hcju/isabelleps huggingface dataset:
ur_closure: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set"   and i :: "'c"   and p :: "('d, 'c) fm" assumes "Hintikka A H"   and "p at i in' H"   and "\<forall>a. p = Nom a \<or> p = (\<^bold>\<diamond> Nom a) \<longrightarrow> a \<in> A" shows "\<forall>a\<in>assign A H i. p at a in' H"
diff_left: fixes a :: "'b \<Rightarrow>\<^sub>L 'a"   and a' :: "'b \<Rightarrow>\<^sub>L 'a"   and b :: "'b" shows "blinfun_apply (a - a') b = blinfun_apply a b - blinfun_apply a' b"
map_add_map_of_foldr: fixes m :: "'b \<Rightarrow> 'c option"   and ps :: "('b \<times> 'c) list" shows "m ++ map_of ps = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) ps m"
HSem_ignores_fresh_restr: fixes S :: "'b set"   and \<Gamma> :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c" assumes "atom ` S \<sharp>* \<Gamma>" shows "(\<lbrace> \<Gamma> \<rbrace>\<rho>) f|` (- S) = \<lbrace> \<Gamma> \<rbrace>\<rho> f|` (- S)"
pi_succ_it: fixes S :: "'G"   and v :: "'V" shows "proper_it (succ_it S v) (succ_it S v)"
that_2: fixes x :: "'a" assumes "xs = [x]" shows "thesis"
inc: fixes n :: "nat" shows "F n \<subseteq> F (Suc n)"
skl1_obs_iagreement_Init: shows "oreach skl1 \<subseteq> l1_iagreement_Init"
pos_mod_sign: fixes l :: "int"   and k :: "int" assumes "0 < l" shows "0 \<le> k mod l"
FAssRed1: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs} := e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e'\<bullet>F{Cs} := e\<^sub>2,s'\<rangle>"
CD_on_range': fixes c :: "'x"   and ds :: "'d set"   and A :: "'x set" assumes "c \<in> CD_on ds A" shows "c \<in> A \<inter> (\<Union>d\<in>ds. Field (Pd d))"
TupleElementCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and elem :: "String.literal"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E TupleElementCall src k elem : \<tau>" obtains \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "k = DotCall" and "\<Gamma> \<turnstile>\<^sub>E src : Tuple \<pi>" and "fmlookup \<pi> elem = Some \<tau>"
image_empty: fixes f :: "'f \<Rightarrow> 'e" shows "f ` {} = {}"
iter_def: fixes nrandom :: "natural"   and sz :: "natural"   and seed :: "natural \<times> natural" shows "Random_Pred.iter nrandom sz seed = Random_Pred.iter' TYPE('b) nrandom sz seed"
span_clauses_2: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'d set" assumes "module scale" shows "(0::'d) \<in> module.span scale S"
bst_join: fixes l :: "('a \<times> 'b) tree"   and a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) tree" assumes "Tree2.bst \<langle>l, (a, b), r\<rangle>" shows "Tree2.bst (join l a r)"
GuardStripSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and R :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P :: "'e set"   and f :: "'f"   and g :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>R c Q,A"   and "P \<subseteq> R"   and "f \<in> F" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Guard f g c Q,A"
rep_eq: fixes x :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "'c set_rbt" shows "RBT_Set2.fold x xa = RBT_Impl.fold (\<lambda>a _. x a) (mapping_rbt.impl_of xa)"
image_def: fixes f :: "'g \<Rightarrow> 'f"   and A :: "'g set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
induced_automorphism_trivial_outside: fixes v :: "'a" assumes "v \<notin> \<Union> X" shows "\
 v = v"
finite_bij_enumerate: fixes S :: "'b set" assumes "finite S" shows "bij_betw (wellorder_class.enumerate S) {..<card S} S"
aff_dim_convex_hull: fixes S :: "'b set" shows "aff_dim (convex hull S) = aff_dim S"
append_Cons: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "(x # xs) \<cdot> ys = x # xs \<cdot> ys"
abs_eq: fixes x :: "'f" shows "Pred\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (tPred x)"
exec_Normal_elim_cases_7: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and r :: "('e \<times> 'e) set"   and s :: "'e"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> t" obtains ta :: "'e" where "t = Normal ta" and "(s, ta) \<in> r"   | "t = Stuck" and "\<forall>t. (s, t) \<notin> r"
bij_betw_imp_surj: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "bij_betw f A UNIV" shows "surj f"
h2: shows "\<Gamma>' \<turnstile> s is t : T"
absorb_def: fixes d :: "('aa, 'ab) distinguisher"   and conv :: "('aa, 'ab, 'y, 'z) converter" shows "absorb d conv = map_gpv fst id (inline run_converter d conv)"
LeftDerivationLadder_implies_LeftDerivation_at_index: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list"   and index :: "nat" assumes "LeftDerivationLadder \<alpha> D L \<gamma>"   and "index < length L" shows "LeftDerivation \<alpha> (take (ladder_n L index) D) (ladder_\<gamma> \<alpha> D L index)"
prems_4: shows "pac p \<inter> set (w @ [b]) = {}"
group_relative_homology_group: fixes p :: "int"   and X :: "'b topology"   and S :: "'b set" shows "Group.group (relative_homology_group p X S)"
coeffs_map_poly_hom: fixes p :: "'b poly" shows "coeffs (map_poly to_fract p) = map to_fract (coeffs p)"
induct: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b"   and a1 :: "'a list"   and a2 :: "'a list"   and a3 :: "'a list" assumes "\<And>f acc. P f acc [] []"   and "\<And>f acc x xs. P f (x # acc) xs [] \<Longrightarrow> P f acc (x # xs) []"   and "\<And>f acc y ys. P f (y # acc) ys [] \<Longrightarrow> P f acc [] (y # ys)"   and "\<And>f acc x xs y ys. \<lbrakk>f x \<le> f y \<Longrightarrow> P f (x # acc) xs (y # ys); \<not> f x \<le> f y \<Longrightarrow> P f (y # acc) (x # xs) ys\<rbrakk> \<Longrightarrow> P f acc (x # xs) (y # ys)" shows "P a0 a1 a2 a3"
simps_7: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n X\<^sub>n \<phi> = suffix 1 \<xi> \<Turnstile>\<^sub>n \<phi>"
int_f: fixes a :: "'n"   and b :: "'n" shows "(\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b"
refine: fixes R_Iec :: "('b \<times> unit) set" shows "(impl_blue_dfs_hash, blue_dfs) \<in> bg_impl_rel_ext R_Iec Id \<rightarrow> \<langle>\<langle>\<langle>Id\<rangle>list_rel \<times>\<^sub>r \<langle>Id\<rangle>list_rel\<rangle>option_rel\<rangle>nres_rel"
simps_2: fixes T :: "Unary_PCF.type"   and U :: "Unary_PCF.type" shows "\<delta> (T \<rightarrow> U) = HMSet (add_mset (\<delta> T) (hmsetmset (\<delta> U)))"
Abs_Store_inverse: fixes y :: "StoreImpl" assumes "y \<in> Store" shows "Rep_Store (Abs_Store y) = y"
cont: shows "continuous_on (U \<times> {a..x}) (\<lambda>(z, t). f' z t)"
state_subset: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set" assumes "s \<Turnstile> s'" shows "s \<subseteq> s'"
err_def': fixes A :: "'b set" shows "err A = insert Err {x. \<exists>y\<in>A. x = OK y}"
factors_mult: fixes G :: "('c, 'd) monoid_scheme"   and fa :: "'c list"   and a :: "'c"   and fb :: "'c list"   and b :: "'c" assumes "Group.monoid G"   and "factors G fa a"   and "factors G fb b"   and "set fa \<subseteq> carrier G"   and "set fb \<subseteq> carrier G" shows "factors G (fa @ fb) (a \<otimes>\<^bsub>G\<^esub> b)"
oexp_def: shows "rt.oexp = {(f, g). f \<in> rt.FINFUNC \<and> g \<in> rt.FINFUNC \<and> ((let m = t.max_fun_diff f g in (f m, g m) \<in> r) \<or> f = g)}"
RID\<^sub>G_def: fixes s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" shows "RID\<^sub>G s \<equiv> dom s \<union> \<Union> (RID\<^sub>L ` ran s)"
irreducible\<^sub>d_dvd_eq: fixes a :: "'c poly"   and b :: "'c poly" assumes "irreducible\<^sub>d a"   and "irreducible\<^sub>d b"   and "a dvd b"   and "monic a"   and "monic b" shows "a = b"
Nats_cases: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and x :: "'d" assumes "class.semiring_1 one times plus zero"   and "x \<in> semiring_1.Nats one plus zero" obtains n :: "nat" where "x = semiring_1.of_nat one plus zero n"
f: shows "f \<in> space M \<rightarrow> \<Omega>"
ThrowNull\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>1\<rangle>"
frac_of_int: fixes z :: "int" shows "frac (of_int z) = (0::'a)"
span_bicategory_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "span_bicategory C prj0 prj1 \<equiv> elementary_category_with_pullbacks C prj0 prj1 \<and> span_vertical_category C"
rAlpha: fixes A\<^sub>P :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and p :: "(name \<times> name) list"   and B :: "('a, 'b, 'c) boundOutput"   and C :: "'d"   and \<Psi>\<^sub>P :: "'b" assumes "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* (p \<bullet> A\<^sub>P)"   and "A\<^sub>P \<sharp>* B"   and "A\<^sub>P \<sharp>* C"   and "set p \<subseteq> set A\<^sub>P \<times> set (p \<bullet> A\<^sub>P)"   and "distinctPerm p"   and "Prop C \<Psi> P M B A\<^sub>P \<Psi>\<^sub>P" shows "Prop C \<Psi> P M B (p \<bullet> A\<^sub>P) (p \<bullet> \<Psi>\<^sub>P)"
assms_1: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<tau>'"
le_TimesL: fixes s :: "'a language"   and r :: "'a language" assumes "\<oo> s" shows "r \<le> Times r s"
ParCR_MtransC: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and c2' :: "('test, 'atom) com"   and s' :: "'state"   and c1 :: "('test, 'atom) com" assumes "(c2, s) \<rightarrow>*c (c2', s')" shows "(c1 | c2, s) \<rightarrow>*c (c1 | c2', s')"
prems_3: shows "s' \<in> set_pmf (action (repcs s cfg))"
grid_split: fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds' :: "nat set"   and ds :: "nat set" assumes "p \<in> grid b (ds' \<union> ds)" shows "\<exists>x\<in>grid b ds. p \<in> grid x ds'"
le_infI: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x a"   and "less_eq x b" shows "less_eq x (inf a b)"
simps: fixes I :: "'a set"   and v :: "nat \<Rightarrow> 'b"   and s :: "('a, 'b) state" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>s s = (v \<Turnstile>\<^sub>t \<T> s \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>b \<B>\<I> s)"
zabs_def: fixes i :: "Quotient_Int.int" shows "\<bar>i\<bar> = (if i < 0 then - i else i)"
IH_1: assumes "x \<in> vars_of t1" shows "t1 \<lhd> \<sigma> \<noteq> t1 \<lhd> \<eta>"
partial_division_of_tagged_division: fixes s :: "('c \<times> 'c set) set"   and i :: "'c set" assumes "s tagged_partial_division_of i" shows "snd ` s division_of \<Union> (snd ` s)"
cPar2B: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi"   and C :: "'a" assumes "Q \<longmapsto> a<x> \<prec> Q'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "x \<noteq> a"   and "\<And>C. F C Q a x Q'" shows "F C (P \<parallel> Q) a x (P \<parallel> Q')"
IH: fixes S :: "'a ltl set" assumes "S \<Turnstile>\<^sub>P af \<phi> xs" shows "S \<Turnstile>\<^sub>P af\<^sub>G \<phi> xs"
eval_elim_cases_9: fixes G :: "prog"   and s :: "st"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (BinOp binop e1 e2)\<succ>\<rightarrow> (v, s')" obtains v1 :: "val" and s1 :: "state" and v2 :: "val" where "v = In1 (eval_binop binop v1 v2)" and "G\<turnstile>Norm s \<midarrow>e1-\<succ>v1\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ>\<rightarrow> (In1 v2, s')"
abst_inj: fixes isom :: "('a, 'b, 'c) tuple_isomorphism"   and x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "isomorphic_tuple isom" shows "(abst isom x = abst isom y) = (x = y)"
distinct_3: fixes x1 :: "int"   and x31 :: "pol"   and x32 :: "nat"   and x33 :: "pol" shows "Pc x1 \<noteq> PX x31 x32 x33"
max_less_iff_conj: fixes x :: "'t"   and y :: "'t"   and z :: "'t" shows "(ord_term_lin.max x y \<prec>\<^sub>t z) = (x \<prec>\<^sub>t z \<and> y \<prec>\<^sub>t z)"
dvd_smult: fixes p :: "'a poly"   and q :: "'a poly"   and a :: "'a" assumes "p dvd q" shows "p dvd Polynomial.smult a q"
cases: fixes a1 :: "'a list set"   and a2 :: "'a list set" assumes "local.bisimilar a1 a2" obtains K :: "'a list set" and L :: "'a list set" where "a1 = K" and "a2 = L" and "K \<subseteq> lists S" and "L \<subseteq> lists S" and "([] \<in> K) = ([] \<in> L)" and "\<And>x. x \<in> S \<Longrightarrow> local.bisimilar (Deriv x K) (Deriv x L)"
lNxt_active: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>the_singleton \<rightarrow> t\<rangle>\<^bsub>n\<^esub> = n"
ln_ln_nonneg: fixes x :: "real" assumes "3 \<le> x" shows "0 \<le> ln (ln x)"
remaining_qty_predict: fixes asset :: "'b"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and v :: "real" assumes "borel_adapt_stoch_proc F (prices Mkt asset)"   and "trading_strategy pf"   and "support_adapt Mkt pf" shows "borel_predict_stoch_proc F (remaining_qty Mkt v pf asset)"
simps_23: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a" shows "local.less_aexp_aux (Times e1 e2) (V v) = False"
assms_2: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "pure (g x) h"
primeI: fixes x :: "'b" assumes "prime_elem x"   and "normalize x = x" shows "prime x"
post_def: shows "post = (\<lambda>(blues0, reds0, onstack0, u0) (blues, reds, onstack, cyc). onstack = onstack0 \<and> gen_dfs_post E reachable onstack0 blues0 u0 blues (cyc \<noteq> NO_CYC) \<and> cyc_post blues reds onstack u0 cyc)"
qAbs_preserves_alpha: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and X' :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "X #= X'"   and "qGood X \<or> qGood X'" shows "qAbs xs x X $= qAbs xs x X'"
A: shows "((s, c), ee, s', c') \<in> ntrp fg"
intros_7: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {P} c { Q}"   and "\<And>s n. P' (s, n) \<Longrightarrow> P (s, n)"   and "\<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n)" shows "\<turnstile>\<^sub>3\<^sub>a {P'} c { Q'}"
foldr_plus_mono: fixes n :: "'lbl"   and m :: "'lbl"   and xs :: "'lbl list" assumes "n \<le> m" shows "foldr (+) xs n \<le> foldr (+) xs m"
greaterThan_eq_iff: fixes x :: "'a"   and y :: "'a" shows "({x<..} = {y<..}) = (x = y)"
prob_space_K: fixes M :: "'b measure"   and K :: "'b \<Rightarrow> 'b measure"   and x :: "'b" assumes "discrete_Markov_process M K"   and "x \<in> space M" shows "prob_space (K x)"
dlex_pm_strict_def: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "dlex_pm_strict s t = strict dlex_pm s t"
induced_vector_additive: fixes v :: "'v"   and v' :: "'v" assumes "v \<in> V"   and "v' \<in> V" shows "induced_vector (v + v') = induced_vector v + induced_vector v'"
simp: shows "mk_rtrancl_main todo fin = (if set_of todo = {} then fin else let (a, tod) = sel todo in if \<exists>b\<in>fin. subsumes b a then mk_rtrancl_main tod fin else mk_rtrancl_main (un (r a) tod) (insert a fin))"
oalist_inv_raw_ConsI: fixes xs :: "('a \<times> 'd) list"   and v :: "'d"   and k :: "'a" assumes "oalist_inv_raw xs"   and "v \<noteq> (0::'d)"   and "xs \<noteq> [] \<Longrightarrow> lt k (fst (hd xs))" shows "oalist_inv_raw ((k, v) # xs)"
d_1: shows "x = real_of_int c / real_of_int d"
setPaireq: fixes x :: "'b"   and y :: "'b"   and a :: "'b"   and b :: "'b" assumes "{x, y} = {a, b}" shows "x = a \<and> y = b \<or> x = b \<and> y = a"
IH: assumes "CT OK"   and "mtype(CT,m,Da) = Ds \<rightarrow> D" shows "\<exists>D0 C0. CT \<turnstile> Da <: D0 \<and> CT \<turnstile> C0 <: D \<and> CT;[xs [\<mapsto>] Ds, this \<mapsto> D0] \<turnstile> e : C0"
smod_word_def: fixes v :: "'b word"   and w :: "'b word" shows "v smod w = word_of_int (sint v smod sint w)"
decreasing_tendsto: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and l :: "'b"   and f :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in F. less_eq l (f n)"   and "\<And>x. less l x \<Longrightarrow> \<forall>\<^sub>F n in F. less (f n) x" shows "filterlim f (topological_space.nhds open l) F"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "ab_group_add_hom hom"   and "inj_group_add_hom hom" shows "inj_ab_group_add_hom hom"
at_most_one_marker: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and cid :: "nat"   and p :: "'d"   and q :: "'d"   and i :: "nat" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "channel cid = Some (p, q)" shows "Marker \<notin> set (msgs (distributed_system.s channel trans send recv init t i) cid) \<or> (\<exists>!j. j < length (msgs (distributed_system.s channel trans send recv init t i) cid) \<and> msgs (distributed_system.s channel trans send recv init t i) cid ! j = Marker)"
simps_2: fixes uv :: "preal" shows "preal_plus preal.PInfty uv = preal.PInfty"
EX_lem0l: fixes x :: "'a"   and f :: "'a set"   and g :: "'a set" assumes "x \<in> EX' f" shows "x \<in> EX' (f \<union> g)"
inj_on_mult_units: fixes x :: "nat"   and q :: "nat" assumes "coprime x q" shows "inj_on (\<lambda>b. x * b mod q) ({..<q} - {0})"
proj_tuple_replicate: fixes bs :: "bool list"   and as :: "'a option list" assumes "\<And>i. i \<in> set bs \<Longrightarrow> \<not> i"   and "length bs = length as" shows "proj_tuple bs as = replicate (length bs) None"
T8: fixes w :: "i"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<forall>x. \<psi> x in w] = (\<forall>x. [\<psi> x in w])"
get_root_node_document: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root"   and "is_document_ptr_kind root" shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root)"
not_final_thread_iff: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" shows "not_final_thread s t = (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (\<not> final x \<or> ln \<noteq> no_wait_locks \<or> (\<exists>w. wset s t = \<lfloor>w\<rfloor>)))"
AE_cong_simp: fixes M :: "'c measure"   and N :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "M = N"   and "\<And>x. x \<in> space N =simp=> P x = Q x" shows "almost_everywhere M P = almost_everywhere N Q"
isolated_da_pre: shows "H.isolated_verts = G.isolated_verts \<union> (if tail G a \<in> H.isolated_verts then {tail G a} else {}) \<union> (if head G a \<in> H.isolated_verts then {head G a} else {})"
refl_symm_trans_closure_is_symm_refl_trans_closure: fixes Rel :: "('a \<times> 'a) set" shows "(symcl (Rel\<^sup>=))\<^sup>+ = (symcl Rel)\<^sup>*"
caller_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan" shows "caller ch \<equiv> \<lambda>s p. (cbit (ch s p), arg (ch s p))"
sum_cls_val_process: fixes pf1 :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and pf2 :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and Mkt :: "('c, 'b) discrete_market" assumes "portfolio pf1"   and "portfolio pf2" shows "\<forall>n w. cls_val_process Mkt (qty_sum pf1 pf2) n w = cls_val_process Mkt pf1 n w + cls_val_process Mkt pf2 n w"
mono_gamma_c: fixes ca :: "'av Abs_State.st option acom"   and ca' :: "'av Abs_State.st option acom" assumes "ca \<sqsubseteq> ca'" shows "local.\<gamma>\<^sub>c ca \<le> local.\<gamma>\<^sub>c ca'"
map2_val_compat_def: fixes x :: "'o"   and f :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'f) list" shows "map2_val_compat x f = (\<forall>zs. oalist_inv_raw x zs \<longrightarrow> oalist_inv_raw x (f zs) \<and> fst ` set (f zs) \<subseteq> fst ` set zs)"
cases: fixes istate :: "'a"   and step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and T :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and a :: "'a" assumes "BD_Security.reachNT istate step T a" obtains "a = istate"   | trn :: "('a, 'b, 'c) trans" where "a = tgtOf trn" and "BD_Security.reachNT istate step T (srcOf trn)" and "step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn)" and "\<not> T trn"
indRelLTPO_impl_TRel_is_weak_barbed_contrasimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_barbed_contrasimulation (indRelLTPO TRel) (STCalWB SWB TWB)" shows "weak_barbed_contrasimulation (TRel\<^sup>+) TWB"
poly_bigtheta: fixes p :: "'a poly"   and F :: "'a filter" assumes "p \<noteq> 0"   and "F \<le> at_infinity" shows "poly p \<in> \<Theta>[F](\<lambda>x. x ^ degree p)"
wf_termi_call_steps: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" shows "wf (termi_call_steps \<Gamma>)"
subset_mset_def: fixes A :: "'f multiset"   and B :: "'f multiset" shows "(A \<subset># B) = (A \<subseteq># B \<and> A \<noteq> B)"