hcju/isabelleps huggingface dataset:
the_gpv_parametric': fixes A :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and C :: "'j \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'n \<Rightarrow> bool" shows "(rel_gpv'' A C R ===> rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R))) the_gpv the_gpv"
suffix_lists: fixes xs :: "'b list"   and ys :: "'b list"   and A :: "'b set" assumes "xs \<le>s ys"   and "ys \<in> lists A" shows "xs \<in> lists A"
ide_def: fixes a :: "'t" shows "S'.ide a \<equiv> a \<cdot>\<acute> a \<noteq> S'.null \<and> (\<forall>f. (f \<cdot>\<acute> a \<noteq> S'.null \<longrightarrow> f \<cdot>\<acute> a = f) \<and> (a \<cdot>\<acute> f \<noteq> S'.null \<longrightarrow> a \<cdot>\<acute> f = f))"
freshImEnvAt_def: fixes xs :: "'varSort"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and ys :: "'varSort"   and y :: "'var" shows "freshImEnvAt xs x rho ys y \<equiv> rho ys y = None \<and> (ys \<noteq> xs \<or> y \<noteq> x) \<or> (\<exists>Y. rho ys y = Some Y \<and> fresh xs x Y)"
det_one: fixes n :: "nat" shows "det (1\<^sub>m n) = (1::'a)"
restrict_indets_def: shows "restrict_indets = poly_subst (\<lambda>x. monomial (1::'c) (restrict_indets_subst x))"
assms_3: shows "op \<notin> set (\<pi> ! k)"
isomorphism_trans: fixes G1 :: "('a, 'c) monoid_scheme"   and G2 :: "('b, 'd) monoid_scheme"   and r2 :: "'b \<Rightarrow> 'b"   and r1 :: "'a \<Rightarrow> 'a"   and G3 :: "('e, 'f) monoid_scheme"   and r3 :: "'e \<Rightarrow> 'e" assumes "\<exists>f\<in>Group.iso G1 G2. \<forall>x\<in>carrier G1. r2 (f x) = f (r1 x)"   and "\<exists>f\<in>Group.iso G2 G3. \<forall>x\<in>carrier G2. r3 (f x) = f (r2 x)" shows "\<exists>f\<in>Group.iso G1 G3. \<forall>x\<in>carrier G1. r3 (f x) = f (r1 x)"
big_d_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "big_d a b y i = map (\<lambda>j. dij a b i j - 1) (filter (\<lambda>j. eij a b i j \<le> y ! j) [0..<length y])"
sqrt_le_iff: fixes y :: "nat"   and x :: "nat" shows "(local.sqrt y \<le> x) = (\<forall>z. z\<^sup>2 \<le> y \<longrightarrow> z \<le> x)"
is_lub_below_iff: fixes S :: "'a set"   and x :: "'a"   and u :: "'a" assumes "S <<| x" shows "(x \<sqsubseteq> u) = S <| u"
simps_1: fixes uu :: "'a"   and x :: "'a" shows "the_default uu (Some x) = x"
assoc_is_natural_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a> \<mu> \<nu> \<tau> = V (\<a> (partial_magma.cod V \<mu>) (partial_magma.cod V \<nu>) (partial_magma.cod V \<tau>)) (H (H \<mu> \<nu>) \<tau>)"
simps: fixes a1 :: "('var \<Rightarrow> 'val) \<times> nat"   and a2 :: "('var \<Rightarrow> 'val) \<times> nat" shows "(a1 \<cc> a2) = ((\<exists>\<pi>' k' \<pi> k \<sigma> \<sigma>'. a1 = (\<sigma>', k') \<and> a2 = (\<sigma>, k) \<and> cs\<^bsup>\<pi>'\<^esup> k' \<prec> cs\<^bsup>\<pi>\<^esup> k \<and> \<pi> = path \<sigma> \<and> \<pi>' = path \<sigma>' \<and> \<pi> (Suc (\<pi> \<exclamdown> cs\<^bsup>\<pi>'\<^esup> k')) \<noteq> \<pi>' (Suc k')) \<or> (\<exists>\<pi>' k' \<pi> k \<sigma> \<sigma>'. a1 = (\<sigma>', k') \<and> a2 = (\<sigma>, k) \<and> cs\<^bsup>\<pi>'\<^esup> k' = cs\<^bsup>\<pi>\<^esup> k \<and> \<pi> = path \<sigma> \<and> \<pi>' = path \<sigma>' \<and> \<sigma>\<^bsup>k\<^esup> \<restriction> local.reads (\<pi> k) \<noteq> \<sigma>'\<^bsup>k'\<^esup> \<restriction> local.reads (\<pi> k)))"
preserves_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "src\<^sub>D (H \<mu>) = H.map\<^sub>0 (src\<^sub>C \<mu>)"
SN: fixes S :: "('a \<times> 'a) set" assumes "SN_ars S" shows "SN S"
ir_If_True_True: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp"   and c\<^sub>1 :: "com"   and c\<^sub>1' :: "com"   and Q\<^sub>1 :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>2 :: "com"   and c\<^sub>2' :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> bval b s \<and> bval b' s') c\<^sub>1 c\<^sub>1' Q\<^sub>1" shows "ir_hoare P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (IF b' THEN c\<^sub>1' ELSE c\<^sub>2') Q\<^sub>1"
natfun_bigoE: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and n0 :: "nat" assumes "f \<in> O(g)"   and "\<And>n. n0 \<le> n \<Longrightarrow> g n \<noteq> (0::'b)" obtains c :: "real" where "0 < c" and "\<And>n. n0 \<le> n \<Longrightarrow> norm (f n) \<le> c * norm (g n)"
C: fixes x :: "nat \<Rightarrow> val" assumes "x \<in> space (state_measure V \<Gamma>)" shows "\<integral>\<^sup>+ y. f (merge V V' (x, \<rho>)) y * indicator X (case_nat y (merge V V' (x, \<rho>))) \<partial>stock_measure t = \<integral>\<^sup>+ y. f (merge V V' (x, \<rho>)) y * indicator X (merge (shift_var_set V) (Suc ` V') (case_nat y x, case_nat undefined \<rho>)) \<partial>stock_measure t"
le_Ord_diff: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<subseteq>\<^sub>\<circ> \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>" obtains \<gamma> :: "V" where "\<alpha> + \<gamma> = \<beta>" and "\<gamma> \<subseteq>\<^sub>\<circ> \<beta>" and "Ord \<gamma>"
spmf_rel_eq: shows "rel_spmf (=) = (=)"
le_inf_iff: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "less_eq x (inf y z) = (less_eq x y \<and> less_eq x z)"
simps_1: fixes x :: "udom"   and f :: "udom \<rightarrow> udom\<cdot>'e resT" shows "bindU\<cdot>(Done\<cdot>x)\<cdot>f = f\<cdot>x"
l3_obs_iagreement_Init: shows "oreach l3 \<subseteq> l2_iagreement_Init"
estep_Cn_def: fixes e :: "nat" shows "estep_Cn e \<equiv> if e2lenls e = e_length (pdec222 (e2i e)) then if e2rv e = 0 then prod_encode (e_cons (triple_encode (pdec122 (e2i e)) (e2ls e) 0) (e2stack e), 0) else prod_encode (e2tail e, e2rv e) else if e2rv e = 0 then if e2lenls e < e_length (pdec222 (e2i e)) then prod_encode (e_cons (triple_encode (e_nth (pdec222 (e2i e)) (e2lenls e)) (e2xs e) 0) (e2stack e), 0) else prod_encode (e2tail e, e2rv e) else prod_encode (e_cons (triple_encode (e2i e) (e2xs e) (e_snoc (e2ls e) (e2rv e - 1))) (e2tail e), 0)"
length_maskedDB_zero: fixes emBits :: "nat"   and maskedDB :: "bit list" assumes "roundup emBits 8 * 8 - emBits \<le> length maskedDB" shows "length (maskedDB_zero maskedDB emBits) = length maskedDB"
collect_pseudominimals_below_in_less_eq_top: fixes P :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'c set"   and A :: "'c set"   and a :: "'a set" assumes "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater (P .\<^bold>\<le> x) f"   and "f ` (P .\<^bold>\<le> x) = Pow A"   and "order.greater_eq {y. pseudominimal_in (P .\<^bold>\<le> x) y} a" shows "x \<succeq> the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a))"
Ccompt: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and X :: "'g"   and Y :: "'g"   and g :: "'h"   and Z :: "'g" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y"   and "g maps\<^bsub>C\<^esub> Y to Z" shows "f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> X to Z"
collapse_7: fixes expr :: "'a expr" assumes "is_Call expr" shows "Call (source expr) (expr.kind expr) (un_Call3 expr) = expr"
Basis_nonneg: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'c" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "less_eq zero i"
roundup_help1_new: fixes l :: "nat" assumes "0 < l" shows "roundup l 160 - 1 + 1 = roundup l 160"
assms: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> B"
new_Marker_in_set_implies_nonregular_occurence: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and i :: "nat"   and p :: "'a"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "Marker \<notin> set (msgs c i)"   and "Marker \<in> set (msgs c' i)"   and "channel i = Some (p, q)" shows "\<not> regular_event ev"
waiting_code_3: fixes w :: "'a" shows "waiting \<lfloor>InWS w\<rfloor> = True"
odd_prime_gt_2_int: fixes p :: "int" assumes "odd p"   and "prime p" shows "2 < p"
Max_wrt_among_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" shows "Max_wrt_among R A = {x \<in> A. R x x \<and> (\<forall>y\<in>A. R x y \<longrightarrow> R y x)}"
ChiStar_def: fixes A :: "('d, 'e, 'f) hierauto" shows "ChiStar A \<equiv> (ChiRel A)\<^sup>*"
i_set_ind_subset_i_set0_ind: shows "i_set_ind \<subseteq> i_set0_ind"
Max_mapping_def: fixes Ps :: "('b \<Rightarrow> 'a option) set"   and x :: "'b" shows "Max_mapping Ps x = (if \<forall>P\<in>Ps. P x \<noteq> None then Some (MAX P\<in>Ps. the (P x)) else None)"
iT_Div_iAll_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<oslash> k. P t) = (\<box> t I. P (t div k))"
Axe: fixes A :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Delta> :: "'b formula list" assumes "A \<in> set \<Gamma>"   and "A \<in> set \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
obs: shows "L - Tmax \<le> exp (- a * t) * (L - T) \<and> exp (- a * t) * (L - T) \<le> L - T"
simps_7: shows "normalized_nnf_match (MatchNot MatchAny) = False"
card_of_Times_mono1: fixes A :: "'a set"   and B :: "'b set"   and C :: "'c set" assumes "|A| \<le>o |B|" shows "|A \<times> C| \<le>o |B \<times> C|"
nextElem_cases: fixes xs :: "'a list"   and d :: "'a"   and x :: "'a"   and y :: "'a" assumes "nextElem xs d x = y" shows "x \<notin> set xs \<and> y = d \<or> xs \<noteq> [] \<and> x = last xs \<and> y = d \<and> x \<notin> set (butlast xs) \<or> (\<exists>us vs. xs = us @ [x, y] @ vs \<and> x \<notin> set us)"
le: shows "radj f (f x \<rightarrow> y) \<le> x \<rightarrow> radj f y"
take_Suc_Cons: fixes n :: "nat"   and x :: "'k"   and xs :: "'k list" shows "take (Suc n) (x # xs) = x # take n xs"
Mp_factorization_m: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset" shows "factorization_m (Mp f) cfs = factorization_m f cfs"
induct: fixes P :: "'b triple \<Rightarrow> bool"   and triple :: "'b triple" assumes "\<And>x1a x2a x3a. P ({x1a} x2a\<succ> {x3a})" shows "P triple"
nat_pow_comm: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier \<G>" shows "x [^] n \<otimes> x [^] m = x [^] m \<otimes> x [^] n"
drop_exp_base: fixes n :: "nat"   and x :: "'a list"   and m :: "nat" shows "drop n (UP.exp_base x m) = UP.exp_base x (m - n)"
exphr_hypreal_of_real_exp_eq: fixes x :: "real" shows "exphr x = hypreal_of_real (exp x)"
sylow_thm: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" assumes "sylow G p a m" shows "\<exists>H. subgroup H G \<and> card H = p ^ a"
assms_2: shows "A \<in>\<^sub>\<circ> Vset i"
analz_Un_EncSet2: fixes H :: "msg set"   and K :: "msg set"   and G :: "msg set" assumes "Enc_keys_clean H"   and "K \<subseteq> range LtK"   and "K \<inter> synth (analz H) = {}" shows "analz (EncSet G K \<union> H) \<subseteq> EncSet G K \<union> analz H"
padic_one_id: fixes p :: "int"   and x :: "nat \<Rightarrow> int" assumes "prime p"   and "x \<in> carrier (padic_int p)" shows "\<one>\<^bsub>padic_int p\<^esub> \<otimes>\<^bsub>padic_int p\<^esub> x = x"
subspace_image: fixes S :: "('a, 'b) vec set"   and A :: "(('a, 'b) vec, 'c) vec" assumes "vec.subspace S" shows "vec.subspace ((*v) A ` S)"
hcomp_simps_4: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "hseq \<nu> \<mu>" shows "cod (\<nu> \<star> \<mu>) = cod \<nu> \<star> cod \<mu>"
obj_is_self_adjoint_1: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.adjoint_pair a a"
ssubst_Pair_rhs: fixes r :: "'e"   and s :: "'f"   and R :: "('e \<times> 'f) set"   and s' :: "'f" assumes "(r, s) \<in> R"   and "s' = s" shows "(r, s') \<in> R"
finite_UNIV_int_def: shows "finite_UNIV = Phantom(int) False"
next_plane0_finalVertex_facesAt_eq: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and v :: "nat" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "Invariants.inv g"   and "v \<in> \<V> g"   and "finalVertex g v" shows "set (facesAt g' v) = set (facesAt g v)"
f_expand_snoc: fixes k :: "nat"   and xs :: "'d message_af list"   and x :: "'d message_af" assumes "0 < k" shows "(xs @ [x]) \<odot> k = xs \<odot> k @ x # NoMsg\<^bsup>k - Suc 0\<^esup>"
assms_1: shows "\<parallel>id\<parallel>\<^bsub>t n\<^esub>"
that: shows "\<I> \<turnstile>g gpv \<surd>"
paths2: shows "\<exists>k\<ge>1. \<forall>as nx. n'' -as\<rightarrow>* nx \<and> k \<le> length as \<longrightarrow> n' \<in> set (sourcenodes as)"
inverse_function_theorem_scaled: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a" assumes "open U"   and "\<And>x. x \<in> U \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"   and "continuous_on U f'"   and "(0::'a) \<in> U"   and "f (0::'a) = (0::'a)"   and "f' (0::'a) = id_blinfun" obtains U' :: "'a set" and V :: "'a set" and g :: "'a \<Rightarrow> 'a" and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where "open U'" and "U' \<subseteq> U" and "(0::'a) \<in> U'" and "open V" and "(0::'a) \<in> V" and "homeomorphism U' V f g" and "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative g' y) (at y)" and "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f' (g y)))" and "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f' (g y)))"
induct: fixes P :: "('c, 'd) Node list \<Rightarrow> bool"   and a0 :: "('c, 'd) Node list" assumes "\<And>a b. P [a, b]"   and "\<And>a b c. P [a, b, c]"   and "\<And>a b c d. P [a, b, c, d]"   and "\<And>a b c v vb vc. P (v # vb # vc) \<Longrightarrow> P (a # b # c # v # vb # vc)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
G2_def: shows "G2 = g ` {a..dist x y}"
space_subgroup_props_5: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and n :: "nat"   and E :: "'a set"   and k :: "'a"   and v :: "'a" assumes "ring R"   and "subfield K R"   and "ring.dimension R n K E"   and "k \<in> K"   and "v \<in> E" shows "k \<otimes>\<^bsub>R\<^esub> v \<in> E"
conv_self_conjugate: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<smile> \<cdot> y = 0) = (x \<cdot> y\<^sup>\<smile> = 0)"
prod_tuple: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "span f g"   and "seq h f"   and "seq k g" shows "(h \<otimes> k) \<cdot> \<langle>f, g\<rangle> = \<langle>h \<cdot> f, k \<cdot> g\<rangle>"
inj_on_diff_nat2: fixes n :: "nat" shows "inj_on ((-) n) {..n}"
Max_in: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A \<in> A"
UINF_as_Sup_image: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> P \<in> A \<bullet> f P) = \<Sqinter>(f ` A)"
normalize_non_0_smult: fixes p :: "'b poly" shows "\<exists>a. a \<noteq> (0::'b) \<and> smult a (primitive_part p) = p"
sinh_real_neg_iff: fixes x :: "real" shows "(sinh x < 0) = (x < 0)"
add_right_imp_eq: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<oplus>\<^sub>a a = c \<oplus>\<^sub>a a" shows "b = c"
Sup_dual_def_var: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a"   and X :: "'a set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "dual (Sup X) = Inf (dual ` X)"
redT_invariant: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "'t \<Rightarrow> 'i option" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "ts_inv P I (thr s) (shr s)" shows "ts_inv P (upd_invs I P \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>) (thr s') (shr s')"
split_slices: fixes w :: "'d word"   and u :: "'b word"   and v :: "'c word" assumes "word_split w = (u, v)" shows "u = slice (size v) w \<and> v = slice 0 w"
bounded_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<le> inf b c) = (a \<le> b \<and> a \<le> c)"
inter_dif_help_lemma: fixes A :: "'b set"   and B :: "'b set"   and S :: "'b set" assumes "A \<inter> B = {}" shows "B - S = B - (S - A)"
nrename_fresh: fixes x :: "name"   and M :: "trm"   and y :: "name" assumes "x \<sharp> M" shows "M[x\<turnstile>n>y] = M"
tendsto_in\<^sub>N_def: fixes N :: "'c quasinorm"   and u :: "nat \<Rightarrow> 'c"   and x :: "'c" shows "tendsto_in\<^sub>N N u x = (\<lambda>n. Norm N (u n - x)) \<longlonglongrightarrow> 0"
A_invariant: fixes a :: "'a"   and b :: "'a" assumes "PA a"   and "A a b" shows "PA b"
rightUniqueRestrictedGraph: fixes f :: "'c \<Rightarrow> 'd"   and P :: "'c \<Rightarrow> bool" shows "runiq {(x, f x) |x. P x}"
monoH: shows "mono (H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
redT_updLs_def: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and las :: "'a lock_actions" shows "redT_updLs ls t las \<equiv> (\<lambda>(l, la). upd_locks l t la) \<circ>$ ($ls, las$)"
QCongANullAcute_def: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "QCongANullAcute a \<equiv> QCongAACute a \<and> (\<forall>A B C. a A B C \<longrightarrow> B Out A C)"
branch_inv_Leaf: fixes b :: "bool list" shows "branch b Leaf = (b = [])"
image_def: fixes f :: "'i \<Rightarrow> 'h"   and A :: "'i set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
simps_2: fixes h :: "'b heap" shows "merge_pairs [h] = h"
Least_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool"   and k :: "'e" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
delete_block_sub_sys: fixes b :: "'a set" shows "sub_incidence_system \<V> (del_block b) \<V> \<B>"
ws_cdecl_Throwable: shows "ws_cdecl tprg (SXcpt Throwable) Object"
Collect_case_prod: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'h \<Rightarrow> bool" shows "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q"
intros_21: fixes t1 :: "t"   and v :: "v"   and t2 :: "t" assumes "types_agree t1 v" shows "\<lparr>[$C v, $Cvtop t2 Reinterpret t1 None]\<rparr> \<leadsto> \<lparr>[$C wasm_deserialise (bits v) t2]\<rparr>"
integrable_cnj: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> complex" assumes "complex_integrable M f" shows "complex_integrable M (\<lambda>x. cnj (f x))"
aux0_def: shows "aux0 = concat (map (\<lambda>(t, rel). if enat (\<tau> \<sigma> ne - t) \<le> right I then [(t, mrtabulate mrs (\<lambda>mr. r\<delta> id rel rels mr \<union> (if t = \<tau> \<sigma> ne then r\<epsilon>_strict n rels mr else {})))] else []) aux)"
FAssThrow2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>2\<rangle>"
WT_elim_cases_5: 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 E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and e :: "expr"   and T :: "ty" assumes "P,E \<turnstile> V:=e :: T" obtains T' :: "ty" where "E V = \<lfloor>T\<rfloor>" and "P,E \<turnstile> e :: T'" and "P \<turnstile> T' \<le> T"
in_RSpan_obtain_same_length_coeffs: fixes R :: "'a set"   and n :: "'b"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and ms :: "'b list" assumes "R_scalar_mult R"   and "n \<in> R_scalar_mult.RSpan R smult ms" shows "\<exists>rs. set rs \<subseteq> R \<and> length rs = length ms \<and> n = scalar_mult.lincomb smult rs ms"
must_acquire_lock_contains_lock: fixes las :: "lock_action list" assumes "must_acquire_lock las" shows "Lock \<in> set las"
edge_succ_in_arcs: fixes a :: "'b" shows "(edge_succ HM a \<in> arcs H) = (a \<in> arcs H)"
\<beta>_boundedness_lt: fixes m :: "int"   and x :: "'a" assumes "m \<le> int (k x)"   and "x \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. u x < real_of_int m}"
seq_2: fixes i :: "nat"   and x :: "'a" shows "0 \<le> U i x"
punit_min_term: shows "extended_ord.punit.min_term = 0"
Lex_is_lexer: shows "\<forall>t\<in>\<TT>. is_lexer (Lex t)"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains v :: "name" and y :: "tm" and u :: "tm" and w :: "tm" and au :: "name" where "atom v \<sharp> (y, u, w, au)" and "atom au \<sharp> (y, u, w)" and "x = (y, u, w)"
multiset_eq_iff: fixes M :: "'b multiset"   and N :: "'b multiset" shows "(M = N) = (\<forall>a. count M a = count N a)"
1: shows "y \<in> F \<and> z \<in> G \<and> y \<sqinter> z \<le> x"
of_nat_less_ennreal_of_nat: fixes n :: "nat"   and x :: "enat" shows "(of_nat n \<le> ennreal_of_enat x) = (of_nat n \<le> x)"
iMin_mono_on2: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set" assumes "SetInterval2.mono_on f I"   and "I \<noteq> {}" shows "iMin (f ` I) = f (iMin I)"
atLeastLessThan_concat: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> n"   and "n \<le> p" shows "prod g {m..<n} * prod g {n..<p} = prod g {m..<p}"