hcju/isabelleps huggingface dataset:
atLeastatMost_empty_iff2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.preorder less_eq less" shows "({} = ord.atLeastAtMost less_eq a b) = (\<not> less_eq a b)"
verit_comp_simplify_3: fixes b' :: "'d"   and a' :: "'d" shows "(\<not> b' \<le> a') = (a' < b')"
continuous_sum: fixes I :: "'a set"   and F :: "'b filter"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> continuous F (f i)" shows "continuous F (\<lambda>x. \<Sum>i\<in>I. f i x)"
inverse_equality: fixes u :: "'a"   and v :: "'a" assumes "u \<cdot> v = \<one>"   and "v \<cdot> u = \<one>"   and "u \<in> G"   and "v \<in> G" shows "inverse u = v"
is_map: shows "is_left_adjoint HHfgh_HfHgh.chine"
insert_Diff_single: fixes a :: "'h"   and A :: "'h set" shows "insert a (A - {a}) = insert a A"
t3_space: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and y :: "'b" assumes "class.t3_space open"   and "topological_space.closed open S"   and "y \<notin> S" shows "\<exists>U V. open U \<and> open V \<and> y \<in> U \<and> S \<subseteq> V \<and> U \<inter> V = {}"
healthy_feasibleD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy t" shows "feasible t"
R1: shows "(tsl s, l') \<in> \<langle>Rk\<rangle>list_rel"
bvars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('b, 'a) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map (set \<circ> bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p) S))"
NOTIN: shows "\<forall>i\<in>{1..m}. Suc j \<notin> A i"
map_iso_def: fixes G :: "('f, 'g) pre_digraph"   and M :: "'g pre_map"   and f :: "('f, 'g, 'h, 'e) digraph_isomorphism" shows "pre_digraph_map.map_iso G M f \<equiv> \<lparr>edge_rev = pre_digraph.wrap_iso_arcs G f (edge_rev M), edge_succ = pre_digraph.wrap_iso_arcs G f (edge_succ M)\<rparr>"
remove_var: fixes v :: "'b"   and s :: "'b trm"   and t :: "'b trm" assumes "v \<notin> vars_of s" shows "v \<notin> vars_of (t \<lhd> [(v, s)])"
cmp_simps_1: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.arr (\<Phi>\<^sub>F (f, g))"
fsafe_Prop_simps: fixes p :: "'c"   and args :: "'c \<Rightarrow> ('a, 'c) trm" shows "fsafe ($\<phi> p args) = (\<forall>x. dsafe (args x))"
arr_char: fixes f :: "'a" shows "R.arr f = right a f"
ex_graph'_def: shows "ex_graph' \<equiv> Lgraph {(home, cloud), (sphone, cloud), (cloud, hospital)} (\<lambda>x. if x = cloud then {''Patient''} else if x = hospital then {''Doctor'', ''Eve''} else {}) ex_creds ex_locs"
memberid_pair_simp: fixes c :: "'a"   and m :: "'b" shows "memberid (c, m) = memberid m"
length_n_lists_elem: fixes ys :: "'c list"   and n :: "nat"   and xs :: "'c list" assumes "ys \<in> set (List.n_lists n xs)" shows "length ys = n"
coeff_pCons: fixes x :: "'b"   and xa :: "'b poly" shows "poly.coeff (pCons x xa) = case_nat x (poly.coeff xa)"
vcgaIsMaximalAux2: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N"   and "\<forall>a\<in>allAllocations ({seller} \<union> N) (set \<Omega>). \<forall>X\<in>Range a. b (seller, X) = (0::'d)" shows "sum b (Outside' {seller} (the_elem (argmax (sum (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>))) (maximalStrictAllocations N (set \<Omega>) b)))) = Max {sum b a |a. a \<in> soldAllocations N (set \<Omega>)}"
p\<^sub>1_simps: shows "trg p\<^sub>1 = src k"
fin: shows "\<not> is_final (steps0 c A (Suc n))"
1: fixes l :: "nat" assumes "n \<le> l" shows "card (level A w (2 * card (nodes A)) l) = 0"
indep_setsI: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'a set set" assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"   and "\<And>A J. \<lbrakk>J \<noteq> {}; J \<subseteq> I; finite J; \<forall>j\<in>J. A j \<in> F j\<rbrakk> \<Longrightarrow> prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))" shows "indep_sets F I"
M_U_soundness: shows "CU.Model wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) \<Phi> (InfModel.intTF any) local.M.intFF local.M.intPF"
my_U_represents_pref: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf"   and q :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>"   and "some_best outcomes \<R> \<succeq>[\<R>] some_worst outcomes \<R> \<and> (some_worst outcomes \<R>, some_best outcomes \<R>) \<notin> \<R>"   and "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "p \<succeq>[\<R>] q = (my_U outcomes \<R> q \<le> my_U outcomes \<R> p)"
generalized_zipping_lemma3: fixes \<rho>2 :: "'e V_rec \<Rightarrow> 'e set" assumes "N\<^bsub>\<V>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {}"   and "EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> N\<^bsub>\<V>1\<^esub>)"   and "BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>" shows "\<forall>\<tau> lambda t1 t2. set \<tau> \<subseteq> E\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> set lambda \<subseteq> V\<^bsub>\<V>\<^esub> \<and> set t1 \<subseteq> E\<^bsub>ES1\<^esub> \<and> set t2 \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> @ t1 \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> @ t2 \<in> Tr\<^bsub>ES2\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES1\<^esub> = t1 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> t1 \<upharpoonleft> C\<^bsub>\<V>1\<^esub> = [] \<and> t2 \<upharpoonleft> C\<^bsub>\<V>2\<^esub> = [] \<longrightarrow> (\<exists>t. \<tau> @ t \<in> Tr\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> t \<upharpoonleft> V\<^bsub>\<V>\<^esub> = lambda \<and> t \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
gr_zeroI: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
PR_CONST_def: fixes x :: "'c" shows "PR_CONST x \<equiv> x"
assms_8: shows "(add_mset z X, add_mset z Y) \<in> mult2_s ns s"
equation_minus_iff: fixes a :: "'b"   and b :: "'b" shows "(a = - b) = (b = - a)"
preserves_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<guillemotleft>\<F>C.\<nu> f : \<F>C.map a \<rightarrow> (\<F>\<^sub>SC.map \<circ> \<F>C.D) b\<guillemotright>"
assms_1: shows "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"
has_field_derivative_cong_ev: fixes x :: "'b"   and y :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and u :: "'b"   and v :: "'b"   and t :: "'b set" assumes "x = y"   and "\<forall>\<^sub>F x in nhds x. x \<in> S \<longrightarrow> f x = g x"   and "u = v"   and "S = t"   and "x \<in> S" shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative v) (at y within t)"
Riemann_lebesgue_cos: fixes f :: "real \<Rightarrow> real" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>n. LINT x|lebesgue_on {- pi..pi}. cos (real n * x) * f x) \<longlonglongrightarrow> 0"
distinct_insort: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "distinct (insort_key f x xs) = (x \<notin> set xs \<and> distinct xs)"
dghm_is_dghm_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<beta>\<^esub> \<BB>"
mod_smult_right: fixes a :: "'a"   and x :: "'a poly"   and y :: "'a poly" assumes "a \<noteq> (0::'a)" shows "x mod Polynomial.smult a y = x mod y"
poly_IVT_neg: fixes a :: "real"   and b :: "real"   and p :: "real Polynomial.poly" assumes "a < b"   and "0 < poly p a"   and "poly p b < 0" shows "\<exists>x>a. x < b \<and> poly p x = 0"
domains_char: fixes f :: "'a" shows "domains f = {x \<in> Obj. x = f}"
induct: fixes P :: "'a tree \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "'a tree"   and a1 :: "'a" assumes "\<And>w\<^sub>b b a. P (Leaf w\<^sub>b b) a"   and "\<And>w w\<^sub>b b w\<^sub>c c a. P (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a"   and "\<And>w v va vb t\<^sub>2 a. \<lbrakk>a \<in> alphabet (Node v va vb) \<Longrightarrow> P (Node v va vb) a; \<lbrakk>a \<notin> alphabet (Node v va vb); a \<in> alphabet t\<^sub>2\<rbrakk> \<Longrightarrow> P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w (Node v va vb) t\<^sub>2) a"   and "\<And>w t\<^sub>1 v va vb a. \<lbrakk>a \<in> alphabet t\<^sub>1 \<Longrightarrow> P t\<^sub>1 a; \<lbrakk>a \<notin> alphabet t\<^sub>1; a \<in> alphabet (Node v va vb)\<rbrakk> \<Longrightarrow> P (Node v va vb) a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 (Node v va vb)) a" shows "P a0 a1"
fixp_above_Field: fixes lub :: "'a set \<Rightarrow> 'a"   and leq :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "bourbaki_witt_fixpoint lub leq f"   and "a \<in> Field leq" shows "bourbaki_witt_fixpoint.fixp_above lub leq f a \<in> Field leq"
dup_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C f" shows "partial_magma.cod C (elementary_category_with_binary_products.tuple C pr0 pr1 f f) = elementary_category_with_binary_products.prod C pr0 pr1 (partial_magma.cod C f) (partial_magma.cod C f)"
subsetClosed': fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and P :: "'d" shows "(set (p \<bullet> xvec) \<subseteq> supp (p \<bullet> P)) = (set xvec \<subseteq> supp P)"
prop_in_f_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[Impossible F \<^bold>\<rightarrow> Indiscriminate F in v]"
homotopic_paths_eq: fixes p :: "real \<Rightarrow> 'b"   and s :: "'b set"   and q :: "real \<Rightarrow> 'b" assumes "path p"   and "path_image p \<subseteq> s"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t" shows "homotopic_paths s p q"
multiset_nonemptyE: fixes A :: "'f multiset" assumes "A \<noteq> {#}" obtains x :: "'f" where "x \<in># A"
superadditive_un_ge_nu1: fixes u :: "nat \<Rightarrow> real"   and n :: "nat" assumes "superadditive u"   and "0 < n" shows "real n * u 1 \<le> u n"
geodesic_segment_between_x_x_1: fixes x :: "'b" shows "geodesic_segment_between {x} x x"
scaleR_minus_right: fixes a :: "real"   and x :: "'a" shows "a *\<^sub>R - x = - (a *\<^sub>R x)"
merges_incr: fixes xs :: "'a list"   and n :: "nat"   and ps :: "(nat \<times> 'a) list" assumes "xs \<in> list n A"   and "\<forall>(p, x)\<in>set ps. p < length xs \<and> x \<in> A" shows "xs [\<sqsubseteq>\<^bsub>r\<^esub>] merges f ps xs"
ResF: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> \<alpha> \<prec> P'"   and "y \<sharp> \<alpha>" shows "<\<nu>y>P \<longmapsto> \<alpha> \<prec> <\<nu>y>P'"
heterogenous_cblinfun_id_def: shows "heterogenous_cblinfun_id \<equiv> CBlinfun (if bounded_clinear heterogenous_identity then heterogenous_identity else (\<lambda>_. 0::'b))"
tagrel_sub': fixes sub :: "'b run"   and r :: "'b run"   and c\<^sub>1 :: "clock"   and c\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool"   and n :: "nat" assumes "sub \<lless> r"   and "sub \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L" shows "R (time (Rep_run r n c\<^sub>1), time (Rep_run r n c\<^sub>2))"
doesnt_read_or_modify_def: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var" shows "doesnt_read_or_modify c x \<equiv> doesnt_read_or_modify_vars c ({x} \<union> \<C>_vars x)"
simps_8: fixes f1 :: "T1 \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "rec_T2 f1 f2 (D x2) = f2 x2"
joinable_vwalk_join_r_iff: fixes p :: "'a list"   and q :: "'a list"   and r :: "'a list" shows "joinable p (q \<oplus> r) = (joinable p q \<or> q = [] \<and> joinable p (tl r))"
continuous_hurwitz_zeta: fixes a :: "real"   and s :: "complex"   and A :: "complex set" assumes "0 < a"   and "s \<noteq> 1" shows "continuous (at s within A) (hurwitz_zeta a)"
GrpE: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'d" assumes "Grp A f x y" obtains "f x = y" and "x \<in> A"
eq_3: shows "H \<turnstile> x3 EQ x3'"
F_def: fixes x :: "'b" shows "F x = f -` {x} \<inter> space M"
Quotient3_residual: shows "Quotient3 alpha_residual abs_residual rep_residual"
terminal_def: fixes subs :: "'b \<Rightarrow> 'b set"   and delta :: "'b" shows "terminal subs delta = (subs delta = {})"
roots_of_rat_poly_main: fixes p :: "rat poly" assumes "p \<noteq> 0" shows "set (roots_of_rat_poly_main p) = {x. poly p x = 0}"
codomain_restrict_chart: fixes S :: "'c set"   and c :: "('c, 'd) chart" assumes "open S" shows "codomain (restrict_chart S c) = codomain c \<inter> inv_chart c -` S"
suffix_first: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "first (s |\<^sub>s n) = s n"
SPEC_rule_weak_nested2: fixes f :: "('b \<times> 'c) nres"   and Q :: "'b \<times> 'c \<Rightarrow> bool"   and P :: "'b \<times> 'c \<Rightarrow> bool"   and g :: "'b \<times> 'c \<Rightarrow> ('b \<times> 'c) nres" assumes "f \<le> SPEC Q"   and "f \<le> SPEC P"   and "\<And>r1 r2. \<lbrakk>Q (r1, r2); P (r1, r2)\<rbrakk> \<Longrightarrow> g (r1, r2) \<le> SPEC P" shows "f \<le> SPEC (\<lambda>r'. g r' \<le> SPEC P)"
neval_S_n: fixes x :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and y :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and n :: "nat"   and z :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "x \<leadsto> y"   and "y \<leadsto>\<^bsup>n\<^esup> z" shows "x \<leadsto>\<^bsup>Suc n\<^esup> z"
sin_cos_squared_add: fixes x :: "'a" shows "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (1::'a)"
correct_game_def: fixes m :: "'challenge" shows "abstract_com.correct_game m = key_gen \<bind> (\<lambda>(ck, vk). commit ck m \<bind> (\<lambda>(c, d). return_spmf (verify vk m c d)))"
poly_eventually_not_zero: fixes p :: "real poly" assumes "p \<noteq> 0" shows "\<forall>\<^sub>F x in at_infinity. poly p x \<noteq> 0"
disc_20: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "\<not> is_Var (StaticOperationCall x61 x62 x63)"
ValidTC_def: fixes p :: "'a set"   and c :: "'a com"   and q :: "'a set" shows "[p]   c   [q] \<equiv> \<forall>s. s \<in> p \<longrightarrow> (\<exists>t. Sem c s t \<and> t \<in> q)"
rep_zero: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "rep_nat zero = 0"
domain_def: fixes R :: "('b, 'c) ring_scheme" shows "domain R \<equiv> cring R \<and> domain_axioms R"
simps_11: fixes f :: "'b"   and v :: "'b" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<langle>v\<^bold>\<rangle> = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> \<^bold>\<langle>v\<^bold>\<rangle>"
exec_move_WhileI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e' :: "'addr expr1" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (while (e) e') h s ta h' s'"
currentD_IN': fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "current \<Gamma>' f"   and "x \<notin> B \<Gamma>'" shows "d_IN f x = 0"
cong_ex_bigtheta: fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f2 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "landau_symbol L L' Lr"   and "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> L F g1) = (f2 \<in> L F g2)"
notifyWatchesLoopPreservedWatches: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)" shows "let state' = notifyWatches_loop literal Wl newWl state in \<forall>c. c \<notin> set Wl \<longrightarrow> getWatch1 state' c = getWatch1 state c \<and> getWatch2 state' c = getWatch2 state c"
l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf get_tag_name get_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs set_shadow_root set_shadow_root_locs ShadowRootClass.known_ptr Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_shadow_root get_shadow_root_locs Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs remove_shadow_root remove_shadow_root_locs ShadowRootClass.known_ptrs Shadow_DOM.get_parent Shadow_DOM.get_parent_locs"
pid_run_sim': fixes r :: "nat \<Rightarrow> (nat, local_state, global_state) Pid_Scheduler.pid_global_config" assumes "pid.is_run r" shows "li'.s1.sa.is_run ((Some \<circ>\<circ> cprog.gc_\<alpha>) prog \<circ> pidgc_\<alpha> \<circ> r)"
degreen_npolyhCN: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh (CN c n p) n0" shows "degreen c n = 0"
additive_wp_Apply: fixes f :: "'a \<Rightarrow> 'a" shows "Transformers.additive (wp (Apply f))"
method_Nat_suc: shows "method Nat suc = Some \<lparr>par = NT, res = Class Nat, lcl = [], bdy = Res :== new Nat;; LAcc Res..pred:==LAcc This\<rparr>"
unsandwich_def: fixes A :: "'o"   and u :: "'o \<Rightarrow> 'a set_arrow" shows "\<sigma>\<^sup>\<leftarrow>(A,u) \<equiv> set_func (u A) (Id A)"
path_from_toI: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "Digraph G"   and "Digraph.walk G xs \<and> distinct xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
matrix_to_iarray_Gauss_Jordan_column_k_1: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and i :: "nat" assumes "k < ncols A"   and "i \<le> nrows A" shows "fst (Gauss_Jordan_column_k (i, A) k) = fst (Gauss_Jordan_column_k_iarrays (i, matrix_to_iarray A) k)"
ext_inject: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a"   and cLengthF' :: "rat"   and cMassF' :: "rat"   and cTimeF' :: "rat"   and cCurrentF' :: "rat"   and cTemperatureF' :: "rat"   and cAmountF' :: "rat"   and cIntensityF' :: "rat"   and more' :: "'a" shows "(\<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = \<lparr>cLengthF = cLengthF', cMassF = cMassF', cTimeF = cTimeF', cCurrentF = cCurrentF', cTemperatureF = cTemperatureF', cAmountF = cAmountF', cIntensityF = cIntensityF', \<dots> = more'\<rparr>) = (cLengthF = cLengthF' \<and> cMassF = cMassF' \<and> cTimeF = cTimeF' \<and> cCurrentF = cCurrentF' \<and> cTemperatureF = cTemperatureF' \<and> cAmountF = cAmountF' \<and> cIntensityF = cIntensityF' \<and> more = more')"
alw_mp: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs) xs" shows "alw \<psi> xs"
gval_foldr_gOr_invalid: fixes l :: "'a gexp list"   and g :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "(gval (fold gOr l g) s = invalid) = (\<exists>g'\<in>set (g # l). gval g' s = invalid)"
gs_def: shows "gs = gram_schmidt0 d (map vec_of_basis_enum S)"
parallel_converter2_eq_\<I>_cong: fixes \<I>1 :: "('y, 'z) \<I>"   and \<I>1' :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and conv1' :: "('y, 'z, 'aa, 'ab) converter"   and \<I>2 :: "('ac, 'ad) \<I>"   and \<I>2' :: "('ae, 'af) \<I>"   and conv2 :: "('ac, 'ad, 'ae, 'af) converter"   and conv2' :: "('ac, 'ad, 'ae, 'af) converter" assumes "\<I>1,\<I>1' \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>2' \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2,\<I>1' \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<sim> conv1' |\<^sub>= conv2'"
ln_gt_zero_iff: fixes x :: "real" assumes "0 < x" shows "(0 < ln x) = (1 < x)"
simps_1: fixes v :: "'V"   and v' :: "'V" shows "is_path v [] v' = (v = v' \<and> v' \<in> V)"
prime_iff_sigma: fixes n :: "nat" shows "prime n = (sigma n = Suc n)"
iter_unguarded: fixes s :: "'a"   and n :: "nat" assumes "\<not> guard s" shows "fuse.iter n s = return_spmf s"
finite_ordLess_var: fixes S :: "'a set" shows "(|S| <o |UNIV| \<or> finite S) = (|S| <o |UNIV|)"
\<alpha>nodes_finite: fixes g :: "'m" assumes "invar g" shows "finite (\<alpha>nodes_aux g)"
ss\<^sub>1: shows "snd ` set ss\<^sub>1 \<subseteq> A"
prems_3: shows "p\<^sub>4 = ell_field_class.add a p\<^sub>1 p\<^sub>3"
arr_dom_iff_arr: fixes f :: "'a2" shows "C2.arr (C2.dom f) = C2.arr f"
of_int_div: fixes d :: "int"   and n :: "int" assumes "d \<noteq> 0"   and "d dvd n" shows "of_int (n div d) = of_int n / of_int d"
minus_unique: fixes a :: "'a"   and b :: "'a" assumes "a + b = (0::'a)" shows "- a = b"
simps: fixes v1 :: "nat"   and p1 :: "(form, nat) out_port"   and v2 :: "nat"   and p2 :: "(form, nat) antecedent" shows "task.edge_end ((v1, p1), v2, p2) = v2"
F: shows "s \<sqinter> (s' \<sqinter> p') \<le> q"
simps: fixes l :: "nat"   and vs :: "(nat \<times> char list) list"   and c :: "call" shows "lambdas (Lambda l vs c) = {Lambda l vs c} \<union> lambdasC c"
ide_eval_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "D.ide \<lbrace>t\<rbrace>"
intro: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool"   and normalise :: "'b ltln \<Rightarrow> 'b ltln" assumes "ltl_equivalence eq"   and "GF_advice_congruent_axioms eq normalise" shows "GF_advice_congruent eq normalise"