hcju/isabelleps huggingface dataset:
new_tv_W: fixes n :: "nat"   and A :: "type_scheme list"   and e :: "expr"   and S :: "nat \<Rightarrow> typ"   and t :: "typ"   and m :: "nat" assumes "new_tv n A"   and "W e A n = Some (S, t, m)" shows "new_tv m S \<and> new_tv m t"
doubleReverse: fixes s1 :: "'c event list"   and t :: "'c event list"   and A :: "'c event set"   and u :: "'c event list" shows "s1 setinterleaves ((t, u), A) \<longrightarrow> rev s1 setinterleaves ((rev t, rev u), A)"
0: shows "r a\<^sub>0 a\<^sub>1"
simps: fixes j :: "nat" shows "local.Dube_aux j = (if j + 2 < n then 2 + (local.Dube_aux (j + 1) choose 2) + (\<Sum>i = j + 3..n - 1. local.Dube_aux i choose Suc (i - j)) else if j + 2 = n then d\<^sup>2 + 2 * d else 2 * d)"
sas_plus_equivalent_to_strips_i_a_XII: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops' :: "('c \<times> 'd) strips_operator list"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op'\<in>set ops'. op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)" shows "execute_parallel_operator (\<phi>\<^sub>S \<Psi> s) ops' = \<phi>\<^sub>S \<Psi> execute_parallel_operator_sas_plus s (map (strips_op_to_sasp \<Psi>) ops')"
ltln_expand_StrongRelease: fixes \<xi> :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> and\<^sub>n \<phi> or\<^sub>n X\<^sub>n (\<phi> M\<^sub>n \<psi>)"
inject_3: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c"   and y31 :: "nat"   and y32 :: "'a"   and y33 :: "'a"   and y34 :: "'b"   and y35 :: "'b"   and y36 :: "'c" shows "(Recv x31 x32 x33 x34 x35 x36 = Recv y31 y32 y33 y34 y35 y36) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34 \<and> x35 = y35 \<and> x36 = y36)"
indaut_wallvertex: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and D :: "'b set"   and v :: "'b" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "v \<in> C \<inter> D" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g v = v"
basis_subset_eq: fixes B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "\<E>.basis B\<^sub>1"   and "\<E>.basis B\<^sub>2"   and "B\<^sub>1 \<subseteq> B\<^sub>2" shows "B\<^sub>1 = B\<^sub>2"
HFinite_FreeUltrafilterNat: fixes X :: "nat \<Rightarrow> 'b" assumes "star_n X \<in> HFinite" shows "\<exists>u. \<forall>\<^sub>F n in \<U>. norm (X n) < u"
a_rcos_module_rev: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and x :: "'e"   and x' :: "'e" assumes "abelian_subgroup H G"   and "x \<in> carrier G"   and "x' \<in> carrier G"   and "x' \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x \<in> H" shows "x' \<in> H +>\<^bsub>G\<^esub> x"
coeff_smult: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "p \<in> carrier P" shows "coeff P (a \<odot>\<^bsub>P\<^esub> p) n = a \<otimes> coeff P p n"
chop_add_length_ge_0: fixes r :: "real_int"   and s :: "real_int"   and t :: "real_int" shows "R_Chop(r,s,t) \<and> 0 < \<parallel>s\<parallel> \<and> 0 < \<parallel>t\<parallel> \<longrightarrow> 0 < \<parallel>r\<parallel>"
as: fixes i :: "nat" assumes "i < Suc (Suc kk)" shows "as ! i = local.dim_gen_eigenspace A ev i"
direct_decomp_perm: fixes A :: "'b set"   and ss1 :: "'b set list"   and ss2 :: "'b set list" assumes "direct_decomp A ss1"   and "ss1 <~~> ss2" shows "direct_decomp A ss2"
listassoc_trans: fixes as :: "'a list"   and bs :: "'a list"   and cs :: "'a list" assumes "as [\<sim>] bs"   and "bs [\<sim>] cs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G"   and "set cs \<subseteq> carrier G" shows "as [\<sim>] cs"
ordertype_le_Ord: fixes \<alpha> :: "V"   and X :: "V set" assumes "Ord \<alpha>"   and "X \<subseteq> elts \<alpha>" shows "ordertype X VWF \<le> \<alpha>"
n0: shows "(deriv ^^ n0) f \<xi> \<noteq> 0"
ideD_2: fixes a :: "'c" assumes "C.ide a" shows "C.dom a = a"
obsne: shows "obs \<sigma> i \<noteq> obs \<sigma>' i"
disc_1: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<forall>r\<in>R. Resumption.resumption.is_Done r" shows "Resumption.resumption.is_Done (resumption_lub R)"
powr_smallo_one: fixes e :: "real" assumes "e < 0" shows "(\<lambda>x. x powr e) \<in> o(\<lambda>_. 1)"
oinvariant_def: fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<times> 'e \<Rightarrow> bool" shows "(A \<Turnstile> (S, U \<rightarrow>) P) = (\<forall>s\<in>oreachable A S U. P s)"
primepow_factors_def: fixes n :: "'a" shows "primepow_factors n = {x. primepow x \<and> x dvd n}"
in_carrier_atlasI: fixes c :: "('a, 'c) chart"   and x :: "'a" assumes "c \<in> src.atlas"   and "x \<in> domain c" shows "x \<in> src.carrier"
accept_def: fixes w :: "nat \<Rightarrow> 'l" shows "b.accept w \<equiv> \<exists>r. b.is_run r \<and> w = b.L \<circ> r"
evaln_Inj_elim: fixes G :: "prog"   and s :: "state"   and t :: "Term.term"   and n :: "nat"   and w :: "vals"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (w, s')" shows "case t of In1 (Inl e) \<Rightarrow> \<exists>v. w = In1 v | In1 (Inr c) \<Rightarrow> w = \<diamondsuit> | In2 e \<Rightarrow> \<exists>v. w = In2 v | In3 e \<Rightarrow> \<exists>v. w = In3 v"
wf_act_adds_are_atoms: fixes P :: "ast_problem"   and effs :: "object ast_effect"   and ae :: "object atom formula" assumes "wf_ast_problem P"   and "ast_problem.wf_effect_inst P effs"   and "ae \<in> set (adds effs)" shows "is_predAtom ae"
msg_zhops_def: fixes m :: "msg" shows "msg_zhops m \<equiv> case m of Rreq hopsc dipc x xa oipc xb sipc \<Rightarrow> hopsc = 0 \<longrightarrow> oipc = sipc | Rrep hopsc dipc x xa sipc \<Rightarrow> hopsc = 0 \<longrightarrow> dipc = sipc | _ \<Rightarrow> True"
assms_2: fixes x :: "real" assumes "x0 \<le> x" shows "cg \<le> \<bar>g x\<bar>"
ex_prefix_of: fixes p :: "'c prefix" shows "\<exists>s. prefix_of p s"
mono_addRPD: fixes r :: "mregex" shows "mono (\<lambda>S. insert r S \<union> (insert r S \<bind> RPD))"
to_tree_order_get_root_node: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and root_ptr :: "(_) object_ptr"   and ptr'' :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to"   and "h \<turnstile> Shadow_DOM.get_root_node ptr' \<rightarrow>\<^sub>r root_ptr"   and "ptr'' \<in> set to" shows "h \<turnstile> Shadow_DOM.get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr"
q: shows "q = rat_of_int (z2 * n1) / rat_of_int (n1 * n2)"
commute: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" shows "linordered_monoid.min a b = linordered_monoid.min b a"
map\<^sub>0_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (\<sigma>.map\<^sub>0 a) = F.map\<^sub>0 a"
path_components_of_disjoint: fixes C :: "'b set"   and X :: "'b topology"   and C' :: "'b set" assumes "C \<in> path_components_of X"   and "C' \<in> path_components_of X" shows "disjnt C C' = (C \<noteq> C')"
crit_pair_cbelow_mono: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and p :: "'t \<Rightarrow>\<^sub>0 'e"   and q :: "'t \<Rightarrow>\<^sub>0 'e"   and G :: "('t \<Rightarrow>\<^sub>0 'e) set" assumes "crit_pair_cbelow_on d m F p q"   and "F \<subseteq> G" shows "crit_pair_cbelow_on d m G p q"
box_borel: fixes a :: "'b"   and b :: "'b" shows "box a b \<in> sets borel"
c_2: fixes n :: "nat" shows "\<bar>\<MM> (real n) - ln (real n)\<bar> \<le> c"
inj_onD: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and x :: "'h"   and y :: "'h" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
preservation_no_await: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"   and "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"   and "no_await c" shows "\<exists>\<Gamma>'''' \<S>'''' P''. \<turnstile> \<Gamma>'''',\<S>'''',P'' {c'} \<Gamma>',\<S>',P' \<and> (tyenv_wellformed mds \<Gamma> \<S> P \<and> pred P mem \<and> (\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem \<le> dma mem x) \<longrightarrow> tyenv_wellformed mds' \<Gamma>'''' \<S>'''' P'' \<and> pred P'' mem' \<and> (\<forall>x\<in>dom \<Gamma>''''. x \<notin> mds' AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma>'''' x)) mem' \<le> dma mem' x))"
induct: fixes P :: "primfun \<times> 'a \<Rightarrow> 'b \<times> 'a \<Rightarrow> bool"   and a0 :: "primfun \<times> 'a"   and a1 :: "'b \<times> 'a" assumes "\<And>x a y b. P (x, a) (y, b)" shows "P a0 a1"
simps: fixes n :: "enat"   and xs :: "'a llist" shows "ldrop n xs = (case n of 0 \<Rightarrow> xs | eSuc n' \<Rightarrow> case xs of []\<^sub>l \<Rightarrow> []\<^sub>l | x #\<^sub>l xs' \<Rightarrow> ldrop n' xs')"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case FO of FO \<Rightarrow> f1 | SO \<Rightarrow> f2) = f1"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "frechet_derivative f (at (g x)) (b1 + b2) = frechet_derivative f (at (g x)) b1 + frechet_derivative f (at (g x)) b2"
list_all2_appendI: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b list"   and b :: "'c list"   and c :: "'b list"   and d :: "'c list" assumes "list_all2 P a b"   and "list_all2 P c d" shows "list_all2 P (a @ c) (b @ d)"
alg_dvd_def: fixes x :: "'b"   and y :: "'b" shows "(x alg_dvd y) = ((x = (0::'b) \<longrightarrow> y = (0::'b)) \<and> algebraic_int (y / x))"
self_domain_iff_ide: fixes a :: "('o, 'a) arr" shows "(a \<in> domains a) = ide a"
measure_count_space: fixes A :: "'d set"   and X :: "'d set" shows "Sigma_Algebra.measure (count_space A) X = (if X \<subseteq> A then real (card X) else 0)"
ul: shows "uniform_limit (sphere z r) (\<lambda>n x. \<Sum>k<n. (w - z) ^ k * (f x / (x - z) ^ Suc k)) (\<lambda>x. f x / (x - w)) sequentially"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (store_sub2 instr curr_win rd 10 addr s1)) \<and> t2 = snd (fst (store_sub2 instr curr_win rd 10 addr s2))"
bs_nat_inv: fixes n :: "nat"   and v :: "exp" assumes "bs_val (BNat n) v" obtains "v = ENat n"
tendsto_asymp_equiv_cong: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "f \<sim>[F] g" shows "(f \<longlongrightarrow> c) F = (g \<longlongrightarrow> c) F"
ID_distr_completion_2: fixes \<E> :: "('b, 'a, 'c) cntxt"   and e :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (\<E> [e]) = LID\<^sub>C \<E> \<union> LID\<^sub>E e"
Abs_inverse: fixes m :: "int" assumes "m \<in> {0..<n}" shows "Rep (Abs m) = m"
distinct_61: fixes x6 :: "Arraytype"   and x7 :: "CTypeId" shows "ArrT x6 \<noteq> CClassT x7"
det_mul: fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" shows "det (A ** B) = det A * det B"
vector_to_cblinfun_adj_comp_vector_to_cblinfun: fixes \<psi> :: "'b"   and \<phi> :: "'b" shows "vector_to_cblinfun \<psi>* o\<^sub>C\<^sub>L vector_to_cblinfun \<phi> = (\<psi> \<bullet>\<^sub>C \<phi>) *\<^sub>C id_cblinfun"
get_ancestors_ptrs_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors" shows "ptr' |\<in>| object_ptr_kinds h"
vsv_ex1_app2: fixes a :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f)" shows "(VLambda A f\<lparr>a\<rparr> = b) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VLambda A f)"
coinduct: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and language :: "'a language"   and language' :: "'a language" assumes "R language language'"   and "\<And>language language'. R language language' \<Longrightarrow> \<oo> language = \<oo> language' \<and> rel_fun (=) R (\<dd> language) (\<dd> language')" shows "language = language'"
root_opt_bst2_bound: fixes i :: "int"   and j :: "int" assumes "opt_bst2_dom (i, j)"   and "i \<le> j" shows "root (opt_bst2 i j) \<in> {i..j}"
trivial_phis_def: fixes g :: "'g" shows "ssa.trivial_phis g = Set.filter (\<lambda>(n, v). uninst_code.ssa.trivial_code v (the (Mapping.lookup (phis g) (n, v)))) (Mapping.keys (phis g))"
path_connectedin_standard_simplex: fixes p :: "nat" shows "path_connectedin (powertop_real UNIV) (standard_simplex p)"
currentD_SAT: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f" shows "(x \<in> SAT \<Gamma> f) = (x \<in> A \<Gamma> \<or> d_IN f x = weight \<Gamma> x)"
distinct_step: fixes s :: "'c list"   and r :: "'c"   and as :: "nat \<times> nat list" shows "distinct (step s r as) = distinct s"
Hash: fixes a :: "cmsg"   and b :: "cmsg" assumes "eq a b" shows "eq (cHash a) (cHash b)"
case_4: fixes f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> 'i \<Rightarrow> 'f"   and f3 :: "'j \<Rightarrow> ('k \<Rightarrow> 'i) \<Rightarrow> 'f"   and f4 :: "('g, 'h, 'j, 'k) resumption \<Rightarrow> 'f"   and x4 :: "('g, 'h, 'j, 'k) resumption" shows "(case Resumption_corec x4 of Terminal_corec x \<Rightarrow> f1 x | Linear_corec x xa \<Rightarrow> f2 x xa | Branch_corec x xa \<Rightarrow> f3 x xa | Resumption_corec x \<Rightarrow> f4 x) = f4 x4"
merge_guards_res_DynCom: fixes c :: "('d, 'e, 'f) com"   and c' :: "'d \<Rightarrow> ('d, 'e, 'f) com" assumes "merge_guards c = DynCom c'" shows "\<exists>c''. c = DynCom c'' \<and> (\<lambda>s. merge_guards (c'' s)) = c'"
map_2: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b bdd"   and x22 :: "'b bdd" shows "bdd_map f (Branch x21 x22) = Branch (bdd_map f x21) (bdd_map f x22)"
zip_Nil: fixes ys :: "'l list" shows "zip [] ys = []"
assms_4: shows "path_image \<gamma> \<subseteq> {w. b < a \<bullet> w}"
simps_1: fixes t :: "nat"   and q :: "nat"   and e :: "nat" shows "excessAtType t q e \<equiv> if e = 0 then if 7 < t + q then squanderTarget else \<b> t q - t * d3_const - q * d4_const else if t + q + e \<noteq> 6 then 0 else if t = 5 then \<a> else squanderTarget"
l_set_val_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_child_nodes type_wf set_val set_val_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_val type_wf set_val set_val_locs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_val_get_child_nodes_axioms set_val_locs get_child_nodes_locs"
x4: shows "item_end x = item_end (inc_dot (length (item_\<alpha> x)) (init_item (item_nonterminal x, item_rhs x) 0))"
ideals_directed: fixes X :: "'a set" assumes "idealp X" shows "directed X"
reduce_system_matrix_equation_preserved: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list" assumes "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "all_list_constr subsets (length qs)"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "satisfy_equation p qs (get_subsets (reduce_system p (qs, M_mat signs subsets, subsets, signs))) (get_signs (reduce_system p (qs, M_mat signs subsets, subsets, signs)))"
1: shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright> \<and> \<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = t\<^sub>1 \<star> \<gamma> \<and> \<a>[u, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> ((\<chi> \<star> s\<^sub>0) \<cdot> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> r\<^sub>0s\<^sub>1.\<phi> \<cdot> (\<omega>.the_\<theta> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] = ((u \<star> the_\<theta>) \<cdot> \<a>[u, u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0, chine] \<cdot> (\<a>[u, u\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>0] \<star> chine) \<cdot> ((\<mu> \<star> t\<^sub>0u\<^sub>1.p\<^sub>0) \<star> chine) \<cdot> (t\<^sub>0u\<^sub>1.\<phi> \<star> chine) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>0, t\<^sub>0u\<^sub>1.p\<^sub>1, chine]) \<cdot> (t\<^sub>0 \<star> \<gamma>)"
part_sum: fixes n :: "nat"   and P :: "nat set set"   and f :: "nat \<Rightarrow> 'a" assumes "part {..<n} P" shows "sum f {..<n} = sum (sum f) P"
surjective: fixes r :: "'a member_scheme" shows "r = \<lparr>access = access r, static = static r, \<dots> = member.more r\<rparr>"
outer_regular_lborel: fixes B :: "'b set"   and e :: "real" assumes "B \<in> sets borel"   and "0 < e" obtains U :: "'b set" where "open U" and "B \<subseteq> U" and "emeasure lborel (U - B) < ennreal e"
ih_2: assumes "getCn state' \<noteq> 1"   and "InvariantUniq (getM (applyExplain (getCl state') state'))"   and "InvariantCFalse (getConflictFlag (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state'))"   and "InvariantCEntailed (getConflictFlag (applyExplain (getCl state') state')) F0 (getC (applyExplain (getCl state') state'))"   and "InvariantClCharacterization (getCl (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantCnCharacterization (getCn (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantClCurrentLevel (getCl (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantGetReasonIsReason (getReason (applyExplain (getCl state') state')) (getF (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) (set (getQ (applyExplain (getCl state') state')))"   and "InvariantEquivalentZL (getF (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) F0"   and "getConflictFlag (applyExplain (getCl state') state')"   and "0 < currentLevel (getM (applyExplain (getCl state') state'))" shows "let state' = applyExplainUIP (applyExplain (getCl state') state') in isUIP (opposite (getCl state')) (getC state') (getM state')"
keys_to_list_zero: shows "keys_to_list 0 = []"
that: fixes b :: "'a" assumes "a \<rightarrow>\<^sup>* b"   and "is_final b" shows "thesis"
plus_commutative: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "a \<squnion>\<^bsub>f\<^esub> b = b \<squnion>\<^bsub>f\<^esub> a"
max: shows "maximum (i \<squnion> j) + 1 = minimum k"
CFG_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set" shows "CFG \<alpha>e \<alpha>n invar inEdges' Entry defs uses \<equiv> graph_Entry \<alpha>e \<alpha>n invar inEdges' Entry \<and> CFG_axioms \<alpha>n invar defs uses"
outstanding_non_volatile_refs_owned_or_read_only_nth_update: fixes i :: "nat"   and \<O> :: "nat set"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "non_volatile_owned_or_read_only False \<S>\<^sub>s\<^sub>b \<O> sb" shows "outstanding_non_volatile_refs_owned_or_read_only \<S>\<^sub>s\<^sub>b (ts\<^sub>s\<^sub>b[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
Times_PlusR: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Times r (Plus s t) = Plus (Times r s) (Times r t)"
Call: fixes G :: "'a triple set"   and P :: "'a assn"   and pn :: "pname"   and Q :: "'a assn"   and s' :: "state"   and X :: "vname"   and a :: "state \<Rightarrow> nat" assumes "G|-{P}. BODY pn .{\<lambda>Z s. Q Z (setlocs s (loc s')[X::=s<Res>])}" shows "G|-{\<lambda>Z s. s' = s \<and> P Z (setlocs s newlocs[Loc Arg::=a s])}. X:=CALL pn(a) .{Q}"
snocr: fixes ys :: "'b list"   and y :: "'b" shows "P [] (ys @ [y])"
EQ_LEQ: fixes G :: "Var \<Rightarrow> L"   and p :: "L"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val"   and pp :: "L" assumes "EQ G p s t"   and "LEQ pp p" shows "EQ G pp s t"
step_a_z: fixes l :: "'d"   and g :: "('e, 'f) cconstraint"   and a :: "'g"   and r :: "'e list"   and l' :: "'d"   and A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and Z :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', (Z \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub> \<inter> {u. u \<turnstile> inv_of A l'}\<rangle>"
set_intros_1: fixes x1 :: "'b" shows "x1 \<in> set_literal (x1\<^sup>+)"
negligible_Int: fixes s :: "'b set"   and t :: "'b set" assumes "negligible s \<or> negligible t" shows "negligible (s \<inter> t)"
finite_field_factorization_main_integer: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "finite_field_factorization_main p (finite_field_ops_integer (integer_of_int p)) f = (c, fs)"   and "square_free_m f" shows "unique_factorization_m f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
that_4: assumes "n = Helper"   and "p = Reg anyP" shows "thesis"
LocalSecretsComposition_neg1_k: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and Keys :: "Keys" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "\<not> ine P (kE Keys)"   and "\<not> ine Q (kE Keys)"   and "kKS Keys \<notin> LocalSecrets P"   and "kKS Keys \<notin> LocalSecrets Q" shows "kKS Keys \<notin> LocalSecrets PQ"
Gmult_assoc: fixes g :: "'g"   and h :: "'g"   and v :: "'w" assumes "g \<in> H"   and "h \<in> H"   and "v \<in> W" shows "g *\<star> h *\<star> v = (g + h) *\<star> v"
x_plus_y_p_le_xp_plus_yp: fixes p :: "real"   and x :: "real"   and y :: "real" assumes "0 < p"   and "p \<le> 1"   and "0 \<le> x"   and "0 \<le> y" shows "(x + y) powr p \<le> x powr p + y powr p"