hcju/isabelleps huggingface dataset:
b_e_type_get_global_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Get_global i" shows "\<exists>t. ts' = ts @ [t] \<and> tg_t (global \<C> ! i) = t"
igSwapInp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'b, 'i, 'j) Iteration.model_scheme"   and zs :: "'e"   and z1 :: "'h"   and z2 :: "'h"   and inp :: "'a \<Rightarrow> 'b option" shows "igSwapInp MOD zs z1 z2 inp \<equiv> lift (igSwap MOD zs z1 z2) inp"
map_of_inject_set: fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" assumes "distinct (map fst xs)"   and "distinct (map fst ys)" shows "(map_of xs = map_of ys) = (set xs = set ys)"
isolation: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<omega> = c\<^sup>\<star> \<sqinter> c\<^sup>\<infinity>"
int_to_rat_def: fixes n :: "int" shows "to_rat n = rat_of_int n"
val_equiv_axiom3: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant"   and v'' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "valuation K v'"   and "v_equiv K v v'"   and "v_equiv K v' v''" shows "v_equiv K v v''"
subst_syntactic_and_axioms: shows "subst_syntactic_and Term.wellformed"
nth_append: fixes xs :: "'d list"   and ys :: "'d list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
boundL: assumes "w \<noteq> NEG_INF"   and "w \<noteq> POS_INF" shows "sint NEG_INF < sint (Rep_bword bw)"
add_mono_thms_linordered_field_4: fixes i :: "'c"   and j :: "'c"   and k :: "'c"   and l :: "'c" assumes "i \<le> j \<and> k < l" shows "i + k < j + l"
hom_div: fixes x :: "real"   and y :: "real" shows "of_real (x / y) = of_real x / of_real y"
indep_in_indep: fixes \<E> :: "'a set"   and X :: "'a set" assumes "X_insert.indep_in \<E> X" shows "indep_in (insert x X) X"
case_1: fixes f1 :: "floatarith \<Rightarrow> floatarith \<Rightarrow> floatarith \<Rightarrow> form \<Rightarrow> 'a"   and f2 :: "floatarith \<Rightarrow> floatarith \<Rightarrow> form \<Rightarrow> 'a"   and f3 :: "floatarith \<Rightarrow> floatarith \<Rightarrow> 'a"   and f4 :: "floatarith \<Rightarrow> floatarith \<Rightarrow> 'a"   and f5 :: "floatarith \<Rightarrow> floatarith \<Rightarrow> floatarith \<Rightarrow> 'a"   and f6 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and f7 :: "form \<Rightarrow> form \<Rightarrow> 'a"   and x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form" shows "(case Bound x11 x12 x13 x14 of Bound x xa xb xc \<Rightarrow> f1 x xa xb xc | Assign x xa xb \<Rightarrow> f2 x xa xb | Less x xa \<Rightarrow> f3 x xa | LessEqual x xa \<Rightarrow> f4 x xa | AtLeastAtMost x xa xb \<Rightarrow> f5 x xa xb | Conj x xa \<Rightarrow> f6 x xa | Disj x xa \<Rightarrow> f7 x xa) = f1 x11 x12 x13 x14"
id_stutter_sampler: fixes \<sigma> :: "nat \<Rightarrow> 'f" shows "stutter_sampler id \<sigma>"
invf_closed: fixes x :: "'a" assumes "x \<in> carrier K - {\<zero>}" shows "x\<^bsup>\<hyphen> K\<^esup> \<in> carrier K"
ps1_def: shows "ps1 = apply_ncrit ncrit data gs bs hs (apply_icrit icrit data gs bs hs (np gs bs hs data))"
lset_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lset (lappend xs ys) \<subseteq> lset xs \<union> lset ys"
\<II>\<nn>\<ll>_invertible: fixes op :: "'op"   and xs :: "'a list"   and opinl :: "'opinl" assumes "\<II>\<nn>\<ll> op xs = Some opinl" shows "\<DD>\<ee>\<II>\<nn>\<ll> opinl = op"
HF: shows "(H::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) \<circ> (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. (c::'a::refinement_lattice) ; x ; (d::'a::refinement_lattice) \<sqinter> d)"
labels_path_increasing: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation' list"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts'" shows "l u \<le> l' u"
index_drop: fixes x :: "'c"   and xs :: "'c list"   and i :: "nat" assumes "x \<in> set xs"   and "index xs x < i" shows "x \<notin> set (drop i xs)"
one_time_pad: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "map_spmf (([\<oplus>]) xs) (spmf_of_set (nlists UNIV n)) = spmf_of_set (nlists UNIV n)"
cis_eq: fixes a :: "real"   and b :: "real" assumes "cis a = cis b" shows "\<exists>x. a - b = 2 * real_of_int x * pi"
IH_2: fixes env :: "('a, bool) mapping" shows "reduced (reduce env t2) (Mapping.keys env)"
vsv_vimage_vinsert: fixes a :: "V"   and Aa :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f)" shows "VLambda Aa f `\<^sub>\<circ> vinsert a A = vinsert (VLambda Aa f\<lparr>a\<rparr>) (VLambda Aa f `\<^sub>\<circ> A)"
simps_1: fixes ys :: "'b list" shows "gcp [] ys = []"
hull_induct: fixes a :: "'c"   and Q :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> Q hull S"   and "\<And>x. x \<in> S \<Longrightarrow> P x"   and "Q {x. P x}" shows "P a"
rbt_height_bheight: fixes t :: "('b \<times> color) tree" assumes "rbt t" shows "real (height t) / 2 \<le> real (bheight t)"
compE2_blocks1: fixes n :: "nat"   and Ts :: "ty list"   and body :: "'a expr1" shows "compE2 (blocks1 n Ts body) = compE2 body"
stable_match: fixes \<mu> :: "('m \<times> 'w) set" assumes "stable \<mu>" shows "match \<mu>"
homotopy_eqv_empty1: fixes S :: "'c set" shows "(S homotopy_eqv {}) = (S = {})"
D_def: fixes x :: "'a \<times> 'b" shows "D x = frechet_derivative (\<lambda>(x, y). f x y) (at x)"
sectionI: fixes g :: "'a"   and f :: "'a" assumes "S.ide (g \<cdot>\<^sub>S f)" shows "S.section f"
filter_pdevs_raw_def: fixes I :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" shows "filter_pdevs_raw I X = (\<lambda>i. if I i (X i) then X i else (0::'b))"
assms_2: shows "s \<cdot> \<sigma> = t \<cdot> \<tau>"
divide_divide_eq_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide a b) c = divide a (times b c)"
absolutely_integrable_negligible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible S" shows "f absolutely_integrable_on S"
fset_of_list_append: fixes xs :: "'c list"   and ys :: "'c list" shows "fset_of_list (xs @ ys) = fset_of_list xs |\<union>| fset_of_list ys"
IH: assumes "wf\<^sub>s\<^sub>t (V \<union> fv t') (S @ S')"   and "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> (V \<union> fv t')" shows "wf\<^sub>s\<^sub>t (V \<union> fv t') (S @ receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S')"
boundedI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "a \<le> c" shows "a \<le> inf b c"
normalize_dvd_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 a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times (normalize a) b = dvd.dvd times a b"
SN_on_all_reducts_SN_on_conv: fixes r :: "('b \<times> 'b) set"   and a :: "'b" shows "SN_on r {a} = (\<forall>b. (a, b) \<in> r \<longrightarrow> SN_on r {b})"
eq_onD: fixes r :: "'a \<Rightarrow> 'b"   and s :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and x :: "'a" assumes "r = s on X"   and "x \<in> X" shows "r x = s x"
minf_4: fixes t :: "'b" shows "\<exists>z. \<forall>x<z. (x \<noteq> t) = True"
measurable_discrete_difference: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and X :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. x \<in> X \<Longrightarrow> g x \<in> space N"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "g \<in> M \<rightarrow>\<^sub>M N"
\<Phi>_def: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>" shows "two_set_categories_bij_betw_Univ.\<Phi> S S' \<AA> \<phi> f = (if partial_magma.arr S f then set_category_given_img.mkArr S' (set_category.some_img S' \<AA>) \<AA> (set_category_data.set S' (set_category.some_img S' \<AA>) (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> (partial_magma.dom S f))) (set_category_data.set S' (set_category.some_img S' \<AA>) (two_set_categories_bij_betw_Univ.\<Phi>o S S' \<AA> \<phi> (partial_magma.cod S f))) (two_set_categories_bij_betw_Univ.\<Phi>a S \<AA> \<phi> f) else partial_magma.null S')"
absorb_iff2: fixes b :: "'q multiset"   and a :: "'q multiset" shows "(b \<subseteq># a) = (a \<inter># b = b)"
vars\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>t_bvars\<^sub>s\<^sub>t: fixes S :: "('d, 'c) strand_step list" shows "vars\<^sub>s\<^sub>t S = fv\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>t S"
arcs_app_inv_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "iso_arcs (inv_iso hom) ` iso_arcs hom ` arcs G = arcs G"
Int_absorb: fixes A :: "'a set" shows "A \<inter> A = A"
list_all2_Cons2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and y :: "'d"   and ys :: "'d list" shows "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
x: shows "x \<in> carrier (G\<lparr>carrier := H\<rparr>)"
\<phi>_def: shows "d\<^sub>0e\<^sub>0.\<phi> \<equiv> transformation_by_components.map (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>)) (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>)) (identity_functor.map (subcategory.comp (\<cdot>) (\<lambda>\<mu>. \<guillemotleft>\<mu> : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>))) (\<lambda>f. (e\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[d\<^sub>1, f \<star> e\<^sub>0, d\<^sub>0]) \<cdot> \<a>[e\<^sub>1, d\<^sub>1, (f \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, e\<^sub>0, d\<^sub>0]) \<cdot> (local.inv \<epsilon>\<^sub>1 \<star> f \<star> local.inv \<epsilon>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src d\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f])"
l2_inv4D: fixes Ra :: "fid_t"   and gny :: "msg"   and B :: "agent"   and A :: "agent"   and Nr :: "msg"   and x :: "l2_state" assumes "hmac \<langle>Number 0, Exp Gen (NonceF (Ra $ nx)), gny, Messages.Agent B, Messages.Agent A\<rangle> (hmac (NonceF (Ra $ ni)) Nr) \<in> parts (extr (bad x) (ik x) (chan x))"   and "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "A \<notin> bad x \<and> B \<notin> bad x"   and "x \<in> l2_inv4" shows "\<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress x Rb) {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgny = Some gny \<and> sklvl1.guessed_frame Rb xnr = Some Nr \<and> sklvl1.guessed_frame Rb xni = Some (NonceF (Ra $ ni)) \<and> sklvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx)))"
grounding_ground: fixes C :: "'a literal multiset"   and M :: "'a literal multiset set" assumes "C \<in> grounding_of_clss M" shows "is_ground_cls C"
surj_const_mult: fixes a :: "'a" assumes "a \<in> carrier G" shows "(\<otimes>) a ` carrier G = carrier G"
deg_rr: shows "degree (smult lc r) \<le> dr"
real_of_nat_div: fixes d :: "nat"   and n :: "nat" assumes "d dvd n" shows "real (n div d) = real n / real d"
subformulas\<^sub>\<mu>_subfrmlsn: fixes \<phi> :: "'b ltln" shows "subformulas\<^sub>\<mu> \<phi> \<subseteq> subfrmlsn \<phi>"
less_imp_le: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "x \<le> y"
vector_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "is_vector y" shows "(x \<cdot> y\<^sup>T) ; z = x ; (y \<cdot> z)"
case_distrib: fixes h :: "'g \<Rightarrow> 'f"   and f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and prod :: "'h \<times> 'i" shows "h (case prod of (x, xa) \<Rightarrow> f x xa) = (case prod of (x1, x2) \<Rightarrow> h (f x1 x2))"
is_sig_red_sing_regI: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_sig_red sing_reg top_tail F p" shows "is_sig_red (\<preceq>\<^sub>t) top_tail F p"
cnf_of_encode_interfering_operator_exclusion_is_ii: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "set (concat (map (\<lambda>(op\<^sub>1, op\<^sub>2). map (\<lambda>k. encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2) [0..<t]) (filter (\<lambda>(op\<^sub>1, op\<^sub>2). index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2) (List.product (\<Pi>\<^sub>\<O>) (\<Pi>\<^sub>\<O>))))) = (\<Union>(op\<^sub>1, op\<^sub>2)\<in>{(op\<^sub>1, op\<^sub>2). (op\<^sub>1, op\<^sub>2) \<in> set (\<Pi>\<^sub>\<O>) \<times> set (\<Pi>\<^sub>\<O>) \<and> index (\<Pi>\<^sub>\<O>) op\<^sub>1 \<noteq> index (\<Pi>\<^sub>\<O>) op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2}. (\<lambda>k. encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2) ` {0..<t})"
BSD_implies_SD_for_modified_view: fixes \<V>' :: "'e V_rec" assumes "BSD \<V>' Tr\<^bsub>(induceES SES)\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>'\<^esub> \<union> N\<^bsub>\<V>'\<^esub>, N = {}, C = C\<^bsub>\<V>'\<^esub>\<rparr>" shows "SD \<V>' Tr\<^bsub>(induceES SES)\<^esub>"
sup_Some: fixes x :: "'a"   and y :: "'a" shows "sup (Some x) (Some y) = Some (sup x y)"
rel_option_bind: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option"   and f :: "'c \<Rightarrow> 'c option"   and g :: "'d \<Rightarrow> 'd option" assumes "rel_option R x y"   and "\<And>a b. R a b \<Longrightarrow> rel_option R (f a) (g b)" shows "rel_option R (x \<bind> f) (y \<bind> g)"
indep_sets_finite: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'a set set" assumes "I \<noteq> {}"   and "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"   and "\<And>i. i \<in> I \<Longrightarrow> space M \<in> F i" shows "indep_sets F I = (\<forall>A\<in>Pi I F. prob (\<Inter> (A ` I)) = (\<Prod>j\<in>I. prob (A j)))"
cases: fixes a1 :: "int"   and a2 :: "(int \<times> Instr) list"   and a3 :: "Expr"   and a4 :: "(int \<times> Instr) list \<times> int" assumes "(a1, a2, a3, a4) \<in> compileExpr" obtains l :: "int" and code :: "(int \<times> Instr) list" and p :: "Prim" and code1 :: "(int \<times> Instr) list" and l1 :: "int" and OUT :: "(int \<times> Instr) list \<times> int" where "a1 = l__" and "a2 = code__" and "a3 = PrimE p__" and "a4 = OUT__" and "(l__, code__, p__, code1__, l1__) \<in> compilePrim" and "OUT__ = (code1__[l1__\<mapsto>vreturn], l1__ + 1)"   | l :: "int" and code :: "(int \<times> Instr) list" and p :: "Prim" and code1 :: "(int \<times> Instr) list" and l1 :: "int" and code2 :: "(int \<times> Instr) list" and l2 :: "int" and x :: "Var" and e :: "Expr" and OUT :: "(int \<times> Instr) list \<times> int" where "a1 = la__" and "a2 = codea__" and "a3 = LetE x__ p e" and "a4 = OUTa__" and "(la__, codea__, p, code1, l1) \<in> compilePrim" and "(code2, l2) = (code1[l1\<mapsto>store x__], l1 + 1)" and "(l2, code2, e, OUTa__) \<in> compileExpr"   | l :: "int" and code :: "(int \<times> Instr) list" and e2 :: "Expr" and codeElse :: "(int \<times> Instr) list" and XXX :: "int" and e1 :: "Expr" and codeThen :: "(int \<times> Instr) list" and YYY :: "int" and OUT :: "(int \<times> Instr) list \<times> int" and x :: "Var" where "a1 = lb__" and "a2 = codeb__" and "a3 = CondE xa__ e1__ e2__" and "a4 = OUTb__" and "(lb__ + 2, codeb__, e2__, codeElse, XXX) \<in> compileExpr" and "(XXX, codeElse, e1__, codeThen, YYY) \<in> compileExpr" and "OUTb__ = (codeThen[lb__\<mapsto>load xa__][lb__ + 1\<mapsto>iftrue XXX], YYY)"   | l :: "int" and code :: "(int \<times> Instr) list" and e2 :: "Expr" and codeCons :: "(int \<times> Instr) list" and lNil :: "int" and e1 :: "Expr" and codeNil :: "(int \<times> Instr) list" and lRes :: "int" and OUT :: "(int \<times> Instr) list \<times> int" and x :: "Var" and FALSE :: "Val" and h :: "Var" and t :: "Var" where "a1 = l" and "a2 = code" and "a3 = MatchE x e1 h t e2" and "a4 = OUT" and "(l + 9, code, e2, codeCons, lNil) \<in> compileExpr" and "(lNil, codeCons, e1, codeNil, lRes) \<in> compileExpr" and "OUT = (codeNil[l\<mapsto>load x][l + 1\<mapsto>unop (\<lambda>v. if v = RVal Nullref then TRUE else FALSE)][l + 2\<mapsto>iftrue lNil][l + 3\<mapsto>load x][l + 4\<mapsto>getfield LIST HD][l + 5\<mapsto>store h][l + 6\<mapsto>load x][l + 7\<mapsto>getfield LIST TL][l + 8\<mapsto>store t], lRes)"
interior_of_maximal: fixes T :: "'a set"   and S :: "'a set"   and X :: "'a topology" assumes "T \<subseteq> S"   and "openin X T" shows "T \<subseteq> X interior_of S"
minus_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> y \<squnion> z) = (x - y \<le> z)"
gen_lossless_gpv_map_id: fixes b :: "bool"   and \<I> :: "('d, 'e) \<I>"   and f :: "'g \<Rightarrow> 'f"   and gpv :: "('g, 'd, 'e) gpv" shows "gen_lossless_gpv b \<I> (map_gpv f id gpv) = gen_lossless_gpv b \<I> gpv"
inject_1: fixes x1 :: "(char list, val, char list \<times> label, char list) edge_kind"   and y1 :: "(char list, val, char list \<times> label, char list) edge_kind" shows "(IEdge x1 = IEdge y1) = (x1 = y1)"
Gauss_Jordan_upt_k_PA_def: fixes A :: "(('d, 'f) vec, 'e) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k_PA A k = (let foldl = foldl Gauss_Jordan_column_k_PA (mat (1::'d), 0, A) [0..<Suc k] in (fst foldl, snd (snd foldl)))"
cond_B_def: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list" assumes "hlde a b" shows "hlde.cond_B a b x = (\<forall>k\<le>length a. take k a \<bullet> take k x \<le> b \<bullet> map (hlde_ops.max_y a b (take k x)) [0..<length b])"
Scons_def: fixes M :: "('c, 'd) node set"   and N :: "('c, 'd) node set" shows "Scons M N \<equiv> Push_Node (Inr 1) ` M \<union> Push_Node (Inr (Suc 1)) ` N"
cut_ge_in_imp: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I \<down>\<ge> t" shows "x \<in> I"
op_map_delete_def: shows "op_map_delete \<equiv> \<lambda>k m. m(k := None)"
14: shows "taq {x. poly p x = 0} q \<equiv> int (card {x. poly p x = 0 \<and> 0 < poly q x}) + - 1 * int (card {x. poly p x = 0 \<and> poly q x < 0})"
cprank_max_cprank: fixes A :: "'b tensor" shows "cprank_max (cprank A) A"
le_less: fixes x :: "'a word"   and y :: "'a word" shows "(x \<le>s y) = (x <s y \<or> x = y)"
ranked_tree_automaton_axioms: shows "ranked_tree_automaton TA A"
strict_prefernce_iff_strict_utility: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "(x \<succeq> y \<and> (y, x) \<notin> relation) = (u y < u x)"
coinduct: fixes G :: "('c, 'd) Graph_scheme"   and X :: "'c llist \<Rightarrow> bool"   and x :: "'c llist" assumes "Digraph G"   and "X x"   and "\<And>x. X x \<Longrightarrow> x = LNil \<or> (\<exists>v. x = LCons v LNil \<and> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)) \<or> (\<exists>Ps v. x = LCons v Ps \<and> \<not> lnull Ps \<and> (X Ps \<or> Digraph.maximal_path G Ps))" shows "Digraph.maximal_path G x"
D_def: shows "D = min (real_of_ereal C) (Min {u n |n. n \<le> N})"
lspasl_p_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj" shows "(heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> heap_sep_algebra.tern_rel plus sep_disj h1 h2 h3) = (heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> h0 = h3)"
RedBinOp: fixes bop :: "bop"   and v\<^sub>1 :: "val"   and v\<^sub>2 :: "val"   and v :: "val"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "binop (bop, v\<^sub>1, v\<^sub>2) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>Val v\<^sub>1 \<guillemotleft>bop\<guillemotright> Val v\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>Val v,s,b\<rangle>"
integrable_divide: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "integrable M f" shows "integrable M (\<lambda>x. f x / y)"
Rep_uexpr: fixes x :: "('b, 'a) uexpr" shows "\<lbrakk>x\<rbrakk>\<^sub>e \<in> UNIV"
TLA2E: fixes P :: "state \<Rightarrow> bool"   and sigma :: "behavior"   and A :: "state \<times> state \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>P"   and "\<turnstile> $P \<and> P$ \<longrightarrow> A" shows "sigma \<Turnstile> \<box>A"
geq_mix_imp_weak_pref: fixes \<alpha> :: "real"   and \<beta> :: "real" assumes "\<alpha> \<in> {0..1}"   and "\<beta> \<in> {0..1}"   and "\<beta> \<le> \<alpha>" shows "mix_pmf \<alpha> local.some_best local.some_worst \<succeq>[\<R>] mix_pmf \<beta> local.some_best local.some_worst"
Hp: shows "G\<lparr>carrier := N <#> N1\<rparr> Mod N1 \<cong> G\<lparr>carrier := N\<rparr> Mod N \<inter> N1"
poly_ge_def: fixes p :: "('d monom \<times> 'e) list"   and q :: "('d monom \<times> 'e) list" shows "p \<ge>p q = (\<forall>\<alpha>. pos_assign \<alpha> \<longrightarrow> eval_poly \<alpha> q \<le> eval_poly \<alpha> p)"
bits_mod_div_trivial: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (modulo a b) b = zero"
Polys_closed_minus: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p - q \<in> P[X]"
value_plus_never_string: fixes a :: "value option"   and b :: "value option"   and x :: "String.literal" shows "value_plus a b \<noteq> Some (value.Str x)"
listrel_eq_len: fixes xs :: "'c list"   and ys :: "'d list"   and r :: "('c \<times> 'd) set" assumes "(xs, ys) \<in> listrel r" shows "length xs = length ys"
Hermite_of_upt_row_preserves_nonzero_rows_ge: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and k :: "nat"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<not> is_zero_row i A"   and "k \<le> mod_type_class.to_nat i" shows "\<not> is_zero_row i (Hermite_of_upt_row_i A k ass res)"
\<T>o_def: fixes e :: "'s"   and b :: "'c" shows "\<T>o e b = S.mkArr (Hom.set (b, a)) (F.SET b) (\<lambda>x. F.FUN (\<psi> (b, a) x) e)"
reduction_inv_9: fixes n :: "nat"   and t' :: "db"   and P :: "bool" assumes "DBNum n \<rightarrow>\<^sub>v t'" shows "P"