hcju/isabelleps huggingface dataset:
abstract_primitive_in_doubt_allow_Allow2: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and p :: "('b, 'c) tagged_packet_scheme"   and disc :: "'b common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "\<not> matches (\<beta>, in_doubt_allow) m Drop p" shows "\<not> matches (\<beta>, in_doubt_allow) (abstract_primitive disc m) Drop p"
irrefl_trancl_rD: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "\<forall>x. (x, x) \<notin> r\<^sup>+"   and "(x, y) \<in> r" shows "x \<noteq> y"
dynlookup_access: fixes emh :: "ref_ty \<times> mhead"   and G :: "prog"   and accC :: "qtname"   and statT :: "ref_ty"   and sig :: "sig"   and dynC :: "qtname" assumes "emh \<in> mheads G accC statT sig"   and "G,statT \<turnstile> dynC valid_lookup_cls_for is_static emh"   and "isrtype G statT"   and "wf_prog G" shows "\<exists>dynM. dynlookup G statT dynC sig = Some dynM \<and> G \<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
bits_div_by_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide a one = a"
that_1: fixes i :: "nat" assumes "i < M"   and "f i = (0::'a)" shows "thesis"
sym_conv_converse_eq: fixes r :: "('c \<times> 'c) set" shows "sym r = (r\<inverse> = r)"
ascii_of_Char: fixes b0 :: "bool"   and b1 :: "bool"   and b2 :: "bool"   and b3 :: "bool"   and b4 :: "bool"   and b5 :: "bool"   and b6 :: "bool"   and b7 :: "bool" shows "ascii_of (Char b0 b1 b2 b3 b4 b5 b6 b7) = Char b0 b1 b2 b3 b4 b5 b6 False"
replicate_eq_list_of_eucl_zero: shows "replicate DIM('e) 0 = list_of_eucl (0::'e)"
cp_4: shows "path_image cp \<subseteq> s - insert p pts"
case_prime: assumes "prime (n div p)" shows "\<exists>qs. prime_factors n = set qs \<and> prod_list qs = n \<and> 2 \<le> length qs"
cond_actions_ok_\<tau>mRed1_inv: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list"   and s1 :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and s1' :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and ct :: "'b conditional_action" assumes "FWdelay_bisimulation_lift_aux final1 r1 final2 r2 \<tau>move1 \<tau>move2"   and "(\<tau>trsys.silent_move (multithreaded_base.redT final1 r1 convert_RA) (\<tau>multithreaded.m\<tau>move \<tau>move1))\<^sup>*\<^sup>* s1 s1'"   and "final_thread.cond_action_ok final1 s1 t ct" shows "final_thread.cond_action_ok final1 s1' t ct"
2: shows "rev v @ [b] @ [a] @ rev u =\<^sub>S rev v @ [a] @ [b] @ rev u"
Cons_1: assumes "b \<in> set bs" shows "Inl (p, b) \<in> set (new_spairs bs p)"
encode_interfering_operator_pair_exclusion_valid: fixes sv :: "sat_plan_variable"   and t :: "nat"   and op\<^sub>1 :: "(nat \<times> nat) strips_operator"   and op\<^sub>2 :: "(nat \<times> nat) strips_operator" assumes "sv \<in> atoms (encode_interfering_operator_pair_exclusion (\<phi> prob_with_noop abs_prob ) t op\<^sub>1 op\<^sub>2)"   and "t < Suc h"   and "op\<^sub>1 \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)"   and "op\<^sub>2 \<in> set ((\<phi> prob_with_noop abs_prob )\<^sub>\<O>)" shows "valid_state_var sv"
assms_2: shows "(v, w') \<in> edges_of_walk xs"
lookup_filter_aux: fixes xs :: "('c \<times> 'd) list"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "('c, 'd) RBT.rbt"   and k :: "'c" assumes "distinct (list.map fst xs)" shows "lookup (List.fold (\<lambda>(k, v) t. if P k v then RBT.insert k v t else t) xs t) k = (case map_of xs k of None \<Rightarrow> lookup t k | Some v \<Rightarrow> if P k v then Some v else lookup t k)"
assms_2: shows "\<forall>i\<in>I. finite (A i)"
linear_scale: fixes f :: "'b \<Rightarrow> 'c"   and r :: "'a"   and b :: "'b" assumes "linear (*a) (*b) f" shows "f (r *a b) = r *b f b"
descendants_types: fixes k :: "'b"   and i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and xs :: "(nat \<times> nat) set"   and v :: "nat"   and v' :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and p :: "('c, 'b) fm" assumes "descendants k i branch xs"   and "(v, v') \<in> xs"   and "branch !. v = Some (ps, a)"   and "ps !. v' = Some p" shows "p = (\<^bold>\<diamond> Nom k) \<or> (\<exists>q. p = (\<^bold>\<not> (\<^bold>@ k q)))"
Int_card2: fixes B :: "'a set"   and A :: "'a set" assumes "finite B" shows "card (A \<inter> B) \<le> card B"
min_add_distrib_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus x (ord.min less_eq y z) = ord.min less_eq (plus x y) (plus x z)"
rebase_of_nat: fixes n :: "nat" shows "@of_nat n = of_nat n"
AE_False: fixes M :: "'c measure" assumes "prob_space M" shows "(AE x in M. False) = False"
plus_list_lub: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list"   and n :: "nat" assumes "set xs \<subseteq> A"   and "set ys \<subseteq> A"   and "set zs \<subseteq> A"   and "length xs = n \<and> length ys = n"   and "xs [\<sqsubseteq>\<^bsub>r\<^esub>] zs \<and> ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs" shows "xs [+f] ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
dom_subid: fixes x :: "('a \<times> 'a) set" shows "rdom x \<subseteq> Id"
good_interp_inverse: fixes y :: "('a, 'b, 'c) interp" assumes "y \<in> {I. is_interp I}" shows "raw_interp (good_interp y) = y"
sep_add_zero: fixes x :: "'a" shows "x + (0::'a) = x"
axioms_2: fixes hom :: "'c \<Rightarrow> 'd" assumes "monoid_add_hom hom" shows "semigroup_add_hom hom"
set_rotate: fixes n :: "nat"   and xs :: "'a list" shows "set (rotate n xs) = set xs"
rec: fixes f :: "('c \<Rightarrow> 'd) \<Rightarrow> 'b"   and x :: "'c \<Rightarrow> 'd" shows "rec_T3 f (E x) = f x"
YM4_parts_knows_Spy: fixes A :: "agent"   and Y :: "msg"   and c :: "msg"   and evs :: "event list" assumes "Gets A \<lbrace>Crypt (shrK A) Y, c\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "c \<in> parts (knows Spy evs)"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> ns_shared" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
coerce_idem: fixes x :: "'a" assumes "DEFL('a) \<sqsubseteq> DEFL('b)" shows "COERCE('b, 'c)\<cdot>(COERCE('a, 'b)\<cdot>x) = COERCE('a, 'c)\<cdot>x"
B: fixes n :: "nat" assumes "1 \<le> n" shows "\<bar>additive_strength f xi + additive_strength (inv f) xi\<bar> \<le> 6 * deltaG TYPE('a) * (1 / real n)"
value_gt_true: fixes a :: "value option"   and b :: "value option" shows "(value_gt a b = true) = (\<exists>x y. a = Some (Num x) \<and> b = Some (Num y) \<and> y < x)"
inverse_arrowsE: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" obtains "C.ide (C g f)" and "C.ide (C f g)"
fundamental_lemma: fixes bad1 :: "'b \<Rightarrow> bool"   and bad2 :: "'c \<Rightarrow> bool"   and A :: "'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'c spmf" assumes "rel_spmf (\<lambda>x y. bad1 x = bad2 y \<and> (\<not> bad2 y \<longrightarrow> A x = B y)) p q" shows "\<bar>Sigma_Algebra.measure (measure_spmf p) {x. A x} - Sigma_Algebra.measure (measure_spmf q) {y. B y}\<bar> \<le> Sigma_Algebra.measure (measure_spmf p) {x. bad1 x}"
invar_delete: fixes m :: "'q"   and a :: "'v" assumes "Q_invar m" shows "Q_invar (Q_delete a m)"
sep_disj_addI2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x ## y + z"   and "y ## z" shows "x + z ## y"
no_deadlock: fixes s :: "'a" assumes "A.reachable s" shows "\<exists>s'. A.step s s'"
iso_inv_block_in_alt: fixes bl :: "'a set" assumes "\<pi> ` bl \<in># \<B>'"   and "bl \<subseteq> \<V>" shows "bl \<in># \<B>"
distinct_41: fixes x61 :: "msg"   and x62 :: "msg"   and x71 :: "key"   and x72 :: "msg" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Crypt x71 x72"
listassoc_refl: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list" assumes "Group.monoid G"   and "set as \<subseteq> carrier G" shows "as [\<sim>]\<^bsub>G\<^esub> as"
A_def: fixes i :: "nat" shows "A i = (if i = 1 then S else T)"
subgroupE_3: fixes H :: "'c set"   and a :: "'c" assumes "H \<le> H"   and "a \<in> H" shows "inv\<^bsub>H\<^esub> a \<in> H"
FunctorComp: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l, 'm) Functor_scheme"   and f :: "'i"   and g :: "'i" assumes "PreFunctor F"   and "f \<approx>>\<^bsub>CatDom F\<^esub> g" shows "F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)"
zdiv_numeral_Bit1: fixes v :: "num"   and w :: "num" shows "numeral (num.Bit1 v) div numeral (num.Bit0 w) = numeral v div numeral w"
LT2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> F \<longrightarrow> \<diamond>G"
simps: fixes fail :: "ennreal"   and \<I> :: "('h, 'i) \<I>"   and f :: "'j \<Rightarrow> ennreal"   and gpv :: "('j, 'h, 'i) gpv" shows "expectation_gpv fail \<I> f gpv = \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv fail \<I> f (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None)"
execute_plan_action_def: fixes P :: "ast_problem"   and \<pi> :: "plan_action"   and M :: "object atom formula set" shows "ast_problem.execute_plan_action P \<pi> M = apply_effect (ground_action.effect (ast_problem.resolve_instantiate P \<pi>)) M"
inf: shows "\<exists>\<^sub>\<infinity>i. (as i, as (Suc i)) \<in> R"
mono_imp_sup_derivable_lfp: fixes p :: "'a" assumes "extreme_bound A (\<sqsubseteq>) (Collect derivable) p" shows "extreme {q \<in> A. f q = q} (\<lambda>x y. y \<sqsubseteq> x) p"
edges_split_face2: fixes f :: "face"   and u :: "nat"   and v :: "nat"   and vs :: "nat list" assumes "pre_split_face f u v vs" shows "\<E> (snd (split_face f u v vs)) = Edges (u # vs @ [v]) \<union> Edges (v # between (vertices f) v u @ [u])"
fields_norec: fixes G :: "prog"   and fd :: "qtname"   and c :: "class"   and fn :: "vname"   and f :: "Decl.field" assumes "class G fd = Some c"   and "ws_prog G"   and "table_of (cfields c) fn = Some f" shows "table_of (DeclConcepts.fields G fd) (fn, fd) = Some f"
split_aform_largest_take_refine: fixes ni :: "nat"   and n :: "nat"   and xi :: "(real \<times> real pdevs) list"   and x :: "real list set"   and d :: "nat" assumes "(ni, n) \<in> nat_rel"   and "(xi, x) \<in> aforms_rel"   and "length xi = d" shows "(RETURN (split_aforms_largest_uncond_take ni xi), split_spec_params d n x) \<in> \<langle>aforms_rel \<times>\<^sub>r aforms_rel\<rangle>nres_rel"
GSubspaceI: fixes U :: "'v set" assumes "FGModule G (\<cdot>) U"   and "U \<subseteq> V" shows "RModule FG (\<cdot>) U \<and> U \<subseteq> V"
bin_ex_rl: fixes bin :: "int" shows "\<exists>w b. w BIT b = bin"
assert_disch4: fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d" assumes "P \<sigma>" shows "\<sigma> \<Turnstile> assert\<^sub>S\<^sub>E P"
create_graph__name_ident: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<le> SPEC expand_name_ident"
coprime_power2: fixes n :: "nat" assumes "n \<noteq> 0"   and "\<And>x. x < n \<Longrightarrow> coprime x n = odd x" shows "\<exists>k. n = 2 ^ k"
app_hd_tl: fixes Cs :: "'a list"   and Cs' :: "'a list" assumes "Cs \<noteq> []"   and "Cs = Cs' @ tl Cs" shows "Cs' = [hd Cs]"
finvimage_def: fixes r :: "V"   and A :: "V" shows "r -`\<^sub>\<bullet> A = r\<inverse>\<^sub>\<bullet> `\<^sub>\<bullet> A"
map_indets_times: fixes f :: "'e \<Rightarrow> 'c"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "map_indets f (p * q) = map_indets f p * map_indets f q"
small_det_csmall_det: fixes P :: "'prog" assumes "\<forall>\<sigma>. small P \<sigma> = {} \<or> (\<exists>\<sigma>'. small P \<sigma> = {\<sigma>'})" shows "\<forall>\<sigma>. csmall P \<sigma> = {} \<or> (\<exists>o'. csmall P \<sigma> = {o'})"
simps_2: fixes x :: "num" shows "Num.inc (num.Bit0 x) = num.Bit1 x"
eta_norm_imp_eta_reds: fixes t :: "Core.term"   and t' :: "Core.term" assumes "eta_norm t = t'" shows "t \<rightarrow>\<^sub>\<eta>\<^sup>* t'"
induct_rulify_fallback_2: fixes A :: "bool"   and B :: "bool" shows "HOL.induct_implies A B \<equiv> A \<longrightarrow> B"
abs_eq: fixes xb :: "'b"   and xa :: "'c"   and x :: "'b \<Rightarrow> 'c option" shows "Mapping.update xb xa (Mapping x) = Mapping (x(xb \<mapsto> xa))"
aux: fixes f :: "'q \<Rightarrow>\<^sub>0 'r" shows "(lookup f = (\<lambda>_. 0::'r)) = (f = 0)"
cmr: fixes S :: "(nat \<Rightarrow> real) set" shows "continuous_map (subtopology (nsphere n) S) (nsphere n) r"
constant_martingale: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and c :: "real" assumes "finite_measure M"   and "\<forall>t. sigma_finite_subalgebra M (F t)"   and "Filtration.filtration M F" shows "martingale M F (\<lambda>n w. c)"
mat_of_rows_carrier_3: fixes n :: "nat"   and vs :: "'c vec list" shows "dim_col (mat_of_rows n vs) = n"
comp_of_ords_of_le_lt: fixes c :: "'a \<Rightarrow> 'a \<Rightarrow> order" shows "comp_of_ords (le_of_comp c) (lt_of_comp c) = c"
cnj_of_nat: fixes x :: "nat" shows "Octonions.cnj (octo_of_nat x) = octo_of_nat x"
satisfies_state_def: fixes v :: "nat \<Rightarrow> 'a"   and s :: "('b, 'a) state" shows "v \<Turnstile>\<^sub>s s \<equiv> v \<Turnstile>\<^sub>b \<B> s \<and> v \<Turnstile>\<^sub>t \<T> s"
ABinds_env_restr: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysis.ABinds Aexp \<Delta>\<cdot>(ae f|` domA \<Delta>) = ArityAnalysis.ABinds Aexp \<Delta>\<cdot>ae"
map_return_pmf: fixes f :: "'d \<Rightarrow> 'b"   and x :: "'d" shows "map_pmf f (return_pmf x) = return_pmf (f x)"
simps: fixes a1 :: "('c, 'd) conf"   and a2 :: "('c, 'd) label"   and a3 :: "('c, 'd) conf" shows "se a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = Skip \<and> a3 = c) \<or> (\<exists>c e. a1 = c \<and> a2 = Assume e \<and> a3 = \<lparr>store = store c, pred = pred c \<union> {adapt_bexp e (store c)}\<rparr>) \<or> (\<exists>sv v c e. a1 = c \<and> a2 = Assign v e \<and> a3 = \<lparr>store = (store c)(v := snd sv), pred = pred c \<union> {\<lambda>\<sigma>. \<sigma> sv = adapt_aexp e (store c) \<sigma>}\<rparr> \<and> fst sv = v \<and> Conf.fresh_symvar sv c))"
group_design_axioms: shows "group_design \<V> \<B> \<G>"
empty_selected_part_def: fixes Sel :: "'a set"   and S :: "'a Literal set set" shows "empty_selected_part Sel S = {C \<in> S. selected_part Sel C = {}}"
lin_dep_iff_in_span: fixes S :: "'c set"   and v :: "'c" assumes "S \<subseteq> carrier V"   and "lin_indpt S"   and "v \<in> carrier V"   and "v \<notin> S" shows "(v \<in> local.module.span S) = lin_dep (S \<union> {v})"
simps_2: fixes uu :: "'a"   and x :: "'a" shows "the_default uu (Some x) = x"
sum_tree_list_children: fixes ts :: "'b tree list"   and k :: "nat" assumes "\<forall>t\<in>set ts. t \<noteq> \<langle>\<rangle>" shows "(\<Sum>t\<leftarrow>ts. k * size t) = (\<Sum>t\<leftarrow>map left ts @ map right ts. k * size t) + k * length ts"
assms_5: shows "revision_step r' s\<^sub>2 ((s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r'' := None)(r' \<mapsto> ls))"
hr_form: fixes R :: "agent"   and S2TTP :: "msg"   and pwd :: "msg"   and hr :: "msg"   and evs :: "event list" assumes "Notes TTP \<lbrace>Agent R, Agent TTP, S2TTP, pwd, hr\<rbrace> \<in> set evs"   and "evs \<in> certified_mail" shows "hr \<in> synth (analz (knows Spy evs)) \<or> (\<exists>clt q S em. hr = Hash \<lbrace>Number clt, Nonce q, response S R q, em\<rbrace>)"
sup_sls_def: shows "sup_sls \<equiv> Abs_semilattice_set sup"
lookup_tuple_in': fixes as :: "'a option list" shows "Mapping.lookup tuple_in' as = safe_max (fst ` {tas \<in> ts_tuple_rel (set (linearize data_in')). valid_tuple tuple_since tas \<and> as = snd tas})"
rev_nth: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "rev xs ! n = xs ! (length xs - Suc n)"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "indRelLPOp a1 a2 = ((\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>S. S \<in>S a1 \<and> S \<in>S a2) \<or> (\<exists>T. T \<in>T a1 \<and> T \<in>T a2) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> indRelLPOp P Q \<and> indRelLPOp Q R))"
invert_side_of_triangle_1: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "C.arr h"   and "f \<cdot>\<^sub>C g = h"   and "C.iso f" shows "C.seq (C.inv f) h \<and> g = C.inv f \<cdot>\<^sub>C h"
Nset2finite_inj: fixes A :: "'e set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e" assumes "finite A"   and "card A = Suc n"   and "surj_to f {i. i \<le> n} A" shows "inj_on f {i. i \<le> n}"
walk_to_path: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "Graph.walk G xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "\<exists>ys. v \<leadsto>ys\<leadsto>\<^bsub>G\<^esub> w \<and> set ys \<subseteq> set xs"
v_eq0_imp_b_eq_0: assumes "\<v> = 0" shows "\<b> = 0"
leg0_composite: shows "fgh.\<nu>\<pi>.leg0 = fgh.\<pi>.leg0 \<cdot> fgh.\<nu>\<pi>.prj\<^sub>0"
measure_Pi_pmf_PiE_dflt: fixes A :: "'b set"   and dflt :: "'c"   and p :: "'b \<Rightarrow> 'c pmf"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A" shows "measure_pmf.prob (Pi_pmf A dflt p) (PiE_dflt A dflt B) = (\<Prod>x\<in>A. measure_pmf.prob (p x) (B x))"
l_set_attribute_get_child_nodes_axioms_def: fixes set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_attribute_get_child_nodes_axioms set_attribute_locs get_child_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_attribute_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
l_attach_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_attach_shadow_root_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel get_shadow_root get_shadow_root_locs get_tag_name get_tag_name_locs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root"
eAbs_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'b, 'a) model"   and xs :: "'varSort"   and x :: "'var"   and eX :: "'b withERR"   and A :: "'a" assumes "eAbs MOD xs x eX = OK A" shows "\<exists>s X. eX = OK X \<and> isInBar (xs, s) \<and> A = igAbs MOD xs x X \<and> igWls MOD s X"
G2'_def: shows "G2' \<equiv> fps_expansion (pre_zeta 1) 1"
lasso_run_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>lasso_run_rel = br run_of_lasso (\<lambda>_. True) O (nat_rel \<rightarrow> R)"
eigenspace_def: fixes a :: "complex"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "eigenspace a A = kernel (A - a *\<^sub>C id_cblinfun)"
simps_1: fixes return :: "unit \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> (unit \<Rightarrow> 'a) \<Rightarrow> 'a"   and tick :: "'a \<Rightarrow> 'a" shows "hanoi_base.hanoi return bind tick 0 = return ()"
Qg_unit_group: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set" assumes "Group G"   and "G \<triangleright> N" shows "(set_rcs G N = {N}) = (carrier G = N)"
set_unsat_simps_3: fixes I :: "'a list"   and s :: "('a, 'b) state" shows "\<T> (set_unsat I s) = \<T> s"
eng_add: fixes h1 :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set"   and h2 :: "'a" assumes "h1 \<in> generate_field R H"   and "h2 \<in> generate_field R H" shows "h1 \<oplus>\<^bsub>R\<^esub> h2 \<in> generate_field R H"
subset_closed_def: fixes C :: "'b set set" shows "subset_closed C \<equiv> \<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C"
box_sem: fixes I :: "('b, 'c, 'a) interp"   and \<alpha> :: "('b, 'c, 'a) hp"   and \<phi> :: "('b, 'c, 'a) formula" shows "fml_sem I ([[\<alpha>]]\<phi>) = {\<nu>. \<forall>\<omega>. (\<nu>, \<omega>) \<in> prog_sem I \<alpha> \<longrightarrow> \<omega> \<in> fml_sem I \<phi>}"
bind_distr_return: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd"   and L :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L"   and "space M \<noteq> {}" shows "distr M N f \<bind> (\<lambda>x. return L (g x)) = distr M L (\<lambda>x. g (f x))"
urel_integer_inj': fixes pp :: "integer"   and x :: "integer"   and z :: "int"   and y :: "integer" assumes "p = int_of_integer pp"   and "urel_integer x z"   and "urel_integer y z" shows "x = y"
alpha_Tree_supp_rel: fixes t1 :: "('a, 'b, 'c, 'd) Tree"   and t2 :: "('a, 'b, 'c, 'd) Tree" assumes "t1 =\<^sub>\<alpha> t2" shows "supp_rel (=\<^sub>\<alpha>) t1 = supp_rel (=\<^sub>\<alpha>) t2"
type_less_eq_x_OclAny_intro: fixes \<tau> :: "'a basic_type" shows "\<tau> \<le> OclAny"
sums_single: fixes i :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "(\<lambda>r. if r = i then f r else (0::'a)) sums f i"
insert_lepoll_cong: fixes A :: "'a set"   and B :: "'b set"   and b :: "'b"   and a :: "'a" assumes "A \<lesssim> B"   and "b \<notin> B" shows "insert a A \<lesssim> insert b B"
mat_erase_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "mat_erase A i j = mat (dim_row A) (dim_col A) (\<lambda>(i', j'). if i' = i \<or> j' = j then 0::'b else A $$ (i', j'))"
finite_UNIV_finite_5_def: shows "finite_UNIV = Phantom(Enum.finite_5) True"
elims_2: fixes x :: "'b val" assumes "is_Bool x" obtains b :: "bool" where "x = Bool b"
inclusion_is_functor: shows "functor (\<cdot>\<^sub>C) \<F>\<^sub>SC.comp \<F>\<^sub>SC.inclusion_of_generators"
fpxs_shift_power: fixes n :: "rat"   and x :: "'b fpxs"   and m :: "nat" shows "fpxs_shift n x ^ m = fpxs_shift (rat_of_nat m * n) (x ^ m)"
inj_on_domain: assumes "inj_on h (carrier R)"   and "domain S" shows "domain R"
simps_4: fixes player0' :: "'c set \<Rightarrow> 'c set"   and verts :: "'c set"   and arcs :: "('c \<times> 'c) set"   and player0 :: "'c set"   and priority :: "'c \<Rightarrow> nat"   and more :: "'d" shows "player0_update player0' \<lparr>verts = verts, arcs = arcs, player0 = player0, priority = priority, \<dots> = more\<rparr> = \<lparr>verts = verts, arcs = arcs, player0 = player0' player0, priority = priority, \<dots> = more\<rparr>"