hcju/isabelleps huggingface dataset:
evtEA: fixes n :: "nat"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'c"   and t' :: "nat \<Rightarrow> 'c"   and \<gamma> :: "(nat \<Rightarrow> 'c) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. active c (t i)"   and "dynamic_component.eval tCMP active c t t' n (dynamic_component.evt \<gamma>)" shows "\<exists>n'\<ge>dynamic_component.nxtAct active c t n. (\<exists>i\<ge>n'. active c (t i) \<and> (\<forall>n''\<ge>dynamic_component.lNAct active c t n'. n'' \<le> dynamic_component.nxtAct active c t n' \<longrightarrow> dynamic_component.eval tCMP active c t t' n'' \<gamma>)) \<or> \<not> (\<exists>i\<ge>n'. active c (t i)) \<and> dynamic_component.eval tCMP active c t t' n' \<gamma>"
termi_call_steps_def: shows "termi_call_steps = {(t, s). body \<down> s \<and> (\<exists>cs. ([body], s) \<rightarrow>\<^sup>* (CALL # cs, t))}"
assms_1: shows "natural_isomorphism (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F' F \<tau>"
tailrec_transform_aux1: fixes s :: "'c"   and m :: "'c nres"   and a :: "'c \<Rightarrow> 'c nres"   and b :: "'c \<Rightarrow> 'd nres" assumes "RETURN s \<le> m" shows "REC (\<lambda>D s. sup (a s \<bind> D) (b s)) s \<le> lfp (\<lambda>x. sup m (x \<bind> a)) \<bind> b"
neq_qf: fixes t :: "tm" shows "qfree (neq t)"
cases: fixes x :: "('f, 'v) Term.term \<times> ('f, 'v) Term.term" obtains s :: "('f, 'v) Term.term" and t :: "('f, 'v) Term.term" where "x = (s, t)"
Disjunctive_def: shows "Disjunctive = {x. \<forall>X. times_abc x (Sup_b X) = Sup_c (times_abc x ` X)}"
upt_add_eq_append: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "[i..<j + k] = [i..<j] @ [j..<j + k]"
iter_narrow_step_ivl_termination: fixes C :: "ivl st option acom" assumes "top_on\<^sub>c C (- vars C)"   and "step_ivl \<top> C \<le> C" shows "\<exists>C'. iter_narrow (step_ivl \<top>) C = Some C'"
prems_1: shows "(e, d) \<in> set (m # M')"
simps_2: fixes t' :: "'b"   and n :: "nat"   and t :: "'b" shows "lock_lock \<lfloor>(t', n)\<rfloor> t = \<lfloor>(t', Suc n)\<rfloor>"
prod_mset_prod_list: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and xs :: "'a list" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one (mset xs) = monoid_mult.prod_list one times xs"
Int_space_eq2: fixes x :: "'a set"   and M :: "'a measure" assumes "x \<in> fmeasurable M" shows "x \<inter> space M = x"
connect_def: fixes d :: "('e, 'f) distinguisher"   and res :: "('e, 'f) resource" shows "connect d res = run_gpv run_resource d res"
bisim1CAS3: fixes P :: "'a prog"   and h :: "'heap"   and e3 :: "'addr expr1"   and e3' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e1 :: "'addr expr1"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and v :: "'addr val"   and v' :: "'addr val" assumes "P,e3,h \<turnstile> (e3', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3),h \<turnstile> (Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e3'), xs) \<leftrightarrow> (stk @ [v', v], loc, length (compE2 e1) + length (compE2 e2) + pc, xcp)"
map_sq_matrix_diag: fixes f :: "'d \<Rightarrow> 'c"   and c :: "'d" assumes "f (0::'d) = (0::'c)" shows "map_sq_matrix f (diag c) = diag (f c)"
is_expansion_aux_cong: fixes F :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "is_expansion_aux F f basis"   and "\<forall>\<^sub>F x in at_top. f x = g x" shows "is_expansion_aux F g basis"
groupF_equality: shows "\<forall>A\<in>set (groupF (\<lambda>wi. (map (\<lambda>d. runFw (getOneIp wi) d c rs) (map getOneIp (getParts rs)), map (\<lambda>s. runFw s (getOneIp wi) c rs) (map getOneIp (getParts rs)))) (getParts rs)). \<forall>a1\<in>set A. \<forall>a2\<in>set A. (map (\<lambda>d. runFw (getOneIp a1) d c rs) (map getOneIp (getParts rs)), map (\<lambda>s. runFw s (getOneIp a1) c rs) (map getOneIp (getParts rs))) = (map (\<lambda>d. runFw (getOneIp a2) d c rs) (map getOneIp (getParts rs)), map (\<lambda>s. runFw s (getOneIp a2) c rs) (map getOneIp (getParts rs)))"
acrrs: shows "all_chains (no_call_to c) \<Gamma> rrs"
Collect_mono: fixes P :: "'f \<Rightarrow> bool"   and Q :: "'f \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
relabel_adm_edges_2: fixes u :: "nat" assumes "relabel_precond f (relabel_effect f l u) u" shows "adm_edges f (relabel_effect f (relabel_effect f l u) u) \<inter> cf.incoming u = {}"
subset_Intr: fixes B :: "'a set"   and C :: "'a set"   and A :: "'a set" assumes "B \<subseteq> C" shows "A \<inter> B \<subseteq> C"
homeomorphism_empty: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "homeomorphism {} {} f g"
fempty_iff: fixes c :: "'m" shows "(c |\<in>| {||}) = False"
List_def: fixes h :: "'c \<Rightarrow> 'c ref"   and x :: "'c ref"   and as :: "'c list" shows "List h x as = Path h x as Null"
replace_s_by_s': fixes A :: "type_scheme list"   and t :: "typ"   and S :: "nat \<Rightarrow> typ"   and e :: "expr" assumes "$ (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) A |- e :: $ (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) t" shows "$ S A |- e :: $ S t"
E: shows "j \<in> offs_set_next ns xs index key mi ma i"
asset_i_asset_ii: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, Crypt (Auth_ShaKey m) (PriKey A)) \<in> s"   and "(Asset n, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s" shows "False"
prefix_snoc: fixes xs :: "'c list"   and ys :: "'c list"   and y :: "'c" shows "prefix xs (ys @ [y]) = (xs = ys @ [y] \<or> prefix xs ys)"
wf\<^sub>M\<^sub>G\<^sub>U_def: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term" shows "wf\<^sub>M\<^sub>G\<^sub>U \<theta> s t \<equiv> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> MGU \<theta> s t \<and> subst_domain \<theta> \<union> range_vars \<theta> \<subseteq> fv s \<union> fv t"
bind_map_pmf: fixes f :: "'f \<Rightarrow> 'e"   and M :: "'f pmf"   and g :: "'e \<Rightarrow> 'd pmf" shows "map_pmf f M \<bind> g = M \<bind> (\<lambda>x. g (f x))"
StaticCastNull: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"
conjugate_neg: fixes conjugate :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate (uminus a) = uminus (conjugate a)"
norm_ge_zero: fixes x :: "'b" shows "0 \<le> norm x"
map_poly_eq_0_iff: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "f (0::'b) = (0::'a)"   and "\<And>x. \<lbrakk>x \<in> set (coeffs p); x \<noteq> (0::'b)\<rbrakk> \<Longrightarrow> f x \<noteq> (0::'a)" shows "(map_poly f p = 0) = (p = 0)"
prems: shows "Square_Matrix.row B i = axis j (1::'a)"
eq: shows "{1..Suc n} = insert (Suc n) {1..n}"
sinvar_mono_imp_negative_delete_edge_mono: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and X :: "('c \<times> 'c) set"   and Y :: "('c \<times> 'c) set"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph G"   and "X \<subseteq> Y"   and "\<not> sinvar (delete_edges G Y) nP" shows "\<not> sinvar (delete_edges G X) nP"
induced_automorphism_def: fixes v :: "'a" shows "\
 v \<equiv> if v \<in> f ` \<Union> X then g v else restrict1 f (g ` \<Union> X) v"
intros_2: fixes xa :: "'b list"   and xb :: "'b list"   and x :: "'b" assumes "prefixList xa xb" shows "prefixList (x # xa) (x # xb)"
IfE: fixes b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and ct :: "com \<times> (char list \<Rightarrow> int)" assumes "(IF b THEN c1 ELSE c2, s) \<rightarrow> ct" obtains "ct = (c1, s)" and "bval b s"   | "ct = (c2, s)" and "\<not> bval b s"
T_def: fixes e :: "((nat \<times> nat) \<times> transition) fset" shows "T e = {t. accepts_trace e 0 <> t}"
map_add_dom_app_simps_1: fixes m :: "'a"   and l2 :: "'a \<Rightarrow> 'b option"   and l1 :: "'a \<Rightarrow> 'b option" assumes "m \<in> dom l2" shows "(l1 ++ l2) m = l2 m"
openin_pullback_topology: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b topology"   and S :: "'a set" shows "openin (pullback_topology A f T) S = (\<exists>U. openin T U \<and> S = f -` U \<inter> A)"
l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, char list option) prog"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog) \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and find_slot :: "bool \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and assigned_slot :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog" shows "l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order find_slot assigned_slot \<equiv> find_slot = l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_find_slot get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order \<and> assigned_slot = l_find_slot\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_assigned_slot get_parent get_shadow_root get_mode get_attribute get_tag_name first_in_tree_order"
cont: shows "continuous_on {0..1} (\<lambda>x. Im (integral {0..x} (\<lambda>x. vector_derivative \<gamma> (at x) / (\<gamma> x - z))))"
viable_market_def: fixes Mkt :: "('a, 'c) discrete_market" shows "viable_market Mkt = (\<forall>p. stock_portfolio Mkt p \<longrightarrow> \<not> arbitrage_process Mkt p)"
trie_update_with_def: fixes ks :: "'b list"   and v :: "'c"   and f :: "'c \<Rightarrow> 'c" shows "trie_update_with ks v f \<equiv> update_with_trie ks (\<lambda>vo. f (case vo of None \<Rightarrow> v | Some v \<Rightarrow> v))"
Inter_lower: fixes B :: "'b set"   and A :: "'b set set" assumes "B \<in> A" shows "\<Inter> A \<subseteq> B"
simps_20: shows "cospan_shape.comp cospan_shape.AT cospan_shape.AT = cospan_shape.Null"
one_product_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "\<one>\<^bsub>product_group I G\<^esub> = (\<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>)"
cmp_simps_4: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi>\<^sub>F (f, g)) = F f \<star>\<^sub>D F g"
stutinv_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" shows "STUTINV F \<equiv> \<forall>\<sigma> \<tau>. \<sigma> \<approx> \<tau> \<longrightarrow> F \<sigma> = F \<tau>"
fls_lr_inverse_mult_ring1_2: fixes x :: "'a"   and f :: "'a fls"   and y :: "'a"   and g :: "'a fls" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)"   and "fls_nth f (fls_subdegree f) * x = (1::'a)"   and "y * fls_nth g (fls_subdegree g) = (1::'a)"   and "fls_nth g (fls_subdegree g) * y = (1::'a)" shows "fls_right_inverse (f * g) (y * x) = fls_right_inverse g y * fls_right_inverse f x"
lexext_rev_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"   and "lexext_rev gt ys xs" shows "lexext_rev gt (map f ys) (map f xs)"
vimage_eq: fixes a :: "'g"   and f :: "'g \<Rightarrow> 'h"   and B :: "'h set" shows "(a \<in> f -` B) = (f a \<in> B)"
entails_transitive: fixes atom_ordering :: "('a \<times> 'a) set"   and S1 :: "'a Literal set set"   and S2 :: "'a Literal set set"   and S3 :: "'a Literal set set" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.entails_formula S1 S2"   and "propositional_atoms.entails_formula S2 S3" shows "propositional_atoms.entails_formula S1 S3"
simps_5: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'e"   and x2 :: "'f" shows "(case Some x2 of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = f2 x2"
is_unit_div_div_commute: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "is_unit b"   and "is_unit c" shows "a div b div c = a div c div b"
lookup_inverse: fixes x :: "'b \<Rightarrow>\<^sub>0 'c" shows "Abs_poly_mapping (lookup x) = x"
range_add_nat: fixes c :: "nat" shows "range (\<lambda>n. n + c) = {c..}"
apd_d_def: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.antidomain_pre_dioid antidomain_op plus times one zero less_eq less" shows "antidomain_pre_dioid.apd_d antidomain_op x = antidomain_op (antidomain_op x)"
lmap_eq_LNil: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "(lmap f xs = LNil) = (xs = LNil)"
ex_unique_reduced_GB_finite: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "\<exists>!G. finite G \<and> is_reduced_GB G \<and> pmdl G = pmdl F"
mono_omega_fun: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "mono (omega_fun f)"
arrI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "C.arr f"
tab_in_hom: shows "\<guillemotleft>tab : r\<^sub>1 \<star> p\<^sub>1 \<Rightarrow> (r \<star> s) \<star> s\<^sub>0 \<star> p\<^sub>0\<guillemotright>"
le_Suc_cases: fixes k :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "\<And>i. i < k \<Longrightarrow> P i"   and "P k" shows "\<forall>i<Suc k. P i"
sum_mset_constant: fixes y :: "'a"   and A :: "'b multiset" shows "(\<Sum>x\<in>#A. y) = of_nat (size A) * y"
select_convs_4: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "obj \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = obj"
rec_power_def: shows "rec_power = rec_swap (PR (constn 1) (CN rec_mult [recf.Id 3 1, recf.Id 3 2]))"
fgroup_by_single: fixes a :: "'d"   and x :: "'e fset"   and f :: "'f \<Rightarrow> 'd \<times> 'e"   and s :: "'f fset"   and y :: "'e fset" assumes "(a, x) |\<in>| fgroup_by f s"   and "(a, y) |\<in>| fgroup_by f s" shows "x = y"
assms_3: shows "\<forall>x\<in>X - {mx}. f x < f mx"
L: shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) (L_pos @ L_neg)"
d_def: shows "d = 1 - inverse (p' - p) * e"
monomial_pmdl: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "is_monomial_set B"   and "p \<in> pmdl B" shows "monomial (lookup p v) v \<in> pmdl B"
psubstEnv_substEnv_compose_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "goodEnv rho'"   and "good Y"   and "freshEnv ys y rho'" shows "rho &[Y / y]_ys &[rho'] = rho &[rho'] &[Y #[rho'] / y]_ys"
pos_mod_sign: fixes b :: "'a"   and a :: "'a" assumes "(0::'a) < b" shows "(0::'a) \<le> a mod b"
netglobalsimp: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> bool"   and s :: "((state \<times> (state, msg, pseqp, pseqp label) seqp) \<times> msg list \<times> (msg list, msg, unit, unit label) seqp) net_state" shows "netglobal P s = P (fst (initmissing (netgmap (\<lambda>(p, q). (fst (id p), snd (id p), q)) s)))"
trans_rtrancl: fixes r :: "('a \<times> 'a) set" shows "trans (r\<^sup>*)"
fill_offs_enum_count_item: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and n :: "nat"   and x :: "'c" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}"   and "0 < n" shows "count (mset (map the (fill xs (offs (enum xs index key n mi ma) 0) index key (length xs) mi ma))) x = count (mset xs) x"
residual_blocks_exclude: fixes b :: "'a set"   and x :: "'a" assumes "b \<in># \<B>\<^sup>R"   and "x \<in> b" shows "x \<notin> bl"
welldef: shows "list_constr (a # I) (length qs) \<longrightarrow> list_constr I (length qs)"
word_mod_def: fixes a :: "'b word"   and b :: "'b word" shows "a mod b = word_of_int (uint a mod uint b)"
listassoc_refl: fixes as :: "'a list" assumes "set as \<subseteq> carrier (Ring_Divisibility.mult_of R)" shows "as [\<sim>]\<^bsub>Ring_Divisibility.mult_of R\<^esub> as"
arg_min_least: fixes S :: "'a set"   and y :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "S \<noteq> {}"   and "y \<in> S" shows "f (arg_min_on f S) \<le> f y"
circ_sup_one_left_unfold: fixes x :: "'a" assumes "reflexive x" shows "x * x\<^sup>\<circ> = x\<^sup>\<circ>"
blocks_def: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and m :: "'a"   and w :: "'b"   and \<mu> :: "('a \<times> 'b) set" assumes "StableMarriage Pm Pw" shows "StableMarriage.blocks Pm Pw m w \<mu> = (w \<in> StableMarriage.m_strictly_prefers Pm m \<mu> \<and> m \<in> StableMarriage.w_strictly_prefers Pw w \<mu>)"
simps_2: fixes m :: "'a simple_match"   and uu :: "simple_action" shows "has_default_policy [SimpleRule m uu] = (m = simple_match_any)"
lex_ord_pair_valI: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and aux :: "order option"   and k :: "'a"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "comparator.oalist_inv_raw comp ys"   and "aux \<noteq> Some Eq"   and "k \<in> fst ` set xs \<union> fst ` set ys"   and "aux = f k (comparator.lookup_pair comp xs k) (comparator.lookup_pair comp ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set xs \<union> fst ` set ys; lt_of_comp comp k' k\<rbrakk> \<Longrightarrow> f k' (comparator.lookup_pair comp xs k') (comparator.lookup_pair comp ys k') = Some Eq" shows "comparator.lex_ord_pair comp f xs ys = aux"
eq_strict_1: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b" assumes "class.Eq_strict eq" shows "eq\<cdot>x\<cdot>\<bottom> = \<bottom>"
order_uminus: fixes x :: "'a"   and p :: "'a poly" shows "order x (- p) = order x p"
fold_random_permutation_fold: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "finite A" shows "fold_random_permutation f x A = map_pmf (\<lambda>xs. fold f xs x) (pmf_of_set (permutations_of_set A))"
scaleC_diff_right: fixes a :: "complex"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>C (x - y) = a *\<^sub>C x - a *\<^sub>C y"
add_self_mod_2: fixes m :: "nat" shows "is_even (m + m)"
id_Rel_def: fixes A :: "V" shows "id_Rel A = incl_Rel A A"
left_gpv_map: fixes f :: "'f \<Rightarrow> 'a"   and g :: "'g \<Rightarrow> 'b"   and gpv :: "('f, 'g, 'd) gpv"   and h :: "'h \<Rightarrow> 'c" shows "left_gpv (map_gpv f g gpv) = map_gpv f (map_sum g h) (left_gpv gpv)"
incrIndexList_help3: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "last ls = nmax - Suc 0"
subtopology_def: fixes S :: "'b set set"   and T :: "'b set set" shows "subtopology S T \<equiv> \<forall>s. (s open\<^bsub>S\<^esub>) = (\<exists>t. t open\<^bsub>T\<^esub> \<and> s = t \<inter> carrier\<^bsub>S\<^esub>)"
called_c_def: fixes rs :: "'a rule list" shows "called_c rs = _Collect c (\<exists>m. Rule m (Call c) \<in> set rs)"
ord_spmf_bind_spmfI1: fixes p :: "'a spmf"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b spmf"   and q :: "'b spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) q" shows "ord_spmf R (p \<bind> f) q"
N1_4: shows "n1 = 0 \<or> \<not> enabled (execConf (n1 - 1)) msg \<or> execMsg (n1 - 1) = msg"
otherwithE: fixes Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and I :: "'b set"   and P :: "('b \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a"   and a :: "'c"   and R :: "('b \<Rightarrow> 'a) \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool" assumes "otherwith Q I P \<sigma> \<sigma>' a"   and "\<lbrakk>P \<sigma> a; \<forall>j. j \<notin> I \<longrightarrow> Q (\<sigma> j) (\<sigma>' j)\<rbrakk> \<Longrightarrow> R \<sigma> \<sigma>' a" shows "R \<sigma> \<sigma>' a"
unbounded: shows "\<forall>x\<in>X1. \<forall>j\<ge>i. unbounded x ([xs !! j]\<^sub>\<R>)"
mset_eq_setD: fixes xs :: "'d list"   and ys :: "'d list" assumes "mset xs = mset ys" shows "set xs = set ys"
compile: shows "rec_ci rec_F = (F_ap, rs_pos, a_md)"
CSHOnextConfig_def: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'f \<Rightarrow> 'g"   and HO :: "'f \<Rightarrow> 'f set"   and SHO :: "'f \<Rightarrow> 'f set"   and coord :: "'f \<Rightarrow> 'f"   and cfg' :: "'f \<Rightarrow> 'g" shows "CSHOnextConfig A r cfg HO SHO coord cfg' \<equiv> \<forall>p. \<exists>\<mu>\<in>SHOmsgVectors A r p cfg (HO p) (SHO p). CnextState A r p (cfg p) \<mu> (coord p) (cfg' p)"
comp_fun_idem_axioms: shows "comp_fun_idem (\<^bold>*)"
simps_1: fixes f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "root (f \<diamondop> x) = root f (root x)"
expr_typing_letE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "expr"   and e2 :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> LET e1 IN e2 : t" obtains t1 :: "pdf_type" where "\<Gamma> \<turnstile> e1 : t1" and "case_nat t1 \<Gamma> \<turnstile> e2 : t"