hcju/isabelleps huggingface dataset:
simps_3: fixes x :: "('f, 'g, 'h) Resumption.resumption"   and f :: "'f \<Rightarrow> ('i, 'g, 'h) Resumption.resumption" assumes "Resumption.resumption.is_Done x"   and "Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))" shows "Resumption.resumption.result (x \<bind> f) = Resumption.resumption.result x \<bind> Resumption.resumption.result \<circ> f"
lq_spref: fixes u :: "'b list"   and v :: "'b list" assumes "u <p v" shows "u\<inverse>\<^sup>>v \<noteq> \<epsilon>"
Invariants_cond_exp_is_integral: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "AE x in M. real_cond_exp M Invariants f x = expectation f"
le_fin_not_member': fixes a :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "[a] \<preceq>\<^sub>F u @ v"   and "a \<notin> set u" shows "[a] \<preceq>\<^sub>F v"
Least_nat_ex: fixes n :: "nat"   and S :: "nat set" assumes "n \<in> S" shows "\<exists>x\<in>S. \<forall>y\<in>S. x \<le> y"
v11_vconverse_app_in_vdomain: fixes y :: "V" assumes "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_inverse" shows "vreal_inverse\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_inverse"
assms_3: shows "a21 * x1 + a22 * x2 = (0::'a)"
inv_implies_DF: fixes ps :: "nat list"   and fs :: "nat list" assumes "inv_dining ps fs" shows "Tr\<^sub>D N (ps, fs) \<noteq> {}"
gfp_cadm_induct: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a" assumes "dual_chain_admissible P"   and "P (Inf {})"   and "mono f"   and "\<And>x. P x \<Longrightarrow> P (f x)" shows "P (gfp f)"
preserves_reflects_arr: fixes f :: "'b \<times> 'b \<times> 'b" shows "arr (\<alpha>' f) = VVV.arr f"
subset_imp: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "Max A \<le> Max B"
assms_2: fixes \<sigma>' :: "nat \<Rightarrow> 'a"   and \<sigma> :: "nat \<Rightarrow> 'a"   and m :: "'c" assumes "\<sigma>' i = \<sigma> i" shows "P (\<sigma>, receive m, \<sigma>')"
qSwap_qSwapped: fixes X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and zs :: "'f"   and x :: "'g"   and y :: "'g" shows "(X, X #[[x \<and> y]]_zs) \<in> qSwapped"
intros: fixes A :: "'c ring"   and B :: "'c ring" assumes "id \<in> ring_hom A B" shows "A \<lesssim> B"
fmap_of_list_surj: shows "surj fmap_of_list"
M_subset_calM: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set"   and M :: "'a set set" assumes "sylow_central G p a m M1 M" shows "M \<subseteq> {s. s \<subseteq> carrier G \<and> card s = p ^ a}"
occurs_on_def: fixes event :: "('a, 'b, 'c) event" shows "occurs_on event = (case event of Trans x11 x12 x13 \<Rightarrow> x11 | Send x21 x22 x23 x24 x25 x26 \<Rightarrow> x22 | Recv x31 x32 x33 x34 x35 x36 \<Rightarrow> x32 | Snapshot x4 \<Rightarrow> x4 | RecvMarker x51 x52 x53 \<Rightarrow> x52)"
sumhr_Suc: fixes m :: "nat star"   and f :: "nat \<Rightarrow> real" shows "sumhr (m, m + 1, f) = (*f* f) m"
dcd_2: shows "n dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> l"
ex231e: fixes x :: "'a" shows "x * top * x * top = x * top"
invariant: fixes s :: "'s"   and D :: "'s pmf" assumes "s \<in> S"   and "D \<in> K s" shows "\<forall>s'\<in>set_pmf D. s' \<in> S"
word_le_make_less: fixes y :: "'b word"   and x :: "'b word" assumes "y \<noteq> - 1" shows "(x \<le> y) = (x < y + 1)"
refine: fixes prog :: "program"   and e :: "edge"   and p :: "pState"   and g :: "gState\<^sub>I" shows "RETURN (applyEdge_impl prog e p g) \<le> applyEdge prog e p g"
smult_l_distr: fixes x :: "'a Matrix.vec"   and a :: "'a"   and b :: "'a" assumes "True"   and "True"   and "x \<in> carrier_vec n" shows "(a + b) \<cdot>\<^sub>v x = a \<cdot>\<^sub>v x + b \<cdot>\<^sub>v x"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post"   and "ProofStepInduct_y_eq_new_last_axioms G v0 v1 P_new sep_size y" shows "ProofStepInduct_y_eq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
rep_eq: fixes x :: "integer" shows "int_of_integer (bin_rest_integer x) = int_of_integer x div 2"
distinct_8: fixes x51 :: "pdf_type"   and x52 :: "pdf_type" shows "PRODUCT x51 x52 \<noteq> UNIT"
filterlim_tendsto_pos_mult_at_top: fixes f :: "'c \<Rightarrow> real"   and c :: "real"   and F :: "'c filter"   and g :: "'c \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_top"
inf_bot_left: fixes x :: "'b multiset" shows "{#} \<inter># x = {#}"
max2_equals1: fixes r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "wo_rel r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(max2 r a b = a) = ((b, a) \<in> r)"
\<eta>o_in_hom: fixes y :: "'b" assumes "Adj.D.ide y" shows "\<guillemotleft>Adj.\<eta>o y : y \<rightarrow>\<^sub>D G (F y)\<guillemotright>"
configuration_token: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "x \<in> configuration (token_run x n) n"
prob_space_empty_dens_ctxt: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "prob_space (dens_ctxt_measure ({}, {}, \<Gamma>, \<lambda>_. 1) (\<lambda>_. undefined))"
subdegree_mult': fixes f :: "'a fps"   and g :: "'a fps" assumes "fps_nth f (subdegree f) * fps_nth g (subdegree g) \<noteq> (0::'a)" shows "subdegree (f * g) = subdegree f + subdegree g"
labeled_list_insert_eq_cases_2: fixes i :: "'c"   and d :: "'d"   and D :: "('c \<times> 'd) list" assumes "(i, d) \<in> set D" shows "List.insert d (unlabel D) = unlabel (List.insert (i, d) D)"
Diagonalize_preserves_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "Ide \<^bold>\<lfloor>a\<^bold>\<rfloor>"
degree_mult_eq: fixes p :: "'c poly"   and q :: "'c poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
norm_pths_1: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (norm (x - y) \<le> 0)"
MGU_is_Unifier: fixes t :: "('c, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and u :: "('c, 'd) Term.term" assumes "MGU \<sigma> t u" shows "Unifier \<sigma> t u"
D_def: shows "D = sum_list (ss @ [s]) `\<rightarrow> C0"
cases: fixes a1 :: "s"   and a2 :: "s_context" assumes "store_typing a1 a2" obtains \<S> :: "s_context" and \<C>s :: "t_context list" and tfs :: "tf list" and ns :: "nat list" and ms :: "nat list" and tgs :: "tg list" and insts :: "inst list" and fs :: "cl list" and tclss :: "cl option list list" and bss :: "mem list" and gs :: "global list" where "a1 = \<lparr>inst = insts, funcs = fs, tab = tclss, mem = bss, globs = gs\<rparr>" and "a2 = \<S>" and "\<S> = \<lparr>s_inst = \<C>s, s_funcs = tfs, s_tab = ns, s_mem = ms, s_globs = tgs\<rparr>" and "list_all2 (inst_typing \<S>) insts \<C>s" and "list_all2 (cl_typing \<S>) fs tfs" and "list_all (tab_agree \<S>) (concat tclss)" and "list_all2 (\<lambda>tcls n. n \<le> length tcls) tclss ns" and "list_all2 mem_agree bss ms" and "list_all2 glob_agree gs tgs"
assms: fixes x :: "'a" assumes "x \<in> space N" shows "f x = g x"
inv_o_cancel: fixes f :: "'d \<Rightarrow> 'e" assumes "inj f" shows "inv f \<circ> f = id"
m1_inv3r_initD: fixes x :: "'a m1_state_scheme"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Ts :: "nat"   and Ta :: "nat"   and nlb :: "atom list" assumes "runs x Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # aNum Ta # nlb)"   and "x \<in> m1_inv3r_init"   and "B \<notin> bad"   and "A \<notin> bad"   and "Kab \<notin> Domain (m1_state.leak x)" shows "\<exists>Ra nla. runs x Ra = Some (Init, [A, B], aKey Kab # aNum Ts # aNum Ta # nla)"
finite_UN: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and M :: "'b measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> null_sets M" shows "\<Union> (A ` I) \<in> null_sets M"
upd_nested_lookup: fixes m :: "('c, ('a, 'b) mapping) mapping"   and d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and X :: "('c \<times> 'a) set"   and a :: "'c" shows "Mapping.lookup (upd_nested m d f X) a = (case Mapping.lookup m a of None \<Rightarrow> if a \<in> fst ` X then Some (upd_set' Mapping.empty d f {b. (a, b) \<in> X}) else None | Some m' \<Rightarrow> Some (upd_set' m' d f {b. (a, b) \<in> X}))"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i \<sqsubset> length xs" shows "xs[i := x] ! i = x"
resOutputLeft: fixes x :: "name"   and a :: "name"   and b :: "name"   and Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "x \<noteq> a"   and "x \<noteq> b"   and "Id \<subseteq> Rel" shows "<\<nu>x>a{b}.P \<leadsto>[Rel] a{b}.(<\<nu>x>P)"
simps_87: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x61 :: "\<I>"   and x62 :: "'c formula" shows "(case Prev x61 x62 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f6 x61 x62"
not_less_iff_gr_or_eq: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(\<not> x <s y) = (y <s x \<or> x = y)"
polynomial_f_def: fixes w :: "nat \<Rightarrow> real" shows "polynomial_f w = det (submatrix (ten2mat (A w)) rows_with_1 rows_with_1)"
conversep_iff: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'b" shows "r\<inverse>\<inverse> a b = r b a"
smult_conv_map_poly: fixes c :: "'b"   and p :: "'b poly" shows "Polynomial.smult c p = map_poly ((*) c) p"
4: shows "\<forall>x. x |\<in>| |dom|f \<longrightarrow> f |@| x |\<in>| |cod|f"
normA: fixes a :: "'a" assumes "a \<in> basisA" shows "norm a = 1"
cases: fixes a1 :: "'b tag_const"   and a2 :: "'b tag_const" assumes "a1 \<le> a2" obtains n :: "'b" and m :: "'b" where "a1 = \<tau>\<^sub>c\<^sub>s\<^sub>t n" and "a2 = \<tau>\<^sub>c\<^sub>s\<^sub>t m" and "n \<le> m"
Ball_starI: fixes as :: "'b list"   and A :: "'b list set" assumes "\<forall>a\<in>set as. [a] \<in> A" shows "as \<in> star A"
set_nths: fixes xs :: "'b list"   and I :: "nat set" shows "set (nths xs I) = {xs ! i |i. i < length xs \<and> i \<in> I}"
roots: fixes i :: "nat" shows "r (\<phi> i) \<in> F"
lfp_unfold: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "\<nu> f = f (\<nu> f)"
simps: fixes I :: "'c set"   and v :: "nat \<Rightarrow> 'd"   and s :: "('c, 'd) state" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>s s = (v \<Turnstile>\<^sub>t \<T> s \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>b \<B>\<I> s)"
insertAssertionResChain: fixes xvec :: "name list"   and \<Psi> :: "'b"   and F :: "'b frame" assumes "xvec \<sharp>* \<Psi>" shows "insertAssertion (\<lparr>\<nu>*xvec\<rparr>F) \<Psi> = \<lparr>\<nu>*xvec\<rparr>insertAssertion F \<Psi>"
lookup_empty: fixes k :: "'b" shows "DAList.lookup DAList.empty k = None"
subtype_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and P :: "POPLmarkRecord.type"   and Q :: "POPLmarkRecord.type"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile> P <: Q"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B" shows "\<up>\<^sub>e 1 0 \<Delta> @ B \<Colon> \<Gamma> \<turnstile> \<up>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> P <: \<up>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> Q"
codomain_smult_zero: fixes r :: "'f" assumes "r \<in> UNIV" shows "r \<star> (0::'w) = (0::'w)"
order_G: shows "order (G\<lparr>carrier := P\<rparr>) = p ^ a * k"
ok: shows "\<exists>s' y. ord_rec_Seq T G s' k y \<and> z = G y \<and> s = insf s' k y"
ifmi'_def: fixes v :: "nat"   and ni1 :: "nat"   and ni2 :: "nat"   and s :: "bdd" shows "ifmi' v ni1 ni2 s \<equiv> oassert (bdd_sane s \<and> bdd_node_valid s ni1 \<and> bdd_node_valid s ni2) \<bind> (\<lambda>_. Some (ifmi v ni1 ni2 s))"
prime_factorization_subset_imp_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "factorial_semiring.prime_factorization zero times one normalize x \<subseteq># factorial_semiring.prime_factorization zero times one normalize y" shows "dvd.dvd times x y"
ord_divides: fixes a :: "nat"   and d :: "nat"   and n :: "nat" shows "[a ^ d = 1] (mod n) = (Pocklington.ord n a dvd d)"
that_3: fixes m :: "nat" assumes "bit w m" shows "m \<le> n"
typeof_f64: fixes v :: "v" assumes "typeof v = T_f64" shows "\<exists>c. v = ConstFloat64 c"
prevElem_nextElem: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "nextElem (rev xs) (last xs) (nextElem xs (hd xs) x) = x"
K: shows "filter (\<lambda>x. key x = k) (nths xs (A - {maxi xs key})) = filter (\<lambda>x. key x = k) (nths (rev vs) (A \<inter> {..<maxi xs key}))"
is_distinguishing_formula_def: fixes x :: "('idx, 'pred, 'act, 'effect) FL_Formula.formula"   and P :: "'state"   and Q :: "'state" shows "x distinguishes P from Q \<equiv> P \<Turnstile> x \<and> \<not> Q \<Turnstile> x"
filterlim_ident: fixes F :: "'b filter" shows "LIM x F. x :> F"
simps: fixes aa :: "'a"   and a :: "'a" shows "(aa \<in> iterates_above a) = (aa = a \<or> (\<exists>x. aa = f x \<and> x \<in> iterates_above a) \<or> (\<exists>M. aa = lub M \<and> M \<in> Chains leq \<and> M \<noteq> {} \<and> (\<forall>x. x \<in> M \<longrightarrow> x \<in> iterates_above a)))"
bisim1Sync8: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and a' :: "'addr"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (insync\<^bsub>V\<^esub> (a) Throw a', xs) \<leftrightarrow> ([xs ! V, Addr a'], xs, 7 + length (compE2 e1) + length (compE2 e2), None)"
fc_eq: fixes x :: "real" assumes "x \<in> D01" shows "(f \<circ> c) x = h x"
g_nonzero: fixes t :: "real" assumes "t \<in> {a..b}" shows "g t \<noteq> 0"
comp_fun_idem_inf: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.semilattice_inf inf less_eq less" shows "comp_fun_idem inf"
tau_steps_not_self: fixes x :: "('idx, 'pred, 'act) formula"   and n :: "nat" shows "(x = tau_steps x n) = (n = 0)"
components_are_iso: fixes a :: "'a" assumes "Left_a.ide a" shows "Left_a.iso (Left_a.L a)"
infinite_hyp_wrong: fixes s :: "nat \<Rightarrow> nat option"   and n :: "nat"   and j :: "nat" assumes "r1_bc s" shows "\<exists>m>n. \<phi> (the (s (r1_bc.psi s j \<triangleright> m))) \<noteq> r1_bc.psi s j"
binomial_finite_char: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat"   and p :: "nat" assumes "x \<in> carrier S"   and "y \<in> carrier S"   and "[p] \<cdot>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub> = \<zero>\<^bsub>S\<^esub>"   and "prime p" shows "(x \<oplus>\<^bsub>S\<^esub> y) [^]\<^bsub>S\<^esub> p = x [^]\<^bsub>S\<^esub> p \<oplus>\<^bsub>S\<^esub> y [^]\<^bsub>S\<^esub> p"
augment_flow_presv_cap: fixes u :: "nat"   and v :: "nat" shows "(0::'capacity) \<le> augment f' (u, v) \<and> augment f' (u, v) \<le> c (u, v)"
divide_left_mono_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less_eq c zero"   and "less zero (times a b)" shows "less_eq (divide c a) (divide c b)"
assms_1: shows "\<Psi> \<hookrightarrow> \<Psi>'"
lookup_present_eq_key'': fixes al :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "distinct (map fst al)" shows "(lookup (\<lambda>x. x = k) al = Some v) = ((k, v) \<in> set al)"
enum_ex_sum_def: fixes P :: "'a + 'b \<Rightarrow> bool" shows "enum_ex P = (enum_ex (\<lambda>x. P (Inl x)) \<or> enum_ex (\<lambda>x. P (Inr x)))"
moebius_circline_comp_inv_right: fixes M :: "moebius"   and H :: "circline" shows "moebius_circline M (moebius_circline (moebius_inv M) H) = H"
advantage_def: fixes key_gen :: "('a \<times> 'b, 'c, 'd) gpv"   and aencrypt :: "'a \<Rightarrow> 'e \<Rightarrow> ('f, 'c, 'd) gpv"   and valid_plains :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and "oracle" :: "'g \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'g) spmf"   and \<sigma> :: "'g"   and \<A> :: "('a \<Rightarrow> (('e \<times> 'e) \<times> 'h, 'c, 'd) gpv) \<times> ('f \<Rightarrow> 'h \<Rightarrow> (bool, 'c, 'd) gpv)" shows "ind_cpa_pk.advantage key_gen aencrypt valid_plains oracle \<sigma> \<A> = \<bar>spmf (run_gpv oracle (ind_cpa_pk.ind_cpa key_gen aencrypt valid_plains \<A>) \<sigma>) True - 1 / 2\<bar>"
elims: fixes x :: "nat"   and xa :: "nat list"   and y :: "nat list" assumes "into_WW x xa = y" obtains k :: "nat" where "x = k__" and "xa = []" and "y = []"   | k :: "nat" and n :: "nat" and ns :: "nat list" where "x = k" and "xa = n # ns" and "y = (k + n) # into_WW (Suc (k + n)) ns"
join_above1: fixes x :: "'b"   and y :: "'b" assumes "compatible x y" shows "x \<sqsubseteq> x \<squnion> y"
update_update: fixes f :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'k \<Rightarrow> 'a option"   and g :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'k \<Rightarrow> 'a option"   and m :: "'m" shows "update f (update g m) = update (g \<circ> f) m"
SUP_mult_left_ennreal: fixes c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal"   and I :: "'a set" shows "c * \<Squnion> (f ` I) = (\<Squnion>i\<in>I. c * f i)"
VDMWhile: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "G \<rhd> c : B"   and "\<forall>s. \<not> evalB b s \<longrightarrow> A s s"   and "\<forall>s r t. evalB b s \<longrightarrow> B s r \<longrightarrow> A r t \<longrightarrow> A s t" shows "G \<rhd> While b c : (\<lambda>s t. A s t \<and> \<not> evalB b t)"
ul_f: shows "uniform_limit (path_image \<gamma>) f l F"
cases: fixes x :: "nat \<times> nat \<times> nat \<times> nat" obtains n :: "nat" and p :: "nat" and q :: "nat" and r :: "nat" where "x = (Suc (Suc n), pa__, q__, r__)"   | p :: "nat" and q :: "nat" and r :: "nat" where "x = (0, pb__, qa__, ra__)"   | p :: "nat" and q :: "nat" and r :: "nat" where "x = (Suc 0, p, q, r)"
hyps_2: fixes b :: "'b" assumes "\<Psi> \<simeq> b" shows "b \<rhd> P \<parallel> !P \<longmapsto> Rs"
word_le_nat_alt: fixes a :: "'a word"   and b :: "'a word" shows "(a \<le> b) = (unat a \<le> unat b)"
U_consts_3: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n false\<^sub>n U\<^sub>n \<phi> = w \<Turnstile>\<^sub>n \<phi>"
lexext_compat_cons: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "lexext gt ys xs" shows "lexext gt (x # ys) (x # xs)"
divide_eq_0_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = zero) = (a = zero \<or> b = zero)"
pmp: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool" assumes "|~ F"   and "|~ F \<longrightarrow> G" shows "|~ G"
Compl_eq_Compl_iff: fixes A :: "'a set"   and B :: "'a set" shows "(- A = - B) = (A = B)"
collapse: fixes char :: "char" shows "Char (digit0 char) (digit1 char) (digit2 char) (digit3 char) (digit4 char) (digit5 char) (digit6 char) (digit7 char) = char"
one_smallo_powr: fixes e :: "real" assumes "0 < e" shows "(\<lambda>_. 1) \<in> o(\<lambda>x. x powr e)"
insert_image: fixes x :: "'e"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "x \<in> A" shows "insert (f x) (f ` A) = f ` A"
assms_4: shows "h \<turnstile> getter \<rightarrow>\<^sub>r x"
kpi: fixes i :: "nat" assumes "i < st.d" shows "ket_plus $ st.encode2 i = 1 / csqrt 2"
coeff_ltrm: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "0 < deg R p" shows "zcf (monom P (p (deg R p)) (deg R p)) = \<zero>"
sub1_preserves_diagnal: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "sub1 mf q k l (r, A) = (r', A')"   and "l < dim_col A"   and "k + l < dim_row A" shows "A' $$ (l, l) = A $$ (l, l)"
iFROM_minus: fixes x :: "nat"   and n :: "nat"   and k :: "nat" assumes "x \<in> [n\<dots>]"   and "k \<le> x - n" shows "x - k \<in> [n\<dots>]"
rcs_fixed: fixes H :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "H \<bullet> a = H" shows "a \<in> H"
term_of_pair_pair: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_of_pair (term_powerprod.pp_of_term pair_of_term v, term_powerprod.component_of_term pair_of_term v) = v"
suffix_singleton_suffix: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "w i ## suffix (Suc i) w = suffix i w"
is_weak_bisimulation_def: fixes R :: "'state \<Rightarrow> 'state \<Rightarrow> bool" shows "is_weak_bisimulation R \<equiv> symp R \<and> (\<forall>P Q \<phi>. R P Q \<and> P \<turnstile> \<phi> \<longrightarrow> (\<exists>Q'. Q \<Rightarrow> Q' \<and> R P Q' \<and> Q' \<turnstile> \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<Rightarrow>\<langle>\<alpha>\<rangle> Q' \<and> R P' Q')))"
closedin_derived_set_of_gen: fixes X :: "'c topology"   and S :: "'c set" assumes "t1_space X" shows "closedin X (X derived_set_of S)"
intro: fixes A :: "('h, 'i, 'j) sa_rec_scheme"   and B :: "('k, 'i, 'l) sa_rec_scheme"   and R :: "('h \<times> 'k) set" assumes "sa A"   and "sa B"   and "simulation R A B"   and "lsimulation_axioms R A B" shows "lsimulation R A B"
rep_eq: fixes x :: "'c \<Longrightarrow> 'd"   and xa :: "('e, 'd) uexpr" shows "(x \<natural> xa) = (\<forall>b b'. \<lbrakk>xa\<rbrakk>\<^sub>e (b' \<triangleleft>\<^bsub>x\<^esub> b) = \<lbrakk>xa\<rbrakk>\<^sub>e b)"
sequent_cost_def: fixes \<Gamma> :: "'b multiset"   and \<Delta> :: "'b multiset" shows "sequent_cost \<Gamma> \<Delta> = Suc (sum_list (sorted_list_of_multiset (image_mset size (\<Gamma> + \<Delta>))))"