hcju/isabelleps huggingface dataset:
sum_list_map_filter': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.monoid_add plus zero" shows "monoid_add.sum_list plus zero (map f (filter P xs)) = monoid_add.sum_list plus zero (map (\<lambda>x. if P x then f x else zero) xs)"
dc2: shows "density_context V V' \<Gamma> (\<lambda>\<sigma>. \<delta> \<sigma> * (if P' \<sigma> then 1 else 0))"
dvd_minus_mod: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "dvd.dvd times b (minus a (modulo a b))"
lconf_upd2: fixes P :: "'b prog"   and h :: "'heap"   and l :: "String.literal \<Rightarrow> 'addr val option"   and E :: "String.literal \<Rightarrow> ty option"   and v :: "'addr val"   and T :: "ty"   and V :: "String.literal" assumes "P,h \<turnstile> l (:\<le>) E"   and "P,h \<turnstile> v :\<le> T" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>) E(V \<mapsto> T)"
sup_conv_inf: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = - (- x \<sqinter> - y)"
project_inject_cancel: fixes x :: "nat" shows "project_Nat_nat (inject_Nat_nat x) = x"
more_arith_simps_10: fixes a :: "'c" shows "- (- a) = a"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set1_blindable\<^sub>m x"   and "b \<in> set1_blindable\<^sub>m y" shows "m a b = m' a b"
assms_1: shows "u \<le>p w \<cdot> z"
summableI_nonneg_bounded: fixes f :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "\<And>n. (0::'c) \<le> f n"   and "\<And>n. sum f {..<n} \<le> x" shows "summable f"
rawpsubst: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubst \<phi> txs \<in> fmla"
alphaInput: fixes c :: "name"   and P :: "pi"   and a :: "name"   and x :: "name" assumes "c \<sharp> P" shows "a<x>.P = a<c>.([(x, c)] \<bullet> P)"
kernel_to_trivial_group: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "trivial_group H" shows "kernel G H h = carrier G"
DERIV_const_ratio_const: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and k :: "real" assumes "a \<noteq> b"   and "\<And>x. (f has_real_derivative k) (at x)" shows "f b - f a = (b - a) * k"
one_left: fixes a :: "'a" shows "(1::'a) * a = a"
safe_distance0_2_def: fixes v\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance0_2 v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> + 1 / 2 * v\<^sub>o\<^sup>2 / a\<^sub>o"
xsy: shows "xys = (1::'a, 0::'a) # xysp"
gso'_array_def: fixes dmusa :: "int iarray iarray"   and fs :: "int vec list" shows "gso_array.gso'_array dmusa fs = gso_array.gsos' 0 (length fs) dmusa (IArray fs) (IArray [])"
sSuc: shows "\<forall>i. (s (Suc i), s i) \<in> r"
separated_subset: fixes V1 :: "nat set"   and V2 :: "nat set"   and g :: "graph" assumes "V1 \<subseteq> V2"   and "separated g V2" shows "separated g V1"
imult_iless_mono1: fixes i :: "enat"   and j :: "enat"   and k :: "enat" assumes "i < j"   and "0 < k"   and "k \<noteq> \<infinity>" shows "i * k \<le> j * k"
exhaust: fixes y :: "level" obtains "y = Lo"   | "y = Hi"
i: shows "i < dim_row (submatrix (submatrix A I UNIV) UNIV J)"
map_pmf_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and M :: "'c pmf" shows "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
Skip: fixes P :: "'d set"   and Q :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and A :: "'d set" assumes "P \<subseteq> Q" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Skip Q,A"
Annihilator_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme" shows "Ann\<^bsub>R\<^esub> M = {\<zero>\<^bsub>M\<^esub>} \<^bsub>R\<ddagger>M\<^esub> carrier M"
f3: shows "low_equal (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1)))))) (snd (fst (write_reg (user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))) (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2))))))"
insert_Nil: fixes x :: "'b" shows "List.insert x [] = [x]"
norm_divide_numeral: fixes a :: "'a"   and w :: "num" shows "norm (a / numeral w) = norm a / numeral w"
last_remdups_adj: fixes xs :: "'c list" shows "last (remdups_adj xs) = last xs"
k_def: shows "k = nat \<lceil>log 2 (real n)\<rceil>"
invertible_iff_is_unit: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "invertible A = (Determinants.det A dvd (1::'a))"
node_local_deliver: fixes i :: "nat"   and \<zeta>\<^sub>i :: "((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool" shows "\<langle>i : \<zeta>\<^sub>i : R\<^sub>i\<rangle>\<^sub>o \<Turnstile>\<^sub>A (S, U \<rightarrow>) globala (\<lambda>(uu_, a, uu_). \<forall>j. j \<noteq> i \<longrightarrow> (\<forall>d. a \<noteq> j:deliver(d)))"
real_sqrt_unique: fixes y :: "real"   and x :: "real" assumes "y\<^sup>2 = x"   and "0 \<le> y" shows "sqrt x = y"
dominates_induce_ss: fixes u :: "'a"   and v :: "'a"   and S :: "'a set"   and T :: "'a set" assumes "u \<rightarrow>\<^bsub>G \<restriction> S\<^esub> v"   and "S \<subseteq> T" shows "u \<rightarrow>\<^bsub>G \<restriction> T\<^esub> v"
deltaG_Bonk_Schramm_extension_def: fixes uu :: "'a Bonk_Schramm_extension itself" shows "deltaG uu = deltaG TYPE('a)"
axioms_2: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "group G composition unit" shows "Group_Theory.group_axioms G composition unit"
f1: shows "prob1 3 (Matrix.mat 8 (Suc 0) (\<lambda>(i, j). [[\<phi> $$ (0, 0) / complex_of_real (sqrt 2), \<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2), \<phi> $$ (Suc 0, 0) / complex_of_real (sqrt 2), \<phi> $$ (0, 0) / complex_of_real (sqrt 2), 0, 0, 0, 0]] ! j ! i)) (Suc 0) = 1 / 2"
assms_2: fixes m :: "'a"   and n :: "'a" assumes "m \<in> A"   and "n \<in> A"   and "R m n" shows "f m \<inter> f n = {}"
save_restore_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (save_restore_instr instr s1)) \<and> t2 = snd (fst (save_restore_instr instr s2))" shows "low_equal t1 t2"
pentagon: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "ide a"   and "ide b"   and "ide c"   and "ide d" shows "(local.prod a (assoc b c d) \<cdot> assoc a (local.prod b c) d) \<cdot> local.prod (assoc a b c) d = assoc a b (local.prod c d) \<cdot> assoc (local.prod a b) c d"
br_list_rel: fixes f :: "'g \<Rightarrow> 'h"   and P :: "'g \<Rightarrow> bool" shows "\<langle>br f P\<rangle>list_rel = br (map f) (list_all P)"
fmeasurable_Diff_D: fixes T :: "'c set"   and S :: "'c set"   and M :: "'c measure" assumes "T - S \<in> fmeasurable M"   and "S \<in> fmeasurable M"   and "S \<subseteq> T" shows "T \<in> fmeasurable M"
bounded_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Min A) = (\<forall>a\<in>A. x \<le> a)"
Sigma_Un_distrib2: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" shows "(SIGMA i:I. A i \<union> B i) = Sigma I A \<union> Sigma I B"
fls_decompose_reg_pr_parts_1: fixes f :: "'a fls" shows "f = fls_prpart_as_fls f + fls_regpart_as_fls f"
R: shows "\<forall>T. open T \<and> T \<subseteq> S \<longrightarrow> open (f ` T)"
nn_integral_multc: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * c \<partial>M = integral\<^sup>N M f * c"
Equiv_Exec_trans: fixes m :: "'d"   and equiv_states12 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and localState1 :: "'g \<Rightarrow> 'e"   and localState2 :: "'h \<Rightarrow> 'f"   and input_fun1 :: "'d \<Rightarrow> 'i message_af"   and input_fun2 :: "'d \<Rightarrow> 'j message_af"   and output_fun1 :: "'g \<Rightarrow> 'k message_af"   and output_fun2 :: "'h \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and trans_fun2 :: "'j message_af \<Rightarrow> 'h \<Rightarrow> 'h"   and k1 :: "nat"   and k2 :: "nat"   and c1 :: "'g"   and c2 :: "'h"   and equiv_states23 :: "'f \<Rightarrow> 'l \<Rightarrow> bool"   and localState3 :: "'m \<Rightarrow> 'l"   and input_fun3 :: "'d \<Rightarrow> 'n message_af"   and output_fun3 :: "'m \<Rightarrow> 'k message_af"   and trans_fun3 :: "'n message_af \<Rightarrow> 'm \<Rightarrow> 'm"   and k3 :: "nat"   and c3 :: "'m"   and equiv_states13 :: "'e \<Rightarrow> 'l \<Rightarrow> bool" assumes "Equiv_Exec m equiv_states12 localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 c1 c2"   and "Equiv_Exec m equiv_states23 localState2 localState3 input_fun2 input_fun3 output_fun2 output_fun3 trans_fun2 trans_fun3 k2 k3 c2 c3"   and "equiv_states13 = (\<lambda>s1 s3. (if s1 = localState1 c1 \<and> s3 = localState3 c3 then equiv_states12 s1 (localState2 c2) \<and> equiv_states23 (localState2 c2) s3 else equiv_states12 s1 (localState2 (f_Exec_Comp trans_fun2 (input_fun2 m # NoMsg\<^bsup>k2 - Suc 0\<^esup>) c2))) \<and> equiv_states23 (localState2 (f_Exec_Comp trans_fun2 (input_fun2 m # NoMsg\<^bsup>k2 - Suc 0\<^esup>) c2)) s3)" shows "Equiv_Exec m equiv_states13 localState1 localState3 input_fun1 input_fun3 output_fun1 output_fun3 trans_fun1 trans_fun3 k1 k3 c1 c3"
next_send: fixes c :: "('p, 's, 'm) configuration"   and i :: "nat"   and p :: "'p"   and q :: "'p"   and s :: "'s"   and s' :: "'s"   and msg :: "'m"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Send i p q s s' msg \<mapsto> c' = (can_occur (Send i p q s s' msg) c \<and> states c' p = s' \<and> (\<forall>r. r \<noteq> p \<longrightarrow> states c' r = states c r) \<and> msgs c' i = msgs c i @ [Msg msg] \<and> (\<forall>i'. i \<noteq> i' \<longrightarrow> msgs c' i' = msgs c i') \<and> no_snapshot_change c c')"
right_inverse: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "Groups.group f z inverse" shows "f a (inverse a) = z"
content_def: fixes p :: "'d poly" shows "content p = gcd_list (coeffs p)"
dbl_simps_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl plus zero = zero"
tendsto_cmult_ennreal: fixes c :: "ennreal"   and l :: "ennreal"   and f :: "'c \<Rightarrow> ennreal"   and F :: "'c filter" assumes "\<not> (c = \<infinity> \<and> l = 0)"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * l) F"
subset: shows "g ` {a..b} \<subseteq> {c..d}"
esl_def: shows "Product.esl = (\<lambda>(A, r\<^sub>A, f\<^sub>A) (B, r\<^sub>B, f\<^sub>B). (A \<times> B, Product.le r\<^sub>A r\<^sub>B, Product.sup f\<^sub>A f\<^sub>B))"
1: fixes n :: "'node"   and v :: "'val" assumes "phis g (n, v) \<noteq> None" shows "\<exists>v'. phis' g (n, v') \<noteq> None \<and> var g v = var' g v'"
CondInv: fixes P :: "'d set"   and Q :: "'d set"   and b :: "'d set"   and P\<^sub>1 :: "'d set"   and P\<^sub>2 :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and A :: "'d set"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "P \<subseteq> Q"   and "Q \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>1 c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>2 c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Cond b c\<^sub>1 c\<^sub>2 Q,A"
assms_1: shows "(a, a') \<in> \<langle>Ra\<rangle>array_rel"
cInf_greatest: fixes X :: "'a multiset set"   and z :: "'a multiset" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> z \<subseteq># x" shows "z \<subseteq># Inf X"
from_nat_neq_cols: fixes i :: "nat"   and A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and k :: "nat" assumes "i < ncols A"   and "k < ncols A"   and "i \<noteq> k" shows "from_nat i \<noteq> from_nat k"
Union_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_fun (rel_set (rel_set A)) (rel_set A) \<Union> \<Union>"
subst_type: fixes x :: "name"   and U :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and u :: "trm" assumes "(x, U) # \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> u : U" shows "\<Gamma> \<turnstile> t[x\<mapsto>u] : T"
lexext_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "lexext gt ys xs" shows "lexext gt' ys xs"
min_cost_le_walk_cost: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and c :: "'b \<Rightarrow> real" assumes "awalk u p v" shows "\<mu> c u v \<le> ereal (awalk_cost c p)"
sym: fixes s :: "'a event list"   and X :: "'a event set"   and t :: "'a event list" shows "setinterleaving (s, X, t) = setinterleaving (t, X, s)"
subtopology_superset: fixes U :: "'b topology"   and V :: "'b set" assumes "topspace U \<subseteq> V" shows "subtopology U V = U"
exec_Normal_elim_cases_8: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,Normal s\<rangle> \<Rightarrow> t" obtains s' :: "('b, 'c) xstate" where "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> s'" and "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow> t"
simple_cfg_on: fixes ct :: "'s \<Rightarrow> 's pmf"   and s :: "'s" assumes "ct \<in> Pi S K" shows "simple ct s \<in> cfg_on s"
nonprimitive_imp_conductor_less: assumes "\<not> primitive_dchar n \<chi>" shows "conductor < n"
surjective: fixes r :: "('c, 'd) rgraph_scheme" shows "r = \<lparr>root = root r, edges = edges r, \<dots> = rgraph.more r\<rparr>"
relabel_u_no_incoming_adm: fixes v :: "nat" assumes "lo u \<noteq> l u" shows "(v, u) \<notin> adm_edges f l"
run_of_lasso_def: fixes L :: "'a lasso" shows "run_of_lasso L \<equiv> lasso_reach L \<frown> (lasso_cycle L)\<^sup>\<omega>"
SatN': fixes b :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> (\<^bold>@ a p)) at b in' set ((ps, a) # branch)"   and "A, n \<turnstile> ((\<^bold>\<not> p) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
irreducible_def: fixes p :: "'b" shows "irreducible p = (p \<noteq> (0::'b) \<and> \<not> p dvd (1::'b) \<and> (\<forall>a b. p = a * b \<longrightarrow> a dvd (1::'b) \<or> b dvd (1::'b)))"
lucas_lehmer_add_in_carrier: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" assumes "0 < m" shows "lucas_lehmer_add m x y \<in> {..<m} \<times> {..<m}"
isNSCont_def: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" shows "isNSCont f a = (\<forall>y. y \<approx> star_of a \<longrightarrow> (*f* f) y \<approx> star_of (f a))"
obs_ids_def: fixes \<pi> :: "nat \<Rightarrow> 'n" shows "obs_ids \<pi> = {k. local.att (\<pi> k) \<noteq> None}"
coeffs_smult: fixes a :: "'b"   and p :: "'b poly" shows "coeffs (Polynomial.smult a p) = (if a = (0::'b) then [] else map ((*) a) (coeffs p))"
continuous_inv_chart: fixes x :: "'a"   and c :: "('b, 'a) chart"   and X :: "'a set" assumes "x \<in> codomain c" shows "continuous (at x within X) (inv_chart c)"
set_integral_add_1: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "set_integrable M A f"   and "set_integrable M A g" shows "set_integrable M A (\<lambda>x. f x + g x)"
alw_sdrop: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream"   and n :: "nat" assumes "alw \<phi> xs" shows "alw \<phi> (sdrop n xs)"
ladder_stepdown_length: fixes L :: "(nat \<times> nat \<times> nat) list" assumes "1 < length L" shows "length (ladder_stepdown L) = length L - 1"
individually_rational_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Y.individually_rational_on ds X = (Y.CD_on ds X = X \<and> Y.CH X = X)"
actions_ok'_iff: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('c \<times> 'a \<Rightarrow>f nat) option) \<times> 'd) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ta :: "('a, 'b, 'f, 'd, 'e, 'g) thread_action" shows "final_thread.actions_ok' s t ta = (lock_ok_las' (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> \<and> thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub> \<and> final_thread.cond_action_oks' s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> \<and> wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> interrupt_actions_ok' (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>)"
simps: fixes xs :: "'a list"   and ar :: "nat"   and n :: "nat"   and x :: "'a" shows "rewrite_fundef_body (Fundef xs ar) n x = Fundef (rewrite xs n x) ar"
sfirst_eq_eSuc: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and n :: "enat" shows "(sfirst P \<omega> = eSuc n) = (\<not> P \<omega> \<and> sfirst P (stl \<omega>) = n)"
measure_spmf_return_spmf: fixes x :: "'c" shows "measure_spmf (return_spmf x) = return (count_space UNIV) x"
ap_spec_add_pairs: fixes np :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and icrit :: "nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and ncrit :: "nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and ocrit :: "nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and comb :: "((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "punit.np_spec np"   and "punit.icrit_spec icrit"   and "punit.ncrit_spec ncrit"   and "punit.ocrit_spec ocrit"   and "\<And>xs ys. set (comb xs ys) = set xs \<union> set ys" shows "punit.ap_spec (punit.add_pairs np icrit ncrit ocrit comb)"
mset_subset_eq_mono_add: fixes A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b multiset"   and D :: "'b multiset" assumes "A \<subseteq># B"   and "C \<subseteq># D" shows "A + C \<subseteq># B + D"
while_option_sim: fixes R :: "'a option \<Rightarrow> 'b option \<Rightarrow> bool"   and s1 :: "'a"   and s2 :: "'b"   and I :: "'a \<Rightarrow> bool"   and b1 :: "'a \<Rightarrow> bool"   and b2 :: "'b \<Rightarrow> bool"   and c1 :: "'a \<Rightarrow> 'a"   and c2 :: "'b \<Rightarrow> 'b" assumes "R (Some s1) (Some s2)"   and "\<And>s1 s2. \<lbrakk>R (Some s1) (Some s2); I s1\<rbrakk> \<Longrightarrow> b1 s1 = b2 s2"   and "\<And>s1 s2. \<lbrakk>R (Some s1) (Some s2); I s1; b1 s1\<rbrakk> \<Longrightarrow> R (Some (c1 s1)) (Some (c2 s2))"   and "R None None"   and "I s1"   and "\<And>s1. \<lbrakk>I s1; b1 s1\<rbrakk> \<Longrightarrow> I (c1 s1)" shows "R (while_option b1 c1 s1) (while_option b2 c2 s2)"
bezout_matrix_not_zero: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a \<noteq> b"   and "A $h a $h j \<noteq> (0::'b)" shows "(bezout_matrix A a b j bezout ** A) $h a $h j \<noteq> (0::'b)"
simps_1: fixes A :: "'a iarray iarray"   and i :: "nat"   and j :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "bezout_iterate_iarrays A 0 i j bezout = A"
choice_axiom_def: shows "choice_axiom \<equiv> [[$\<alpha> vid1 \<union>\<union> $\<alpha> vid2]]Pc pid1 \<leftrightarrow> ([[$\<alpha> vid1]]Pc pid1 && [[$\<alpha> vid2]]Pc pid1)"
index_by_def: fixes f :: "'d \<Rightarrow> 'c"   and l :: "'d list" shows "index_by f l \<equiv> map_of (map (\<lambda>x. (f x, x)) l)"
assms: shows "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
un: fixes a :: "'b"   and b :: "'b" shows "set_of (un a b) = set_of a \<union> set_of b"
lead_coeff_prod: fixes f :: "'b \<Rightarrow> 'a poly"   and A :: "'b set" shows "Polynomial.lead_coeff (prod f A) = (\<Prod>x\<in>A. Polynomial.lead_coeff (f x))"
simps_1: fixes V :: "'a set" shows "wf\<^sub>s\<^sub>t V [] = True"
map_upt_len_conv: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "map (\<lambda>i. f (xs ! i)) [0..<length xs] = map f xs"
intro: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan G"   and "DFS_invar G (Tarjan_def.tarjan_params G) s" shows "Tarjan_invar G s"
ExcessTab_final: fixes p :: "nat \<times> nat"   and g :: "graph" assumes "p \<in> set (ExcessTable g (vertices g))" shows "finalVertex g (fst p)"
node_ptr_kinds_eq: shows "node_ptr_kinds h' = node_ptr_kinds h"
comp_ide_self: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.ide a" shows "a \<cdot>\<^sub>S a = a"