hcju/isabelleps huggingface dataset:
cut_ge_finite: fixes I :: "'b set"   and t :: "'b" assumes "finite I" shows "finite (I \<down>\<ge> t)"
hcomp_in_vhom: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a"   and \<nu> :: "'a"   and h :: "'a"   and k :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<rightarrow>\<^sub>B k\<guillemotright>"   and "src\<^sub>B h = trg\<^sub>B f" shows "\<guillemotleft>\<nu> \<star>\<^sub>B \<mu> : h \<star>\<^sub>B f \<rightarrow>\<^sub>B k \<star>\<^sub>B g\<guillemotright>"
obj_src: fixes \<mu> :: "'a" assumes "S.arr \<mu>" shows "S.obj (S.src \<mu>)"
productive_in_N: fixes N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "production N C \<noteq> {}" shows "C \<in> N"
b: shows "X \<in> obj\<^bsub>C\<^esub>"
fls_uminus_subdegree: fixes f :: "'a fls" shows "fls_subdegree (- f) = fls_subdegree f"
trss_c_cases: fixes s :: "'f list"   and c :: "'f list multiset"   and w :: "('g, 'h) label list"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c), w, s', c') \<in> trcl (trss fg)" obtains csp :: "'f list multiset" where "c' = csp + c" and "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"
ide_is_iso: fixes a :: "'c local.term set" assumes "\<F>\<^sub>SC.ide a" shows "\<F>\<^sub>SC.iso a"
vdw_1_right: fixes k :: "nat" shows "vdw k 1 1"
acc_point_res: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "residue (p ^ k) (acc_point s (Suc k)) = acc_point s k"
simps_1: 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 cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.upd_rev_cs kind cs [] = cs"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_on_left_moving m rs (l, r) = (wadjust_on_left_moving_O m rs (l, r) \<or> wadjust_on_left_moving_B m rs (l, r))"
sup_commute: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x y = sup y x"
h2: shows "\<forall>csa. (csa \<in> consistent_sign_vectors (fst (factorize_polys qs)) UNIV \<and> has_no_zeros csa) = List.member (find_consistent_signs_at_roots (coprime_r (cast_rat_list (fst (factorize_polys qs)))) (cast_rat_list (fst (factorize_polys qs)))) csa"
fset_of_list_simps_1: shows "fset_of_list [] = {||}"
term_ok_theory_cond_code: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and ax :: "Core.term list" assumes "exe_sig_conds (ExeSignature cto tao sa)" shows "(\<forall>p\<in>set ax. term_ok (translate_theory (ExeTheory (ExeSignature cto tao sa) ax)) p \<and> typ_of p = Some propT) = (\<forall>p\<in>set ax. exeterm_ok' (ExeSignature cto tao sa) p \<and> typ_of p = Some propT)"
atomsPropagate': fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and x :: "signs \<times> predicate \<times> vbl list"   and n :: "nat" assumes "branch subs gamma f"   and "x \<in> set (fst (f n))" shows "x \<in> set (fst (f (Suc n)))"
newman': fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "wfP R\<inverse>\<inverse>"   and "\<And>a b c. \<lbrakk>R a b; R a c\<rbrakk> \<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"   and "R\<^sup>*\<^sup>* a b"   and "R\<^sup>*\<^sup>* a c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
eq_set_tree_empty: fixes t :: "('a \<times> 'b) tree" shows "(Tree2.set_tree t = {}) = (t = \<langle>\<rangle>)"
Main_no_return_source: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>Main\<^esub>f'" shows "False"
ex_un_null: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "partial_magma C" shows "\<exists>!n. \<forall>f. C n f = n \<and> C f n = n"
step: fixes f :: "'d"   and g :: "'f" assumes "P (U1 f) (U2 g)" shows "P (U1 (F f)) (U2 (G g))"
add_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and rs1_val :: "word32"   and operand2 :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s1)) \<and> t2 = snd (fst (add_instr_sub1 instr_name result rs1_val operand2 s2))" shows "low_equal t1 t2"
vconn_implies_hpar_3: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom \<mu> f g" shows "Span.src g = Span.src f"
fimage_fempty: fixes f :: "'b \<Rightarrow> 'a" shows "f |`| {||} = {||}"
evenE: fixes 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 divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.semiring_parity plus minus zero times one divide modulo"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" obtains b :: "'c" where "a = times (numeral.numeral one plus (num.Bit0 num.One)) b"
nt_inv0: fixes a :: "'a"   and n :: "int" assumes "a \<in> carrier A" shows "-\<^sub>a n\<triangleright>a\<^bsub>A\<^esub> = (- n)\<triangleright>a\<^bsub>A\<^esub>"
orthogonal_matrix_transpose: fixes A :: "(('a, 'b) vec, 'b) vec" shows "orthogonal_matrix (Finite_Cartesian_Product.transpose A) = orthogonal_matrix A"
comp_arr_inv': fixes f :: "'a" assumes "C.iso f" shows "f \<cdot>\<^sub>C C.inv f = C.cod f"
elem_set_implies_incl_in: fixes a :: "'s"   and b :: "'s" assumes "a \<in> local.set b" shows "incl_in a b"
assms: fixes i :: "'a" assumes "i \<in> A" shows "convex (B i)"
distrib_imp2: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" assumes "\<And>x y z. (x \<or> y \<and> z) = ((x \<or> y) \<and> (x \<or> z))" shows "(x \<and> (y \<or> z)) = (x \<and> y \<or> x \<and> z)"
relhoare_conseq_right: fixes pre :: "('a \<times> 'b) set"   and Ra :: "('a \<times> 'a) set"   and Rc :: "('b \<times> 'b) set"   and post' :: "('a \<times> 'b) set"   and post :: "('a \<times> 'b) set" assumes "{pre} Ra, Rc {> post'}"   and "post' \<subseteq> post" shows "{pre} Ra, Rc {> post}"
elims: fixes x :: "char list \<times> fterm list"   and y :: "char list \<times> hterm list" assumes "hatom_of_fatom x = y" obtains p :: "char list" and ts :: "fterm list" where "x = (p, ts)" and "y = (p, hterms_of_fterms ts)"
tl_drop: fixes n :: "nat"   and xs :: "'k list" shows "tl (drop n xs) = drop n (tl xs)"
fun_upds_notin: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and ys :: "'b list" assumes "x \<notin> set xs" shows "f(xs [:=] ys) x = f x"
distrib_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
ocondition_cong: fixes c :: "'d \<Rightarrow> bool"   and c' :: "'d \<Rightarrow> bool"   and l :: "'d \<Rightarrow> 'e"   and l' :: "'d \<Rightarrow> 'e"   and r :: "'d \<Rightarrow> 'e"   and r' :: "'d \<Rightarrow> 'e" assumes "c = c'"   and "\<And>s. c' s \<Longrightarrow> l s = l' s"   and "\<And>s. \<not> c' s \<Longrightarrow> r s = r' s" shows "ocondition c l r = ocondition c' l' r'"
inv_approx_preserv: fixes S :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "inv_approx S X (\<Delta> + 1)"   and "x \<in> V - X" shows "inv_approx (insert x S) (X \<union> neighbors x \<union> {x}) (\<Delta> + 1)"
index_eq_iff: fixes xs :: "'b list"   and x :: "'b"   and i :: "nat" shows "(index xs x = i) = (i \<le> length xs \<and> (\<forall>j<i. xs ! j \<noteq> x) \<and> (i < length xs \<longrightarrow> xs ! i = x))"
3: fixes g :: "'g"   and p :: "'val" assumes "unnecessaryPhi g p"   and "\<not> uninst_code.ssa.trivial uses phis g p"   and "p \<in> uninst_code.ssa.allVars uses phis g" obtains q :: "'val" where "uninst_code.ssa.phiArg phis g p q" and "unnecessaryPhi g q" and "\<not> uninst.def_dominates (\<lambda>g. Mapping.lookup (phis g)) g q p"
liftn_strip_context: fixes t :: "dB"   and k :: "nat"   and n :: "nat" assumes "\<forall>i\<in>frees t. i < k \<or> k + n \<le> i" shows "liftn n (strip_context n t k) k = t"
secure_refinement_simple_def: fixes \<R>\<^sub>A :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set"   and \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and abs_steps :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat) \<Rightarrow> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat) \<Rightarrow> nat" shows "secure_refinement_simple \<R>\<^sub>A \<R> abs_steps \<equiv> closed_others \<R> \<and> preserves_modes_mem \<R> \<and> new_vars_private \<R> \<and> simple_refinement_safe \<R>\<^sub>A \<R> abs_steps \<and> bisim_simple \<R>\<^sub>A \<and> (\<forall>c\<^sub>1\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A c\<^sub>1\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>c\<^sub>1\<^sub>C' mds\<^sub>C' mem\<^sub>1\<^sub>C'. eval_abv\<^sub>C \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<exists>c\<^sub>1\<^sub>A' mds\<^sub>A' mem\<^sub>1\<^sub>A'. A.neval \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A (abs_steps \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>)))"
mod_ring_rel_def: fixes x :: "int"   and x' :: "'a mod_ring" shows "mod_ring_rel x x' = (x = to_int_mod_ring x')"
int_and_lt0: fixes x :: "int"   and y :: "int" shows "(x AND y < 0) = (x < 0 \<and> y < 0)"
pw_bind_nofail: fixes M :: "'f nres"   and f :: "'f \<Rightarrow> 'e nres" shows "nofail (M \<bind> f) = (nofail M \<and> (\<forall>x. inres M x \<longrightarrow> nofail (f x)))"
map_of_map_default: fixes k :: "'b"   and v :: "'c"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('b \<times> 'c) list" shows "map_of (map_default k v f xs) = (map_of xs)(k := case map_of xs k of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (f v'))"
simps_3: fixes G :: "prog"   and tn :: "qtname" shows "isrtype G (ClassT tn) = is_class G tn"
empty_iff: fixes c :: "'h" shows "(c \<in> {}) = False"
PO_rhoare_defs_2: fixes r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
Equality: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and a :: "poscheckvariant" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S"   and "Unifier \<I> t t'" shows "sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> (S @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
sign_simps_34: fixes bm :: "'d"   and am :: "'d"   and ck :: "'d" shows "bm * (am * ck) = am * (bm * ck)"
negligible_convex_frontier: fixes S :: "'b set" assumes "convex S" shows "negligible (frontier S)"
homeomorphism_cont2: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" assumes "homeomorphism S T f g" shows "continuous_on T g"
trivial_FGModule: fixes smult :: "('a, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>a\<in>group_ring. smult a (0::'b) = (0::'b)" shows "FGModule G smult 0"
nmi: shows "\<not> Ifm (a # bs) (plusinf p)"
compactin_imp_Bolzano_Weierstrass: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "compactin X S"   and "infinite T \<and> T \<subseteq> S" shows "S \<inter> X derived_set_of T \<noteq> {}"
eval_embed_true: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "P s" shows "\<guillemotleft> P \<guillemotright> s = 1"
vars_mset_subst_db: fixes i :: "nat"   and x :: "'b"   and s :: "('s, 'b) tm" shows "vars_mset (subst_db i x s) = {#y \<in># vars_mset s. y \<noteq> x#}"
THE_NRES_def: shows "THE_NRES = case_option SUCCEED RETURN"
Inf_top_conv_1: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf A = top) = (\<forall>x\<in>A. x = top)"
Or\<^sub>n_def: fixes \<Phi> :: "'b ltln set" shows "Or\<^sub>n \<Phi> \<equiv> SOME \<phi>. fold_graph Or_ltln false\<^sub>n \<Phi> \<phi>"
mr_cos_sprod_one: fixes H :: "'a set"   and X :: "'a set" assumes "submodule R M H"   and "X \<in> set_mr_cos M H" shows "mr_cos_sprod M H 1\<^sub>r\<^bsub>R\<^esub> X = X"
preallocated_def: fixes h :: "heap" shows "preallocated h \<equiv> \<forall>C\<in>sys_xcpts. \<exists>fs. h (addr_of_sys_xcpt C) = \<lfloor>(C, fs)\<rfloor>"
inv_a_2: shows "a \<otimes> inv a = \<one>"
card_bind_constant: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c set"   and k :: "nat" assumes "finite S"   and "\<forall>X\<in>S. finite (f X)"   and "disjoint_family_on f S"   and "\<And>x. x \<in> S \<Longrightarrow> card (f x) = k" shows "card (S \<bind> f) = card S * k"
dens: shows "dens_ctxt_\<alpha> ([], [], \<Gamma>, CReal 1) \<turnstile>\<^sub>d e \<Rightarrow> (\<lambda>\<rho> x. ennreal (eval_cexpr f \<rho> x))"
total_one_closed: shows "times_top_class.total (1::'a)"
elims: fixes x :: "recf"   and y :: "nat" assumes "arity x = y" obtains "x = Z" and "y = 1"   | "x = S" and "y = 1"   | m :: "nat" and n :: "nat" where "x = recf.Id m n__" and "y = m"   | n :: "nat" and f :: "recf" and gs :: "recf list" where "x = Cn na__ fa__ gs" and "y = na__"   | n :: "nat" and f :: "recf" and g :: "recf" where "x = Pr nb__ fb__ g" and "y = Suc nb__"   | n :: "nat" and f :: "recf" where "x = Mn n f" and "y = n"
Gauss_Jordan_upt_k_iarrays_def: fixes A :: "'a iarray iarray"   and k :: "nat" shows "Gauss_Jordan_upt_k_iarrays A k = snd (foldl Gauss_Jordan_column_k_iarrays (0, A) [0..<Suc k])"
minimize_wrt_subset: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "set (minimize_wrt P xs) \<subseteq> set xs"
F_set_conv_fold: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "Min (set xs) = Finite_Set.fold min (hd xs) (set (tl xs))"
Q'Chain: shows "\<Psi> \<otimes> \<one> \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q''"
assms_3: shows "1 / p + 1 / q = 1"
ins_bal_i_Bal: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Bal t'"   and "bal_i n (height t + d)" shows "bal_i (n + 1) (height t' + d)"
confinement: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and l :: "nat" assumes "(c, s) \<Rightarrow> t"   and "l \<turnstile> c" shows "s = t (< l)"
conv_star_conv: fixes x :: "'a" shows "x\<^sup>\<star> \<le> x\<^sup>T\<^sup>\<star>\<^sup>T"
pr_tuple_1: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "span f g"   and "cod f = a"   and "cod g = b" shows "\<pp>\<^sub>1[a, b] \<cdot> \<langle>f, g\<rangle> = f"
valuesum_def: fixes m :: "('d, 'c) mapping" shows "valuesum m = (\<Sum>k\<in>Mapping.keys m. the (Mapping.lookup m k))"
spmf_of_set_singleton: fixes x :: "'c" shows "spmf_of_set {x} = return_spmf x"
ipurge_fail_aux_t_failures: fixes P :: "'b process"   and I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and X :: "'b set"   and U :: "'c set" assumes "secure P I D"   and "(xs, X) \<in> failures P" shows "ipurge_fail_aux_t I D U xs X \<in> failures P"
meet_below2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "y \<sqsubseteq> z" shows "x \<sqinter> y \<sqsubseteq> z"
nchop_def: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<equiv> i = j \<squnion> k \<and> (j = bot \<or> k = bot \<or> consec j k)"
nat_pow_Suc: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and n :: "nat" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
Q_def: shows "Q \<equiv> \<lambda>E d. M E d - V \<subseteq> R d - V"
nvL_tL: fixes \<sigma> :: "'tp"   and l :: "('fsym, 'b) lit" assumes "\<not> unprot \<sigma>" shows "\<forall>x\<in>nvL (tL l). TE.tpOfV x \<noteq> \<sigma>"
locallyI: fixes S :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "\<And>w x. \<lbrakk>openin (top_of_set S) w; x \<in> w\<rbrakk> \<Longrightarrow> \<exists>u v. openin (top_of_set S) u \<and> P v \<and> x \<in> u \<and> u \<subseteq> v \<and> v \<subseteq> w" shows "locally P S"
independent_span_bound: fixes T :: "'d set"   and S :: "'d set" assumes "finite T"   and "cindependent S"   and "S \<subseteq> cspan T" shows "finite S \<and> card S \<le> card T"
run_owner_iii_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "case run_owner_iii n s of (s', S, A, B, C, x) \<Rightarrow> {Asset n} \<times> {Crypt (Auth_ShaKey n) (PriKey S), \<lbrace>Num 2, PubKey B\<rbrace>} \<union> {Owner n} \<times> {\<lbrace>Num 1, PubKey A\<rbrace>, \<lbrace>Num 3, PubKey C\<rbrace>} \<subseteq> s' \<and> (Asset n, PubKey B) \<notin> s'"
primitive_part_def: fixes p :: "'b poly" shows "primitive_part p = map_poly (\<lambda>x. x div content p) p"
in_these_eq: fixes x :: "'b"   and A :: "'b option set" shows "(x \<in> Option.these A) = (Some x \<in> A)"
\<Delta>_eq_\<rho>: shows "(\<mu> \<star> r\<^sub>0) \<cdot> \<rho> = \<rho>"
sub1: fixes A :: "'d"   and k :: "'d"   and a :: "'d"   and b :: "'d" assumes "A \<equiv> k + a" shows "A - b \<equiv> k + (a - b)"
l_cancel_one: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier S"   and "a \<in> carrier S" shows "(x \<oplus>\<^bsub>S\<^esub> a = x) = (a = \<zero>\<^bsub>S\<^esub>)"
bij_betw_multiset_number_partition_with_size: fixes n :: "nat"   and k :: "nat" shows "bij_betw count {N. number_partition n N \<and> size N = k} {p. p partitions n \<and> sum p {..n} = k}"
LstSeqP_OrdP: fixes H :: "fm set"   and r :: "tm"   and k :: "tm"   and y :: "tm" assumes "H \<turnstile> LstSeqP r k y" shows "H \<turnstile> OrdP k"
proper_interval_set_aux_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "proper_interval_set_aux_fusion g1 g2 s1 s2 = proper_interval_set_aux (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
enumdom_empty_domain: fixes i :: "nat"   and y :: "nat" assumes "\<And>x. \<phi> i x \<up>" shows "eval r_enumdom [i, y] \<up>"
igSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igSwapCls MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbs MOD \<and> igSwapIGOp MOD"
add_le_less_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less_eq a b"   and "less c d" shows "less (plus a c) (plus b d)"
collapse_2: fixes literal :: "'a literal" assumes "is_neg literal" shows "Neg (atm_of literal) = literal"
isotone_def: fixes A :: "('a, 'b) gorder_scheme"   and B :: "('c, 'd) gorder_scheme"   and f :: "'a \<Rightarrow> 'c" shows "isotone A B f \<equiv> weak_partial_order A \<and> weak_partial_order B \<and> (\<forall>x\<in>carrier A. \<forall>y\<in>carrier A. le A x y \<longrightarrow> le B (f x) (f y))"
cr_envT_stateTE: fixes s :: "'f"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m1 :: "('f, 'g) envT"   and m2 :: "('f, 'h) stateT" assumes "cr_envT_stateT s M m1 m2" obtains f :: "'f \<Rightarrow> 'g" and g :: "'f \<Rightarrow> 'h" where "m1 = EnvT f" and "m2 = StateT g" and "(eq_onp ((=) s) ===> M) f g"
split_face_edges_f12: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and vs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and vs1 :: "nat list" assumes "pre_split_face f ram1 ram2 vs"   and "(f12, f21) = split_face f ram1 ram2 vs"   and "vs \<noteq> []"   and "vs1 = between (vertices f) ram1 ram2"   and "vs1 \<noteq> []" shows "\<E> f12 = {(hd vs, ram1), (ram1, hd vs1), (last vs1, ram2), (ram2, last vs)} \<union> Edges (rev vs) \<union> Edges vs1"
prod_as_plus: fixes X :: "'b \<Longrightarrow> 'd"   and Y :: "'c \<Longrightarrow> 'e" shows "X \<times>\<^sub>L Y = X ;\<^sub>L fst\<^sub>L +\<^sub>L Y ;\<^sub>L snd\<^sub>L"
preq: fixes V :: "'a set" assumes "V \<subseteq> S" shows "identity S \<^sup>\<inverse> S V = V"
inject: fixes x1 :: "nat"   and x2 :: "'b"   and x3 :: "'b tree list"   and y1 :: "nat"   and y2 :: "'b"   and y3 :: "'b tree list" shows "(Node x1 x2 x3 = Node y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
tMap_tFirst: fixes f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace" shows "tFirst (tMap f t) = f (tFirst t)"
SUP_least: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and u :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
Compl_in_sets_lebesgue: fixes A :: "'c set" shows "(- A \<in> sets lebesgue) = (A \<in> sets lebesgue)"
triangle_num_eq_0_iff: fixes n :: "nat" shows "(triangle_num n = 0) = (n = 0)"
ctr_simps_UNIV_I: fixes x :: "'e" shows "x \<in> UNIV"