hcju/isabelleps huggingface dataset:
lossless_R1: fixes m1 :: "bool \<times> bool"   and m2 :: "bool" shows "lossless_spmf (R1 m1 m2)"
a: shows " Lam [a].t \<longrightarrow>\<^sub>1 t'"
read_tmps_distinct_nth_update: fixes i :: "nat"   and sb :: "'a memref list"   and p :: "'a"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "distinct_read_tmps sb" shows "read_tmps_distinct (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
intros_1: fixes env :: "nat \<Rightarrow> LambdaType.type"   and x :: "nat"   and T :: "LambdaType.type" assumes "env x = T" shows "env \<turnstile> dB.Var x : T"
map_vals_of_spec: fixes es :: "'a exp list"   and vs :: "val list" assumes "map_vals_of es = \<lfloor>vs\<rfloor>" shows "es = map Val vs"
lconf_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v\<^sub>m :: "'a \<Rightarrow> val option"   and T\<^sub>m :: "'a \<Rightarrow> ty option" shows "P,h \<turnstile> v\<^sub>m (:\<le>)\<^sub>w T\<^sub>m \<equiv> \<forall>V v. v\<^sub>m V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. T\<^sub>m V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
lex_p_eq: fixes n' :: "nat"   and x' :: "nat"   and n :: "nat"   and x :: "nat" shows "(((n', x'), n, x) \<in> lex_p) = (n' < n \<or> n' = n \<and> x' < x)"
distrib_comp_sum_right: fixes T :: "'c \<Rightarrow> 'b"   and T' :: "'c \<Rightarrow> 'b"   and S :: "'a \<Rightarrow> 'c" shows "T + T' \<circ> S = (T \<circ> S) + (T' \<circ> S)"
unifies_all_pairs_iff: fixes xs :: "('c, 'd) Term.term list"   and \<sigma> :: "'d \<Rightarrow> ('c, 'e) Term.term" shows "(\<forall>p\<in>set (pairs xs). fst p \<cdot> \<sigma> = snd p \<cdot> \<sigma>) = (\<forall>a\<in>set xs. \<forall>b\<in>set xs. a \<cdot> \<sigma> = b \<cdot> \<sigma>)"
\<section>: shows "(\<Sum>(i, j)\<in>(SIGMA i:{..m + n}. {m + n - i<..m + n}). a i * x ^ i * (b j * x ^ j)) = (0::'a)"
submap_imp_state_succ_submap_a: fixes s1 :: "('c, 'd) fmap"   and s2 :: "('c, 'd) fmap"   and s3 :: "('c, 'd) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "s2 \<subseteq>\<^sub>f s3" shows "s1 \<subseteq>\<^sub>f s3"
poly_monom_mult: fixes p :: "'b"   and i :: "nat"   and q :: "'b poly"   and j :: "nat"   and y :: "'b" shows "poly (monom p i * q ^ j) y = poly (monom p j * [:y:] ^ i) (poly q y)"
skl1_inv3I: fixes x :: "skl1_state" assumes "\<And>A B Rb Ra gny Nr. \<lbrakk>guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>; in_progressS (progress x Rb) {xny, xni, xnr, xgnx, xgny, xsk}; sklvl1.guessed_frame Rb xgny = Some gny; sklvl1.guessed_frame Rb xnr = Some Nr; sklvl1.guessed_frame Rb xni = Some (NonceF (Ra $ ni)); sklvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx))); guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; progress x Ra = Some {xnx, xgnx, xni}; can_signal x A B\<rbrakk> \<Longrightarrow> signalsInit x (Commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>) < signalsInit x (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>)" shows "x \<in> skl1_inv3"
sorted_sort_id: fixes xs :: "'a list" assumes "sorted xs" shows "sort xs = xs"
a_get_disconnected_nodes_locs_def: fixes document_ptr :: "(_) document_ptr" shows "a_get_disconnected_nodes_locs document_ptr = {preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr disconnected_nodes)}"
simps_1: fixes e :: "char list \<Rightarrow> 'b option"   and x :: "char list" shows "Env e!x = e x"
smult_in_span: fixes W :: "'b vec set"   and n :: "nat"   and x :: "'b vec"   and c :: "'b" assumes "W \<subseteq> carrier_vec n"   and "x \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W" shows "c \<cdot>\<^sub>v x \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W"
comm_neutral: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and a :: "'e" assumes "comm_monoid f z" shows "f a z = a"
terminates_elim_cases_8: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and p :: "'d"   and s :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>Call p \<down> s" obtains f :: "'f" where "sb__ = Fault f"   | bdy :: "('e, 'd, 'f) com" and s :: "'e" where "sb__ = Normal sc__" and "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile>bdy \<down> Normal sc__"   | s :: "'e" where "sb__ = Normal sd__" and "\<Gamma> p = None"   | "sb__ = Stuck"   | s :: "'e" where "sb__ = Abrupt s"
FFI: assumes "n = Falseif" shows "ospec (ffi s) (\<lambda>(r, s'). (r, ff) \<in> Q s \<and> I' s')"
UNIV_def: shows "UNIV = rel_ext (\<lambda>x. True)"
simps: fixes a :: "'a list" shows "Specialisation_Examples.sorted a = (a = [] \<or> (\<exists>xs x. a = x \<Colon> xs \<and> (\<forall>y\<in>set xs. x \<le> y) \<and> Specialisation_Examples.sorted xs))"
lunit_def: fixes f :: "'a" shows "Left.lunit f \<equiv> THE \<mu>'. Left.in_hom \<mu>' (Left.L f) f \<and> Left.L \<mu>' = Left.comp (\<i>[trg \<mu>] \<star> f) (local.inv \<a>[trg \<mu>, trg \<mu>, f])"
g: shows "g \<in> N \<rightarrow>\<^sub>M T"
in_HomD_5: fixes F :: "'a set"   and a :: "'a"   and b :: "'a"   and f :: "'a" assumes "F \<in> Hom a b"   and "f \<in> F" shows "B.is_left_adjoint f"
comp_ide_self: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "a \<cdot> a = a"
ceq: shows "c = c' * (\<Prod>(a, i)\<in>set fs'. lead_coeff a ^ Suc i)"
sublist_at_altdef: fixes xs :: "'a list"   and ys :: "'a list"   and i :: "nat" shows "sublist_at xs ys i = (\<exists>ps ss. ys = ps @ xs @ ss \<and> i = length ps)"
hyps_2: shows "set (bts_dfs val rs) \<union> set |q| = set |bts_augment rs q|"
setops\<^sub>s\<^sub>s\<^sub>t_cons: fixes a :: "('c, 'd) stateful_strand_step"   and A :: "('c, 'd) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t (a # A) = setops\<^sub>s\<^sub>s\<^sub>t\<^sub>p a \<union> setops\<^sub>s\<^sub>s\<^sub>t A"
fps_binomial_minus_one: shows "fps_binomial (- (1::'a)) = inverse (1 + fps_X)"
b_pr0_def: shows "b_pr0 = local.comp b (pr0 (local.dom (\<Lambda> a b c g)) (local.dom b))"
A: shows "trnl\<^sub>\<eta> v \<in> hom (f \<star> v) u \<rightarrow> hom v (g \<star> u)"
3: fixes t :: "real"   and k :: "real" assumes "t \<in> {A..}" shows "cmod (indicat_real {A..k} t *\<^sub>R local.laplace_integrand s t) \<le> M * exp (- (Re s - c) * t)"
eSuc_inject: fixes m :: "eint"   and n :: "eint" shows "(m + 1 = n + 1) = (m = n)"
prj_simps_4: shows "C.arr \<pi>\<rho>.prj\<^sub>0"
sep_conj_sep_emptyE: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and s :: "'c"   and Q :: "'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "P s"   and "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_empty zero) s \<Longrightarrow> sep_algebra.sep_conj plus sep_disj Q R s" shows "sep_algebra.sep_conj plus sep_disj Q R s"
plus_adds_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus (plus a b) c" shows "plus.adds plus b c"
a_very_costrict: fixes x :: "'a" shows "(ad x = (1::'a)) = (x = 0)"
r_nth_inbounds_def: shows "r_nth_inbounds \<equiv> let r = Pr 1 (recf.Id 1 0) (Cn 3 r_pdec2 [recf.Id 3 1]) in Cn 3 r_ifeq [recf.Id 3 0, recf.Id 3 2, Cn 3 r [recf.Id 3 2, recf.Id 3 1], Cn 3 r_pdec1 [Cn 3 r [recf.Id 3 2, recf.Id 3 1]]]"
assms: shows "q2 \<in> io_targets M q1 io"
cases: fixes ars :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "('b \<times> 'a) list" assumes "seqp ars a1 a2" obtains a :: "'a" where "a1 = a__" and "a2 = []"   | a :: "'a" and \<alpha> :: "'b" and b :: "'a" and ss :: "('b \<times> 'a) list" where "a1 = a" and "a2 = (\<alpha>, b) # ss" and "ars a \<alpha> b" and "seqp ars b ss"
ccField_ccBinds: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and G :: "CoCalls" shows "ccField (ccBinds \<Gamma>\<cdot>(ae, G)) \<subseteq> fv \<Gamma>"
varsA_substA: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and at :: "('c, 'd) atm" shows "varsA (substA \<pi> at) = (\<Union>x\<in>varsA at. vars (\<pi> x))"
assms_2: shows "g \<circ> h \<le> h \<circ> f"
check_eqv_complete: fixes \<Sigma> :: "'a list"   and n :: "nat"   and r :: "'a atom rexp"   and s :: "'a atom rexp" assumes "lang \<Sigma> n r = lang \<Sigma> n s"   and "M2L_Equivalence_Checking.wf \<Sigma> n r"   and "M2L_Equivalence_Checking.wf \<Sigma> n s" shows "slow.check_eqvRE \<Sigma> n r s"
Cons_2: shows "sorted_descending (map ofe_prio (a # ft))"
fold_append: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and xs :: "'h list"   and ys :: "'h list" shows "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
range_tmI: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and tm :: "taylor_model" assumes "x \<in>\<^sub>i interval_of (insertion e (map_poly real_of_float (tm_poly tm))) + real_interval (tm_bound tm)" shows "x \<in>\<^sub>i range_tm e tm"
inject_2: fixes x2 :: "'b nmultiset multiset"   and y2 :: "'b nmultiset multiset" shows "(MSet x2 = MSet y2) = (x2 = y2)"
advantage_def: fixes "oracle" :: "'c \<Rightarrow> 'a \<Rightarrow> (bool list \<times> 'c) spmf"   and \<sigma> :: "'c"   and \<A> :: "('a \<Rightarrow> ((bool list \<times> bool list) \<times> 'd, 'a, bool list) gpv) \<times> ('a \<times> bool list \<Rightarrow> 'd \<Rightarrow> (bool, 'a, bool list) gpv)" shows "ind_cpa.advantage oracle \<sigma> \<A> = \<bar>spmf (run_gpv oracle (ind_cpa.ind_cpa \<A>) \<sigma>) True - 1 / 2\<bar>"
subst_closed: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and f :: "'b"   and s :: "'c" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "closed f" shows "subst s f = f"
inv_f_eq_iff: fixes x :: "'b"   and y :: "'a" shows "(inv (inv (inv hom)) x = y) = (x = inv (inv hom) y)"
ext_inject: fixes nothing :: "unit"   and more :: "'a"   and nothing' :: "unit"   and more' :: "'a" shows "(\<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing', \<dots> = more'\<rparr>) = (nothing = nothing' \<and> more = more')"
WR0_iff: fixes y :: "hf" shows "WR 0 y = (y = 0)"
elims_3: fixes x :: "'d ifex"   and xa :: "'d \<Rightarrow> bool" assumes "\<not> val_ifex x xa" obtains s :: "'d \<Rightarrow> bool" where "x = Falseif" and "xa = s__"   | n :: "'d" and t1 :: "'d ifex" and t2 :: "'d ifex" and s :: "'d \<Rightarrow> bool" where "x = IF n t1 t2" and "xa = s" and "\<not> (if s n then val_ifex t1 s else val_ifex t2 s)"
H_assign: fixes P :: "(char list \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'b) \<Rightarrow> 'b" shows "PRE (\<lambda>s. P (s(v := e s))) (v ::= e) POST P"
partner_symmetric: fixes R :: "fid_t"   and R' :: "fid_t" assumes "partner_runs R R'" shows "partner_runs R' R"
map_lift_spmf: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and p :: "'g spmf" shows "map_gpv f g (lift_spmf p) = lift_spmf (map_spmf f p)"
pro_betw: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x \<in> prolongation a b) = [[a b x]]"
run_reachable: fixes i :: "nat" shows "A.reachable (s i)"
Scons_D: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "Scons M N \<in> sexp" shows "M \<in> sexp \<and> N \<in> sexp"
assms_5: shows "igSubstAbsIPresIGWlsAbs MOD"
execn_to_execn_normalize: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Language.normalize c,s\<rangle> =n\<Rightarrow> t"
icrit_specD: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and m :: "nat"   and data :: "nat \<times> 'e"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "icrit_spec crit"   and "dickson_grading d"   and "fst ` (set gs \<union> set bs \<union> set hs) \<subseteq> dgrad_p_set d m"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "p \<in> set hs"   and "q \<in> set gs \<union> set bs \<union> set hs"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "crit data gs bs hs p q" shows "crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs \<union> set hs)) (fst p) (fst q)"
simps_1: fixes x :: "rat" shows "info_2 (Rational x) = Inl x"
focus_Polys_Compl: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set" assumes "p \<in> P[- X]" shows "focus X p = monomial p 0"
B_def: fixes i :: "'a" shows "B i = (\<Sum>\<^sub>ax\<in>S i. norm (f x))"
V_BCLIM_R1: shows "V\<^bsub>BC-LIM\<^esub> \<subseteq> \<R>"
lset_llist_of_tllist: fixes xs :: "('c, 'd) tllist" shows "lset (llist_of_tllist xs) = tset xs"
converseE: fixes yx :: "'b \<times> 'c"   and r :: "('c \<times> 'b) set" assumes "yx \<in> r\<inverse>" obtains x :: "'c" and y :: "'b" where "yx = (y, x)" and "(x, y) \<in> r"
spec_imp_SafeStickyNrecVacantUpto: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw SafeStickyNrecVacantUpto s"
Limit_vtimes_in_VsetI: fixes \<alpha> :: "V"   and A :: "V"   and B :: "V" assumes "Limit \<alpha>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "A \<times>\<^sub>\<circ> B \<in>\<^sub>\<circ> Vset \<alpha>"
arg_min_def: fixes less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and P :: "'d \<Rightarrow> bool" shows "ord.arg_min less f P = (SOME x. ord.is_arg_min less f P x)"
canonical_fun_def: fixes U :: "'a set"   and x :: "'b" shows "stx.canonical_fun U x = stx.class_of U x"
process_prop_1_def: fixes P :: "('c list \<times> 'c set) set \<times> 'c list set" shows "process_prop_1 P \<equiv> ([], {}) \<in> fst P"
abs: fixes C_info :: "(name, dt_def) fmap"   and \<Gamma> :: "(name, value) fmap"   and cs :: "(Term_Class.term \<times> sterm) list" shows "special_constants.veval' C_info \<Gamma> (Sabs cs) (Vabs cs \<Gamma>)"
prod_rel_id_simp: shows "Id \<times>\<^sub>r Id = Id"
simps_3: fixes x :: "'b"   and hs :: "'b hp list" shows "cost Del_min [Some (Hp x hs)] = 1 + T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>2 (pass\<^sub>1 hs) + T\<^sub>p\<^sub>a\<^sub>s\<^sub>s\<^sub>1 hs"
nat_k_skip_induct: fixes k :: "nat"   and P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "1 \<le> k"   and "\<forall>i<k. P i"   and "\<forall>n\<^sub>0. P n\<^sub>0 \<longrightarrow> P (n\<^sub>0 + k)" shows "P n"
sorted_repeat_poly_list_rel_with0_wrt_def: fixes S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "('a \<times> char list multiset) set" shows "sorted_repeat_poly_list_rel_with0_wrt S R = {(xs, ys). (xs, ys) \<in> \<langle>R \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel \<and> sorted_wrt S (map fst xs)}"
rem_implicit_pres_ops_lookup_op_Some_1': fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and n :: "char list"   and p :: "(nat \<times> nat) list"   and vp :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and e :: "nat" assumes "ast_problem.lookup_operator prob name = Some (n, p, vp, e)" shows "ast_problem.lookup_operator (rem_implicit_pres_ops prob) name = Some (rem_implicit_pres (n, p, vp, e))"
cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "o[F](f) = o[F](g)"
sq_mtx_inv_def: shows "sq_mtx_inv \<equiv> map_fun to_vec to_mtx matrix_inv"
override_on_def: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
interior_of_maximal: fixes T :: "'d set"   and S :: "'d set"   and X :: "'d topology" assumes "T \<subseteq> S"   and "openin X T" shows "T \<subseteq> X interior_of S"
span_span: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set" assumes "module scale" shows "module.span scale (module.span scale A) = module.span scale A"
project_set_Union: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and A :: "'f set set" shows "project_set h (\<Union> A) = \<Union> (project_set h ` A)"
assertion_fun_dual: fixes x :: "'b \<Rightarrow> 'b" assumes "x \<in> assertion_fun" shows "(x \<circ> \<top>) \<sqinter> dual_fun x = x"
compl_coset: fixes xs :: "'d list" shows "- List.coset xs = set xs"
preserves_arr: fixes f :: "'a" assumes "arr f" shows "arr (src f)"
has_dom_perm_swap: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'c"   and S :: "'c set" shows "has_dom (perm_swap a b f) S = has_dom f ((a \<rightleftharpoons>\<^sub>F b) ` S)"
Uint8_def: shows "Uint8 \<equiv> map_fun int_of_integer Abs_uint8 word_of_int"
simps_5: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and m :: "'b"   and x :: "nat" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs (Actual_out (m, x)) = (\<exists>a Q p f ins outs. valid_edge a \<and> kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f \<and> targetnode a = m \<and> (p, ins, outs) \<in> set procs \<and> x < length outs)"
\<G>_suffix: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and i :: "nat" shows "\<G> \<phi> w \<subseteq> \<G> \<phi> (suffix i w)"
M_times_1: fixes x :: "int"   and y :: "int" shows "pl.M (pl.M x * y) = pl.M (x * y)"
sup_lift_eq_lift: fixes x :: "V"   and u :: "V"   and v :: "V" assumes "x \<squnion> lift x u = x \<squnion> lift x v" shows "lift x u = lift x v"
l: shows "simplicial_simplex p (standard_simplex q) (oriented_simplex p l)"
elims_2: fixes x :: "(char list \<Rightarrow> 'av) option"   and xa :: "char list set" assumes "top_on\<^sub>o x xa" obtains S :: "char list \<Rightarrow> 'av" and X :: "char list set" where "x = Some S" and "xa = Xa__" and "top_on\<^sub>s S Xa__"   | X :: "char list set" where "x = None" and "xa = X"
fun_upd_same: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'d" shows "(f(x := y)) x = y"
insert_commute: fixes x :: "'e"   and y :: "'e"   and A :: "'e set" shows "insert x (insert y A) = insert y (insert x A)"
LLL_d\<mu>_bound: fixes i :: "nat"   and j :: "nat" assumes "i < m"   and "j < i" shows "\<bar>d\<mu> fs i j\<bar> \<le> int (N ^ (2 * (m - 1)) * 2 ^ (m - 1) * m)"
Suc: fixes x :: "'s" shows "ennreal (p x y (n + m)) = \<integral>\<^sup>+ x. ennreal (p x x n * p x y m) \<partial>count_space UNIV"
sym_converse: fixes r :: "('a \<times> 'a) set" shows "sym (r\<inverse>) = sym r"
map_fst_zip_take: fixes xs :: "'d list"   and ys :: "'e list" shows "map fst (zip xs ys) = take (min (length xs) (length ys)) xs"
s: fixes x :: "real" assumes "x \<noteq> pi / 2"   and "norm (x - pi / 2) < s" shows "norm (cos x / sin x - 0) < inverse y"
less_cons: fixes s :: "'b list"   and t :: "'b list"   and a :: "'b" assumes "s < t" shows "a # s < a # t"
5: shows "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` {dual\<^sub>s\<^sub>t \<A>}) \<union> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` {dual\<^sub>s\<^sub>t \<A>}))"
divided_differences: fixes k :: "nat"   and i :: "nat" assumes "k \<le> n"   and "i \<le> k" shows "(\<Sum>j\<leftarrow>[0..<Suc k - i]. local.xij_f i (i + j) * local.pprod k i (i + j)) = local.f k"
\<alpha>_pos: shows "(1::'a) \<le> \<alpha>"
prems_2: shows "is_ord (MKT n l (MKT rn rl rr rh) h)"
prems: shows "{$$} \<turnstile> e, eP, eV : \<tau>"
refl_onD: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b" assumes "refl_on A r"   and "a \<in> A" shows "a \<succeq>[r] a"
reflexion_is_sound: fixes C :: "'a eclause"   and D :: "'a eclause"   and \<sigma> :: "('a \<times> 'a trm) list"   and k :: "inferences"   and C' :: "'a literal set" assumes "finite (cl_ecl C)"   and "reflexion C D \<sigma> k C'" shows "clause_entails_clause (cl_ecl C) (cl_ecl D)"
x: fixes z :: "'a" assumes "z \<in> set ys"   and "P z" shows "False"
set_ConsD: fixes y :: "'d"   and x :: "'d"   and xs :: "'d list" assumes "y mem (x # xs)" shows "y = x \<or> y mem xs"
and_zero_eq: 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 push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and a zero = zero"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and y :: "name" assumes "(\<Psi>', R, S) \<in> Rel"   and "y \<sharp> \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<prec> b) = (a = ordered_powerprod_lin.min a b \<and> a \<noteq> b)"
FW_not_empty: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat" assumes "u \<turnstile>\<^bsub>v,n\<^esub> FW M' n" shows "u \<turnstile>\<^bsub>v,n\<^esub> M'"