hcju/isabelleps huggingface dataset:
simps_7: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r X\<^sub>r \<phi> = suffix 1 \<xi> \<Turnstile>\<^sub>r \<phi>"
pullSummand: fixes P :: "pi"   and Q :: "pi" assumes "P \<in> summands Q"   and "uhnf Q" shows "\<exists>Q'. P \<oplus> Q' \<equiv>\<^sub>e Q \<and> summands Q' = summands Q - {P' |P'. P' \<in> summands Q \<and> P' \<equiv>\<^sub>e P} \<and> uhnf Q'"
simps_5: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> (\<phi> R\<^sub>n \<psi>) w = \<G>\<F> \<phi> w \<union> \<G>\<F> \<psi> w"
atomize_imp: fixes A :: "bool"   and B :: "bool" shows "(A \<Longrightarrow> B) \<equiv> A \<longrightarrow> B"
minus_eq: fixes x :: "'c"   and y :: "'c" shows "x \<ominus>\<^bsub>S\<^esub> y = x \<oplus>\<^bsub>S\<^esub> \<ominus>\<^bsub>S\<^esub> y"
hd: shows "\<forall>a\<in>set as. hd_coeff_is1 a"
o_substL: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" shows "substL \<pi>1 \<circ> substL \<pi>2 = substL (subst \<pi>1 \<circ> \<pi>2)"
enum_strict_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "(t.enum i < t.enum j) = (i < j)"
set_tag_name_get_shadow_root_is_l_set_tag_name_get_shadow_root: shows "l_set_tag_name_get_shadow_root ShadowRootClass.type_wf set_tag_name set_tag_name_locs get_shadow_root get_shadow_root_locs"
star_sup_1: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> * (y * x\<^sup>\<star>)\<^sup>\<star>"
underS_Under_trans: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and C :: "'b set" assumes "trans r"   and "antisym r"   and "a \<in> underS r b"   and "b \<in> Under r C" shows "a \<in> UnderS r C"
diff_fun_between_chartsD: fixes c :: "('a, 'b) chart"   and x :: "'a" assumes "c \<in> atlas"   and "x \<in> domain c" shows "k-smooth_on (codomain c) (a \<circ> inv_chart c)"
merge_eq_appendI: fixes xs :: "'b list"   and ys :: "'b list"   and cmp :: "'b comparator" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> compare cmp x y \<noteq> Greater" shows "Sorting_Algorithms.merge cmp xs ys = xs @ ys"
"1_2": fixes x :: "'a \<times> 'a list"   and xa :: "'a"   and y :: "'a list"   and finl :: "'a list" assumes "set todo \<noteq> {}"   and "x = (case todo of h # t \<Rightarrow> (h, t))"   and "(xa, y) = x"   and "\<not> (\<exists>b\<in>fin. subsumes b xa)"   and "insert xa fin = set finl" shows "set (mk_rtrancl_list_main subsumes r (r xa @ y) finl) = mk_rtrancl_main (r xa @ y) (insert xa fin)"
is_GB_cofactor_boundE_Polys: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set"   and b :: "nat"   and X :: "'x set" assumes "is_GB_cofactor_bound F b"   and "F \<subseteq> P[X]" obtains G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a) set" where "punit.is_Groebner_basis G" and "ideal G = ideal F" and "G \<subseteq> P[X]" and "\<And>g. g \<in> G \<Longrightarrow> \<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))"
emeasure_spmf_single: fixes p :: "'b spmf"   and x :: "'b" shows "emeasure (measure_spmf p) {x} = ennreal (spmf p x)"
simps_4: fixes f :: "'c"   and v :: "'c local.term"   and va :: "'c local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> v \<^bold>\<otimes> va = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> v \<^bold>\<otimes> va"
CongruenceD: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Ring_Congruence" shows "\<exists>k\<in>I. a = b + k"
res_c: shows "dmu_array_row_main_cost (fs !! i) i (iarray_append dmus (IArray [fs !! i \<bullet> fs !! 0])) 0 = (res, c_main)"
paths_second_disjoint: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and xs :: "'c list"   and ys :: "'c list" assumes "DisjointPaths G v0 v1 paths"   and "xs \<in> paths"   and "ys \<in> paths"   and "xs \<noteq> ys" shows "hd (tl xs) \<noteq> hd (tl ys)"
monoid_add_hom_axioms: fixes i :: "nat" shows "monoid_add_hom (\<lambda>p. coeff p i)"
natpermute_contain_maximal: fixes n :: "nat"   and k :: "nat" shows "{xs \<in> natpermute n (k + 1). n \<in> set xs} = (\<Union>i\<in>{0..k}. {(replicate (k + 1) 0)[i := n]})"
inorder_split_min: fixes t :: "'c bro"   and h :: "nat"   and a :: "'c"   and t' :: "'c bro" assumes "t \<in> T h" shows "(split_min t = None) = (inorder t = []) \<and> (split_min t = Some (a, t') \<longrightarrow> inorder t = a # inorder t')"
impl_partition: fixes NI :: "msg set"   and I :: "msg set"   and K :: "msg set" assumes "NI \<subseteq> payload"   and "I \<subseteq> valid" shows "I \<subseteq> extractable K I \<union> {implConfid A B M |A B M. (A, B) \<in> UNIV - broken K \<and> M \<in> payload} \<union> {implSecure A B M |A B M. (A, B) \<in> UNIV - broken K \<and> M \<in> payload}"
removeNones_in: fixes as :: "'a option list"   and a :: "'a"   and bs :: "'a option list" shows "removeNones (as @ Some a # bs) = removeNones as @ a # removeNones bs"
abs_eq: fixes x :: "('a \<times> 'b) list" shows "fmap_of_list x = map_of_fmap (map_of x)"
stream_enc_welldef: fixes \<Sigma> :: "'b list"   and w :: "'b list"   and I :: "(nat + nat set) list"   and w' :: "'b list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.stream_enc \<Sigma> (w, I) = formula.stream_enc \<Sigma> (w', I')"   and "formula.wf_formula \<Sigma> (length I) \<phi>"   and "formula.wf_interp_for_formula \<Sigma> (w, I) \<phi>"   and "formula.wf_interp_for_formula \<Sigma> (w', I') \<phi>" shows "formula.satisfies \<Sigma> (w, I) \<phi> = formula.satisfies \<Sigma> (w', I') \<phi>"
not_call_simps_7: fixes l :: "'d"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}send(fmsg) . p)"
mult_mono': fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d"   and "bot \<le> a"   and "bot \<le> c" shows "a * c \<le> b * d"
ucast_s2: fixes w :: "word32" assumes "w AND 128 = 0" shows "get_S w = 0"
integrable_monotone_convergence: fixes M :: "'a measure"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and u :: "'a \<Rightarrow> real"   and x :: "real" assumes "\<And>i. integrable M (f i)"   and "AE x in M. incseq (\<lambda>n. f n x)"   and "AE x in M. (\<lambda>i. f i x) \<longlonglongrightarrow> u x"   and "(\<lambda>i. integral\<^sup>L M (f i)) \<longlonglongrightarrow> x"   and "u \<in> borel_measurable M" shows "integrable M u"
smult_monom_mult: fixes a :: "'b"   and b :: "'b"   and n :: "nat"   and f :: "'b poly" shows "smult a (monom b n * f) = monom (a * b) n * f"
filterlim_sequentially_Suc: fixes f :: "nat \<Rightarrow> 'c"   and F :: "'c filter" shows "(LIM x sequentially. f (Suc x) :> F) = filterlim f F sequentially"
simps_4: fixes p :: "fm"   and q :: "fm" shows "qfree (And p q) = (qfree p \<and> qfree q)"
bind_empty: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e measure" assumes "space M = {}" shows "M \<bind> f = count_space {}"
init_opar_comp: fixes s :: "((nat \<Rightarrow> 'd) \<times> 'e, 'g seq_action) automaton"   and i :: "nat"   and t :: "('f, 'g seq_action) automaton" shows "init (s \<langle>\<langle>\<^bsub>i\<^esub> t) = extg ` (init s \<times> init t)"
intro: fixes empty :: "'a"   and up :: "'b \<Rightarrow> 'c list \<Rightarrow> 'a \<Rightarrow> 'a"   and map_of :: "'a \<Rightarrow> 'b \<Rightarrow> 'c list"   and M :: "'a \<Rightarrow> bool"   and qle :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and subsumed :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and I :: "'c \<Rightarrow> bool"   and key :: "'c \<Rightarrow> 'b" assumes "maps empty up map_of M"   and "quasi_order qle"   and "set_mod_maps_axioms qle subsumed I key" shows "set_mod_maps empty up map_of M qle subsumed I key"
first_in_tree_order_def: shows "first_in_tree_order \<equiv> l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order"
Aheap_nonrec_simp: fixes x :: "Vars.var"   and e :: "exp"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "Aheap_nonrec x e\<cdot>i = esing x\<cdot>(ABind_nonrec x e\<cdot>i)"
assms_6: shows "(g \<longlongrightarrow> C) at_top"
l7_22_aux_R2: fixes A2 :: "'p"   and C :: "'p"   and A1 :: "'p"   and B1 :: "'p"   and B2 :: "'p"   and M1 :: "'p"   and M2 :: "'p" assumes "A2 \<noteq> C"   and "Bet A1 C A2"   and "Bet B1 C B2"   and "Cong C A1 C B1"   and "Cong C A2 C B2"   and "M1 Midpoint A1 B1"   and "M2 Midpoint A2 B2"   and "C A1 Le C A2" shows "Bet M1 C M2"
pp_of_term_of_pair: fixes t :: "'a"   and k :: "nat" shows "pp_of_term (term_of_pair (t, k)) = t"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_shadow_root get_shadow_root_locs get_host get_host_locs" shows "l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf"
ESem_graph_aux_def: shows "ESem_graph_aux \<equiv> lfp (\<lambda>p x1 x2. (\<exists>ESem_aux x e. x1 = Lam [x]. e \<and> x2 = (\<Lambda> \<rho>. tick\<cdot>(Fn\<cdot>(\<Lambda> v. ((\<lambda>uu_. uu_) \<circ> ESem_aux) e\<cdot>((\<rho> f|` fv (Lam [x]. e))(x := v))))) \<and> HOL.induct_forall (\<lambda>xa. HOL.induct_forall (\<lambda>xb. p e (((\<lambda>uu_. uu_) \<circ> ESem_aux) e)))) \<or> (\<exists>ESem_aux e x. x1 = App e x \<and> x2 = (\<Lambda> \<rho>. tick\<cdot>(Fn_project\<cdot>(((\<lambda>uu_. uu_) \<circ> ESem_aux) e\<cdot>\<rho>)\<cdot>(\<rho> x))) \<and> HOL.induct_forall (\<lambda>xa. p e (((\<lambda>uu_. uu_) \<circ> ESem_aux) e))) \<or> (\<exists>ESem_aux x. x1 = Var x \<and> x2 = (\<Lambda> \<rho>. tick\<cdot>(\<rho> x))) \<or> (\<exists>ESem_aux as body. x1 = Terms.Let as body \<and> x2 = (\<Lambda> \<rho>. tick\<cdot>(((\<lambda>uu_. uu_) \<circ> ESem_aux) body\<cdot>(has_ESem.HSem ((\<lambda>uu_. uu_) \<circ> ESem_aux) as\<cdot>(\<rho> f|` fv (Terms.Let as body))))) \<and> HOL.induct_forall (\<lambda>x. p body (((\<lambda>uu_. uu_) \<circ> ESem_aux) body)) \<and> HOL.induct_forall (\<lambda>x. HOL.induct_forall (\<lambda>e. HOL.induct_implies (e \<in> snd ` set as) (p e (((\<lambda>uu_. uu_) \<circ> ESem_aux) e))))) \<or> (\<exists>ESem_aux b. x1 = Bool b \<and> x2 = (\<Lambda> \<rho>. tick\<cdot>(B\<cdot>(Discr b)))) \<or> (\<exists>ESem_aux scrut e1 e2. x1 = (scrut ? e1 : e2) \<and> x2 = (\<Lambda> \<rho>. tick\<cdot>(B_project\<cdot>(((\<lambda>uu_. uu_) \<circ> ESem_aux) scrut\<cdot>\<rho>)\<cdot>(((\<lambda>uu_. uu_) \<circ> ESem_aux) e1\<cdot>\<rho>)\<cdot>(((\<lambda>uu_. uu_) \<circ> ESem_aux) e2\<cdot>\<rho>))) \<and> HOL.induct_forall (\<lambda>x. p scrut (((\<lambda>uu_. uu_) \<circ> ESem_aux) scrut)) \<and> HOL.induct_forall (\<lambda>x. p e1 (((\<lambda>uu_. uu_) \<circ> ESem_aux) e1)) \<and> HOL.induct_forall (\<lambda>x. p e2 (((\<lambda>uu_. uu_) \<circ> ESem_aux) e2))))"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'c"   and generat :: "('d, 'e, 'f) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. generat = Pure x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. generat = IO x21 x22 \<longrightarrow> P (f2 x21 x22)))"
e_lam_intro2: fixes f :: "(val \<times> val) fset"   and x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v2 :: "val"   and v1 :: "val" assumes "VFun f \<in> E (ELam x e) \<rho>"   and "v2 \<in> E e ((x, v1) # \<rho>)" shows "VFun (finsert (v1, v2) f) \<in> E (ELam x e) \<rho>"
integrableI_bounded_set: fixes A :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and B :: "real" assumes "A \<in> sets M"   and "f \<in> borel_measurable M"   and "emeasure M A < \<infinity>"   and "AE x\<in>A in M. norm (f x) \<le> B"   and "AE x in M. x \<notin> A \<longrightarrow> f x = (0::'d)" shows "integrable M f"
fminus_fsubset_conv: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "(A |-| B |\<subseteq>| C) = (A |\<subseteq>| B |\<union>| C)"
bool_to_ternary_Unknown: fixes t :: "bool" shows "(bool_to_ternary t = TernaryUnknown) = False"
ide_leg1: shows "ide r\<^sub>0s\<^sub>1.p\<^sub>1"
simps_2: fixes verts :: "'d set"   and arcs :: "'c set"   and tail :: "'c \<Rightarrow> 'd"   and head :: "'c \<Rightarrow> 'd"   and more :: "'e" shows "arcs \<lparr>verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = arcs"
enabled_def: fixes K :: "'c \<Rightarrow> 'c pmf" shows "MC_syntax.enabled K \<equiv> gfp (\<lambda>p x1 x2. \<exists>\<omega> s. x1 = s \<and> x2 = \<omega> \<and> p (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
minus_sin_cos_eq: fixes x :: "'b" shows "- sin x = cos (x + of_real pi / (2::'b))"
simps_3: fixes f :: "'a itrm"   and x :: "'a itrm" shows "norm (f \<diamondop> x) = norm_nn (norm f) (norm x)"
unique_poincare_line_general: fixes u :: "complex_homo"   and v :: "complex_homo"   and l :: "circline" assumes "u \<noteq> v"   and "u \<noteq> inversion v"   and "u \<in> circline_set l"   and "v \<in> circline_set l"   and "is_poincare_line l" shows "l = poincare_line u v"
Fake_parts_sing_imp_Un: fixes c :: "msg"   and X :: "msg"   and H :: "msg set" assumes "c \<in> parts {X}"   and "X \<in> synth (analz H)" shows "c \<in> synth (analz H) \<union> parts H"
RHS: shows "to_fun (X_poly_minus R b \<otimes>\<^bsub>P\<^esub> g) a = (a \<ominus> b) \<otimes> to_fun g a"
isom_gch_units_transpTr1: fixes E :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> ('g set, 'h) Group_scheme"   and i :: "nat"   and j :: "nat" assumes "Ugp E"   and "Gchain n g"   and "i \<le> n"   and "j \<le> n"   and "g j \<cong> E"   and "i \<noteq> j" shows "insert j ({i. i \<le> n \<and> g i \<cong> E} - {i, j}) = {i. i \<le> n \<and> g i \<cong> E} - {i}"
inEx: fixes x :: "'e"   and A :: "'e set" assumes "x \<in> A" shows "\<exists>y\<in>A. y = x"
eq: shows "L2_set (octo_proj (x + y)) {..7} = L2_set (\<lambda>i. octo_proj x i + octo_proj y i) {..7}"
scale_eq_0_iff: fixes a :: "complex"   and x :: "'d" shows "(a *\<^sub>C x = (0::'d)) = (a = 0 \<or> x = (0::'d))"
refl_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<in> r))"
mult_mono': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq c d"   and "less_eq zero a"   and "less_eq zero c" shows "less_eq (times a c) (times b d)"
summable_add: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "summable f"   and "summable g" shows "summable (\<lambda>n. f n + g n)"
not_eq_newpkt_rreq: fixes hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "\<not> eq_newpkt (Rreq hops dip dsn dsk oip osn sip)"
Class_commutes_with_composition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "additive.Class a [+] additive.Class b = additive.Class (a + b)"
lub_least: fixes M :: "'a set"   and z :: "'a" assumes "M \<in> Chains leq"   and "M \<noteq> {}"   and "\<And>x. x \<in> M \<Longrightarrow> (x, z) \<in> leq" shows "(lub M, z) \<in> leq"
a_inv: fixes xa :: "int" shows "\<ominus>\<^bsub>\<Z>\<^esub> xa = - xa"
N_of_Liminf_state_fin: shows "N_of_state (Liminf_state Sts') = N_of_state (Liminf_state Sts)"
lincomb_insert2: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "a \<in> insert v S \<rightarrow> carrier K"   and "v \<notin> S"   and "v \<in> carrier V" shows "V.module.lincomb a (insert v S) = a v \<odot>\<^bsub>V\<^esub> v \<oplus>\<^bsub>V\<^esub> V.module.lincomb a S"
forallM_index_def: fixes P :: "'c \<Rightarrow> nat \<Rightarrow> 'd + unit"   and xs :: "'c list" shows "forallM_index P xs = forallM_index_aux P 0 xs"
emeasure_pmf_of_list: fixes xs :: "('b \<times> real) list"   and A :: "'b set" assumes "pmf_of_list_wf xs" shows "emeasure (measure_pmf (pmf_of_list xs)) A = ennreal (sum_list (map snd (filter (\<lambda>x. fst x \<in> A) xs)))"
gle: fixes x :: "'a"   and n :: "nat" assumes "x \<in> S" shows "\<bar>g n x\<bar> \<le> B"
expr_sem_op_eq_distr: fixes oper :: "pdf_operator"   and e :: "expr"   and t' :: "pdf_type"   and \<rho> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile> oper $$ e : t'"   and "free_vars e \<subseteq> V \<union> V'"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> (oper $$ e)) = distr (dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure t') (op_sem oper)"
select_append': fixes ys :: "'a list"   and zs :: "'a list"   and k :: "nat" assumes "\<forall>y\<in>set ys. \<forall>z\<in>set zs. y \<le> z"   and "k < length ys + length zs" shows "select k (ys @ zs) = (if k < length ys then select k ys else select (k - length ys) zs)"
size_mset: fixes xs :: "'d list" shows "size (mset xs) = length xs"
succ_inj: fixes r :: "('c \<times> 'c) set"   and i :: "'c"   and j :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}"   and "aboveS r j \<noteq> {}" shows "(succ r i = succ r j) = (i = j)"
finite_measure_compl: fixes M :: "'c measure"   and S :: "'c set" assumes "finite_measure M"   and "S \<in> sets M" shows "Sigma_Algebra.measure M (space M - S) = Sigma_Algebra.measure M (space M) - Sigma_Algebra.measure M S"
h1: shows "\<forall>i\<in>{..<p}. f i = 1"
lens_equiv_iff_lens_equiv': fixes X :: "'a \<Longrightarrow> 'b"   and Y :: "'c \<Longrightarrow> 'b" assumes "vwb_lens X"   and "vwb_lens Y" shows "X \<approx>\<^sub>L Y = X \<approx>\<^sub>L' Y"
partial_order_onD_3: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" assumes "partial_order_on A r" shows "antisym r"
elims_3: fixes x :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and xa :: "('a \<times> 'c) list \<times> 'o"   and xb :: "('a \<times> 'd) list \<times> 'o" assumes "\<not> prod_ord_raw x xa xb" obtains f :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" and xs :: "('a \<times> 'c) list" and ox :: "'o" and ys :: "('a \<times> 'd) list \<times> 'o" where "x = f" and "xa = (xs, ox)" and "xb = ys" and "\<not> prod_ord_pair ox f xs (sort_oalist_aux ox ys)"
dropWhile_ConsD: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" assumes "dropWhile P xs = y # ys" shows "\<not> P y"
integ_exis_horiz: shows "\<forall>(k, \<gamma>)\<in>two_chain_horizontal_boundary two_chain. line_integral_exists F {i} \<gamma>"
Ssegment_unique1: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Worder D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b" shows "\<not> ord_equiv (SIod D (Ssegment D b)) (SIod D (Ssegment D a))"
SSS: fixes r :: "'d rexp"   and r' :: "'d rexp" assumes "(~)\<^sup>*\<^sup>* r r'" shows "(~)\<^sup>*\<^sup>* (Star r) (Star r')"
eConseq: fixes A :: "triple set"   and P' :: "state \<Rightarrow> assn"   and e :: "expr"   and Q' :: "state \<Rightarrow> vassn"   and P :: "assn"   and Q :: "vassn" assumes "\<forall>Z. A \<turnstile>\<^sub>e {P' Z} e {Q' Z}"   and "\<forall>s v t. (\<forall>Z. P' Z s \<longrightarrow> Q' Z v t) \<longrightarrow> P s \<longrightarrow> Q v t" shows "A \<turnstile>\<^sub>e {P} e {Q}"
spmf_map: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c spmf"   and x :: "'b" shows "spmf (map_spmf f p) x = Sigma_Algebra.measure (measure_spmf p) (f -` {x})"
star_closed_unfold: fixes x :: "('a \<times> ('b \<times> 'a) list) set" assumes "t_star x = x" shows "x = t_one \<union> t_prod x x"
linear_surjective_right_inverse: fixes f :: "(('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real" assumes "\<forall>x. x \<in> sub.tangent_space p \<longrightarrow> f x \<in> tangent_space p"   and "linear_on (sub.tangent_space p) (tangent_space p) (*\<^sub>R) (*\<^sub>R) f"   and "f ` sub.tangent_space p = tangent_space p" shows "\<exists>g. (\<forall>x. x \<in> tangent_space p \<longrightarrow> g x \<in> sub.tangent_space p) \<and> linear_on (tangent_space p) (sub.tangent_space p) (*\<^sub>R) (*\<^sub>R) g \<and> (\<forall>x\<in>tangent_space p. f (g x) = x)"
front_tickFree_mono: fixes t :: "'a event list"   and r :: "'a event list" shows "front_tickFree (t @ r) \<and> r \<noteq> [] \<longrightarrow> tickFree t \<and> front_tickFree r"
ccSup_empty: 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" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
a_Var: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and x :: "Syntax.var"   and \<tau> :: "ty" assumes "\<Gamma> $$ x = Some \<tau>" shows "\<Gamma> \<turnstile> Var x, Var x, Var x : \<tau>"
select_convs_5: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "thread \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = thread"
fct_preimage_sigma_sets: fixes X :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "X \<in> space M \<rightarrow> space N" shows "sigma_sets (space M) {X -` B \<inter> space M |B. B \<in> sets N} = {X -` B \<inter> space M |B. B \<in> sets N}"
that: fixes character_data_ptr :: "(_) character_data_ptr" assumes "cast character_data_ptr = node_ptr" shows "thesis"
basis_exists: fixes V :: "'d set" obtains B :: "'d set" where "B \<subseteq> V" and "cindependent B" and "V \<subseteq> cspan B" and "card B = cdim V"
simps_39: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "mmonitorable_exec (formula.Or \<phi> \<psi>) = (fv \<phi> = fv \<psi> \<and> mmonitorable_exec \<phi> \<and> mmonitorable_exec \<psi>)"
set_ConsD: fixes y :: "'a"   and x :: "'a"   and xs :: "'a list" assumes "y \<in> set (x \<Colon> xs)" shows "y = x \<or> y \<in> set xs"
v0_edge_w0: shows "v0 \<rightarrow> w0"
sorted_nth_less: fixes xs :: "'a list"   and i :: "nat"   and j :: "nat" assumes "sortedP xs"   and "i < j"   and "j < length xs" shows "le (xs ! i) (xs ! j)"
bind_UNION: fixes A :: "'i set"   and f :: "'i \<Rightarrow> 'h set" shows "A \<bind> f = \<Union> (f ` A)"
hyps: assumes "\<forall>x. a dvd A x" shows "a dvd (\<Sum>x<n. A x * D x)"
associative_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "associative f \<equiv> \<forall>x y z. f x (f y z) = f (f x y) z"
r_prefixes_def: shows "r_prefixes \<equiv> r_swap (Pr 1 r_singleton_encode r_prefixes_aux)"