hcju/isabelleps huggingface dataset:
insert_eq_iff: fixes a :: "'a"   and A :: "'a set"   and b :: "'a"   and B :: "'a set" assumes "\<not> a \<in> A"   and "\<not> b \<in> B" shows "(insert a A = insert b B) = (if a = b then A = B else \<exists>C. A = insert b C \<and> \<not> b \<in> C \<and> B = insert a C \<and> \<not> a \<in> C)"
contra_subsetD: fixes A :: "'a set"   and B :: "'a set"   and c :: "'a" assumes "A \<subseteq> B"   and "c \<notin> B" shows "c \<notin> A"
cases: fixes x :: "key" obtains A :: "agent" where "x = pubK A__"   | A :: "agent" where "x = priK A"   | v :: "fresh_t" where "x = sesK v"   | va :: "agent" where "x = shrK va"
ind: assumes "abc_list_crsp (lm1 @ 0 \<up> n) lm2" shows "abc_list_crsp lm1 lm2"
lin_dep_def: fixes S :: "'c set" shows "lin_dep S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier K \<and> lincomb a A = \<zero>\<^bsub>V\<^esub> \<and> v \<in> A \<and> a v \<noteq> \<zero>\<^bsub>K\<^esub>)"
cases: fixes x :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> 'a option \<times> 'b option" obtains P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" where "x = (P__, None, None)"   | P :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and a :: "'a" and c :: "'b" where "x = (P, Some a, Some c)"   | uu_ :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and v :: "'a" where "x = (uu___, Some v__, None)"   | uu_ :: "'a \<Rightarrow> 'b \<Rightarrow> assn" and v :: "'b" where "x = (uu_, None, Some v)"
PredicatePairFunction: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<forall>x1 x2. \<exists>y1 y2. P x1 x2 y1 y2" shows "\<exists>f. \<forall>x1 x2. \<exists>y1 y2. f x1 x2 = (y1, y2) \<and> P x1 x2 (fst (f x1 x2)) (snd (f x1 x2))"
dens'': shows "has_density (distr (distr M lborel extract_real_pair) borel (\<lambda>(x, y). f x y)) lborel (g (\<delta> \<circ> RealPairVal))"
execn_Normal_elim_cases_1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,Fault f\<rangle> =n\<Rightarrow> t" obtains "t = Fault f"
fresh_fmdrop_in_fmdom: fixes x :: "'a"   and B :: "('a, 'b) fmap"   and y :: "atom" assumes "x \<in> fmdom' B"   and "y \<sharp> B"   and "y \<sharp> x" shows "y \<sharp> fmdrop x B"
nemp_pow': fixes t :: "'b list"   and m :: "nat" assumes "t \<^sup>@ m \<noteq> \<epsilon>" shows "t \<noteq> \<epsilon>"
boundE: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c" assumes "bound X r b" obtains "\<And>x. x \<in> X \<Longrightarrow> r x b"
Diff_eq_empty_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A\<setminus>B = {}) = (A \<subseteq> B)"
b_Inf_M_Suc: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j \<le> k" shows "b k (j, i) \<lless> {\<Sqinter> (M (Suc k))}"
sum_diff: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<Sum>i\<le>n. f (i + 1) - f i) = f (n + 1) - f 0"
\<R>\<^sub>3_weak_bisim: fixes \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" shows "weak_bisim (\<R>\<^sub>3 \<Gamma>' \<S>' P') (\<R> \<Gamma>' \<S>' P')"
span_Un: fixes S :: "('b \<times> 'c) set"   and T :: "('b \<times> 'c) set" shows "p.span (S \<union> T) = {x + y |x y. x \<in> p.span S \<and> y \<in> p.span T}"
H': assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U" shows "P"
returnU_stateT_def: shows "returnU = (\<Lambda> x. StateT\<cdot>(\<Lambda> s. return\<cdot>(x, s)))"
simps_1: fixes x :: "name"   and y :: "name"   and t' :: "lam" shows "Var x[y::=t'] = (if x = y then t' else Var x)"
C_TApp: fixes E :: "trm \<Rightarrow> trm"   and T :: "POPLmarkRecord.type" assumes "ctxtp E" shows "ctxtp (\<lambda>t. E t \<bullet>\<^sub>\<tau> T)"
rel_interior_convex_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f"   and "convex S" shows "f ` rel_interior S = rel_interior (f ` S)"
span_card_ge_dim: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set"   and V :: "'b set" assumes "vector_space scale"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "finite B" shows "vector_space.dim scale V \<le> card B"
res: shows "local.select_correct_factor init polys = ((q, ri), l, r)"
first_in_tree_order_def: shows "Shadow_DOM.first_in_tree_order \<equiv> l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order Shadow_DOM.to_tree_order"
is_loc_connective_on_def: fixes A :: "'b set"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "is_loc_connective_on A ord r = (\<forall>a\<in>A. \<forall>b1 b2. r a b1 \<and> r a b2 \<longrightarrow> cbelow_on A ord a (\<lambda>a b. r a b \<or> r b a) b1 b2)"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflexive A less_eq"   and "antisymmetric A less_eq" shows "pseudo_ordered_set A less_eq"
subcocycle_add: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and v :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u"   and "subcocycle v" shows "subcocycle (\<lambda>n x. u n x + v n x)"
\<eta>_map_simp: fixes y :: "'b" assumes "B.ide y" shows "transformation_by_components.map (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) B.map FG.\<eta>o y = \<phi> y (F y)"
tllist_all2_conv_all_tnth: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist" shows "tllist_all2 P Q xs ys = (tlength xs = tlength ys \<and> (\<forall>n. enat n < tlength xs \<longrightarrow> P (tnth xs n) (tnth ys n)) \<and> (tfinite xs \<longrightarrow> Q (terminal xs) (terminal ys)))"
emeasure_ge_1_iff: fixes M :: "'a pmf"   and A :: "'a set" shows "(1 \<le> emeasure (measure_pmf M) A) = (emeasure (measure_pmf M) A = 1)"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child)|\<^sub>r"
3: shows "\<^bold>\<not> (E ((\<box>a) \<^bold>\<cdot> a))"
simps_4: fixes a :: "MIR.num" shows "bound0 (Le a) = numbound0 a"
IH_1: shows "\<guillemotleft>ACI_norm_alt r\<guillemotright> = \<guillemotleft>r\<guillemotright>"
rref_upt_condition3_explicit: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and i :: "'c" assumes "reduced_row_echelon_form_upt_k A k"   and "i < i + (1::'c)"   and "\<not> is_zero_row_upt_k i k A"   and "\<not> is_zero_row_upt_k (i + (1::'c)) k A" shows "(LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ (i + (1::'c)) $ n \<noteq> (0::'a))"
reads_restrict: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'e"   and n :: "'a"   and \<sigma>' :: "'b \<Rightarrow> 'e" assumes "IFC prob" shows "(\<sigma> \<restriction> IFC_def.reads prob n = \<sigma>' \<restriction> IFC_def.reads prob n) = (\<forall>x\<in>IFC_def.reads prob n. \<sigma> x = \<sigma>' x)"
embed_bool_idem: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" shows "\<guillemotleft> P \<guillemotright> s * \<guillemotleft> P \<guillemotright> s = \<guillemotleft> P \<guillemotright> s"
infdist_le2: fixes a :: "'b"   and A :: "'b set"   and x :: "'b"   and d :: "real" assumes "a \<in> A"   and "dist x a \<le> d" shows "infdist x A \<le> d"
e: shows "R = Ax \<union> R1 \<union> p_e R2 M N \<union> R3"
STA_Suc: fixes A :: "'b set"   and n :: "nat"   and branch :: "(('c, 'b) fm list \<times> 'b) list" assumes "A, n \<turnstile> branch" shows "A, Suc n \<turnstile> branch"
rPar2: fixes \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> !P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* C"   and "distinct (bn \<alpha>)"   and "\<And>C. Prop C \<Psi> (!P) (\<alpha> \<prec> P')" shows "Prop C \<Psi> (P \<parallel> !P) (\<alpha> \<prec> P \<parallel> P')"
finite_subcls1: 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" shows "finite (subcls1 P)"
fake_bot_def: shows "\<bottom>\<bottom> = - (\<rho> \<squnion> - \<rho>)"
simplePath_empty_conv: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat" shows "Graph.isSimplePath c s [] t = (s = t)"
subterms_inclusion_subset: fixes E1 :: "'a trm set"   and E2 :: "'a trm set"   and E2' :: "'a trm set" assumes "subterms_inclusion E1 E2"   and "E2 \<subseteq> E2'" shows "subterms_inclusion E1 E2'"
neutral: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'a" assumes "\<forall>x\<in>A. g x = (0::'a)" shows "sum_0 g A = (0::'a)"
cMismatch: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and b :: "name"   and c :: "name"   and C :: "'a" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "\<And>C. F C P a x P'"   and "b \<noteq> c" shows "F C ([b\<noteq>c]P) a x P'"
sc_minus_am: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and a :: "'f"   and m :: "'e" assumes "R module M"   and "a \<in> carrier R"   and "m \<in> carrier M" shows "-\<^sub>a\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> m = a \<cdot>\<^sub>s\<^bsub>M\<^esub> (-\<^sub>a\<^bsub>M\<^esub> m)"
hyps: assumes "k < ncols A" shows "span (columns A) = span (columns (Gram_Schmidt_upt_k A k))"
simps_2: fixes worlds :: "'c set"   and relations :: "'b \<Rightarrow> ('c \<times> 'c) set"   and valuation :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "relations \<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> = relations"
sig_crit'_ConsD: fixes b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat" assumes "sig_crit' (b # bs) p"   and "sig_of_pair p \<prec>\<^sub>t lt b" shows "sig_crit' bs p"
strict_upper_solution: fixes y :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real \<Rightarrow> real"   and T :: "real set"   and X :: "real set"   and w :: "real \<Rightarrow> real"   and w' :: "real \<Rightarrow> real"   and t0 :: "real"   and t :: "real" assumes "(y solves_ode f) T X"   and "(w has_vderiv_on w') T"   and "\<And>t. \<lbrakk>t \<in> T; t0 < t\<rbrakk> \<Longrightarrow> f t (w t) < w' t"   and "y t0 \<le> w t0"   and "y t0 = w t0 \<Longrightarrow> f t0 (y t0) < w' t0"   and "t0 < t"   and "t0 \<in> T"   and "t \<in> T"   and "is_interval T" shows "y t < w t"
k1: fixes x :: "'c" shows "k (1, x) = g' x"
meet_left: fixes x :: "nat"   and y :: "nat" shows "gcd x y dvd x"
mat_delete_dim_1: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "dim_row (mat_delete A i j) = dim_row A - 1"
assms_1: shows "prefs_from_table_wf agents alts xs"
omit_oob: fixes ps :: "('b, 'c) fm list"   and v :: "nat"   and xs :: "nat set" assumes "length ps \<le> v" shows "omit ({v} \<union> xs) ps = omit xs ps"
purgeIdle_replicate_Idle: fixes n :: "nat"   and st :: "'St" shows "purgeIdle (replicate n (Idle st)) = []"
lowner_le_imp_trace_le: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "A \<le>\<^sub>L B" shows "trace A \<le> trace B"
test_inf_idempotent: fixes x :: "'a" assumes "test x" shows "x \<sqinter> x = x"
assms_5: shows "(\<phi>\<^sub>S \<Psi> s) (v, a') = Some True"
mono_mono_dgr: fixes D :: "'a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "dmono D" shows "mono_mono (dgr D)"
elims_2: fixes x :: "('b \<times> color) tree" assumes "invh x" obtains "xa__ = \<langle>\<rangle>"   | l :: "('b \<times> color) tree" and x :: "'b" and c :: "color" and r :: "('b \<times> color) tree" where "xa__ = \<langle>l, (x, c), r\<rangle>" and "bheight l = bheight r \<and> invh l \<and> invh r"
poly_eq_iff: fixes p :: "linear_poly"   and q :: "linear_poly" shows "(p = q) = (\<forall>v. coeff p v = coeff q v)"
run_eval_def: shows "run_eval \<equiv> SOME f. \<forall>env e s. evaluate True env s e (f env e s)"
in_span: fixes S :: "rat vec set"   and V :: "(rat vec, 'a) partial_object_scheme"   and v :: "rat vec" assumes "S \<subseteq> carrier V"   and "v \<in> gs.span S" shows "\<exists>a A. A \<subseteq> S \<and> True \<and> gs.lincomb a A = v"
simps_1: fixes r :: "'c rexp"   and s :: "'c rexp" shows "toplevel_inters (rexp.Inter r s) = toplevel_inters r \<union> toplevel_inters s"
scaleC_left_mono: fixes x :: "'a"   and y :: "'a"   and a :: "complex" assumes "x \<le> y"   and "0 \<le> a" shows "a *\<^sub>C x \<le> a *\<^sub>C y"
elims_2: fixes x :: "cell list \<times> cell list \<Rightarrow> bool"   and xa :: "nat \<times> cell list \<times> cell list" assumes "x holds_for xa" obtains P :: "cell list \<times> cell list \<Rightarrow> bool" and s :: "nat" and l :: "cell list" and r :: "cell list" where "x = P" and "xa = (s, l, r)" and "P (l, r)"
cancel_semigroup_axioms: shows "cancel_semigroup f\<^sub>a"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>i a. P (Dvd i a)"   and "\<And>i a. P (NDvd i a)"   and "\<And>p. P p \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)"   and "\<And>v. P (Closed v)"   and "\<And>v. P (NClosed v)" shows "P a0"
cryptinverts_hilf2: fixes p :: "nat"   and m :: "nat"   and k :: "nat"   and q :: "nat" assumes "prime p" shows "m * m ^ (k * pred p * pred q) mod p = m mod p"
simps: fixes bs :: "atom set"   and x :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'a" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
sgn_rat_def: fixes a :: "rat" shows "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
is_Insert_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "is_Insert stateful_strand_step = (\<exists>x41 x42. stateful_strand_step = insert\<langle>x41,x42\<rangle>)"
succ_of_list_def: fixes l :: "(nat \<times> nat) list" shows "succ_of_list l \<equiv> let m = fold (\<lambda>(u, v) g. case g u of None \<Rightarrow> g(u \<mapsto> {v}) | Some s \<Rightarrow> g(u \<mapsto> insert v s)) l Map.empty in (\<lambda>u. case m u of None \<Rightarrow> {} | Some s \<Rightarrow> s)"
sum_outgoing_extend: fixes u :: "nat"   and g :: "nat \<times> nat \<Rightarrow> 'b" assumes "\<And>v. \<lbrakk>v \<in> V; (u, v) \<notin> E\<rbrakk> \<Longrightarrow> g (u, v) = (0::'b)" shows "sum g (outgoing u) = (\<Sum>v\<in>V. g (u, v))"
le_iff_le_annos: fixes C1 :: "'a acom"   and C2 :: "'a acom" shows "(C1 \<le> C2) = (strip C1 = strip C2 \<and> (\<forall>i<length (annos C1). annos C1 ! i \<le> annos C2 ! i))"
foundedInsert: fixes subs :: "'b \<Rightarrow> 'b set"   and delta :: "'b"   and P :: "'b \<Rightarrow> bool"   and n :: "nat"   and B :: "(nat \<times> 'b) set" assumes "\<not> terminal subs delta" shows "founded subs P (insert (n, delta) B) = founded subs P B"
fps_normalize_def: fixes f :: "'b fps" shows "normalize f = (if f = 0 then 0 else fps_X ^ subdegree f)"
irreducible_m_dvdm_prod_list: fixes p :: "int"   and a :: "int poly"   and xs :: "int poly list" assumes "poly_mod_prime p"   and "poly_mod.irreducible_m p a"   and "poly_mod.dvdm p a (prod_list xs)" shows "\<exists>b\<in>set xs. poly_mod.dvdm p a b"
onl_invariant_sterms: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and A :: "('e \<times> ('b, 'c, 'a, 'd) seqp, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and P :: "'e \<times> 'd \<Rightarrow> bool"   and \<xi> :: "'e"   and p :: "('b, 'c, 'a, 'd) seqp"   and p' :: "('b, 'c, 'a, 'd) seqp"   and l :: "'d" assumes "wellformed \<Gamma>"   and "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"   and "(\<xi>, p) \<in> reachable A I"   and "p' \<in> sterms \<Gamma> p"   and "l \<in> labels \<Gamma> p'" shows "P (\<xi>, l)"
cblinfun_fixes_range: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and \<psi> :: "'b" assumes "A o\<^sub>C\<^sub>L B = B"   and "\<psi> \<in> space_as_set (B *\<^sub>S \<top>)" shows "A *\<^sub>V \<psi> = \<psi>"
cong: fixes times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e" shows "comm_monoid_mult.prod times one \<equiv> comm_monoid_mult.prod times one"
AbstrLevels_A9_A92: fixes i :: "AbstrLevelsID" assumes "sA9 \<in> AbstrLevel i" shows "sA92 \<notin> AbstrLevel i"
reduced_row_echelon_form_def: fixes A :: "(('a, 'b) vec, 'c) vec" shows "reduced_row_echelon_form A = reduced_row_echelon_form_upt_k A (ncols A)"
strict: shows "\<forall>i. f (Suc i) <\<cdot> f i"
emeasure_Collect_T: fixes P :: "('b \<times> 'a) stream \<Rightarrow> bool"   and s :: "'b \<times> 'a" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (T s) {x \<in> space (T s). P x} = \<integral>\<^sup>+ t. emeasure (T t) {x \<in> space (T t). P (t ## x)} \<partial>measure_pmf (K s)"
X: shows "topspace X \<subseteq> \<Union> (T ` I)"
xy1: shows "det2 a11 a13 a21 a23 * x1 + det2 a12 a13 a22 a23 * y1 = (0::'a)"
div_self: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide a a = one"
less: shows "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"
mult_1_left: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.monoid_mult one times" shows "times one a = a"
NOT_mask_AND_mask: fixes w :: "'b"   and n :: "nat" shows "(w AND mask n) AND NOT (mask n) = (0::'b)"
FSwSbImorph_def: fixes h :: "'e \<Rightarrow> 'f"   and hA :: "'g \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'e, 'g, 'j) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'i, 'f, 'h, 'k) model_scheme" shows "FSwSbImorph h hA MOD MOD' \<equiv> FSwImorph h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
surjective: fixes r :: "'a Config_scheme" shows "r = \<lparr>schedule = schedule r, cycleLength = cycleLength r, \<dots> = Config.more r\<rparr>"
assms_1: shows "(z1, z2) = calc_ideal_point2_cvec A B"
transfer_return: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(R ===> local.rel_state R) State_Monad.return return"
pref_locally_linear: fixes s :: "'b llist"   and x :: "'b llist"   and t :: "'b llist" assumes "s \<le> x"   and "t \<le> x" shows "s \<le> t \<or> t \<le> s"
simps: fixes a1 :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and a2 :: "typ" shows "wf_type a1 a2 = ((\<exists>\<Sigma> \<kappa> Ts. a1 = \<Sigma> \<and> a2 = Ty \<kappa> Ts \<and> type_arity \<Sigma> \<kappa> = Some (length Ts) \<and> (\<forall>T\<in>set Ts. wf_type \<Sigma> T)) \<or> (\<exists>\<Sigma> S a. a1 = \<Sigma> \<and> a2 = Tv a S \<and> wf_sort (subclass (osig \<Sigma>)) S))"
d_idempotent: fixes x :: "'a" shows "d (d x) = d x"
rule_charn3: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int" assumes "wellformed_policy1 p"   and "allNetsDistinct p"   and "singleCombinators p"   and "wellformed_policy3Pr p"   and "applied_rule_rev Cp x p = \<lfloor>DenyAllFromTo c d\<rfloor>"   and "AllowPortFromTo a b po \<in> set p" shows "x \<notin> dom (Cp (AllowPortFromTo a b po))"
eq_name_eq_effs: fixes name :: "char list"   and name' :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "name = name'"   and "(name, pres, effs, layer) \<in> set ast\<delta>"   and "(name', pres', effs', layer') \<in> set ast\<delta>" shows "effs = effs'"
add_mono_le_Fin: fixes x1 :: "'a extended"   and y1 :: "'a"   and x2 :: "'a extended"   and y2 :: "'a" assumes "x1 \<le> Fin y1"   and "x2 \<le> Fin y2" shows "x1 + x2 \<le> Fin (y1 + y2)"
hyp: assumes "(v, y) \<in> r\<^sup>*" shows "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
psubst_exi: fixes \<phi> :: "'fmla"   and y :: "'var"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "y \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "y \<notin> snd ` set txs"   and "y \<notin> \<Union> (FvarsT ` fst ` set txs)"   and "distinct (map snd txs)" shows "psubst (exi y \<phi>) txs = exi y (psubst \<phi> txs)"
finite_greaterThanAtMost_int: fixes l :: "int"   and u :: "int" shows "finite {l<..u}"
simps_2: fixes s :: "('f, 'g) locks \<times> (('g \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option) \<times> 'h) \<times> ('g \<Rightarrow> 'i wait_set_status option) \<times> 'g set"   and t :: "'g" shows "cond_action_ok s t Yield = True"
col_of_dagger: fixes j :: "nat"   and M :: "complex Matrix.mat" assumes "j < dim_row M" shows "Matrix.col M\<^sup>\<dagger> j = Matrix.vec (dim_col M) (\<lambda>i. cnj (M $$ (j, i)))"
wffp_to_wfp: fixes \<pi> :: "(('St, 'U, 'C) state \<times> ('U, 'C, 'Out) Noninterference.aprop set) stream"   and sl :: "('St, 'U, 'C) state list" assumes "\<pi> = map (\<lambda>s. (s, L s)) sl @- sconst (toSink (last sl), L (toSink (last sl)))"   and "wffp sl" shows "wfp UNIV \<pi> \<and> (\<forall>i<length sl. sl ! i = fst (\<pi> !! i)) \<and> (\<forall>i\<ge>length sl. fst (\<pi> !! i) = toSink (last sl)) \<and> stateOf \<pi> = hd sl"
sep_substate_def: fixes x :: "'b"   and y :: "'b" shows "x \<preceq> y \<equiv> \<exists>z. x ## z \<and> x + z = y"
bit_1_iff: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit one n = (one \<noteq> zero \<and> n = 0)"
prems_2: fixes i :: "hf" assumes "i \<^bold>\<in> A \<triangleleft> x" shows "Ord i"
isMaxim_def: fixes A :: "('b \<times> 'c) set"   and b :: "'b \<times> 'c" shows "st.isMaxim A b \<equiv> b \<in> A \<and> (\<forall>a\<in>A. (a, b) \<in> s *o t)"
inter_in_inters: fixes l :: "Lines"   and m :: "Lines" shows "Pascal_Property.inter l m \<in> inters l m"
data_in_cong: fixes tas :: "nat \<times> 'a option list" assumes "tas \<in> ts_tuple_rel (set (linearize data_in))" shows "valid_tuple tuple_since' tas = valid_tuple tuple_since tas"