hcju/isabelleps huggingface dataset:
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'd"   and x :: "('c, 'd) phi \<Rightarrow> 'd" assumes "eq_onp (\<lambda>xf. True) xa xa"   and "eq_onp (\<lambda>xf. True) x x" shows "Abs_lifted_distrib_lattice_top xa \<squnion> Abs_lifted_distrib_lattice_top x = Abs_lifted_distrib_lattice_top (\<lambda>f. xa f \<squnion> x f)"
scene_indep_def: shows "(\<bowtie>\<^sub>S) \<equiv> map_fun Rep_scene (map_fun Rep_scene id) (\<lambda>F G. \<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. G (F s\<^sub>1 s\<^sub>2) s\<^sub>3 = F (G s\<^sub>1 s\<^sub>3) s\<^sub>2)"
\<Psi>_simp_2: fixes y :: "'d"   and x :: "'c" assumes "D.ide y"   and "C.ide x" shows "\<Psi> (y, x) = S.mkArr (HomD.set (y, G x)) (HomC.set (F y, x)) (\<phi>C (F y, x) \<circ> \<psi> x \<circ> \<psi>D (y, G x))"
mk_monoid_simps_3: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and more :: "'b" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
AssertionStatImp_def: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.AssertionStatImp SImp \<Psi> \<Psi>' \<equiv> \<forall>\<Phi>. SImp \<Psi> \<Phi> \<longrightarrow> SImp \<Psi>' \<Phi>"
simps: fixes G :: "prog"   and accclass :: "qtname"   and a1 :: "qtname \<times> memberdecl"   and a2 :: "qtname" shows "G \<turnstile> a1 of a2 accessible_from accclass = ((\<exists>membr class. a1 = membr \<and> a2 = class \<and> G \<turnstile> membr member_of class \<and> G \<turnstile> Class class accessible_in pid accclass \<and> G \<turnstile> membr in class permits_acc_from accclass) \<or> (\<exists>membr class C new old supr. a1 = membr \<and> a2 = class \<and> G \<turnstile> membr member_of class \<and> G \<turnstile> Class class accessible_in pid accclass \<and> membr = (C, mdecl new) \<and> G \<turnstile> (C, new) overrides\<^sub>S old \<and> G\<turnstile>class\<prec>\<^sub>C supr \<and> G \<turnstile>Method old of supr accessible_from accclass))"
app_vconverse_def: fixes r :: "V" shows "r\<inverse>\<^sub>\<circ> = set {\<langle>b, a\<rangle> |a b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}"
R\<^sub>0_def: shows "rs.R\<^sub>0 = \<lbrakk>\<lbrakk>tab\<^sub>0 r\<rbrakk>\<rbrakk>"
irreducible_normalize_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "comm_semiring_1.irreducible times one zero (normalize x) = comm_semiring_1.irreducible times one zero x"
SUP_apply: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'b" shows "(\<Squnion>(f ` A)) x = (SUP y\<in>A. f y x)"
asTerm_pick: fixes X :: "('i, 'j, 'varSort, 'var, 'k) qTerm set" assumes "good X" shows "asTerm (pick X) = X"
ide_char: fixes f :: "'c \<times> 'c" shows "CopxC.ide f = (Cop.ide (fst f) \<and> C.ide (snd f))"
GammaForall: fixes A :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term"   and G :: "('a, 'b) form list" assumes "\<stileturn> A[t/0] # G" shows "\<stileturn> Forall A # G"
valid_index_weave_2: fixes is1 :: "nat list"   and ds :: "nat list"   and A :: "nat set"   and is2 :: "nat list" assumes "is1 \<lhd> nths ds A"   and "is2 \<lhd> nths ds (- A)" shows "nths (weave A is1 is2) A = is1"
fst_image_mp: fixes A :: "('a \<times> 'b) set"   and B :: "'a set"   and x :: "'a"   and y :: "'b" assumes "fst ` A \<subseteq> B"   and "(x, y) \<in> A" shows "x \<in> B"
separating_hyperplane_closed_0_inset: fixes S :: "'a set" assumes "convex S"   and "closed S"   and "S \<noteq> {}"   and "(0::'a) \<notin> S" obtains a :: "'a" and b :: "real" where "a \<in> S" and "a \<noteq> (0::'a)" and "0 < b" and "\<And>x. x \<in> S \<Longrightarrow> b < a \<bullet> x"
lookup_map_values: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and m :: "('c, 'd) mapping"   and k :: "'c" shows "Mapping.lookup (Mapping.map_values f m) k = map_option (f k) (Mapping.lookup m k)"
\<epsilon>_def: shows "\<epsilon> = (SOME \<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>)"
set_mult_subset_generate: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> carrier G"   and "B \<subseteq> carrier G" shows "A <#> B \<subseteq> generate G (A \<union> B)"
valid_graph_def: fixes G :: "('c, 'd) graph" shows "valid_graph G \<equiv> fst ` edges G \<subseteq> nodes G \<and> snd ` snd ` edges G \<subseteq> nodes G"
op_op2_refinement: fixes pat :: "[:'r:]"   and usvs :: "[:'r:] \<times> [:'r:]"   and x :: "'r" assumes "pat \<noteq> \<bottom>" shows "rep2\<cdot>pat\<cdot>(op\<cdot>pat\<cdot>usvs\<cdot>x) = op2\<cdot>pat\<cdot>(rep2\<cdot>pat\<cdot>usvs)\<cdot>x"
valid_eqvt: fixes P :: "'state"   and x :: "('b, unit, ('act, 'pred) S_action) formula"   and p :: "perm" assumes "P \<Turnstile>\<^sub>S x" shows "p \<bullet> P \<Turnstile>\<^sub>S p \<bullet> x"
list_all2_set1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b list"   and as :: "'c list" assumes "list_all2 P xs as" shows "\<forall>x\<in>set xs. \<exists>xa\<in>set as. P x xa"
hom_int_pow: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and n :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "hom_boundary (int n) (nsphere n) (equator n) (x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> n) = hom_boundary (int n) (nsphere n) (equator n) x [^]\<^bsub>homology_group (int n - 1) (subtopology (nsphere n) (equator n))\<^esub> n"
characteristic_iso_2: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>f \<star> \<iota> : f \<star> a \<star> a \<Rightarrow>\<^sub>S f \<star> a\<guillemotright>"
transpose_matrix_def: shows "transpose_matrix \<equiv> Abs_matrix \<circ> transpose_infmatrix \<circ> Rep_matrix"
chain_epredI: fixes Y :: "enat set" assumes "Complete_Partial_Order.chain (\<le>) Y" shows "Complete_Partial_Order.chain (\<le>) (epred ` (Y \<inter> {x. x \<noteq> 0}))"
1: shows "a \<in> carrier (R\<lparr>carrier := S\<rparr>)"
tangent_space_def: fixes p :: "'c" shows "src.manifold_eucl.dest.tangent_space p = {X. src.manifold_eucl.dest.is_derivation X p \<and> extensional0 src.manifold_eucl.dest.diff_fun_space X}"
is_tangent_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c point"   and q :: "'c point" assumes "ell_field R" shows "ell_field.is_tangent R p q = (p \<noteq> Infinity \<and> p = q \<and> p \<noteq> ell_field.opp R q)"
d\<mu>_impl_cost_def: fixes fs :: "int vec list" shows "d\<mu>_impl_cost fs = dmu_array_cost (IArray fs) (IArray []) 0"
dlex_fun_lin: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "dlex_fun s t \<or> dlex_fun t s"
well_def_wp_sublinear: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "well_def prog" shows "sublinear (wp prog)"
assms_2: shows "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"
sub_inc_sysI: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "incidence_system \<U> \<A>"   and "\<U> \<subseteq> \<V>'"   and "\<A> \<subseteq># \<B>'" shows "sub_incidence_system \<U> \<A> \<V>' \<B>'"
distinct_17: fixes x21 :: "nat"   and x22 :: "nat"   and x51 :: "nat"   and x52 :: "nat" shows "x21:deliver(x22) \<noteq> disconnect(x51, x52)"
is_sig_red_top_cong: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg (=) F p"   and "rep_list q \<noteq> 0"   and "lt p = lt q"   and "punit.lt (rep_list p) = punit.lt (rep_list q)" shows "is_sig_red sing_reg (=) F q"
intensional_rews_2: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'d \<Rightarrow> 'c"   and w :: "'d" shows "(f<x>) w \<equiv> f (x w)"
one_neq_neg_one: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd" assumes "class.ring_char_0 one times plus zero minus uminus" shows "one \<noteq> uminus one"
assms: shows "(s, a, s') \<in> automaton.trans (closed (pnet np n))"
sdp_intros_8: fixes S :: "'a \<Rightarrow> 'b set"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> S s \<Longrightarrow> sub_distrib_pconj (p a)"   and "\<And>s a. a \<in> S s \<Longrightarrow> healthy (wp (p a))"   and "\<And>s a. a \<in> S s \<Longrightarrow> nearly_healthy (wlp (p a))"   and "\<And>s. S s \<noteq> {}" shows "sub_distrib_pconj (SetDC p S)"
p_fresh'': shows "(pi \<bullet> supp p) \<sharp>* Base u"
stable_on_CH: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "CH X = X"
val_Rxa_gt_a: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v) - {\<zero>}"   and "y \<in> carrier (Vr K v)"   and "y \<in> Vr K v \<diamondsuit>\<^sub>p x" shows "v x \<le> v y"
elims: fixes x :: "('a, 'b, 'c) formula list \<times> ('a, 'b, 'c) formula list"   and y :: "('a, 'b, 'c) formula" assumes "seq2fml x = y" obtains ante :: "('a, 'b, 'c) formula list" and succ :: "('a, 'b, 'c) formula list" where "x = (ante, succ)" and "y = (foldr (&&) ante TT \<rightarrow> foldr (||) succ FF)"
assert_SE_def: fixes P :: "'d \<Rightarrow> bool" shows "assert\<^sub>S\<^sub>E P = (\<lambda>\<sigma>. if P \<sigma> then Some (True, \<sigma>) else None)"
parPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs" assumes "P \<leadsto><Rel> Q"   and "(P, Q) \<in> Rel"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<parallel> U, T \<parallel> U) \<in> Rel'" shows "P \<parallel> R \<leadsto><Rel'> Q \<parallel> R"
F_def: shows "F = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0}}"
simps: fixes n :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "heuristicPicker n L F = (case let (l, e, g) = aquireData n L in case getBest l L of None \<Rightarrow> case F of [] \<Rightarrow> case getBest g L of None \<Rightarrow> case getBest e L of None \<Rightarrow> None | Some v \<Rightarrow> Some (v, qe_eq_repeat) | Some v \<Rightarrow> Some (v, gen_qe) | a # list \<Rightarrow> case getBest e L of None \<Rightarrow> None | Some v \<Rightarrow> Some (v, qe_eq_repeat) | Some v \<Rightarrow> Some (v, luckyFind') of None \<Rightarrow> None | Some (var, step) \<Rightarrow> if n < var then None else Some (var, step))"
length_nns_n: fixes kdt :: "'b kdt"   and ps :: "(real, 'b) vec list"   and n :: "nat"   and p :: "(real, 'b) vec" assumes "set_kdt kdt \<union> set ps - set (nearest_nbors n ps p kdt) \<noteq> {}" shows "length (nearest_nbors n ps p kdt) = n"
minus_to_eq: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group G"   and "abelian_group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<ominus>\<^bsub>G\<^esub> y = \<zero>\<^bsub>G\<^esub>" shows "x = y"
conjugateI2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<sqinter> conjugate f y = \<bottom>" shows "f x \<sqinter> y = \<bottom>"
diffs0_imp_equal: fixes m :: "nat"   and n :: "nat" assumes "m - n = 0"   and "n - m = 0" shows "m = n"
simps_2: fixes ci :: "'b"   and v :: "'b"   and va :: "'b list"   and xi :: "'b"   and xis :: "'b list" shows "horner_composition (ci # v # va) (xi # xis) = horner_composition (v # va) xis * [:- xi, 1::'b:] + [:ci:]"
case_distinction_on_context: fixes m :: "name"   and t :: "ty"   and \<Gamma> :: "(name \<times> ty) list"   and n :: "name"   and U :: "ty"   and T :: "ty" assumes "valid ((m, t) # \<Gamma>)"   and "(n, U) \<in> set ((m, T) # \<Gamma>)" shows "(n, U) = (m, T) \<or> (n, U) \<in> set \<Gamma> \<and> n \<noteq> m"
t'_3: shows "\<forall>k<length xs. \<exists>t. t \<in>\<^sub>A frontier (c_imp c (case xs ! k of (s, l, t) \<Rightarrow> t)) \<and> t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) (take (k + 1) xs)))"
divide_eq1: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)" shows "(f \<in> L F (\<lambda>x. g x / h x)) = ((\<lambda>x. f x * h x) \<in> L F g)"
IH: assumes "x \<noteq> y" shows "remove1 y (Sorting_Algorithms.insort cmp x zs) = Sorting_Algorithms.insort cmp x (remove1 y zs)"
phis'_wf: fixes g :: "'g"   and n :: "'node"   and v :: "'var \<times> 'node \<times> Def"   and args :: "('var \<times> 'node \<times> Def) list" assumes "phis' g (n, v) = Some args" shows "length (predecessors g n) = length args"
interv: fixes e :: "real" assumes "0 < e" shows "\<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
span_closed: fixes U :: "'a vec set"   and v :: "'a vec" assumes "U \<subseteq> mat_kernel A"   and "v \<in> KA.span U" shows "v \<in> mat_kernel A"
integrable_density: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integrable (density M (\<lambda>x. ennreal (g x))) f = integrable M (\<lambda>x. g x *\<^sub>R f x)"
hyp3: shows "fst (snd (foldl Gauss_Jordan_column_k_PA (mat (1::'a), 0, A) [0..<Suc k])) = fst (snd (foldl Gauss_Jordan_column_k_iarrays_PA (mat_iarray (1::'a) (nrows_iarray (matrix_to_iarray A)), 0, matrix_to_iarray A) [0..<Suc k]))"
poly_fun_def: fixes f :: "int \<Rightarrow> int" shows "poly_fun f = (\<exists>p. \<forall>a. rat_of_int (f a) = poly p (rat_of_int a))"
measurable_lborel2: fixes M :: "'c measure" shows "lborel \<rightarrow>\<^sub>M M = borel \<rightarrow>\<^sub>M M"
fin: shows "finite (set (ftv T\<^sub>1 - ftv \<Gamma>))"
degree_polysub: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "Reflected_Multivariate_Polynomial.degree (p -\<^sub>p q) \<le> max (Reflected_Multivariate_Polynomial.degree p) (Reflected_Multivariate_Polynomial.degree q)"
epiI: fixes f :: "'a" assumes "arr f"   and "\<And>g g'. seq g f \<and> seq g' f \<and> g \<cdot> f = g' \<cdot> f \<Longrightarrow> g = g'" shows "epi f"
WCFG_Entry_no_targetnode: fixes prog :: "cmd"   and n :: "w_node"   and et :: "(char list \<Rightarrow> val option) edge_kind" assumes "prog \<turnstile> n -et\<rightarrow> (_Entry_)" shows "False"
strong_del_block_alt_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection" shows "incidence_system.str_del_block block_collection b = {#bl - b. bl \<in># removeAll_mset b block_collection#}"
abs_0: shows "\<bar>0::'a\<bar> = (0::'a)"
simps_3: fixes f :: "Core.term"   and t :: "Core.term"   and n :: "nat" shows "lift (f $ t) n = lift f n $ lift t n"
assms_1: shows "local.ind_cpa.ibounded_by \<A> (enat q)"
set_nths_subset: fixes xs :: "'b list"   and I :: "nat set" shows "set (nths xs I) \<subseteq> set xs"
op_induct: fixes P :: "[:'r:] \<times> 's \<Rightarrow> bool"   and usvs :: "[:'r:] \<times> 's" assumes "\<And>usvs. (\<And>usvs'. lt\<cdot>(slength\<cdot>(cfst\<cdot>usvs'))\<cdot>(slength\<cdot>(cfst\<cdot>usvs)) = TT \<Longrightarrow> P usvs') \<Longrightarrow> P usvs" shows "P usvs"
r_nth_inbounds_prim: shows "prim_recfn 3 r_nth_inbounds"
publicKey_in_initState: fixes b :: "keymode"   and A :: "agent"   and B :: "agent" shows "Key (publicKey b A) \<in> initState B"
intros: fixes c :: "'form"   and is' :: "nat list"   and "is" :: "nat list"   and i :: "nat"   and ip :: "('form, 'var) antecedent" assumes "c \<in> set conclusions"   and "is' \<in> (#) 0 ` it_paths (it' c)"   and "prefix (is @ [i]) is'"   and "ip = in_port_at (c, is) i" shows "scope' (c, is) ip (c, is')"
NextRecvUpdD_1: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c0 :: "('d, 'c, 'a) Combined.configuration"   and c1 :: "('d, 'c, 'a) Combined.configuration"   and p :: "'d"   and q :: "'d" assumes "dataflow_topology summary results_in"   and "dataflow_topology.NextRecvUpd' c0 c1 p q" shows "init c0 q"
is_const_def: fixes e :: "e" shows "is_const e = (case e of $C x \<Rightarrow> True | $_ \<Rightarrow> False | _ \<Rightarrow> False)"
ivl_rel_def: fixes S :: "('d \<times> 'e) set" shows "\<langle>S\<rangle>ivl_rel = (S \<times>\<^sub>r S) O br set_of_ivl top"
fans_def: fixes subs :: "'b \<Rightarrow> 'b set" shows "fans subs = (\<forall>x. finite (subs x))"
emb2: fixes as :: "letter list"   and bs :: "letter list"   and a :: "letter" assumes "emb as bs" shows "emb (a # as) (a # bs)"
P1: shows "A' Midpoint P' P \<and> Col A B A' \<and> (A B Perp P' P \<or> P = P')"
get_parent_reads: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads Shadow_DOM.get_parent_locs (Shadow_DOM.get_parent node_ptr) h h'"
globEA: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and n' :: "nat" assumes "\<exists>i\<ge>n. active c (t i)"   and "dynamic_component.eval tCMP active c t t' n (dynamic_component.glob \<gamma>)"   and "dynamic_component.lNAct active c t n \<le> n'" shows "dynamic_component.eval tCMP active c t t' n' \<gamma>"
H: shows "\<Gamma> \<turnstile> Rcd fs : RcdT fTs"
order_refl: fixes x :: "'a \<times> ('a, 'b) operation" shows "weak_hb x x"
cB: shows "char_poly B = char_poly B1 * char_poly B4"
trail_arc_balanceE: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "trail u p v" obtains "\<And>w. \<lbrakk>u = v \<or> w \<noteq> u \<and> w \<noteq> v; w \<in> verts G\<rbrakk> \<Longrightarrow> arc_balance w p = 0" and "u \<noteq> v \<Longrightarrow> arc_balance u p = - 1" and "u \<noteq> v \<Longrightarrow> arc_balance v p = 1"
is_sentence_take: fixes y :: "symbol list"   and n :: "nat" assumes "is_sentence y" shows "is_sentence (take n y)"
vec_of_list_index: fixes xs :: "'c list"   and j :: "nat" shows "vec_of_list xs $ j = xs ! j"
distinctPermSimps_2: fixes a :: "name"   and b :: "name"   and p :: "(name \<times> name) list" shows "distinctPerm ((a, b) # p) = (distinctPerm p \<and> a \<noteq> b \<and> a \<sharp> p \<and> b \<sharp> p)"
skip_R1: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> \<R>" shows "(\<lambda>x. f (x + n)) \<in> \<R>"
less_add_same_cancel1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less a (plus a b) = less zero b"
distinct_6: fixes name' :: "name"   and lam' :: "lam"   and lam1 :: "lam"   and lam2 :: "lam" shows "Lam [name'].lam' \<noteq> App lam1 lam2"
simps: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" shows "Optimal_BST.opt_bst_wpl w i j = (if j < i then (\<langle>\<rangle>, 0) else argmin snd (map (\<lambda>k. let (t1, c1) = Optimal_BST.opt_bst_wpl w i (k - 1); (t2, c2) = Optimal_BST.opt_bst_wpl w (k + 1) j in (\<langle>t1, k, t2\<rangle>, c1 + c2 + w i j)) [i..j]))"
htai_prodWR_def: fixes H1 :: "(nat, nat) hashedTa"   and H2 :: "(nat, nat) hashedTa" shows "htai_prodWR H1 H2 \<equiv> hta_reindex (hta_prodWR H1 H2)"
IndP_HPairE: fixes x :: "tm"   and H :: "fm set"   and A :: "fm" shows "insert (IndP (HPair (HPair Zero (HPair Zero Zero)) x)) H \<turnstile> A"
l8_14_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<not> Tarski_neutral_dimensionless.Perp Bet Cong A B A B"
extensionalD_def: fixes d :: "'b"   and A :: "'a set" shows "extensionalD d A = {f. \<forall>x. x \<notin> A \<longrightarrow> f x = d}"
execute_len: fixes a :: "'b array"   and h :: "heap" shows "execute (Array.len a) h = Some (Array.length h a, h)"
kripkeI: fixes M :: "('b, 'd, 'c) KripkeStructure" assumes "\<And>a. relations M a \<subseteq> worlds M \<times> worlds M" shows "kripke M"
ffb_ad_fun: fixes f :: "'a \<Rightarrow> 'b set"   and X :: "'b set" shows "fb\<^sub>\<F> f X = {x. kad (f \<circ>\<^sub>K kad (set2fun X)) x \<noteq> {}}"
nat_mult_leD2: fixes a :: "nat"   and b :: "nat" assumes "real a * real b \<le> x"   and "0 < a" shows "real b \<le> x"
nonabort_j1: shows "chaos \<sqsubseteq> (j\<^sub>1::'a::refinement_lattice)"
42: shows "bprv (exi yy (imp (subst \<phi> \<langle>\<psi>\<rangle> xx) (cnj (subst \<phi> (Var yy) xx) (SS \<langle>\<chi>\<rangle> (Var yy)))))"
linear_pseudo_hoop_algebra_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.linear_pseudo_hoop_algebra less_eq less times inf imp_l one imp_r \<equiv> class.linorder less_eq less \<and> class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r"
measure_measure_pmf_not_zero: fixes p :: "'b pmf"   and s :: "'b set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "measure_pmf.prob p s \<noteq> 0"
fact1: fixes V :: "variable set" assumes "snd (usubstappp \<sigma> V \<alpha>) \<noteq> undefg" shows "BVG (the (snd (usubstappp \<sigma> V \<alpha>))) \<subseteq> fst (usubstappp \<sigma> V \<alpha>)"
Cauchy_theorem_flat: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and f :: "complex \<Rightarrow> complex"   and k :: "real" assumes "continuous_on (convex hull {a, b, c}) f"   and "c - a = k *\<^sub>R (b - a)" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
comm_monoid_axioms: shows "comm_monoid (*) (1::'k)"
divmod_integer_code'': fixes k :: "integer"   and l :: "integer" shows "divmod_integer k l = (if k = 0 then (0, 0) else if 0 < l then if 0 < k then divmod_abs' k l else case divmod_abs' (- k) l of (r, s) \<Rightarrow> if s = 0 then (- r, 0) else (- r - 1, l - s) else if l = 0 then (0, k) else apsnd uminus (if k < 0 then divmod_abs' (- k) (- l) else case divmod_abs' k (- l) of (r, s) \<Rightarrow> if s = 0 then (- r, 0) else (- r - 1, - l - s)))"
strict_implies_not_eq: fixes a :: "'b DBMEntry"   and b :: "'b DBMEntry" assumes "a \<prec> b" shows "a \<noteq> b"
intros_1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and T :: "ty" shows "P \<turnstile> T \<le> T"
t_def: fixes x :: "'n" shows "t x = (THE f. X $ x = (\<Sum>a\<in>UNIV. f a *s Y $ a))"
Max_ne_def: fixes X :: "'a set" shows "Max_ne X = FOLD_bin_ne max X"
indep_card_eq_dim_span: fixes B :: "'b set" assumes "vs1.independent B" shows "finite B \<and> card B = vs1.dim (vs1.span B)"
arith_extra_simps_5: fixes a :: "'d" shows "(0::'d) + a = a"