hcju/isabelleps huggingface dataset:
bisim1AAss2: fixes P :: "'a prog"   and h :: "'heap"   and i :: "'addr expr1"   and i' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and a :: "'addr expr1"   and e :: "'addr expr1"   and v :: "'addr val" assumes "P,i,h \<turnstile> (i', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (Val v\<lfloor>i'\<rceil> := e, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 a) + pc, xcp)"
resultant_impl_main_def: fixes div_exp :: "'a \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and G1 :: "'a poly"   and G2 :: "'a poly" shows "resultant_impl_main div_exp G1 G2 = (if G2 = 0 then if degree G1 = 0 then 1::'a else (0::'a) else case subresultant_prs div_exp G1 G2 of (Gk, hk) \<Rightarrow> if degree Gk = 0 then hk else (0::'a))"
distinct_4: fixes name' :: "name"   and lam' :: "lam"   and name :: "name" shows "(Lam [name'].lam') \<noteq> Var name"
monofunE: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "monofun f"   and "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
POR_result_rel_def: shows "POR_result_rel \<equiv> \<lambda>i. case i of Three \<Rightarrow> ValFF | _ \<Rightarrow> ValTT"
assms_3: assumes "a \<in># C"   and "A = B + (C - {#a#})" shows "P"
subbdt_ex: fixes lst :: "dag"   and stp :: "ref"   and rst :: "dag"   and t :: "dag"   and var :: "ref \<Rightarrow> nat"   and bdt1 :: "bdt" assumes "Node lst stp rst \<le> t"   and "bdt t var = Some bdt1" shows "\<exists>bdt2. bdt (Node lst stp rst) var = Some bdt2"
surj_imp_inv_eq: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b" assumes "surj f"   and "\<And>x. g (f x) = x" shows "inv f = g"
sign_changes'_def: shows "sign_changes' \<equiv> \<lambda>ps x. \<Sum>ps'\<leftarrow>split_sign_changes ps x. sign_changes ps' x"
simps: fixes r :: "'c \<Rightarrow> 'a \<times> 'b"   and a :: "'a \<times> 'b \<Rightarrow> 'c" shows "abst (Tuple_Isomorphism r a) = a"
a0: fixes m :: "nat"   and i :: "nat" assumes "m < 2 ^ n"   and "0 \<le> m"   and "i \<le> n" shows "bin_rep_aux n m ! i = m mod 2 ^ (n - i) div 2 ^ (n - 1 - i)"
sizelemmas_3: fixes A :: "formula"   and z :: "signs" shows "size A < size (FAll z A)"
intro: fixes \<U> :: "'b set"   and \<A> :: "'b set multiset"   and \<V> :: "'b set"   and \<B> :: "'b set multiset" assumes "sub_incidence_system \<U> \<A> \<V> \<B>"   and "design \<V> \<B>" shows "sub_design \<U> \<A> \<V> \<B>"
bind_simps_2: fixes x :: "'d"   and xs :: "'d list"   and f :: "'d \<Rightarrow> 'c list" shows "List.bind (x # xs) f = f x @ List.bind xs f"
strongdlHPPB_aux: fixes d :: "'d"   and PP :: "nat set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set"   and i :: "nat"   and V :: "('exp, 'id) MWLsCom list"   and V' :: "('exp, 'id) MWLsCom list"   and m1 :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m2 :: "'id \<Rightarrow> 'val"   and m1' :: "'id \<Rightarrow> 'val" assumes "SdlHPPB d PP R"   and "i < length V"   and "(V, V') \<in> R"   and "SRabbr (V ! i) m1 \<alpha> p m2"   and "m1 \<sim>\<^bsub>d,htchLocSet PP\<^esub> m1'" shows "\<exists>p' \<alpha>' m2'. SRabbr (V' ! i) m1' \<alpha>' p' m2' \<and> stepResultsinR p p' R \<and> (\<alpha>, \<alpha>') \<in> R \<and> dhequality_alternative d PP (pp (V ! i)) m2 m2'"
components_snd_sym_preproc_aux_subset: fixes vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list" assumes "set vs \<subseteq> Keys (set fs)" shows "component_of_term ` Keys (set (snd (sym_preproc_aux gs ks (vs, fs)))) \<subseteq> component_of_term ` Keys (set gs \<union> set fs)"
list_of_vec_0: fixes n :: "nat" shows "list_of_vec (0\<^sub>v n) = replicate n (0::'b)"
dres_order_simps_2: fixes x :: "'b dres" shows "(dFAIL \<le> x) = (x = dFAIL)"
filter_filter: fixes P :: "'g \<Rightarrow> bool"   and Q :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "filter P (filter Q xs) = filter (\<lambda>x. Q x \<and> P x) xs"
DFAs_axioms: shows "DFAs alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1 alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2 letter_eq"
polymap_apply: fixes v :: "'v"   and p :: "'f poly" assumes "v \<in> V" shows "polymap p v = lincomb' (coeffs p) (map (\<lambda>S. S v) (map endpow [0..<Suc (degree p)]))"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "Left_a.in_hom f a' a"   and "Left_a.iso (H\<^sub>L a f)" shows "Left_a.iso f"
\<F>\<G>_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "\<F>\<G> \<psi> w \<subseteq> \<F>\<G> \<phi> w"
assms_3: shows "\<zero>\<^bsub>L\<^esub> = indexed_const \<zero>"
rbt_comp_insert_def: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order" shows "RBT_Comparator_Impl.rbt_comp_insert c = RBT_Comparator_Impl.rbt_comp_insert_with_key c (\<lambda>_ _ nv. nv)"
mult_eq': shows "(\<otimes>\<^bsub>G\<^esub>) = (\<lambda>x y. x * y mod p)"
ord_eq_less_trans: fixes a :: "'b"   and b :: "'b"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b" assumes "a = b"   and "less b c" shows "less a c"
Ej_def: fixes a :: "nat list"   and b :: "nat list"   and j :: "nat"   and x :: "nat list" shows "hlde_ops.Ej a b j x = {Linear_Diophantine_Equations.eij a b i j - 1 |i. i < length x \<and> Linear_Diophantine_Equations.dij a b i j \<le> x ! i}"
PseudoHA_HAInitState: fixes A :: "('d, 'e, 'f) seqauto"   and D :: "'f data" shows "HAInitState (PseudoHA A D) = InitState A"
concat_def: shows "concat = foldr\<cdot>append\<cdot>[]"
ladder_j_cut: fixes index :: "nat"   and L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "index < length L" shows "ladder_j (ladder_cut L n) index = ladder_j L index"
sigma_sets_subset: fixes a :: "'a set set" assumes "a \<subseteq> M" shows "sigma_sets \<Omega> a \<subseteq> M"
StuckProp: fixes c :: "('a, 'b, 'c) com"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "c \<noteq> Skip"   and "redex c = c" shows "\<Gamma>\<turnstile> (c, Stuck) \<rightarrow> (Skip, Stuck)"
dropWhile_eq_drop: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "dropWhile P xs = drop (length (takeWhile P xs)) xs"
closure_complement: fixes T :: "'b set set"   and A :: "'b set" assumes "topology T" shows "closure\<^bsub>T\<^esub> (carrier\<^bsub>T\<^esub> - A) = carrier\<^bsub>T\<^esub> - core\<^bsub>T\<^esub> A"
sqrtu_gt_0: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "0 < sqrtu x n"
nop_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (nop_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
is_natural_transformation: shows "natural_transformation A B G G \<sigma>'\<sigma>.map"
that: fixes t :: "dbtm"   and A :: "dbfm" assumes "u = \<lbrakk>quot_dbtm t\<rbrakk>e"   and "x = \<lbrakk>quot_dbfm A\<rbrakk>e"   and "x' = \<lbrakk>quot_dbfm (subst_dbfm t (decode_Var v) A)\<rbrakk>e" shows "thesis"
deleted_not_domA: fixes x :: "'c"   and \<Gamma> :: "('c \<times> 'd) list" shows "x \<notin> domA (delete x \<Gamma>)"
has_vderiv_on_union_closed: fixes f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and s :: "real set"   and t :: "real set" assumes "(f has_vderiv_on g) s"   and "(f has_vderiv_on g) t"   and "closed s"   and "closed t" shows "(f has_vderiv_on g) (s \<union> t)"
MC_T_not_sconst: fixes f :: "'s \<Rightarrow> 'b"   and u :: "'b"   and X :: "'s cfg set"   and r :: "ennreal"   and cfg :: "'s cfg"   and x :: "'s" assumes "\<forall>l. \<forall>cfg\<in>\<Union> (cfg_on ` f -` {u}) \<inter> X. emeasure (measure_pmf (action cfg)) (f -` {u}) \<le> r"   and "r < 1"   and "cfg \<in> cfg_on x"   and "almost_everywhere (MC.T cfg) (pred_stream (\<lambda>x. x \<in> X))" shows "AE \<omega> in MC.T cfg. smap (f \<circ> state) \<omega> \<noteq> sconst u"
g_def: shows "g = f \<oplus> (j \<circ> map_prod Some Some \<circ> prod.swap)"
is_root: shows "is_root (p \<otimes>\<^bsub>poly_ring R\<^esub> q) x"
canceled_def: fixes l :: "(bool \<times> 'c) list" shows "canceled l = (\<not> Domainp cancels_to_1 l)"
psubset_contr: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subset> B"   and "B \<subset> A" shows "False"
sp_append: fixes F :: "'a \<Rightarrow> 'b fundef option"   and xs :: "('c, 'd, 'a, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr list"   and ys :: "('c, 'd, 'a, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr list"   and \<Sigma> :: "Unboxed.type option list" shows "Subx.sp F (xs @ ys) \<Sigma> = Subx.sp F xs \<Sigma> \<bind> Subx.sp F ys"
eliminate_entries_gen_def: fixes v :: "nat \<Rightarrow> 'a"   and A :: "'a mat"   and I :: "nat"   and J :: "nat" shows "local.eliminate_entries_gen v A I J = mat (dim_row A) (dim_col A) (\<lambda>(i, j). if i \<noteq> I then minus (A $$ (i, j)) (times (v i) (A $$ (I, j))) else A $$ (i, j))"
compact_uniformly_continuous: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on S f"   and "compact S" shows "uniformly_continuous_on S f"
ord_Zp_not_equal_imp_notequal_1: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp"   and "ord_Zp x \<noteq> ord_Zp y" shows "x \<noteq> y"
siterate: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "siterate f x = x ## siterate f (f x)"
OclANY_def: fixes x :: "'d state \<times> 'd state \<Rightarrow> Sequence('e)" shows "x->any\<^sub>S\<^sub>e\<^sub>q() = (\<lambda>\<tau>. if x \<tau> = invalid \<tau> then \<bottom> else case \<lceil>\<lceil>Rep_Sequence\<^sub>b\<^sub>a\<^sub>s\<^sub>e (x \<tau>)\<rceil>\<rceil> of [] \<Rightarrow> \<bottom> | a # list \<Rightarrow> hd (a # list))"
assms_3: shows "f : c \<mapsto>\<^bsub>dg_prod J \<AA>\<^esub> d"
lc\<^sub>1_def: shows "lc\<^sub>1 \<equiv> \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>"
coeff_pcompose_semiring_closed: fixes R :: "'b set"   and p :: "'b poly"   and q :: "'b poly"   and i :: "nat" assumes "(0::'b) \<in> R"   and "\<And>x y. \<lbrakk>x \<in> R; y \<in> R\<rbrakk> \<Longrightarrow> x + y \<in> R"   and "\<And>x y. \<lbrakk>x \<in> R; y \<in> R\<rbrakk> \<Longrightarrow> x * y \<in> R"   and "\<And>i. coeff p i \<in> R"   and "\<And>i. coeff q i \<in> R" shows "coeff (p \<circ>\<^sub>p q) i \<in> R"
face_of_polytope_polytope: fixes S :: "'b set"   and F :: "'b set" assumes "polytope S"   and "F face_of S" shows "polytope F"
that: fixes \<pi> :: "'a \<Rightarrow> 'a" assumes "\<pi> permutes agents"   and "prefs_from_table xs \<circ> \<pi> = prefs_from_table ys" shows "thesis"
nn_integral_0_iff: fixes u :: "'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (emeasure M {x \<in> space M. u x \<noteq> 0} = 0)"
convex_on_subset: fixes t :: "'a set"   and f :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "convex_on t f"   and "S \<subseteq> t" shows "convex_on S f"
simps: fixes a1 :: "boolean_unop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'a OCL_Types.type" shows "boolean_unop_type a1 a2 a3 = (\<exists>\<tau>. a1 = NotOp \<and> a2 = \<tau> \<and> a3 = \<tau> \<and> \<tau> \<le> Boolean[?])"
lub_singleton: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.ccpo Sup less_eq less" shows "lub_singleton Sup"
non_qAFresh_ordLess_var: fixes XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "Ball XS qGood"   and "|XS| <o |UNIV|" shows "|{uu_. \<exists>x X. uu_ = x \<and> X \<in> XS \<and> \<not> qAFresh xs x X}| <o |UNIV|"
DirProd_list_carrier: fixes G :: "('c, 'd) monoid_scheme"   and Gs :: "('c, 'd) monoid_scheme list" shows "carrier (DirProd_list (G # Gs)) = (\<lambda>(x, xs). x # xs) ` (carrier G \<times> carrier (DirProd_list Gs))"
SA_States_disjunct2: fixes A :: "('d, 'e, 'f) seqauto"   and C :: "'d set"   and B :: "('d, 'e, 'f) seqauto" assumes "States A \<inter> C = {}"   and "States B \<subseteq> C" shows "B \<noteq> A"
isometry_onD: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "isometry_on X f"   and "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) = dist x y"
list_update_id': fixes v :: "'b"   and xs :: "'b list"   and i :: "nat" assumes "v = xs ! i" shows "xs[i := v] = xs"
preserves_iso: fixes f :: "'d" assumes "local.iso f" shows "local.iso (\<rho>' f)"
count_roots_below_code: fixes p :: "real poly"   and a :: "real" shows "count_roots_below p a = (let q = pderiv p in if p = 0 then 0 else if poly p a \<noteq> 0 \<or> poly q a \<noteq> 0 then let ps = sturm p in sign_changes_neg_inf ps - sign_changes ps a else let ps = sturm_squarefree p in sign_changes_neg_inf ps - sign_changes ps a)"
A: shows "A \<in> carrier_mat CARD('nr) CARD('nc)"
globalize_local_consts: fixes is' :: "nat list"   and "is" :: "nat list"   and f :: "'var \<Rightarrow> 'var"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree" assumes "is' \<in> it_paths (globalize is f t)" shows "subst_lconsts (iSubst (tree_at (globalize is f t) is')) \<subseteq> fresh_at_path (globalize is f t) is' \<union> range f"
correct_state_def: fixes P :: "jvm_prog"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list" shows "correct_state P \<Phi> \<equiv> \<lambda>(xp, h, frs, sh). case xp of None \<Rightarrow> case frs of [] \<Rightarrow> True | f # fs \<Rightarrow> P \<turnstile> h \<surd> \<and> P,h \<turnstile>\<^sub>s sh \<surd> \<and> conf_clinit P sh frs \<and> (let (stk, loc, C, M, pc, ics) = f in \<exists>b Ts T mxs mxl\<^sub>0 is xt \<tau>. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = (mxs, mxl\<^sub>0, is, xt) in C \<and> \<Phi> C M ! pc = \<lfloor>\<tau>\<rfloor> \<and> conf_f P h sh \<tau> is f \<and> conf_fs P h sh \<Phi> C M (length Ts) T fs) | \<lfloor>x\<rfloor> \<Rightarrow> frs = []"
exhaust_sel: fixes rule :: "'b rule" obtains "rule = Rule (get_match rule) (get_action rule)"
find_match_rel: fixes cs :: "(Term_Class.term \<times> pterm) list"   and cs' :: "(Term_Class.term \<times> sterm) list"   and t :: "pterm"   and t' :: "sterm" assumes "list_all2 (rel_prod (=) (\<lambda>p. (=) (pterm_to_sterm p))) cs cs'"   and "pterm_to_sterm t = t'" shows "rel_option (rel_prod srelated'.P_env (rel_prod (=) (\<lambda>p. (=) (pterm_to_sterm p)))) (find_match cs t) (find_match cs' t')"
assms_7: shows "(G ===> (=)) defs1 defs2"
exhaust: fixes y :: "pr_operation'" obtains "y = RELABEL'"   | "y = NONSAT_PUSH'"   | x3 :: "nat \<times> nat" where "y = SAT_PUSH' x3"
pp_of_term_splus: fixes t :: "'a"   and v :: "'t" shows "pp_of_term (t \<oplus> v) = t + pp_of_term v"
has_integral_vec1_D_cbox: fixes f :: "(real, 1) vec \<Rightarrow> 'c"   and y :: "'c"   and a :: "(real, 1) vec"   and b :: "(real, 1) vec" assumes "(f \<circ> vec has_integral y) {a $ 1..b $ 1}" shows "(f has_integral y) (cbox a b)"
and_evaluate_def: fixes A :: "bool"   and B :: "bool" shows "and_evaluate A B = return_spmf (A \<and> B)"
lm105: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and Y :: "'a set" assumes "\<forall>x\<in>X. f x = g x"   and "Y \<subseteq> X" shows "sum f Y = sum g Y"
projection_bounded_clinear: fixes M :: "'a set" assumes "closed_csubspace M" shows "bounded_clinear (projection M)"
nth_mem: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "xs ! n \<in> set xs"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<le> a \<squnion>1 b"
monad_state_altc_stateT: fixes return :: "'b \<times> 'c \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'd) \<Rightarrow> 'd"   and altc :: "'e cset \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "monad_altc return bind altc" shows "monad_state_altc (return_state return) (bind_state bind) get_state put_state (altc_state altc)"
Span_finite_dimension: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "R.finite_dimension K (R.Span K Us)"
split_max: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and i :: "'d"   and j :: "'d" assumes "class.linorder less_eq less" shows "P (ord.max less_eq i j) = ((less_eq i j \<longrightarrow> P j) \<and> (\<not> less_eq i j \<longrightarrow> P i))"
sel_1: fixes x1 :: "'a extended"   and x2 :: "'a"   and x3 :: "'a extended"   and x4 :: "'b" shows "P (InsertMessage x1 x2 x3 x4) = x1"
eucl_truncate_up: fixes x :: "'a"   and e :: "nat" shows "x \<le> eucl_truncate_up e x"
Ch_singular: shows "\<forall>h X. inj_on Xd (Ch h X)"
hom_simps_2: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "local.eval (x \<otimes>\<^bsub>K [X]\<^esub> y) x = local.eval x x \<otimes> local.eval y x"
surjective: fixes r :: "('a, 'b) gorder_scheme" shows "r = \<lparr>carrier = carrier r, eq = (.=\<^bsub>r\<^esub>), le = (\<sqsubseteq>\<^bsub>r\<^esub>), \<dots> = gorder.more r\<rparr>"
whisker_right: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.seq \<nu> \<mu>" shows "\<nu> \<bullet> \<mu> \<circ> f = (\<nu> \<circ> f) \<bullet> (\<mu> \<circ> f)"
p_def: fixes t :: "real" shows "ego2.p t = ego.q \<delta> + v\<^sub>e * t + 1 / 2 * a\<^sub>e * t\<^sup>2"
vimage_def: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'d set" shows "f -` B \<equiv> {x. f x \<in> B}"
interior_open: fixes S :: "'c set" assumes "open S" shows "interior S = S"
lemma_4_8_iv: fixes a :: "'a"   and b :: "'a" shows "(a r\<rightarrow> b) r\<rightarrow> b r\<rightarrow> a = b r\<rightarrow> a"
match: fixes pat :: "term"   and t :: "'a"   and env :: "(name, 'a) fmap" assumes "match pat t = Some env"   and "P t" shows "fmpred (\<lambda>_. P) env"
NN_neg_unique_xx': fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (imp (NN \<langle>\<phi>\<rangle> (Var xx')) (eql \<langle>neg \<phi>\<rangle> (Var xx')))"
associatedD2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" shows "dvd.dvd times b a"
mult_le_cancel_left_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less c zero" shows "less_eq (times c a) (times c b) = less_eq b a"
read_only_share_unowned: fixes \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "weak_sharing_consistent \<O> sb"   and "a \<notin> \<O> \<union> all_acquired sb"   and "a \<in> read_only (share sb \<S>)" shows "a \<in> read_only \<S>"
IH: fixes n :: "nat" assumes "wf_formula n \<phi>" shows "lang n (rexp_of n \<phi>) - {[]} = lang n (rexp.Inter (rexp_of_alt n \<phi>) (ENC n (FOV \<phi>))) - {[]}"
subset_code_1: fixes xs :: "'b list"   and B :: "'b set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
l_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "\<one> \<otimes> x = x"
rsuml_lsumr: fixes x :: "'h + 'i + 'j" shows "rsuml (lsumr x) = x"
simps_1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq [] = True"
pprt_move_matrix: fixes A :: "'a matrix"   and j :: "int"   and i :: "int" shows "pprt (move_matrix A j i) = move_matrix (pprt A) j i"
A: shows "LIM x inf F (principal {z. f z \<noteq> 0}). Re (g x) * - ln (cmod (f x)) :> at_top"
le_list_update: fixes xs :: "nat list"   and ys :: "nat list"   and i :: "nat"   and z :: "nat" assumes "xs \<le>\<^sub>v ys"   and "i < length ys"   and "z \<le> ys ! i" shows "xs[i := z] \<le>\<^sub>v ys"
aff_dim_UNIV: shows "aff_dim UNIV = int DIM('c)"
map_disj_def: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>0 \<bottom> h\<^sub>1 \<equiv> dom h\<^sub>0 \<inter> dom h\<^sub>1 = {}"
elims_3: fixes x :: "'a option" assumes "\<not> isSome x" obtains "x = None"
extend_basis_superset: fixes B :: "'b set" assumes "independent B" shows "B \<subseteq> extend_basis B"
inline1_unfold: fixes callee :: "'e \<Rightarrow> 'i \<Rightarrow> ('h \<times> 'e, 'f, 'g) gpv"   and gpv :: "('d, 'i, 'h) gpv"   and s :: "'e" shows "inline1 callee gpv s = the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1 callee (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))"
int: shows "((\<lambda>x. ln (x + 3) / (x + 3)\<^sup>2) has_integral (ln 3 + 1) / 3) {0..}"
causal_program_history_Prog\<^sub>s\<^sub>b: fixes is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list"   and mis :: "instr list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b" assumes "causal_program_history is\<^sub>s\<^sub>b sb" shows "causal_program_history (is\<^sub>s\<^sub>b @ mis) (sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis])"
simps_4: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha> :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota> :: "'d"   and V :: "'b \<Rightarrow> 'd"   and t :: "'b local.term"   and u :: "'b local.term" assumes "evaluation_map C D T \<alpha> \<iota> V" shows "evaluation_map.eval D T \<alpha> \<iota> V (t \<^bold>\<cdot> u) = D (evaluation_map.eval D T \<alpha> \<iota> V t) (evaluation_map.eval D T \<alpha> \<iota> V u)"
DF_gt0: fixes k :: "nat"   and i :: "nat" shows "0 < \<Sqinter> (DF k i)"
SC_WbisT_imp_SC_Wbis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.SC_WbisT tval aval indis c" shows "PL_Indis.SC_Wbis tval aval indis c"
swap_uniform_limit: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and x :: "'c"   and S :: "'c set"   and F :: "'b filter"   and l :: "'d"   and h :: "'c \<Rightarrow> 'd" assumes "\<forall>\<^sub>F n in F. (f n \<longlongrightarrow> g n) (at x within S)"   and "(g \<longlongrightarrow> l) F"   and "uniform_limit S f h F"   and "F \<noteq> bot" shows "(h \<longlongrightarrow> l) (at x within S)"
and_not_mask: fixes w :: "'a word"   and n :: "nat" shows "w AND NOT (mask n) = w >> n << n"
inj_on_linepath: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "inj_on (linepath a b) {0..1}"
assms_1: shows "D \<cdot> \<sigma> \<subset># C"
iso_cancel_left: fixes f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "D.iso f"   and "f \<cdot>\<^sub>D g = f \<cdot>\<^sub>D g'"   and "D.seq f g" shows "g = g'"
SNa_induct: fixes M :: "trm"   and P :: "trm \<Rightarrow> bool" assumes "SNa M"   and "\<And>M'. \<lbrakk>SNa M'; \<forall>M''. M' \<longrightarrow>\<^sub>a M'' \<longrightarrow> P M''\<rbrakk> \<Longrightarrow> P M'" shows "P M"