hcju/isabelleps huggingface dataset:
vsv_appE: fixes a :: "V"   and b :: "V" assumes "(xs #\<^sub>\<circ> x)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (xs #\<^sub>\<circ> x)" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x"
local_4: shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : ts _> ts'"
A: shows "\<forall>\<^sub>F x in at_top. 0 < 1 - b - hb / ln x powr (1 + e)"
interaction_bounded_by_inline1: fixes out' :: "'call"   and out :: "'call'"   and c' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s"   and p :: "enat"   and q :: "enat"   and input :: "'ret'"   and x :: "'ret" assumes "Inr (out', out, c', rpv) \<in> set_spmf (local.inline1' gpv s)"   and "interaction_bounded_by consider' gpv p"   and "\<And>s x. consider' x \<Longrightarrow> interaction_bounded_by consider (callee s x) q"   and "\<And>s x. \<not> consider' x \<Longrightarrow> interaction_bounded_by consider (callee s x) 0" shows "if consider' out' then (if consider out then q \<noteq> 0 \<and> interaction_bounded_by consider (c' input) (q - 1) else interaction_bounded_by consider (c' input) q) \<and> p \<noteq> 0 \<and> interaction_bounded_by consider' (rpv x) (p - 1) else \<not> consider out \<and> interaction_bounded_by consider (c' input) 0 \<and> interaction_bounded_by consider' (rpv x) p"
assms: shows "\<D>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A"
sublist_Nil_right: fixes xs :: "'b list" shows "xs \<le>f \<epsilon> = (xs = \<epsilon>)"
simps: fixes A :: "'b mat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b" shows "diagonal_to_Smith_PQ_JNF A bezout = diagonal_to_Smith_aux_PQ_JNF [0..<min (dim_row A) (dim_col A) - 1] bezout (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A))"
rel_refl_strong: fixes x :: "('d, 'e, 'f) generat"   and R1a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R3a :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> generat_pures x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> generat_outs x \<Longrightarrow> R2a z2 z2"   and "\<And>z3. z3 \<in> generat_conts x \<Longrightarrow> R3a z3 z3" shows "rel_generat R1a R2a R3a x x"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
swap_permute_swap: fixes x :: "'a"   and \<pi> :: "perm"   and y :: "'a"   and t :: "'b" assumes "atom x \<sharp> \<pi>"   and "atom y \<sharp> \<pi>" shows "(x \<leftrightarrow> y) \<bullet> \<pi> \<bullet> (x \<leftrightarrow> y) \<bullet> t = \<pi> \<bullet> t"
semimetric_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real" shows "semimetric dist \<equiv> (\<forall>x y. 0 \<le> dist x y) \<and> (\<forall>x y. (dist x y = 0) = (x = y)) \<and> (\<forall>x y. dist x y = dist y x)"
is_finite_Float_def: fixes x :: "('a, 'b) IEEE.float itself"   and f :: "Float.float" shows "is_finite_Float x f = (is_normal_Float x f \<or> is_denormal_Float x f \<or> f = 0)"
Int_Un_distrib: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" shows "A \<inter> (B \<union> C) = A \<inter> B \<union> A \<inter> C"
complete_direct_successors': fixes \<S> :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list" shows "\<lbrakk> \<S> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<subseteq> \<Union> (HeronConf_interpretation ` \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<S>)"
card_seteq: fixes B :: "'q set"   and A :: "'q set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
constant_function_def: fixes S :: "'c set"   and a :: "'d" shows "constant_function S a = (\<lambda>x\<in>S. a)"
le_fininf_le_inf_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a stream"   and w\<^sub>3 :: "'a stream" assumes "traces ind"   and "traces.le_fininf ind w\<^sub>1 w\<^sub>2"   and "traces.le_inf ind w\<^sub>2 w\<^sub>3" shows "traces.le_fininf ind w\<^sub>1 w\<^sub>3"
wf_strictly_generalizes: shows "wfP (\<prec>\<cdot>)"
method_def2: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "method P C M = (D, b, Ts, T, m)"
cobounded2: fixes b :: "'b"   and a :: "'b" shows "b \<le> max a b"
simps_2: fixes i :: "nat"   and c :: "int" shows "isnorm (Pinj i (Pc c)) = False"
knows_Spy_pubEK_i: fixes b :: "bool"   and A :: "agent"   and evs :: "event list" shows "Key (publicKey b A) \<in> knows Spy evs"
image_set: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "f ` list.set xs = list.set (map f xs)"
ext_inject: fixes nodes :: "'a set"   and edges :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c"   and nodes' :: "'a set"   and edges' :: "('a \<times> 'b \<times> 'a) set"   and more' :: "'c" shows "(\<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes', edges = edges', \<dots> = more'\<rparr>) = (nodes = nodes' \<and> edges = edges' \<and> more = more')"
pdevs_apply_binop_pdevs: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'c pdevs"   and y :: "'d pdevs"   and i :: "nat" shows "pdevs_apply (binop_pdevs f x y) i = (if pdevs_apply x i = (0::'c) \<and> pdevs_apply y i = (0::'d) then 0::'b else f (pdevs_apply x i) (pdevs_apply y i))"
sup_least: fixes A :: "'a set"   and z :: "'a" assumes "A \<subseteq> carrier L"   and "z \<in> carrier L"   and "\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z" shows "\<Squnion>A \<sqsubseteq> z"
case1: fixes a :: "'b fps"   and b :: "'b fps" assumes "1 \<le> subdegree a" shows "fps_shift 1 (a * b) = fps_shift 1 a * b"
simps_1: fixes v :: "val" shows "eval_unop UPlus v = Intg (the_Intg v)"
open_countable_basisE: fixes p :: "'c set \<Rightarrow> bool"   and B :: "'c set set"   and X :: "'c set" assumes "countable_basis p B"   and "p X" obtains B' :: "'c set set" where "B' \<subseteq> B" and "X = \<Union> B'"
squarefree_multD_1: fixes a :: "'b"   and b :: "'b" assumes "squarefree (a * b)" shows "squarefree a"
upper_asymptotic_density_def: fixes A :: "nat set" shows "upper_asymptotic_density A = real_of_ereal (limsup (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x)))"
simps_2: fixes u :: "'a"   and e :: "'b"   and es :: "'b list"   and v :: "'a" shows "cas u (e # es) v = (tail G e = u \<and> cas (head G e) es v)"
span_mono: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "vs2.span A \<subseteq> vs2.span B"
simps_2: fixes u :: "'v"   and l :: "enat"   and ls :: "enat list"   and v :: "'v" shows "path u (l # ls) v = (\<exists>uh. l = w (u, uh) \<and> path uh ls v)"
iMODb_sub: fixes k :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" shows "k \<ominus> [ r, mod m, c ] = (if r + m * c \<le> k then [ k - r - m * c, mod m, c ] else if r \<le> k then [ (k - r) mod m, mod m, (k - r) div m ] else {})"
simps_2: fixes c :: "'a"   and cs :: "'a list"   and p :: "('a, 'b) form" shows "sub_consts (c # cs) p = sub_consts cs (p[App c []/length cs])"
simps_2: fixes n :: "nat" shows "make_vert_block (Suc n) = vert # make_vert_block n"
5: shows "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
remove_fset: fixes x :: "'b"   and xs :: "'b fset" shows "fset (remove_fset x xs) = fset xs - {x}"
is_order_A: shows "partial_order \<X>\<^bsub>F\<^esub>"
bigtheta_trans1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> O[F](h)"
stability: fixes j :: "nat" assumes "t j \<le> t0 + e1'" shows "dist (euler' e g x0' t j) (euler f x0 t j) \<le> sqrt (real DIM('a)) * (B + 1) / 2 * (exp (B' * e1' + 1) - 1) * max_stepsize j"
index_mat_multcol_5: fixes k :: "nat"   and a :: "'d"   and A :: "'d mat" shows "dim_col (multcol k a A) = dim_col A"
assms: shows "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"
P: shows "IMP_LIST Cns ((P, P') \<in> T \<rightarrow> bool_rel)"
U_def: shows "U = {uu_. \<exists>p v. uu_ = v \<and> cf.isSimplePath u p v}"
\<mu>_small_def: fixes fs :: "int vec list"   and i :: "nat" shows "\<mu>_small fs i = (\<forall>j<i. \<bar>gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j\<bar> \<le> 1 / 2)"
assms_2: shows "interaction_bounded_by' (\<lambda>_. True) \<A> q"
succ_\<omega>_iff: fixes n :: "V" shows "(ZFC_in_HOL.succ n \<in>\<^sub>\<circ> \<omega>) = (n \<in>\<^sub>\<circ> \<omega>)"
sleq_antisym: fixes x :: "'a"   and y :: "'a" assumes "sleq x y"   and "sleq y x" shows "x = y"
lfinite_code_2: fixes x :: "'b"   and xs :: "'b llist" shows "lfinite (x % xs) = lfinite xs"
cpx_vec_length_def: fixes v :: "complex vec" shows "\<parallel>v\<parallel> \<equiv> sqrt (\<Sum>i<dim_vec v. (cmod (v $ i))\<^sup>2)"
I_maximal: fixes J :: "'a set" assumes "ideal J R"   and "I \<subseteq> J"   and "J \<subseteq> carrier R" shows "J = I \<or> J = carrier R"
size_correct: fixes s :: "('a, unit) HashMap.hashmap" assumes "my_invar s" shows "hs.size s = card (my_\<alpha> s)"
bm: shows "bounded ((\<lambda>w. w - z) ` path_image \<gamma>)"
strongAppend: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "Q \<leadsto>[Rel'] R"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''" shows "P \<leadsto>\<^sup>^<Rel''> R"
rowvector_def: fixes v :: "('a, 'b) Finite_Cartesian_Product.vec" shows "rowvector v = (\<chi>i j. v $h j)"
simps: fixes i :: "nat"   and j :: "nat" shows "encode12 (i, j) = digit_decode dims (weave vars (digit_encode dims1 i) (digit_encode dims2 j))"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "'i" shows "P5 (s1, s2) = (\<forall>p. status s1 p \<noteq> Sleep \<and> initialized s1 \<and> initVals s1 = {})"
hseq_char': fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "B.hseq \<nu> \<mu> = (\<nu> \<star> \<mu> \<noteq> B.null)"
r7': fixes t :: "trm"   and t' :: "trm"   and s :: "trm"   and x :: "name" assumes "t \<mapsto> t'" shows "s to x in t \<mapsto> s to x in t'"
assms: shows "0 < Re b \<or> Re b = 0 \<and> 0 \<le> Im b"
prj_tuple_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "\<p>\<^sub>1[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = h"
mcurrents_def: fixes m1 :: "'state_t option"   and m2 :: "'state_t option" shows "mcurrents m1 m2 \<equiv> B2 m1 m2 (\<lambda>s t. current s = current t)"
finsum_insert: fixes G :: "('c, 'd) ring_scheme"   and F :: "'e set"   and a :: "'e"   and f :: "'e \<Rightarrow> 'c" assumes "abelian_monoid G"   and "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finsum G f (insert a F) = f a \<oplus>\<^bsub>G\<^esub> finsum G f F"
Unit_def: fixes C :: "'at Literal set" shows "Unit C = (card C = 1)"
vinsert_identD: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "vinsert x A = vinsert x B"   and "x \<notin>\<^sub>\<circ> A"   and "x \<notin>\<^sub>\<circ> B" shows "A = B"
add_cancel_left_left: fixes b :: "'a"   and a :: "'a" shows "(b + a = a) = (b = (0::'a))"
full_pmdl_def: fixes K :: "'k set" shows "full_pmdl K = {p. component_of_term ` keys p \<subseteq> K}"
induct: fixes P :: "real poly \<Rightarrow> real poly \<Rightarrow> bool"   and a0 :: "real poly"   and a1 :: "real poly" assumes "\<And>p q. (degree q \<noteq> 0 \<Longrightarrow> P q (- (p mod q))) \<Longrightarrow> P p q" shows "P a0 a1"
app3_list: fixes t1 :: "('c, 'd) FingerTreeStruc"   and xs :: "('c, 'd) Node list"   and t2 :: "('c, 'd) FingerTreeStruc" shows "toList (app3 t1 xs t2) = toList t1 @ nlistToList xs @ toList t2"
d1_1: shows "the (delay i (f \<triangleright> n\<^sub>1)) = Suc (GREATEST j. j < e_length (f \<triangleright> n\<^sub>1) \<and> 0 < the (eval r_result1 [e_length (f \<triangleright> n\<^sub>1), i, e_take (Suc j) (f \<triangleright> n\<^sub>1)]))"
rep_eq: shows "semilattice_set_apply min_sls = min"
Range_Int_subset: fixes A :: "('f \<times> 'e) set"   and B :: "('f \<times> 'e) set" shows "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
same_state_implies_same_result_state: fixes c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and d :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration" assumes "states c p = states d p"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'" shows "states d' p = states c' p"
insert: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "ListMem x xs" shows "ListMem x (y # xs)"
abss_\<S>D: fixes s :: "'s \<times> ('c \<Rightarrow> real)" assumes "abss s \<in> \<S>" obtains l :: "'s" and u :: "'c \<Rightarrow> real" where "s = (l, u)" and "u \<in> [u]\<^sub>\<R>" and "[u]\<^sub>\<R> \<in> \<R>"
field_iff_prime: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "principal_domain R"   and "a \<in> carrier R - {\<zero>\<^bsub>R\<^esub>}" shows "field (R Quot PIdl\<^bsub>R\<^esub> a) = ring_prime\<^bsub>R\<^esub> a"
xa_1: shows "x \<in> aform_err e (fst X, snd X)"
hyps: assumes "abc_notfinal (abc_steps_l (0, lm) A n) A" shows "abc_steps_l (0, lm) (A [+] B) n = abc_steps_l (0, lm) A n"
length_sum_iarray: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b iarray" assumes "finite S"   and "S \<noteq> {}" shows "IArray.length (sum f S) = Max {IArray.length (f x) |x. x \<in> S}"
ok_0_progress: fixes mr :: "mregex"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "ok 0 mr" shows "progress_regex \<sigma> P (from_mregex mr []) j = j"
Dynkin_system_axioms: shows "Dynkin_system \<Omega> M"
amult_one_l: fixes x :: "ant" shows "1 * x = x"
add_im_sheaf_def: shows "add_im_sheaf \<equiv> \<lambda>V. +\<^bsub>f \<^sup>\<inverse> S V\<^esub>"
chooseNext_eliminated: shows "chooseNext g \<notin> step.allDefs g (defNode g (chooseNext g))"
consistent_sign_vec_semantics_R: fixes fml :: "nat fml"   and ls :: "real poly list"   and x :: "real" assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length ls" shows "lookup_sem fml (map (\<lambda>p. poly p x) ls) = lookup_sem fml (consistent_sign_vec ls x)"
ih1: assumes " t1 \<longrightarrow>\<^sub>1 t'" shows " t' \<longrightarrow>\<^sub>1 t1'"
conc_forms_def: shows "conc_forms = fset_from_list conclusions"
representation_zero: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and basis :: "'d set" assumes "module scale" shows "module.representation scale basis (0::'d) = (\<lambda>b. 0::'c)"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "CC.seq g f = (seq (fst g) (fst f) \<and> seq (snd g) (snd f))"
eq_neutr_iff: fixes a :: "'b"   and b :: "'b" shows "(inf a b = top) = (a = top \<and> b = top)"
R\<^sub>C_ofD: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and \<R>\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and \<R> :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and P :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and c\<^sub>1\<^sub>C :: "'g"   and mds\<^sub>C :: "Mode \<Rightarrow> 'e set"   and mem\<^sub>1\<^sub>C :: "'e \<Rightarrow> 'd"   and c\<^sub>2\<^sub>C :: "'g"   and mds\<^sub>C' :: "Mode \<Rightarrow> 'e set"   and mem\<^sub>2\<^sub>C :: "'e \<Rightarrow> 'd" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"   and "sifum_refinement.secure_refinement dma\<^sub>C \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of \<R>\<^sub>A \<R> P"   and "(((c\<^sub>1\<^sub>C, mds\<^sub>C), mem\<^sub>1\<^sub>C), (c\<^sub>2\<^sub>C, mds\<^sub>C'), mem\<^sub>2\<^sub>C) \<in> sifum_refinement.R\<^sub>C_of dma\<^sub>C \<C>_vars\<^sub>C \<R>\<^sub>A \<R> P" shows "(\<exists>c\<^sub>1\<^sub>A c\<^sub>2\<^sub>A. (((c\<^sub>1\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>1\<^sub>C), (c\<^sub>1\<^sub>C, mds\<^sub>C), mem\<^sub>1\<^sub>C) \<in> \<R> \<and> (((c\<^sub>2\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>2\<^sub>C), (c\<^sub>2\<^sub>C, mds\<^sub>C), mem\<^sub>2\<^sub>C) \<in> \<R> \<and> (((c\<^sub>1\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>1\<^sub>C), (c\<^sub>2\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>2\<^sub>C) \<in> \<R>\<^sub>A) \<and> mds\<^sub>C' = mds\<^sub>C \<and> sifum_security_init.low_mds_eq dma\<^sub>C \<C>_vars\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C \<and> (((c\<^sub>1\<^sub>C, mds\<^sub>C), mem\<^sub>1\<^sub>C), (c\<^sub>2\<^sub>C, mds\<^sub>C'), mem\<^sub>2\<^sub>C) \<in> P"
I_l_closed: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and a :: "'a"   and x :: "'a" assumes "ideal I R"   and "a \<in> I"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> a \<in> I"
abs_eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(dlist x = dlist y) = dlist_eq x y"
span_Diff_zero: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "pmdl (B - {0}) = pmdl B"
eq_uptoI: fixes g :: "nat \<Rightarrow> 'a"   and B :: "(nat \<Rightarrow> 'a) set"   and i :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> B"   and "\<And>j. j < i \<Longrightarrow> f j = g j" shows "g \<in> eq_upto B f i"
af_\<mu>LTL: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool"   and \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "ltl_equivalence eq"   and "\<phi> \<in> \<mu>LTL" shows "w \<Turnstile>\<^sub>n \<phi> = (\<exists>i. eq (af \<phi> (prefix i w)) true\<^sub>n)"
delete_index_ge_length: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "delete_index n xs = xs"
intros_1: fixes t :: "real"   and t' :: "real"   and s :: "'a"   and S :: "'a set"   and s' :: "'a"   and \<omega> :: "(real \<times> 'a) stream" assumes "t < t'"   and "s \<in> S" shows "trace_in S t s ((t', s') ## \<omega>)"
expand: fixes gpv :: "('h, 'i, 'j) gpv"   and gpv' :: "('h, 'i, 'j) gpv" assumes "the_gpv gpv = the_gpv gpv'" shows "gpv = gpv'"
summable_minus: assumes "summable f" shows "summable (\<lambda>n. - f n)"
simps_3: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and g22 :: "'c \<Rightarrow> 'c" assumes "\<not> p a" shows "lhd (unfold_llist p g21 g22 a) = g21 a"
K2_isometry_endpoint_in_S: fixes a :: "proj2"   and b :: "proj2"   and J :: "cltn2" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "apply_cltn2 (endpoint_in_S a b) J = endpoint_in_S (apply_cltn2 a J) (apply_cltn2 b J)"
a_rcos_const: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme"   and h :: "'a" assumes "abelian_subgroup H G"   and "h \<in> H" shows "H +>\<^bsub>G\<^esub> h = H"
zero_le: fixes x :: "'k multiset" shows "{#} \<subseteq># x"
span_bicategory_axioms: shows "span_bicategory (\<cdot>) prj0 prj1"
downset_inj: shows "inj (ord.downset (\<lambda>x y. y \<le> x))"
wt_lbv_wt_method: fixes P :: "jvm_prog"   and Ts :: "ty list"   and p :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and b :: "staticb"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and xt :: "ex_table"   and cert :: "(ty list \<times> ty err list) option err list"   and "is" :: "bytecode" assumes "start_context P Ts p C"   and "wt_lbv P C b Ts T\<^sub>r mxs mxl\<^sub>0 xt cert is" shows "\<exists>\<tau>s. wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s"
showsp_prod_def: fixes s1 :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and s2 :: "nat \<Rightarrow> 'b \<Rightarrow> char list \<Rightarrow> char list"   and p :: "nat" shows "showsp_prod s1 s2 p = pshowsp_prod p \<circ> map_prod (s1 1) (s2 1)"
boundary: shows "abs_Gromov_completion u \<in> Gromov_boundary"
rfb_rfd_demorgan: fixes R :: "('d \<times> 'c) set" shows "\<partial> \<circ> fb\<^sub>\<R> R = fd\<^sub>\<R> R \<circ> \<partial>"
in_nlists_Suc_iff: fixes xs :: "'c list"   and A :: "'c set"   and n :: "nat" shows "(xs \<in> nlists A (Suc n)) = (\<exists>x xs'. xs = x # xs' \<and> x \<in> A \<and> xs' \<in> nlists A n)"
positive_scale: fixes A :: "complex mat"   and n :: "nat"   and c :: "real" assumes "A \<in> carrier_mat n n"   and "positive A"   and "0 \<le> c" shows "positive (complex_of_real c \<cdot>\<^sub>m A)"
out_trivial: fixes A :: "'p"   and P :: "'p" assumes "A \<noteq> P" shows "P Out A A"
map_pmf_eq_return_pmf_iff: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b pmf"   and x :: "'a" shows "(map_pmf f p = return_pmf x) = (\<forall>y\<in>set_pmf p. f y = x)"
prob0_jozsa_algo_of_balanced: assumes "is_balanced" shows "prob0_fst_qubits n jozsa_algo = 0"
unit_simps_1: fixes a :: "'a" assumes "C.obj a" shows "C.arr \<i>\<^sub>C[a]"
Guard: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P :: "('b, 'a, 'c) ann"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set"   and r :: "'b set"   and g :: "'b set"   and f :: "'c" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A"   and "r \<inter> g \<subseteq> pre P"   and "r \<inter> - g \<noteq> {} \<longrightarrow> f \<in> F" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnRec r P) (Guard f g c) Q, A"
elims: fixes x :: "opinl"   and xa :: "Unboxed.type option list"   and y :: "opubx option" assumes "Op_example.ubx x xa = y" obtains "x = OpAddNum" and "xa = [Some Ubx1, Some Ubx1]" and "y = Some OpAddNumUbx"   | uv_ :: "Unboxed.type option list" where "x = OpMulNum" and "xa = uv_" and "y = None"   | uu_ :: "opinl" where "x = uu___" and "xa = []" and "y = None"   | uu_ :: "opinl" and va :: "Unboxed.type option list" where "x = uua___" and "xa = None # va__" and "y = None"   | uu_ :: "opinl" and va :: "Unboxed.type option list" where "x = uub___" and "xa = Some Ubx2 # va" and "y = None"   | uu_ :: "opinl" and v :: "Unboxed.type option" where "x = uuc___" and "xa = [v__]" and "y = None"   | uu_ :: "opinl" and v :: "Unboxed.type option" and vc :: "Unboxed.type option list" where "x = uud___" and "xa = vb__ # None # vc__" and "y = None"   | uu_ :: "opinl" and v :: "Unboxed.type option" and vc :: "Unboxed.type option list" where "x = uue___" and "xa = vd__ # Some Ubx2 # vc" and "y = None"   | uu_ :: "opinl" and v :: "Unboxed.type option" and vb :: "Unboxed.type option" and vd :: "Unboxed.type option" and ve :: "Unboxed.type option list" where "x = uu_" and "xa = v # vb # vd # ve" and "y = None"
l_count_def: fixes x :: "'b"   and xs :: "'b list" shows "l_count x xs \<equiv> length (filter (\<lambda>x'. x' = x) xs)"
R_def: fixes m :: "int" assumes "residues m" shows "residue_ring m \<equiv> residue_ring m"