hcju/isabelleps huggingface dataset:
solution_cs: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state"   and v :: "nat \<Rightarrow> rat" assumes "checked_simplex' cs J s"   and "solution_simplex' s = v" shows "(J, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set cs"
Pt: shows "P\<^sup>T \<in> carrier_mat m m"
finite_inf_linorder_ne_ex: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}" shows "\<exists>x\<in>S. Inf (f ` S) = f x"
closed_Inter: fixes K :: "'b set set" assumes "\<forall>S\<in>K. closed S" shows "closed (\<Inter> K)"
assms_3: shows "\<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d)"
sdrop_cycle_eq: fixes u :: "'a list" assumes "u \<noteq> []" shows "sdrop (length u) (cycle u) = cycle u"
hd_rev: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "hd (rev xs) = last xs"
update_convs_3: fixes new' :: "'b ltlr set \<Rightarrow> 'b ltlr set"   and name :: "nat"   and incoming :: "nat set"   and new :: "'b ltlr set"   and old :: "'b ltlr set"   and "next" :: "'b ltlr set"   and more :: "'c" shows "new_update new' \<lparr>name = name, incoming = incoming, new = new, old = old, next = next, \<dots> = more\<rparr> = \<lparr>name = name, incoming = incoming, new = new' new, old = old, next = next, \<dots> = more\<rparr>"
algebraic_int_sum: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> algebraic_int (f x)" shows "algebraic_int (sum f A)"
igConsIPresIGWls_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gConsPresGWls MOD" shows "igConsIPresIGWls (fromMOD MOD)"
elem_set: fixes x :: "'e"   and xo :: "'e option" shows "(x \<in> set_option xo) = (xo = Some x)"
units_of_def: fixes G :: "('c, 'd) monoid_scheme" shows "units_of G = \<lparr>carrier = Units G, mult = (\<otimes>\<^bsub>G\<^esub>), one = \<one>\<^bsub>G\<^esub>\<rparr>"
solved_def: shows "task1_1.solved = (\<forall>c. c |\<in>| task1_1.conc_forms \<longrightarrow> (\<exists>\<Gamma> t. fst (tree.root t) = (\<Gamma> \<turnstile> c) \<and> \<Gamma> |\<subseteq>| task1_1.ass_forms \<and> task1_1.wf t \<and> tfinite t))"
abs_Tree\<^sub>\<alpha>_def: shows "abs_Tree\<^sub>\<alpha> \<equiv> quot_type.abs (=\<^sub>\<alpha>) Abs_Tree\<^sub>\<alpha>"
weaken_proves': fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and A :: "Core.term" assumes "\<Theta>,\<Gamma> \<turnstile> B"   and "term_ok \<Theta> A"   and "typ_of A = Some propT"   and "A \<notin> \<Gamma>"   and "finite \<Gamma>" shows "\<Theta>,insert A \<Gamma> \<turnstile> B"
integral_map_pmf: fixes g :: "'c \<Rightarrow> real"   and M :: "'c pmf"   and f :: "real \<Rightarrow> real" shows "measure_pmf.expectation (map_pmf g M) f = measure_pmf.expectation M (\<lambda>x. f (g x))"
h1a: shows "((\<Sum>i = 0..<d + 1. f i * C ^ (d - i)) * C ^ (d mod 2) < 0) = ((\<Sum>i = 0..<d + 1. f i / C ^ i) < 0)"
transfer_prover_start: fixes x :: "'c"   and x' :: "'c"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d" assumes "x = x'"   and "Transfer.Rel R x' y" shows "Transfer.Rel R x y"
Derives1_nonterminal_head': fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and v1 :: "symbol list"   and N :: "symbol"   and v2 :: "symbol list" assumes "Derives1 u i r (v1 @ [N] @ v2)"   and "is_nonterminal N"   and "derives v1 []" shows "\<exists>u' M. u = M # u' \<and> is_nonterminal M"
adm_subst: fixes t :: "'c \<Rightarrow> 'd"   and P :: "'d \<Rightarrow> bool" assumes "cont t"   and "adm P" shows "adm (\<lambda>x. P (t x))"
walk_in_V: fixes xs :: "'a list" assumes "H.walk xs" shows "set xs \<subseteq> V\<^bsub>H\<^esub>"
substitute_contains_arg: fixes t :: "'b ttree"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set" shows "paths t \<subseteq> paths (substitute f T t)"
alist_and_NegPos_map_getNeg_getPos_matches: fixes spts :: "'b negation_type list"   and \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and C :: "'b \<Rightarrow> 'c"   and a :: "action"   and p :: "'d" shows "((\<forall>m\<in>set (getNeg spts). matches \<gamma> (MatchNot (Match (C m))) a p) \<and> (\<forall>m\<in>set (getPos spts). matches \<gamma> (Match (C m)) a p)) = matches \<gamma> (alist_and (NegPos_map C spts)) a p"
StepOutMsg: fixes cfg1 :: "('p, 'v, 's) configuration"   and v :: "bool"   and cfg2 :: "('p, 'v, 's) configuration" shows "cfg1 \<turnstile> <\<bottom>, outM v> \<mapsto> cfg2 = False"
co_path_simps_1: fixes e1 :: "'b \<times> 'b"   and e :: "'b \<times> 'b"   and w :: "'b"   and es :: "('b \<times> 'b) list" assumes "e1 \<noteq> (fst e, w)"   and "e1 \<noteq> (snd e, w)" shows "co_path e w (e1 # es) = e1 # co_path e w es"
single_nz_val_carrier: fixes n :: "nat"   and i :: "nat"   and v :: "'a" shows "single_nz_val n i v \<in> carrier_vec n"
radical_Int_subset: fixes A :: "'c set"   and B :: "'c set" shows "\<surd>(A \<inter> B) \<subseteq> \<surd>A \<inter> \<surd>B"
trnr\<^sub>\<eta>_comp: fixes v :: "'a"   and \<mu> :: "'a"   and \<nu> :: "'a" assumes "ide v"   and "seq \<mu> \<nu>"   and "src \<mu> = trg f" shows "T0.trnr\<^sub>\<eta> v (\<mu> \<cdot> \<nu>) = (\<mu> \<star> f) \<cdot> T0.trnr\<^sub>\<eta> v \<nu>"
oracle_hash_def: fixes k :: "'key" shows "oracle_hash k = (\<lambda>L y. let t = hash k y; L = insert y L in return_spmf (t, L))"
c_K33_list_def: shows "c_K33_list \<equiv> ([0..5], concat (map (\<lambda>x. concat (map (\<lambda>y. if even x = odd y then [(x, y)] else []) [0..5])) [0..5]))"
continuous_open_preimage: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "open S"   and "open T" shows "open (S \<inter> f -` T)"
assms_6: shows "\<forall>i<n. msg (Suc 0) (nReturn i)"
f_Exec_Stream_Init_length: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "length (f_Exec_Comp_Stream_Init trans_fun xs c) = Suc (length xs)"
Cauchy_Schwarz_ineq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(inner x y)\<^sup>2 \<le> inner x x * inner y y"
induct: fixes P :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> action \<Rightarrow> 'b match_expr \<Rightarrow> bool"   and a0 :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a1 :: "action"   and a2 :: "'b match_expr" assumes "\<And>uu_ uv_. P uu_ uv_ MatchAny"   and "\<And>uw_ ux_. P uw_ ux_ (MatchNot MatchAny)"   and "\<And>\<beta> \<alpha> a A. P (\<beta>, \<alpha>) a (Match A)"   and "\<And>\<beta> \<alpha> a A. P (\<beta>, \<alpha>) a (MatchNot (Match A))"   and "\<And>\<beta> \<alpha> a m. P (\<beta>, \<alpha>) a m \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot (MatchNot m))"   and "\<And>\<beta> \<alpha> a m1 m2. \<lbrakk>P (\<beta>, \<alpha>) a m1; P (\<beta>, \<alpha>) a m2\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchAnd m1 m2)"   and "\<And>\<beta> \<alpha> a m1 m2. \<lbrakk>P (\<beta>, \<alpha>) a (MatchNot m1); P (\<beta>, \<alpha>) a (MatchNot m2); \<not> (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchAny \<or> remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchAny) \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot m1); \<lbrakk>\<not> (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchAny \<or> remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchAny); remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchNot MatchAny\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot m2); \<lbrakk>\<not> (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchAny \<or> remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchAny); remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) \<noteq> MatchNot MatchAny\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot m2); \<lbrakk>\<not> (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchAny \<or> remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchAny); remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) \<noteq> MatchNot MatchAny; remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchNot MatchAny\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot m1); \<lbrakk>\<not> (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchAny \<or> remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchAny); remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) \<noteq> MatchNot MatchAny; remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) \<noteq> MatchNot MatchAny\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot m1); \<lbrakk>\<not> (remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) = MatchAny \<or> remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) = MatchAny); remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m1) \<noteq> MatchNot MatchAny; remove_unknowns_generic (\<beta>, \<alpha>) a (MatchNot m2) \<noteq> MatchNot MatchAny\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot m2)\<rbrakk> \<Longrightarrow> P (\<beta>, \<alpha>) a (MatchNot (MatchAnd m1 m2))" shows "P a0 a1 a2"
elims: fixes x :: "game option"   and y :: "game option" assumes "Dualo x = y" obtains \<alpha> :: "game" where "x = Agame \<alpha>" and "y = Agame \<alpha>^d"   | "x = undefg" and "y = undefg"
forall_\<o>_def: shows "forall \<equiv> forall\<^sub>\<o>"
subgroup_Inter: fixes S :: "'a set set" assumes "\<forall>I\<in>S. subgroup I" shows "subgroup (\<Inter> S)"
swap_poly: fixes a :: "nat"   and L :: "'a list"   and b :: "nat"   and p :: "'a mpoly" assumes "a < length L"   and "b < length L" shows "insertion (nth_default (0::'a) L) p = insertion (nth_default (0::'a) (swap_list a b L)) (swap_poly a b p)"
meet_right: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<sqsubseteq> y"
elims_2: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and x :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "WBT.wbt \<Delta>1 \<Delta>2 x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> nat) tree" and uu_ :: "'b" and n :: "nat" and r :: "('b \<times> nat) tree" where "x = \<langle>l, (uu_, n), r\<rangle>" and "n = size l + size r + 1 \<and> WBT.balanced1 \<Delta>1 \<Delta>2 l r \<and> WBT.balanced1 \<Delta>1 \<Delta>2 r l \<and> WBT.wbt \<Delta>1 \<Delta>2 l \<and> WBT.wbt \<Delta>1 \<Delta>2 r"
suminf_offset: fixes f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "summable (\<lambda>j. f (j + i))" shows "suminf f = (\<Sum>j. f (j + i)) + sum f {..<i}"
eq_refl: fixes less_eq :: "'n \<Rightarrow> 'n \<Rightarrow> bool"   and less :: "'n \<Rightarrow> 'n \<Rightarrow> bool"   and x :: "'n"   and y :: "'n" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
devExt_shift: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat"   and n'' :: "nat" shows "devExt t n nid (n' + n'') 0 = devExt t n nid n' n''"
wf_envs_def: fixes \<sigma> :: "('c \<times> 'd) trace"   and j :: "nat"   and P :: "'c \<Rightarrow> nat option"   and P' :: "'c \<Rightarrow> nat option"   and V :: "'c \<Rightarrow> (nat \<Rightarrow> 'd set) option"   and db :: "('c, 'd set list) mapping" shows "wf_envs \<sigma> j P P' V db = (dom V = dom P \<and> Mapping.keys db = dom P \<union> {p. p \<in> fst ` \<Gamma> \<sigma> j} \<and> rel_mapping (\<le>) P P' \<and> pred_mapping (\<lambda>i. i \<le> j) P \<and> pred_mapping (\<lambda>i. i \<le> Suc j) P' \<and> (\<forall>p\<in>Mapping.keys db - dom P. the (Mapping.lookup db p) = [{ts. (p, ts) \<in> \<Gamma> \<sigma> j}]) \<and> (\<forall>p\<in>dom P. list_all2 (\<lambda>i X. X = the (V p) i) [the (P p)..<the (P' p)] (the (Mapping.lookup db p))))"
closure_of_mono: fixes S :: "'a set"   and T :: "'a set"   and X :: "'a topology" assumes "S \<subseteq> T" shows "X closure_of S \<subseteq> X closure_of T"
U: shows "{x \<in> topspace X. f x \<in> V} = U"
m_def: shows "m = e_length (e_butlast (local.prefixes j (Suc (Suc n)))) - 1"
gram_schmidt0_result_1: fixes ws :: "complex vec list" assumes "set ws \<subseteq> carrier_vec d" shows "set (gram_schmidt0 d ws) \<subseteq> carrier_vec d"
nonpos_Reals_inverse_I: fixes a :: "'a" assumes "a \<in> \<real>\<^sub>\<le>\<^sub>0" shows "inverse a \<in> \<real>\<^sub>\<le>\<^sub>0"
abs_eq: fixes xa :: "int extended \<times> int extended"   and x :: "int extended \<times> int extended" shows "abs_ivl xa + abs_ivl x = abs_ivl (plus_rep xa x)"
complete_def: fixes S :: "'b set" shows "complete S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l))"
precompact_neighborhoodE: fixes x :: "'b"   and charts :: "('b, 'c) chart set" assumes "x \<in> manifold.carrier charts" obtains C :: "'b set" where "x \<in> C" and "open C" and "compact (closure C)" and "closure C \<subseteq> manifold.carrier charts"
ChiRel_OneAncestor_notmem: fixes S :: "'a"   and T :: "'a"   and U :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<noteq> T"   and "(S, U) \<in> ChiRel A" shows "(T, U) \<notin> ChiRel A"
assms_1: shows "\<^bold>g [^] a = \<^bold>g [^] b"
Compl_set_less_eq_aux_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and ao :: "'a option"   and s1 :: "'d"   and s2 :: "'e" shows "Compl_set_less_eq_aux_fusion g1 g2 ao s1 s2 = Compl_set_less_eq_aux ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
conj_refine: fixes c\<^sub>0 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice" assumes "(c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)"   and "(c\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)" shows "(c\<^sub>0::'a::refinement_lattice) \<iinter> (c\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)"
t: shows "t \<in> B \<rightarrow>\<^sub>M tree_sigma M"
uc_unc: fixes x :: "'a" shows "x \<parallel> U \<parallel> x \<parallel> U = x \<parallel> U"
vunionI2: fixes a :: "V"   and B :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> B" shows "a \<in>\<^sub>\<circ> A \<union>\<^sub>\<circ> B"
dom_mkcan: fixes a :: "'c local.term set"   and b :: "'c local.term set" assumes "ide a"   and "ide b"   and "\<^bold>\<lfloor>DOM a\<^bold>\<rfloor> = \<^bold>\<lfloor>COD b\<^bold>\<rfloor>" shows "local.dom (mkcan a b) = a"
stable_pair_on_CD_on_CH: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH" shows "Contracts.CD_on Pd ds (fst XD_XH) = Contracts.CH Ch (snd XD_XH)"
simps_8: fixes a :: "Cooper.num" shows "bound0 (NEq a) = numbound0 a"
quasinorm_intersection_subset_2: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" shows "N1 \<inter>\<^sub>N N2 \<subseteq>\<^sub>N N2"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) F"
mon_n_def: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and n :: "'b" shows "mon_n fg n \<equiv> mon fg (proc_of fg n)"
ereal_Sup_uminus_image_eq: fixes S :: "ereal set" shows "Sup (uminus ` S) = - Inf S"
notin_set_nthsI: fixes x :: "'a"   and xs :: "'a list"   and I :: "nat set" assumes "x \<notin> set xs" shows "x \<notin> set (nths xs I)"
enat_diff_cancel_left: fixes m :: "enat"   and x :: "enat"   and y :: "enat" assumes "m \<le> x"   and "m \<le> y" shows "(x - m = y - m) = (x = y)"
function_mult_eval_car: fixes R :: "('c, 'd) ring_scheme"   and a :: "'e"   and S :: "'e set"   and f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "ring_functions R"   and "a \<in> S"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "(f \<otimes>\<^bsub>function_ring S R\<^esub> g) a = f a \<otimes>\<^bsub>R\<^esub> g a"
set_removeAll: fixes x :: "'a"   and xs :: "'a list" shows "set (removeAll x xs) = set xs - {x}"
expand_superset: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "{a. g a \<noteq> (0::'b)} \<subseteq> A" shows "Sum_any g = sum g A"
cnvalidI: fixes \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and n :: "nat"   and F :: "'e set"   and c :: "('d, 'f, 'e) com"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "\<And>s t. \<lbrakk>\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A; \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
eq_successI: fixes st' :: "status" assumes "st' \<noteq> FAILED"   and "st' \<noteq> FOUND" shows "st' = SUCCESS"
Pi_split_domain: fixes x :: "'e \<Rightarrow> 'f"   and I :: "'e set"   and J :: "'e set"   and X :: "'e \<Rightarrow> 'f set" shows "(x \<in> Pi (I \<union> J) X) = (x \<in> Pi I X \<and> x \<in> Pi J X)"
gamma_differentiable: shows "\<forall>x\<in>{a..b}. \<gamma> differentiable at x"
dependent_finite: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "module scale"   and "finite S" shows "module.dependent scale S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> (0::'a)) \<and> (\<Sum>v\<in>S. scale (u v) v) = (0::'b))"
at_fresh: fixes a :: "'a"   and b :: "'a" assumes "at TYPE('a)" shows "a \<sharp> b = (a \<noteq> b)"
Sup_to_Inf: shows "Sup = uminus \<circ> Inf \<circ> (`) uminus"
reachable_mode_states_def: fixes gc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" shows "abs.reachable_mode_states gc \<equiv> {mdss. \<exists>cms' mem' sched. abs.meval_sched_abv gc sched (cms', mem') \<and> map snd cms' = mdss}"
jkbpCSn_jkbpCS_subset: fixes n :: "nat" shows "jkbpCS\<^bsub>n\<^esub> \<subseteq> jkbpCS"
assms_7: shows "(red F)\<^sup>*\<^sup>* (spoly p q) 0"
abs_apply_terms_def: fixes M :: "(('f, 'g, 'h) prot_fun, 'i) Term.term set"   and \<alpha> :: "nat \<times> bool \<Rightarrow> 'h set" shows "M \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t \<alpha> \<equiv> (\<lambda>t. t \<cdot>\<^sub>\<alpha> \<alpha>) ` M"
p_subid_interr: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x \<cdot> z \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> z \<cdot> 1\<^sub>\<pi>) = x \<parallel> y \<cdot> z \<cdot> 1\<^sub>\<pi>"
mk_projector_orthog_correct: fixes S :: "'b list" assumes "is_ortho_set (set S)"   and "distinct S" shows "mk_projector_orthog (length canonical_basis) (map vec_of_basis_enum S) = mat_of_cblinfun (Proj (ccspan (set S)))"
leB: fixes z :: "complex" assumes "z \<in> frontier S" shows "cmod (f z) \<le> B"
as0_18: shows "Cs0 \<cdot>\<cdot>cl \<eta>s0 = Cs"
simps: fixes io :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and V :: "'d list set" shows "N io M V = {V'' \<in> Perm V M. map fst (mcp' io V'') = mcp' (map fst io) V}"
concls_of_inferences_between_singleton_eq_Bin_ord_resolve_rename: fixes D :: "'a literal multiset"   and C :: "'a literal multiset" shows "concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) {D} C) = Bin_ord_resolve_rename C C \<union> Bin_ord_resolve_rename C D \<union> Bin_ord_resolve_rename D C"
nvw: shows "non_volatile_writes_unshared \<S> (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
AbstTerm_abst_dbtm: fixes i :: "name"   and n :: "nat"   and e :: "(name, hf) finfun"   and t :: "dbtm" shows "AbstTerm (q_Var i) (ord_of n) \<lbrakk>quot_dbtm t\<rbrakk>e \<lbrakk>quot_dbtm (abst_dbtm i n t)\<rbrakk>e"
L: shows "((\<lambda>x. EM_remainder' n f (real_of_int a) (real_of_int x)) \<longlongrightarrow> L) at_top"
lift_orient_lit: fixes L :: "'a literal"   and t :: "'a trm"   and s :: "'a trm"   and p :: "sign"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "orient_lit_inst L t s p \<sigma>" shows "orient_lit (subst_lit L \<sigma>) (t \<lhd> \<sigma>) (s \<lhd> \<sigma>) p"
remdups_adj_singleton_iff: fixes xs :: "'a list" shows "(length (remdups_adj xs) = Suc 0) = (xs \<noteq> [] \<and> xs = replicate (length xs) (hd xs))"
map_closed: fixes a :: "'a" assumes "a \<in> G" shows "kernel.Class a \<in> kernel.Partition"
invertible_def: fixes u :: "'a" assumes "u \<in> R" shows "additive.invertible u = (\<exists>v\<in>R. u + v = \<zero> \<and> v + u = \<zero>)"
ennreal_top_neq_one: shows "\<top> \<noteq> 1"
argmaxList_def: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'c list" shows "argmaxList f l = map ((!) l) (maxpositions (map f l))"
i_satisfies_atom_set_mono: fixes as :: "('a \<times> 'b atom) set"   and as' :: "('a \<times> 'b atom) set"   and v :: "'a set \<times> (nat \<Rightarrow> 'b)" assumes "as \<subseteq> as'"   and "v \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as'" shows "v \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as"
mem_cone_hull: fixes x :: "'a"   and S :: "'a set"   and c :: "real" assumes "x \<in> S"   and "0 \<le> c" shows "c *\<^sub>R x \<in> cone hull S"
inject: fixes x1 :: "coname"   and x2 :: "trm"   and y1 :: "coname"   and y2 :: "trm" shows "(<x1>:x2 = <y1>:y2) = ([x1].x2 = [y1].y2)"
intPI_def: shows "intPI \<equiv> MonotProblem.intPI GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf GE_\<Phi>"
power_rd_le_sq: fixes x :: "'a"   and n :: "nat" shows "proto_dioid.power_rd 1\<^sub>\<sigma> (\<cdot>) sup (Sup {}) x n \<le> proto_dioid.power_sq 1\<^sub>\<sigma> (\<cdot>) sup x n"
hbase_def: shows "hbase \<equiv> \<lambda>b. {x. hbasep b x}"
DDH_advantage_bound: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.DDH_advantage \<A> \<le> local.ddh.advantage \<A> + local.ddh.advantage (local.ddh.DDH_\<A>' \<A>)"
c_fst_le_c_sum: fixes u :: "nat" shows "c_fst u \<le> c_sum u"
vec_of_list_Nil: shows "vec_of_list [] = vNil"
Seed_holds_faceListAt_len: fixes p :: "nat" shows "faceListAt_len Seed\<^bsub>p\<^esub>"
has_integral_cong: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and i :: "'c" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "(f has_integral i) s = (g has_integral i) s"
equal_fmap_def: shows "equal_class.equal \<equiv> fmrel equal_class.equal"
induct: fixes x :: "'b list \<times> 'b list \<times> 'b list"   and X :: "'b list \<times> 'b list \<times> 'b list"   and P :: "'b list \<times> 'b list \<times> 'b list \<Rightarrow> bool" assumes "x \<in> l_sort_set X"   and "P X"   and "\<And>x xs ys. \<lbrakk>(x # xs, ys, []) \<in> l_sort_set X; P (x # xs, ys, [])\<rbrakk> \<Longrightarrow> P (xs, [], ys @ [x])"   and "\<And>x xs ys z zs. \<lbrakk>(x # xs, ys, z # zs) \<in> l_sort_set X; P (x # xs, ys, z # zs); x \<le> z\<rbrakk> \<Longrightarrow> P (xs, [], ys @ x # z # zs)"   and "\<And>x xs ys z zs. \<lbrakk>(x # xs, ys, z # zs) \<in> l_sort_set X; P (x # xs, ys, z # zs); \<not> x \<le> z\<rbrakk> \<Longrightarrow> P (x # xs, ys @ [z], zs)" shows "P x"