD' :: "('m, 'n, 'g, 'o, 'p, 'j, 'q, 'd, 'r) model_scheme" shows "ipresIGFreshAbs hA MOD MOD' \<equiv> \<forall>ys y us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igFreshAbs MOD ys y A \<longrightarrow> igFreshAbs MOD' ys y (hA A)"
finite_zcount_pos: fixes M :: "'b zmultiset" shows "finite {t. 0 < zcount M t}"
af_subsequence_M: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and n :: "nat" shows "af (\<phi> M\<^sub>n \<psi>) (prefix (Suc n) w) = af \<psi> (prefix (Suc n) w) and\<^sub>n af \<phi> (prefix (Suc n) w) or\<^sub>n af (\<phi> M\<^sub>n \<psi>) (w [1 \<rightarrow> Suc n])"
submodule_sc_closed: fixes H :: "'a set"   and a :: "'b"   and h :: "'a" assumes "submodule R M H"   and "a \<in> carrier R"   and "h \<in> H" shows "a \<cdot>\<^sub>s h \<in> H"
forest_axioms_def: fixes G :: "('a, 'b) graph" shows "forest_axioms G \<equiv> \<forall>(a, w, b)\<in>edges G. \<not> nodes_connected (delete_edge a w b G) a b"
abs_hRe_le_hcmod: fixes x :: "complex star" shows "\<bar>hRe x\<bar> \<le> hcmod x"
ipPartitioningAddSubset2: fixes a :: "'b set"   and ts :: "'b set set" shows "ipPartition {a} (addSubsetSet a ts)"
card_of_ordIso_subst: fixes A :: "'a set"   and B :: "'a set" assumes "A = B" shows "ordIso2 (card_of A) (card_of B)"
And_propos: fixes xs :: "'c ltl list" shows "propos (And xs) = \<Union> {propos x |x. x \<in> set xs}"
poly_add_normalize_aux: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_add p1 p2 = poly_add (local.normalize p1) p2"
union_eq_empty: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(M + N = \<Empt>) = (M = \<Empt> \<and> N = \<Empt>)"
BuildSeq_mono: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf"   and B' :: "hf \<Rightarrow> bool"   and C' :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "BuildSeq B C s k y"   and "\<And>x. B x \<Longrightarrow> B' x"   and "\<And>x y z. C x y z \<Longrightarrow> C' x y z" shows "BuildSeq B' C' s k y"
br'_inner_invar_def: fixes rules :: "('d, 'e) ta_rule set"   and q :: "'d" shows "br'_inner_invar rules q \<equiv> \<lambda>(Q, W, rcm) it (Q', W', rcm'). Q' = Q \<union> {lhs r |r. r \<in> rules - it \<and> the (rcm r) \<le> 1} \<and> W' = W - {q} \<union> ({lhs r |r. r \<in> rules - it \<and> the (rcm r) \<le> 1} - Q) \<and> (\<forall>r. rcm' r = (if r \<in> rules - it then Some (the (rcm r) - 1) else rcm r))"
lookup_inject: fixes e :: "name list"   and n :: "nat"   and x :: "name"   and y :: "name" shows "(lookup e n x = lookup e n y) = (x = y)"
rootE: fixes u :: "'b list"   and r :: "'b list" assumes "u \<in> r*" obtains k :: "nat" where "r \<^sup>@ k = u"
step_beta: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l', Approx\<^sub>\<beta> Z'\<rangle>"
suc_least_AboveS: fixes a :: "'a"   and B :: "'a set" assumes "a \<in> local.AboveS B" shows "(local.suc B, a) \<in> r"
sum_list_Suc: fixes f :: "'b \<Rightarrow> nat"   and xs :: "'b list" shows "(\<Sum>x\<leftarrow>xs. Suc (f x)) = sum_list (map f xs) + length xs"
section_if_inj_1: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and f :: "'b" assumes "set_category S \<AA>"   and "partial_magma.arr S f"   and "inj_on (set_category_given_img.Fun S (set_category.some_img S \<AA>) f) (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f))"   and "set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f) = {} \<longrightarrow> set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S f) = {}" shows "category.section S f"
diffs_cos_coeff: shows "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
Sksplit1: shows "{0..<Suc k} \<union> {Suc k} = {0..<Suc (Suc k)}"
rotoinversion_matrix_def: fixes Q :: "(('a, 'b) vec, 'b) vec" shows "rotoinversion_matrix Q = (orthogonal_matrix Q \<and> det Q = - (1::'a))"
min_algs1: fixes i :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" assumes "i \<in> Field SucFbd" shows "fst (min_algs s1 s2 i) = min_G1 (min_algs s1 s2) i \<union> s1 ` F1in UNIV (min_G1 (min_algs s1 s2) i) (min_G2 (min_algs s1 s2) i)"
compact_fip: fixes U :: "'a set" shows "compact U = (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter> B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter> A \<noteq> {})"
assoc'_simps_2: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "src\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = src\<^sub>D h"
mset_a: shows "tree_to_multiset a = {#(val a, prio a)#}"
simps_3: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and bs :: "('c \<times> ('d, 'c) Term.term) list" shows "unify ((Var x, t) # E) bs = (if t = Var x then unify E bs else if x \<in> vars_term t then None else unify (subst_list (subst x t) E) ((x, t) # bs))"
FAddSA_dom_dom_States: fixes S :: "'d"   and F :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> dom F"   and "S \<notin> States SA" shows "dom (F [f+] (S, SA)) = dom F \<union> States SA"
face_of_polyhedron_subset_explicit: fixes F :: "'b set set"   and S :: "'b set"   and a :: "'b set \<Rightarrow> 'b"   and b :: "'b set \<Rightarrow> real"   and C :: "'b set" assumes "finite F"   and "S = affine hull S \<inter> \<Inter> F"   and "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> (0::'b) \<and> h = {x. a h \<bullet> x \<le> b h}"   and "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter> F'"   and "C face_of S"   and "C \<noteq> {}"   and "C \<noteq> S" obtains h :: "'b set" where "h \<in> F" and "C \<subseteq> S \<inter> {x. a h \<bullet> x = b h}"
extend_indets_comp_restrict_indets: shows "extend_indets \<circ> restrict_indets = dehomogenize None"
fold_rw': shows "foldl Gauss_Jordan_column_k (0, A) [0..<Suc k] = (fst (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]), snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k]))"
basis_in_iff_rank_of: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "X \<subseteq> \<E>" shows "X_insert.basis_in \<E> X = (X_insert.rank_of X = card X \<and> X_insert.rank_of X = X_insert.rank_of \<E>)"
in_abs_intervalI2: fixes lx :: "'b"   and ux :: "'b"   and xa :: "'b" assumes "min \<bar>lx\<bar> \<bar>ux\<bar> \<le> xa"   and "xa \<le> max \<bar>lx\<bar> \<bar>ux\<bar>"   and "lx \<le> ux"   and "(0::'b) \<le> lx \<or> ux \<le> (0::'b)" shows "xa \<in> abs ` {lx..ux}"
c_dom_sources: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and xps :: "('f \<times> 'g) list" shows "c_sources I (c_dom D) u xps = c_sources I D u (map fst xps)"
SOUND_Aux: fixes b :: "bool"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and K :: "nat" assumes "(b, G, C, m, l, A, B, I) \<in> SP_Judgement"   and "G_validn K G"   and "MST_validn K" shows "(b \<longrightarrow> \<Turnstile>\<^sub>K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I ) \<and> (\<not> b \<longrightarrow> \<Turnstile>\<^sub>Suc K \<lbrace> A \<rbrace> C , m , l \<lbrace> B \<rbrace> I )"
join_right: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "y \<sqsubseteq> x \<squnion> y"
make\<Pi>\<^sub>2_inverse: fixes y :: "\<upsilon> \<Rightarrow> \<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool" assumes "y \<in> UNIV" shows "eval\<Pi>\<^sub>2 (make\<Pi>\<^sub>2 y) = y"
stepA: shows "(y + r) \<cdot> p \<cdot> l \<le> (p \<cdot> l + !p \<cdot> l) \<cdot> (y + r)\<^sup>\<star>"
scaleC: fixes r :: "complex"   and x :: "'c" shows "g (r *\<^sub>C x) = cnj r *\<^sub>C g x"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "(('fsym, 'tp) efsym, 'psym) lit list set" shows "satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. satC \<xi> c"
independent_insert: fixes a :: "'b"   and S :: "'b set" shows "local.independent (insert a S) = (if a \<in> S then local.independent S else local.independent S \<and> a \<notin> local.span S)"
assms_1: shows "[M]\<^bsub>v,n\<^esub> \<subseteq> V"
kcomp_def: fixes f :: "'d \<Rightarrow> 'f set"   and g :: "'f \<Rightarrow> 'e set" shows "f \<circ>\<^sub>K g = \<mu> \<circ> \<P> g \<circ> f"
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq> y"   and "y \<preceq> z" shows "x \<preceq> z"
disc_2: fixes xs :: "'c llist"   and ys :: "'d llist" assumes "\<not> lnull xs"   and "\<not> lnull ys" shows "\<not> lnull (lzip xs ys)"
fmsubset_alt_def: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" shows "(m \<subseteq>\<^sub>f n) = fmpred (\<lambda>k v. n $$ k = Some v) m"
CSP1_CSP2_commute: shows "CSP1 \<circ> CSP2 = CSP2 \<circ> CSP1"
S2_def: fixes h :: "'a"   and e :: "nat" shows "S2 h e = sample_uniform (order \<G>) \<bind> (\<lambda>c. let a = \<^bold>g [^] c \<otimes> inv (h [^] e) in return_spmf (a, c))"
non_empty_path_root_iff_aux: fixes r :: "'a"   and x :: "'a" assumes "path_root r x"   and "x \<noteq> 0" shows "r \<le> (x + x\<^sup>T) ; 1"
value_struct_rel_def: fixes Q :: "value \<Rightarrow> value \<Rightarrow> bool" shows "value_struct_rel Q \<equiv> (\<forall>t\<^sub>1 t\<^sub>2. Q t\<^sub>1 t\<^sub>2 \<longrightarrow> veq_structure t\<^sub>1 t\<^sub>2) \<and> (\<forall>name ts name' us. Q (Vconstr name ts) (Vconstr name' us) = (name = name' \<and> list_all2 Q ts us))"
good_item_simps_9: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and delta :: "'c" assumes "goodInp inp"   and "goodBinp binpa" shows "good (Op delta inp binpa)"
atleast_free_Suc: fixes m :: "'a \<Rightarrow> 'b option"   and n :: "nat" shows "atleast_free m (Suc n) = (\<exists>a. m a = None \<and> (\<forall>b. atleast_free (m(a \<mapsto> b)) n))"
tr: shows "set tr \<subseteq> IA \<times> UNIV <+> IU \<times> UNIV"
wtA_cOfFax_unprot: fixes f :: "'fsym" assumes "wtFsym f"   and "\<not> unprot (resOf f)" shows "GE.wtA (cOfFax f)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a') F \<Phi>'"
ab_ba: fixes a :: "'a" assumes "a \<in> A" shows "b_to_a (a_to_b a) = a"
frontier_of_subtopology_subset: fixes U :: "'c set"   and X :: "'c topology"   and S :: "'c set" shows "U \<inter> subtopology X U frontier_of S \<subseteq> X frontier_of S"
env_len_def: fixes env :: "'b list set"   and l :: "nat" shows "env_len env l = (\<forall>xs\<in>env. length xs = l)"
min_eqL: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "min a b = a"
HAStates_Chi_NoCycles_trans: fixes s :: "'a"   and a :: "('a, 'b, 'c) hierauto"   and t :: "'a"   and u :: "'a" assumes "s \<in> HAStates a"   and "t \<in> HAStates a"   and "u \<in> HAStates a"   and "t \<in> Chi a s"   and "u \<in> Chi a t" shows "s \<notin> Chi a u"
assms_3: shows "A $ a $ j \<noteq> (0::'a)"
Special_Solutions_def: shows "Special_Solutions = {local.sij i j |i j. i < m \<and> j < n}"
f_def: shows "f = Polynomial.smult (lead_coeff w) v"
circ_simulate_absorb: fixes y :: "'a"   and x :: "'a" assumes "y * x \<le> x" shows "y\<^sup>\<circ> * x \<le> x \<squnion> y\<^sup>\<circ> * bot"
GSMP_disjoint_fst_specific_not_snd_specific: fixes l :: "'b"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list"   and l' :: "'b"   and Sec :: "('fun, 'var) Term.term set"   and m :: "('fun, 'var) Term.term" assumes "GSMP_disjoint (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec"   and "l \<noteq> l'"   and "proj_specific l m \<A> Sec" shows "\<not> proj_specific l' m \<A> Sec"
Rep_Node_cases: fixes y :: "(nat \<Rightarrow> 'a + nat) \<times> ('b + nat)" assumes "y \<in> Node" obtains x :: "('b, 'a) node" where "y = Rep_Node x"
mstep_def: fixes tdb :: "(String.literal, event_data list set list) mapping \<times> nat"   and st :: "('msaux, 'muaux) mstate" shows "mstep tdb st = (let (xs, m) = meval (mstate_n st) (snd tdb) (fst tdb) (mstate_m st) in (List.enumerate (mstate_i st) xs, \<lparr>mstate_i = mstate_i st + length xs, mstate_m = m, mstate_n = mstate_n st\<rparr>))"
Int_empty_left: fixes B :: "'f set" shows "{} \<inter> B = {}"
assms_3: shows "\<pi> =\<^bsub>j\<^esub> \<pi>'"
Im_double: fixes z :: "complex" shows "Im (2 * z) = 2 * Im z"
startfin_1: shows "(a $ 1 \<le> pathstart f $ 1 \<and> pathstart f $ 1 \<le> b $ 1) \<and> a $ 2 \<le> pathstart f $ 2 \<and> pathstart f $ 2 \<le> b $ 2"
boxInit_act: fixes F :: "state \<times> state \<Rightarrow> bool" shows "\<box>F \<equiv> \<box>Init F"
simps_3: fixes prec :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form prec (Less a b) bs ss = (case (approx prec a bs, approx prec b bs) of (None, b) \<Rightarrow> False | (Some ivl, None) \<Rightarrow> False | (Some ivl, Some ivl') \<Rightarrow> float_plus_up prec (upper ivl) (- lower ivl') < 0)"
PutElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and d :: "Class"   and F :: "Field" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (putfield d F)" shows "\<exists>a Flds v ops. h\<down>a = Some (d, Flds) \<and> os = v # RVal (Loc a) # ops \<and> n = Suc 0 \<and> t = (ops, S, h[a\<mapsto>(d, Flds[F\<mapsto>v])]) \<and> ll = l + 1"
real_alg_2'': fixes p :: "int poly"   and ri :: "root_info"   and l :: "rat"   and r :: "rat" assumes "degree p \<noteq> 1" shows "real_alg_2'' ri p l r = real_alg_2' ri p l r"
rel_nabla_bin_coinduct: fixes P :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" assumes "P \<subseteq> relfdia X P \<union> rdom Q" shows "P \<subseteq> rel_nabla_bin X Q"
FnCall_def: fixes F :: "nat"   and Xs :: "exp list" shows "FnCall F Xs = Abs_Exp (\<Union>Us\<in>listset (map Rep_Exp Xs). exprel `` {FNCALL F Us})"
LIST_FOREACH'_eq: fixes tsl' :: "'d list nres"   and c' :: "'c \<Rightarrow> bool"   and f' :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>' :: "'c" shows "LIST_FOREACH (\<lambda>_ _. True) tsl' c' f' \<sigma>' = LIST_FOREACH' tsl' c' f' \<sigma>'"
wf_action_translations_def: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and J :: "nat \<Rightarrow> ('b, 'a) justifying_execution" shows "wf_action_translations E J = (\<forall>n. wf_action_translation E (J n))"
d_mu_Ints: fixes l :: "nat"   and k :: "nat" assumes "l \<le> k"   and "k < length (map of_int_hom.vec_hom fs)" shows "fs.gs.d (Suc l) * fs.gs.\<mu> k l \<in> \<int>"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier C" shows "inv\<^bsub>C\<^esub> x \<in> carrier C"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_irrefl_axioms ext" shows "ext_irrefl ext"
Collect_conv_if: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "{x. x = a \<and> P x} = (if P a then {a} else {})"
P_lnull_ltl_deadend_v0: assumes "lnull (ltl P')" shows "deadend v0"
line: shows "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<forall>b\<in>basis. line_integral_exists F {b} \<gamma>2"
out_diff1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet A B C" shows "B \<noteq> A"
corrKey_def: shows "corrKey \<equiv> keySetup\<inverse> `` bad"
condensation_edge_impl_path: fixes a :: "'val set"   and b :: "'val set"   and g :: "'g"   and P :: "'val set"   and \<phi>\<^sub>a :: "'val"   and \<phi>\<^sub>b :: "'val" assumes "(a, b) \<in> condensation_edges g P"   and "\<phi>\<^sub>a \<in> a"   and "\<phi>\<^sub>b \<in> b" shows "(\<phi>\<^sub>a, \<phi>\<^sub>b) \<in> (induced_phi_graph g P)\<^sup>*"
set: fixes x1 :: "'b match_expr"   and x2 :: "action" shows "set_rule (Rule x1 x2) = set_match_expr x1"
mult_bit: shows "to_bl (of_bl [z] * y) = map ((\<and>) z) (to_bl y)"