hcju/isabelleps huggingface dataset:
sup1: fixes A :: "'a"   and k :: "'a"   and a :: "'a"   and b :: "'a" assumes "A \<equiv> sup k a" shows "sup A b \<equiv> sup k (sup a b)"
power_one_right: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and a :: "'q" assumes "class.monoid_mult one times" shows "power.power one times a 1 = a"
w3: fixes x :: "'a"   and t :: "complex" assumes "x \<in> {x. f x = (0::'b)}" shows "t *\<^sub>C x \<in> {x. f x = (0::'b)}"
sign_r_pos_add: fixes p :: "real poly"   and x :: "real"   and q :: "real poly" assumes "poly p x = 0"   and "poly q x \<noteq> 0" shows "sign_r_pos (p + q) x = sign_r_pos q x"
trivial_phis_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g" shows "uninst_code.ssa.trivial_phis p g = Set.filter (\<lambda>(n, v). uninst_code.ssa.trivial_code v (the (Mapping.lookup (p g) (n, v)))) (Mapping.keys (p g))"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "comm_monoid f z" shows "comm_monoid_set f z"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "pred_option P = (\<lambda>x. Ball (set_option x) P)"
deadlocked'D2: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "deadlocked' s"   and "not_final_thread s t" obtains "t \<in> deadlocked s"
pos_fun_distr: fixes R :: "'a \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and S' :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun R S OO rel_fun R' S' \<le> rel_fun (R OO R') (S OO S')"
mult_ac_2: fixes a :: "'c"   and b :: "'c" shows "a * b = b * a"
map_ide_simp: fixes a :: "'b \<times> 'b \<times> 'b" assumes "VVV.ide a" shows "\<alpha>' a = local.inv \<a>[fst a, fst (snd a), snd (snd a)]"
map_ident: fixes t :: "'a multiset" shows "{#x. x \<in># t#} = t"
mod_nat_sub: fixes x :: "nat"   and z :: "nat"   and y :: "nat" assumes "x < z" shows "(x - y) mod z = x - y"
STL: shows "(tsl, it_to_sorted_list ordR) \<in> Rs\<^sup>k \<rightarrow>\<^sub>a list_assn Rk"
epsclosure_start_step_star: fixes A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and q :: "bool list"   and R :: "(bool list \<times> bool list) set" shows "((start (RegExp2NAe.star A), q) \<in> R\<^sup>*) = (q = start (RegExp2NAe.star A) \<or> (\<exists>r. (start (RegExp2NAe.star A), r) \<in> R \<and> (r, q) \<in> R\<^sup>*))"
r3: shows "R3 = T3 OO T3\<inverse>\<inverse>"
Call: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and ps :: "expr list"   and vs :: "val list"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and C :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and M :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and pns' :: "char list list"   and body' :: "expr"   and Ds :: "char list list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "expr"   and Cs' :: "char list list"   and vs' :: "val list"   and l\<^sub>2' :: "char list \<Rightarrow> val option"   and new_body :: "expr"   and e' :: "expr"   and h\<^sub>3 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>3 :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, S)\<rfloor>"   and "P \<turnstile> last Cs has least M = (Ts', T', pns', body') via Ds"   and "P \<turnstile> (C,Cs @\<^sub>p Ds) selects M = (Ts, T, pns, body) via Cs'"   and "length vs = length pns"   and "P \<turnstile> Ts Casts vs to vs' "   and "l\<^sub>2' = [this \<mapsto> Ref (a, Cs'), pns [\<mapsto>] vs']"   and "new_body = (case T' of Class D \<Rightarrow> \<lparr>D\<rparr>body | _ \<Rightarrow> body)"   and "P,E(this \<mapsto> Class (last Cs'), pns [\<mapsto>] Ts) \<turnstile> \<langle>new_body,(h\<^sub>2, l\<^sub>2')\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>3)\<rangle>" shows "P,E \<turnstile> \<langle>Call e None M ps,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, l\<^sub>2)\<rangle>"
gf_G_pos: fixes z :: "real"   and x :: "'a"   and y :: "'a"   and K :: "'a \<Rightarrow> 'a pmf" assumes "0 < z"   and "z < 1"   and "(x, y) \<in> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*" shows "0 < MC_syntax.gf_G K x y z"
instantiate_vars_same_typ: fixes insts :: "((variable \<times> typ) \<times> Core.term) list"   and B :: "Core.term"   and lev :: "nat" assumes "list_all (\<lambda>((idx, ty), t). typ_of t = Some ty) insts"   and "\<not> loose_bvar B lev" shows "fold (\<lambda>((idx, ty), t) B. subst_bv1 (bind_fv2 (idx, ty) lev B) lev t) insts B = fold (\<lambda>single. subst_term [single]) insts B"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "\<ll> f = S.null"
reds_preserves_hconf: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and es :: "'addr expr list"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and es' :: "'addr expr list"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and E :: "String.literal \<Rightarrow> ty option"   and Ts :: "ty list" assumes "extTA,P,t \<turnstile> \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>"   and "P,E,hp s \<turnstile> es [:] Ts"   and "hconf (hp s)" shows "hconf (hp s')"
mtf2_def: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "mtf2 n x xs = (if x \<in> set xs then swaps [index xs x - n..<index xs x] xs else xs)"
lm150: fixes R :: "('a \<times> 'b) set"   and f :: "('a \<times> 'b) set" assumes "R \<subseteq> f"   and "runiq f"   and "Domain f = Domain R" shows "runiq R"
mono: fixes xs :: "'a llist \<times> 'b llist" shows "mono_llist (\<lambda>lzip. F lzip xs)"
axioms_1: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e" assumes "module_pair s1 s2" shows "module s1"
infinite_small_llength: fixes xs :: "'c llist"   and n :: "nat" assumes "\<not> lfinite xs" shows "enat n < llength xs"
a4: shows "stateful = set (filter_compliant_stateful_ACS \<lparr>nodes = V, edges = E\<rparr> M edgesList)"
pair_wf_digraph_axioms: shows "pair_wf_digraph G"
max_comm: fixes x :: "'a"   and y :: "'a" shows "max x y = max y x"
inner_sum_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (comm_monoid_add.sum plus zero f A) = (\<Sum>y\<in>A. inner x (f y))"
u: assumes "e \<noteq> (0::'a)" shows "u = z ^ 3"
simps: fixes v1 :: "'vertex"   and p1 :: "('form, 'var) out_port"   and v2 :: "'vertex"   and p2 :: "('form, 'var) antecedent" shows "edge_begin ((v1, p1), v2, p2) = v1"
homotopic_into_contractible: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "continuous_on S g"   and "g ` S \<subseteq> T"   and "contractible T" shows "homotopic_with_canon (\<lambda>h. True) S T f g"
simps: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.height_tree (BinomialHeapStruc.Node e a r ts) = BinomialHeapStruc.height_queue ts"
closed: shows "\<forall>(s, \<mu>)\<in>set K. \<forall>(t, uu_)\<in>set \<mu>. t \<in> set S"
le_Inf_iff: fixes b :: "'a"   and A :: "'a set" shows "(b \<le> \<Sqinter> A) = (\<forall>a\<in>A. b \<le> a)"
relcomp_Image: fixes X :: "('e \<times> 'f) set"   and Y :: "('f \<times> 'd) set"   and Z :: "'e set" shows "(X ; Y) `` Z = Y `` X `` Z"
get_cont_def: fixes f :: "'s \<Rightarrow> ('b, 'm) contT" shows "local.get_cont f = ContT (\<lambda>k. get (\<lambda>s. run_cont (f s) k))"
coeff_monom_mult: fixes c :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "c \<in> carrier R"   and "p \<in> carrier P" shows "coeff P (monom P c n \<otimes>\<^bsub>P\<^esub> p) (m + n) = c \<otimes> coeff P p m"
inject: fixes x2 :: "'c set"   and y2 :: "'c set" shows "(RES x2 = RES y2) = (x2 = y2)"
isFDERIV_approx: fixes vs :: "real list"   and VS :: "float interval option list"   and prec :: "nat"   and n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list" assumes "bounded_by vs VS"   and "isFDERIV_approx prec n xs fas VS" shows "isFDERIV n xs fas vs"
assms: fixes x :: "nat" assumes "x < count_weights False (remove_weights m)" shows "f x = extract_weights False m x"
tendsto_Lim: fixes net :: "'d filter"   and f :: "'d \<Rightarrow> 'e"   and l :: "'e" assumes "net \<noteq> bot"   and "(f \<longlongrightarrow> l) net" shows "Lim net f = l"
jordan_nf_block_size_order_bound: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and n :: "nat"   and a :: "'a" assumes "jordan_nf A n_as"   and "(n, a) \<in> set n_as" shows "n \<le> order a (char_poly A)"
qOp_alpha_iff: fixes delta :: "'g"   and inp :: "'c \<Rightarrow> ('c, 'd, 'e, 'f, 'g) qTerm option"   and binp :: "'d \<Rightarrow> ('c, 'd, 'e, 'f, 'g) qAbs option"   and X' :: "('c, 'd, 'e, 'f, 'g) qTerm" shows "(qOp delta inp binp #= X') = (\<exists>inp' binp'. X' = qOp delta inp' binp' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
Stirling_n_0: fixes n :: "nat" shows "Stirling n 0 = (if n = 0 then 1 else 0)"
sequentially_imp_eventually_nhds_within: fixes s :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))" shows "eventually P (inf (nhds a) (principal s))"
aneq: assumes "a \<noteq> 0" shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. 0 < a * x\<^sup>2 + b * x + c"
le_supI1: fixes sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x a" shows "less_eq x (sup a b)"
lossless_complete_game: fixes h :: "'pub_input"   and w :: "'witness"   and e :: "'challenge" assumes "\<forall>h w. lossless_spmf (init h w)"   and "\<forall>r w e. lossless_spmf (response r w e)" shows "lossless_spmf (completeness_game h w e)"
pmf_map_outside: fixes x :: "'a"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b pmf" assumes "x \<notin> f ` set_pmf M" shows "pmf (map_pmf f M) x = 0"
x_def: shows "x = abs_Bonk_Schramm_extension y"
simps_2: fixes x :: "'d"   and xs :: "'d list" shows "subseqs (x # xs) = (let xss = subseqs xs in map ((#) x) xss @ xss)"
distinct_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "cycle (map f xs) = (cycle xs \<and> inj_on f (set xs))"
vfinite_csucc: fixes A :: "V" assumes "vfinite A" shows "csucc (vcard A) = ZFC_in_HOL.succ (vcard A)"
try_swap_def: fixes blue :: "'a"   and brown :: "'a"   and green :: "'a"   and red :: "'a"   and p :: "'b"   and c\<^sub>1 :: "'a"   and c\<^sub>2 :: "'a"   and w :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "distinct [blue, brown, green, red]"   and "finite UNIV" shows "Blue_Eyes.try_swap blue p c\<^sub>1 c\<^sub>2 w x = (if c\<^sub>1 = blue \<or> c\<^sub>2 = blue \<or> x \<noteq> p then w x else Fun.swap c\<^sub>1 c\<^sub>2 id (w x))"
roifex_Falseif_unique: fixes b :: "'b ifex" assumes "ro_ifex b"   and "\<forall>ass. \<not> val_ifex b ass" shows "b = Falseif"
functor_abbrev_def: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" shows "Ftor F : A \<longrightarrow> B \<equiv> Functor F \<and> CatDom F = A \<and> CatCod F = B"
has_integral_factor_content: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and a :: "'a"   and b :: "'a" shows "(f has_integral i) (cbox a b) = (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of cbox a b \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<le> e * content (cbox a b)))"
some_gcd_connect: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "somegcd mk_monoid a b = some_gcd a b"
exhaust: fixes y :: "('b, 'c) arr" obtains "y = Null"   | x21 :: "'b" and x22 :: "'b" and x23 :: "'c" where "y = MkArr x21 x22 x23"
inv_in_hom: fixes f :: "'s"   and a :: "'s"   and b :: "'s" assumes "local.iso f"   and "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>local.inv f : b \<rightarrow> a\<guillemotright>"
mmonitorable_def: fixes \<phi> :: "Formula.formula" shows "mmonitorable \<phi> = (safe_formula \<phi> \<and> Formula.future_bounded \<phi>)"
keys_cmul: fixes c :: "int"   and a :: "'b \<Rightarrow>\<^sub>0 int" shows "keys (frag_cmul c a) \<subseteq> keys a"
cycles_def: fixes p :: "'state" shows "cycles p \<equiv> {w. path w p \<and> target w p = p}"
cprank_def: fixes A :: "'a tensor" shows "cprank A = (LEAST n. cprank_max n A)"
simps: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and a1 :: "(('d \<times> 'e) \<times> 'f) list"   and a2 :: "'f" shows "path A a1 a2 = ((\<exists>p. a1 = [] \<and> a2 = p) \<or> (\<exists>a p r. a1 = a # r \<and> a2 = p \<and> snd a \<in> succ A (fst a) p \<and> path A r (snd a)))"
lebesgue_integral_const: fixes M :: "'c measure"   and a :: "'b" shows "LINT x|M. a = Sigma_Algebra.measure M (space M) *\<^sub>R a"
simps_2: fixes x :: "'c \<Rightarrow> real"   and y :: "'c \<Rightarrow> real"   and xs :: "('c \<Rightarrow> real) stream"   and i :: "nat" shows "dur (x ## y ## xs) (Suc i) = elapsed x y + dur (y ## xs) i"
case_enat_def: shows "case_enat \<equiv> rec_enat"
sum_character: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" assumes "character G \<chi>" shows "sum \<chi> (carrier G) = (if \<chi> = principal_char G then of_nat (order G) else 0)"
rbt_comp_inter_with_key_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" shows "local.rbt_comp_inter_with_key f t1 t2 = (case RBT_Impl.compare_height t1 t1 t2 t2 of compare.LT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). map_option (\<lambda>w. (k, f k v w)) (local.rbt_comp_lookup t2 k)) (RBT_Impl.entries t1)) | compare.GT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). map_option (\<lambda>w. (k, f k w v)) (local.rbt_comp_lookup t1 k)) (RBT_Impl.entries t2)) | compare.EQ \<Rightarrow> rbtreeify (local.comp_sinter_with f (RBT_Impl.entries t1) (RBT_Impl.entries t2)))"
subst_subst: fixes i :: "nat"   and j :: "nat"   and t :: "dB"   and v :: "dB"   and u :: "dB" assumes "i < j + 1" shows "t[lift v i/Suc j][u[v/j]/i] = t[u/i][v/j]"
scc_disj: fixes G :: "('c, 'd) pre_digraph"   and c :: "('c, 'd) pre_digraph"   and d :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "c \<in> pre_digraph.sccs G"   and "d \<in> pre_digraph.sccs G"   and "c \<noteq> d" shows "verts c \<inter> verts d = {}"
step: shows "(sc, sc') \<in> olv_round r S v D Ob"
gauss_int_norm_i: shows "gauss_int_norm \<i>\<^sub>\<int> = 1"
F: shows "envObs a s \<in> simObs a ` set (KBP.k_succs a (runJP k_mkAlgAuto t a))"
FSwImorph_def: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('b, 'c, 'd, 'e, 'f, 'k, 'h, 'j, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.FSwImorph Delta h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> FixSyn.ipresIGCons Delta h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD'"
eq_iff: fixes a :: "'a::order"   and b :: "'a::order" shows "((a::'a::order) = (b::'a::order)) = (a \<le> b \<and> b \<le> a)"
env_restr_eqvt: fixes \<pi> :: "perm"   and d :: "'a set"   and m :: "'a \<Rightarrow> 'b" shows "\<pi> \<bullet> m f|` d = (\<pi> \<bullet> m) f|` (\<pi> \<bullet> d)"
cexpr_typing_cong': fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and \<Gamma>' :: "nat \<Rightarrow> pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e : t"   and "\<And>x. x \<in> free_vars e \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "\<Gamma>' \<turnstile>\<^sub>c e : t"
implication_frontier_iff_implied_frontier_alt_vacant: fixes c :: "('loc, 't) configuration"   and b :: "'t"   and loc :: "'loc" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "worklists_vacant_to c b" shows "(b \<in>\<^sub>A frontier (c_imp c loc)) = (b \<in>\<^sub>A implied_frontier_alt c loc)"
split_beta': fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
only_touches_def: fixes i :: "'c"   and branch :: "(('d, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set" shows "only_touches i branch xs \<equiv> \<forall>(v, v')\<in>xs. \<forall>ps a. branch !. v = Some (ps, a) \<longrightarrow> i = a"
untilIA: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n' :: "nat"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"   and "\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<Longrightarrow> \<exists>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<and> eval c t t' n'' \<gamma> \<and> (\<forall>n'''\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. n''' < \<langle>c \<Leftarrow> t\<rangle>\<^bsub>n''\<^esub> \<longrightarrow> (\<exists>n''''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'''\<^esub>. n'''' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'''\<^esub> \<and> eval c t t' n'''' \<gamma>'))"   and "\<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<Longrightarrow> eval c t t' n' \<gamma> \<and> (\<forall>n''\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. n'' < n' \<longrightarrow> (\<exists>i\<ge>n''. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> (\<exists>n'''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n''\<^esub>. n''' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n''\<^esub> \<and> eval c t t' n''' \<gamma>') \<or> \<not> (\<exists>i\<ge>n''. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> eval c t t' n'' \<gamma>')" shows "eval c t t' n (\<gamma>' \<UU>\<^sub>b \<gamma>)"
c: shows "singular_chain p (subtopology X (topspace X \<inter> S)) c"
measurable_smap: fixes X :: "'a \<Rightarrow> 'b"   and N :: "'a measure"   and M :: "'b measure" assumes "X \<in> N \<rightarrow>\<^sub>M M" shows "smap X \<in> stream_space N \<rightarrow>\<^sub>M stream_space M"
list_all3_list_all2D: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and zs :: "'f list" assumes "list_all3 P xs ys zs" shows "length xs = length ys \<and> list_all2 (\<lambda>(x, y). P x y) (zip xs ys) zs"
eq_1_size: fixes kdt :: "'b kdt" shows "(1 = KD_Tree.size_kdt kdt) = (\<exists>p. kdt = Leaf p)"
p'_has_real_derivative: fixes a :: "real"   and v :: "real"   and t :: "real"   and s :: "real set" shows "(movement.p' a v has_real_derivative movement.p'' a t) (at t within s)"
inf_sup: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'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 a :: "'b"   and b :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup" shows "plus (minus a (inf a b)) b = sup a b"
simps_1: shows "list_disj_Uni [] = FalseFUni"
awalk_Nil_iff: fixes u :: "'a"   and v :: "'a" shows "G.awalk u [] v = (u = v \<and> u \<in> verts G)"
prems_2: shows "\<forall>a\<in>{(i, t, s)} \<union> set (dl # D) \<union> set Di. case a of (j, p) \<Rightarrow> \<forall>a\<in>{(i, t, s)} \<union> set (dl # D) \<union> set Di. case a of (k, q) \<Rightarrow> (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"
top: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and x :: "'a" assumes "class.SL_top le join Top" shows "le x Top"
assms: shows "\<exists>!x. x \<in> U \<and> P x"
length_output_witness': fixes rs :: "nat list" shows "dim_vec (tensors_from_net (witness'_l rs)) = rs ! 0"
inv_op_closed1: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and h :: "'e" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "h \<in> H" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> h \<otimes>\<^bsub>G\<^esub> x \<in> H"
sum_mset_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" assumes "class.semiring_0 plus zero times" shows "times c (comm_monoid_add.sum_mset plus zero (image_mset f M)) = comm_monoid_add.sum_mset plus zero {#times c (f x). x \<in># M#}"
ord0_linear: fixes x :: "ord0"   and y :: "ord0" shows "x \<le> y \<or> y \<le> x"
iwlsFSwSTR_iterSTR_ipresVar: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "ipresVar (iterSTR MOD) MOD"
bij_betw_id: fixes A :: "'b set" shows "bij_betw id A A"
is_pure_def: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "is_pure P \<equiv> \<exists>P'. \<forall>x x'. P x x' = \<up> (P' x x')"
evaln_elim_cases_4: fixes G :: "prog"   and s :: "st"   and l :: "jump"   and c :: "stmt"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (l\<bullet> c)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains s1 :: "state" where "x = \<diamondsuit>" and "s' = abupd (absorb l) s1" and "G\<turnstile>Norm s \<midarrow>c\<midarrow>n\<rightarrow> s1"
g1: shows "conf (code tp) (bl2wc (<lm>)) stpa = trpl_code (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"