hcju/isabelleps huggingface dataset:
inv\<^sub>2E_3: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and T' :: "nat \<Rightarrow> nat"   and A' :: "nat \<Rightarrow> nat set" assumes "inv\<^sub>2 T A j"   and "lb T' A' j" shows "real (makespan T) \<le> 3 / 2 * real (makespan T')"
succ_rtran_def: fixes G :: "'b graph"   and v :: "'b" shows "FiniteGraph.succ_rtran G v = {e2. (v, e2) \<in> (edges G)\<^sup>*}"
simps_2: fixes y :: "bit" shows "(\<one> \<oplus>\<^sub>b y) = (\<not>\<^sub>b y)"
deriv_def: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "deriv f x \<equiv> SOME D. (f has_field_derivative D) (at x)"
functions_of_is_inj_on: fixes A :: "'c set"   and B :: "'d set"   and P :: "'c set set" assumes "finite A"   and "finite B"   and "partition_on A P"   and "card P \<le> card B"   and "\<forall>X\<in>P. card X = 1" shows "univ (\<lambda>f. inj_on f A) (functions_of P A B)"
lunit'_in_hom_2: fixes f :: "'c" assumes "C.ide f" shows "\<guillemotleft>\<l>\<^sub>C\<^sup>-\<^sup>1[f] : f \<Rightarrow>\<^sub>C trg\<^sub>C f \<star>\<^sub>C f\<guillemotright>"
sranI: fixes b :: "'c"   and m :: "'d \<Rightarrow> 'c set"   and a :: "'d" assumes "b \<in> m a" shows "b \<in> sran m"
assms: shows "(f has_vector_derivative f') (at x0 within S)"
a1: shows " M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2"
list_fast_rec_correct: fixes ts :: "'b tree list"   and k :: "nat"   and xs :: "'b list" assumes "length ts = 2 ^ k"   and "\<forall>i<2 ^ k. braun_list (ts ! i) (take_nths i k xs)" shows "list_fast_rec ts = xs"
A'_def: fixes ws :: "nat \<Rightarrow> real" shows "A' ws = ten2mat (A ws)"
assms: shows "eSwap MOD zs z1 z2 eX = OK Y"
assms_5: shows "\<exists>n w. b n w \<noteq> 0"
parallel_decomposition: fixes V :: "'com list"   and V' :: "'com list"   and d :: "'d" assumes "V \<approx>\<^bsub>d\<^esub> V'" shows "\<forall>i<length V. [V ! i] \<approx>\<^bsub>d\<^esub> [V' ! i]"
norm_triangle_half_r: fixes y :: "'c"   and x1 :: "'c"   and e :: "real"   and x2 :: "'c" assumes "norm (y - x1) < e / 2"   and "norm (y - x2) < e / 2" shows "norm (x1 - x2) < e"
distinct_3: fixes x31 :: "Syntax.var"   and x32 :: "Syntax.term" shows "Unit \<noteq> Lam x31 x32"
m_comm: fixes x :: "rat vec"   and y :: "rat vec" assumes "x \<in> Rn"   and "y \<in> Rn" shows "x + y = y + x"
oInv_equality': fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal"   and x :: "ordinal" assumes "normal F"   and "a = F x" shows "oInv F a = x"
f: shows "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"
dual_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual one = one"
prod_mset_dvd_prod_mset_primes_iff: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "\<And>x. x \<in># A \<Longrightarrow> prime x"   and "\<And>x. x \<in># B \<Longrightarrow> prime x" shows "(\<Prod>\<^sub># A dvd \<Prod>\<^sub># B) = (A \<subseteq># B)"
case_2: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x2 :: "('b \<Rightarrow> 'c option) \<Rightarrow> bool" shows "(case (x2)\<^sub>\<surd> of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f2 x2"
less_Def_def: fixes x :: "Def"   and y :: "Def" shows "(x < y) = (x = SimpleDef \<and> y = PhiDef)"
split_int_le_real: fixes a :: "int"   and b :: "real" shows "(real_of_int a \<le> b) = (a \<le> \<lfloor>b\<rfloor> \<or> a = \<lfloor>b\<rfloor> \<and> real_of_int \<lfloor>b\<rfloor> < b)"
smc_is_arrE: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" obtains "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" and "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" and "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a" and "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
compatible_above1: fixes x :: "'b"   and y :: "'b" assumes "compatible x y" shows "compatible x (x \<squnion> y)"
ccpo_Sup_below_iff: fixes Y :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain orda Y" shows "orda (luba Y) x = (\<forall>y\<in>Y. orda y x)"
elims: fixes x :: "sterm"   and y :: "name option" assumes "unconst x = y" obtains name :: "name" where "x = Sconst name" and "y = Some name"   | v :: "name" where "x = Svar v__" and "y = None"   | v :: "(Term_Class.term \<times> sterm) list" where "x = Sabs va__" and "y = None"   | v :: "sterm" and va :: "sterm" where "x = v $\<^sub>s va" and "y = None"
rbt_map_entry_rbt_sorted: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b"   and t :: "('a, 'b) rbt" shows "ord.rbt_sorted less (ord.rbt_map_entry less k f t) = ord.rbt_sorted less t"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<sqsubseteq> b"
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "com"   and x4 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> bool" assumes "ir_hoare x1 x2 x3 x4"   and "\<And>Q Pa c'. (\<And>t t'. Q t t' \<Longrightarrow> \<exists>s'. Pa t s' \<and> (c', s') \<Rightarrow> t') \<Longrightarrow> P Pa SKIP c' Q"   and "\<And>Pa b c\<^sub>1 c' Q c\<^sub>2. \<lbrakk>ir_hoare (\<lambda>s s'. Pa s s' \<and> bval b s) c\<^sub>1 c' Q; P (\<lambda>s s'. Pa s s' \<and> bval b s) c\<^sub>1 c' Q\<rbrakk> \<Longrightarrow> P Pa (IF b THEN c\<^sub>1 ELSE c\<^sub>2) c' Q"   and "\<And>Pa b c\<^sub>2 c' Q c\<^sub>1. \<lbrakk>ir_hoare (\<lambda>s s'. Pa s s' \<and> \<not> bval b s) c\<^sub>2 c' Q; P (\<lambda>s s'. Pa s s' \<and> \<not> bval b s) c\<^sub>2 c' Q\<rbrakk> \<Longrightarrow> P Pa (IF b THEN c\<^sub>1 ELSE c\<^sub>2) c' Q"   and "\<And>Pa c c' Q d R. \<lbrakk>ir_hoare Pa c c' Q; P Pa c c' Q; ir_hoare Q d SKIP R; P Q d SKIP R\<rbrakk> \<Longrightarrow> P Pa (c;; d) c' R"   and "\<And>Pa x e c' Q. \<lbrakk>ir_hoare (\<lambda>t t'. \<exists>v. Pa (t(x := v)) t' \<and> t x = aval e (t(x := v))) SKIP c' Q; P (\<lambda>t t'. \<exists>v. Pa (t(x := v)) t' \<and> t x = aval e (t(x := v))) SKIP c' Q\<rbrakk> \<Longrightarrow> P Pa (x ::= e) c' Q"   and "\<And>Pa b c' Q c. \<lbrakk>ir_hoare (\<lambda>s s'. Pa s s' \<and> \<not> bval b s) SKIP c' Q; P (\<lambda>s s'. Pa s s' \<and> \<not> bval b s) SKIP c' Q\<rbrakk> \<Longrightarrow> P Pa (WHILE b DO c) c' Q"   and "\<And>Pa b c c' Q. \<lbrakk>ir_hoare (\<lambda>s s'. Pa s s' \<and> bval b s) (c;; WHILE b DO c) c' Q; P (\<lambda>s s'. Pa s s' \<and> bval b s) (c;; WHILE b DO c) c' Q\<rbrakk> \<Longrightarrow> P Pa (WHILE b DO c) c' Q"   and "\<And>Pa b c c' Q. \<lbrakk>\<And>n. ir_hoare (\<lambda>s s'. Pa n s s' \<and> bval b s) c SKIP (Pa (Suc n)); \<And>n. P (\<lambda>s s'. Pa n s s' \<and> bval b s) c SKIP (Pa (Suc n)); ir_hoare (\<lambda>s s'. \<exists>n. Pa n s s') (WHILE b DO c) c' Q; P (\<lambda>s s'. \<exists>n. Pa n s s') (WHILE b DO c) c' Q\<rbrakk> \<Longrightarrow> P (Pa 0) (WHILE b DO c) c' Q"   and "\<And>Pa c c' Q P' Q'. \<lbrakk>ir_hoare Pa c c' Q; P Pa c c' Q; \<And>s s'. Pa s s' \<Longrightarrow> P' s s'; \<And>s s'. Q' s s' \<Longrightarrow> Q s s'\<rbrakk> \<Longrightarrow> P P' c c' Q'"   and "\<And>P\<^sub>1 c c' Q\<^sub>1 P\<^sub>2 Q\<^sub>2. \<lbrakk>ir_hoare P\<^sub>1 c c' Q\<^sub>1; P P\<^sub>1 c c' Q\<^sub>1; ir_hoare P\<^sub>2 c c' Q\<^sub>2; P P\<^sub>2 c c' Q\<^sub>2\<rbrakk> \<Longrightarrow> P (\<lambda>s s'. P\<^sub>1 s s' \<or> P\<^sub>2 s s') c c' (\<lambda>t t'. Q\<^sub>1 t t' \<or> Q\<^sub>2 t t')"   and "\<And>Pa c c' Q. \<lbrakk>ir_hoare (flip Pa) c c' (flip Q); P (flip Pa) c c' (flip Q)\<rbrakk> \<Longrightarrow> P Pa c' c Q" shows "P x1 x2 x3 x4"
\<P>_wellformed: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat" assumes "p \<in> \<P> k u" shows "wellformed_tokens p"
Maclaurin_sin_expansion3: fixes n :: "nat"   and x :: "real" assumes "0 < n"   and "0 < x" shows "\<exists>t>0. t < x \<and> sin x = (\<Sum>m<n. sin_coeff m * x ^ m) + sin (t + 1 / 2 * real n * pi) / fact n * x ^ n"
ring_hom_def: fixes R :: "('g, 'i) ring_scheme"   and S :: "('h, 'j) ring_scheme" shows "ring_hom R S = {h \<in> carrier R \<rightarrow> carrier S. (\<forall>x y. x \<in> carrier R \<and> y \<in> carrier R \<longrightarrow> h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y \<and> h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y) \<and> h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>}"
suntil_implies_until: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and \<psi> :: "'b stream \<Rightarrow> bool"   and \<omega> :: "'b stream" assumes "(\<phi> suntil \<psi>) \<omega>" shows "(\<phi> until \<psi>) \<omega>"
queue_invar_simps_1: shows "BinomialHeapStruc.queue_invar []"
type_arrow_elim: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T1 :: "Syntax.type"   and T2 :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T1 \<rightarrow> T2" obtains x :: "nat" where "t = `x" and "\<Gamma> x = T1 \<rightarrow> T2"   | ta :: "trm" and T1a :: "Syntax.type" and s :: "trm" where "t = ta__ \<degree> s" and "\<Gamma> , \<Delta> \<turnstile>\<^sub>T ta__ : T1a \<rightarrow> T1 \<rightarrow> T2" and "\<Gamma> , \<Delta> \<turnstile>\<^sub>T s : T1a"   | ta :: "trm" where "t = \<lambda>T1:ta" and "\<Gamma>\<langle>0:T1\<rangle> , \<Delta> \<turnstile>\<^sub>T ta : T2"   | c :: "cmd" where "t = \<mu>T1 \<rightarrow> T2:c" and "\<Gamma> , \<Delta>\<langle>0:T1 \<rightarrow> T2\<rangle> \<turnstile>\<^sub>C c"
nn_integral_finite_imp_abs_sumable_on': fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space A < \<infinity>"   and "\<And>x. 0 \<le> f x" shows "f abs_summable_on A"
induced_function_integral_restr_nonergodic_1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets MA"   and "integrable MA f" shows "integrable (restrict_space MA A) (A.induced_function A f)"
split_beta_bezout: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'b"   and b :: "'b" assumes "is_bezout_ext bezout" shows "bezout a b = (fst (bezout a b), fst (snd (bezout a b)), fst (snd (snd (bezout a b))), fst (snd (snd (snd (bezout a b)))), snd (snd (snd (snd (bezout a b)))))"
rref_upt_condition4_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and i :: "'d"   and j :: "'d" assumes "reduced_row_echelon_form_upt_k A k"   and "\<not> is_zero_row_upt_k i k A"   and "i \<noteq> j" shows "A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'b)) = (0::'b)"
f_2: shows "\<integral>\<^sup>+ x. ennreal (case x of (None, None) \<Rightarrow> pmf q None | (None, Some y') \<Rightarrow> spmf q y' - spmf p y' | (Some x', None) \<Rightarrow> 0 | (Some x', Some y') \<Rightarrow> if x' = y' then spmf p x' else 0) \<partial>count_space UNIV = 1"
fresh_ineq_at_base: fixes a :: "atom"   and b :: "'b" assumes "a \<noteq> atom b" shows "a \<sharp> b"
is_pot_ordI: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order" assumes "\<And>u v. snd u < snd v \<Longrightarrow> lt_of_nat_term_order to u v" shows "is_pot_ord_pprod to"
valids_def: fixes D :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set" shows "|\<Turnstile> D \<equiv> \<forall>(P, c, Q)\<in>D. \<Turnstile> {P} c {Q}"
abs_conv_abscissa_diff_le: fixes f :: "'b fds"   and g :: "'b fds" shows "abs_conv_abscissa (f - g) \<le> max (abs_conv_abscissa f) (abs_conv_abscissa g)"
ap_sum_comp: fixes h :: "('h \<Rightarrow> 'e) + 'f"   and g :: "('g \<Rightarrow> 'h) + 'f"   and x :: "'g + 'f" shows "ap_plus (ap_plus (ap_plus (pure (\<circ>)) h) g) x = ap_plus h (ap_plus g x)"
leD: fixes y :: "'c word"   and x :: "'c word" assumes "y \<le>s x" shows "\<not> x <s y"
int_of_rat_def: fixes x :: "rat" shows "int_of_rat x \<equiv> fst (quotient_of x)"
assms_1: shows "f = \<lbrace>t\<rbrace>"
locally_similar_trans: fixes \<rho> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g" assumes "locally_similar \<rho> \<sigma>"   and "locally_similar \<sigma> \<tau>" shows "locally_similar \<rho> \<tau>"
disc_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "\<not> lnull (LCons x21 x22)"
While_sound: fixes pre :: "'b set"   and rely :: "('b \<times> 'b) set"   and b :: "'b set"   and post :: "'b set"   and P :: "'b com"   and guar :: "('b \<times> 'b) set" assumes "stable pre rely"   and "pre \<inter> - b \<subseteq> post"   and "stable post rely"   and "\<Turnstile> P sat [pre \<inter> b, rely, guar, pre]"   and "\<forall>s. (s, s) \<in> guar" shows "\<Turnstile> While b P sat [pre, rely, guar, post]"
vimage_UNIV: fixes f :: "'b \<Rightarrow> 'c" shows "f -` UNIV = UNIV"
card_cartesian_product: fixes A :: "'d set"   and B :: "'e set" shows "card (A \<times> B) = card A * card B"
trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "suffix b a"   and "suffix c b" shows "suffix c a"
cos_int_2pin: fixes n :: "int" shows "cos (2 * pi * real_of_int n) = 1"
all_if_distrib: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<forall>x. if x = a then P x else Q x) = (P a \<and> (\<forall>x. x \<noteq> a \<longrightarrow> Q x))"
unx_some_lemmas_5: fixes u :: "'a"   and z :: "name" assumes "unfree u = Some z" shows "unconst u = None"
assms: shows "\<langle>a, f a\<rangle> \<in>\<^sub>\<circ> VLambda A f"
not_finals_ConsI2: fixes e :: "'a exp"   and v :: "val"   and es :: "'a exp list" assumes "e = Val v"   and "\<not> finals es" shows "\<not> finals (e # es)"
partial_order_on_acyclic: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "partial_order_on A r" shows "acyclic (r - Id)"
dghm_ObjMap_app_in_HomCod_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" shows "\<phi> i\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>"
domain_inter_vimage: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "f \<in> A \<rightarrow> B" shows "A \<inter> f -` B = A"
Sigma_fm_All2: fixes A :: "fm"   and i :: "name"   and t :: "tm" assumes "Sigma_fm A"   and "atom i \<sharp> t" shows "Sigma_fm (All2 i t A)"
if\<^sub>S\<^sub>E_False: fixes c :: "'a \<Rightarrow> ('b \<times> 'a) option"   and d :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "_if_SE (\<lambda>x. False) c d = d"
fundfoldpair_unique_half_chamber_systems_chamber_ng_f: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a"   and C :: "'a set" assumes "(f, g) \<in> fundfoldpairs"   and "(f', g') \<in> fundfoldpairs"   and "Abs_induced_automorph f' g' = Abs_induced_automorph f g"   and "chamber C"   and "C \<notin> g \<turnstile> \<C>" shows "C \<in> f' \<turnstile> \<C>"
space_P: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "space (P J) = space (Pi\<^sub>M J M)"
v'_def: shows "v' = \<lparr>ext = Abs_real_int (left (ext v1), right (ext v3)), lan = lan v, own = own v\<rparr>"
intros_2: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b"   and ys :: "'b list"   and xs :: "'b list" assumes "gt y x" shows "lexext gt (y # ys) (x # xs)"
post_Inf_acom: fixes M :: "'a set acom set"   and c :: "com" assumes "\<forall>C\<in>M. strip C = c" shows "post (Inf_acom c M) = \<Inter> (post ` M)"
mgenerator2Tr3_2: fixes n :: "nat"   and j :: "nat"   and k :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "int" assumes "0 < n"   and "j \<le> n"   and "k \<le> n"   and "j \<noteq> k"   and "distinct_pds K n P" shows "(\<nu>\<^bsub>K P j\<^esub>) (mprod_exp K (K_gamma k) (Kb\<^bsub>K n P\<^esub>) n\<^bsub>K\<^esub>\<^bsup>m\<^esup>) = ant m"
supt_supteq_conv: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" shows "s \<rhd> t = (s \<unrhd> t \<and> s \<noteq> t)"
Casts_conf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and Ts' :: "ty list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option" assumes "wf_C_prog P"   and "P \<turnstile> Ts Casts vs to vs' "   and "P,E \<turnstile> es [::] Ts'"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h, l)\<rangle>"   and "P,E \<turnstile> s \<surd>"   and "P \<turnstile> Ts' [\<le>] Ts" shows "\<forall>i<length Ts. P,h \<turnstile> vs' ! i :\<le> Ts ! i"
Sup_bot_conv_2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(bot = Sup A) = (\<forall>x\<in>A. x = bot)"
is_minimal_basis_trd_is_minimal_basis: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_minimal_basis (set (x # xs))"   and "x \<notin> set xs" shows "is_minimal_basis (set (trd xs x # xs))"
antisym: fixes x :: "'a \<Rightarrow>\<^sub>0 nat"   and y :: "'a \<Rightarrow>\<^sub>0 nat" assumes "x \<preceq> y"   and "y \<preceq> x" shows "x = y"
is_extensional: fixes f :: "'b" assumes "\<not> C'.arr f" shows "C'.map f = C'.null"
list_slice2_nth_nth_eq2: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat"   and n :: "nat" assumes "m = length xs div k"   and "n < length xs mod k" shows "list_slice2 xs k ! m ! n = xs ! (m * k + n)"
image_group_def: fixes f :: "'e \<Rightarrow> 'd"   and G :: "('e, 'f) monoid_scheme" shows "image_group f G \<equiv> \<lparr>carrier = f ` carrier G, mult = \<lambda>a b. f (inv_into (carrier G) f a \<otimes>\<^bsub>G\<^esub> inv_into (carrier G) f b), one = f \<one>\<^bsub>G\<^esub>\<rparr>"
inv: fixes a :: "'b \<times> 'a" assumes "DopxC.ide a" shows "S.inverse_arrows (\<Phi> a) (\<Psi> a)"
sup2E: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c" assumes "sup A B x y" obtains "A x y"   | "B x y"
IF2rel_def: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "IF2rel R = (BNF_Def.Grp (IF2in {(x, y). R x y}) (IF2map fst))\<inverse>\<inverse> OO BNF_Def.Grp (IF2in {(x, y). R x y}) (IF2map snd)"
split_option_bind: fixes P :: "'a option \<Rightarrow> bool"   and res :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "P (option_bind res f) = ((res = None \<longrightarrow> P None) \<and> (\<forall>s. res = Some s \<longrightarrow> P (f s)))"
PolysI: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "keys p \<subseteq> .[X]" shows "p \<in> P[X]"
widens_def: fixes G :: "prog"   and Ts :: "ty list"   and Ts' :: "ty list" shows "G\<turnstile>Ts[\<preceq>]Ts' = list_all2 (widen G) Ts Ts'"
to_tree_order_child2: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r nodes"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<noteq> ptr"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child \<in> set nodes" obtains parent :: "(_) object_ptr" where "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r Some parent" and "parent \<in> set nodes"
eq_set_inc: fixes a :: "'h"   and A :: "'h set"   and B :: "'h set" assumes "a \<in> A"   and "A = B" shows "a \<in> B"
wadjust_correctness: fixes m :: "nat"   and rs :: "nat"   and ln :: "nat"   and rn :: "nat" shows "let P = \<lambda>(len, st, l, r). st = 0; Q = \<lambda>(len, st, l, r). wadjust_inv st m rs (l, r); f = \<lambda>stp. (Suc (Suc rs), steps0 (Suc 0, Bk # Oc \<up> Suc m, Bk # Oc # Bk \<up> ln @ Bk # Oc \<up> Suc rs @ Bk \<up> rn) t_wcode_adjust stp) in \<exists>n. P (f n) \<and> Q (f n)"
Limsup_obtain': fixes c :: "'c"   and F :: "'d filter"   and u :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool" assumes "c < Limsup F u"   and "eventually P F" shows "\<exists>n. P n \<and> c < u n"
phl_seq: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "p \<cdot> x \<le> x \<cdot> r"   and "r \<cdot> y \<le> y \<cdot> q" shows "p \<cdot> (x \<cdot> y) \<le> x \<cdot> y \<cdot> q"
det_linear_rows_finsum_lemma: fixes S :: "nat set"   and T :: "nat set"   and c :: "nat \<Rightarrow> 'b vec"   and n :: "nat"   and a :: "nat \<Rightarrow> nat \<Rightarrow> 'b vec" assumes "finite S"   and "finite T"   and "c \<in> {0..<n} \<rightarrow> carrier_vec n"   and "T \<subseteq> {0..<n}"   and "a \<in> T \<rightarrow> S \<rightarrow> carrier_vec n" shows "det (mat\<^sub>r n n (\<lambda>i. if i \<in> T then finsum_vec TYPE('b) n (a i) S else c i)) = (\<Sum>f | (\<forall>i\<in>T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i). det (mat\<^sub>r n n (\<lambda>i. if i \<in> T then a i (f i) else c i)))"
continuous_map_subset_aux2: fixes X :: "'b topology"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'c topology" assumes "\<forall>S\<subseteq>topspace X. f ` (X closure_of S) \<subseteq> Y closure_of f ` S" shows "continuous_map X Y f"
rev_exhaust: fixes xs :: "'a list" obtains "xs = []"   | ys :: "'a list" and y :: "'a" where "xs = append ys [y]"
sign_simps_11: fixes ab :: "'b"   and bb :: "'b"   and cb :: "'b" shows "ab - (bb - cb) = ab + cb - bb"
fmran'I: fixes m :: "'b \<rightharpoonup>\<^sub>f 'a"   and x :: "'b"   and y :: "'a" assumes "fmlookup m x = Some y" shows "y \<in> fmran' m"
subprob_space_axioms: fixes M :: "'b pmf" shows "subprob_space (measure_pmf M)"