hcju/isabelleps huggingface dataset:
fpairs_eqD: fixes r :: "V"   and x :: "V" assumes "fpairs r = r"   and "x \<in>\<^sub>\<circ> r" shows "\<exists>a b. x = [a, b]\<^sub>\<circ>"
IH: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and \<zeta> :: "'b"   and R :: "nat set" assumes "st = (\<sigma>, NodeS ii \<zeta> R)" shows "P \<sigma> \<zeta> R"
simps_2: fixes h :: "'b hp" shows "pass\<^sub>1 [h] = [h]"
weakest_intuitionistic: fixes P :: "'a \<Rightarrow> bool" shows "\<nexists>Q. (\<forall>h. ((\<lambda>s. True) \<longrightarrow>* P) h \<longrightarrow> Q h) \<and> intuitionistic Q \<and> Q \<noteq> ((\<lambda>s. True) \<longrightarrow>* P) \<and> (\<forall>h. Q h \<longrightarrow> P h)"
rel_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (llist_all2 R)"
function_factors_right: fixes g :: "'e \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f" shows "(\<forall>x. \<exists>y. g y = f x) = (\<exists>h. f = g \<circ> h)"
weak_convex_pref_def: fixes Pr :: "('b \<times> 'b) set" shows "weak_convex_pref Pr = (\<forall>x y. x \<succeq>[Pr] y \<longrightarrow> (\<forall>\<alpha> \<beta>. \<alpha> + \<beta> = 1 \<and> 0 < \<alpha> \<and> 0 < \<beta> \<longrightarrow> \<alpha> *\<^sub>R x + \<beta> *\<^sub>R y \<succeq>[Pr] y))"
lnorm: shows "lnorm (lebesgue_on S) 1 (f - g) < e"
minus_fract_cancel: fixes a :: "'b"   and b :: "'b" shows "Fract (- a) (- b) = Fract a b"
primes_M_bigo: shows "primes_M \<in> O(ln)"
assms_3: shows "\<forall>n\<^sub>0. P n\<^sub>0 \<longrightarrow> P (n\<^sub>0 + k)"
cr_transaction\<^sub>m_def: shows "cr_transaction\<^sub>m \<equiv> \<lambda>x y. x = the_Transaction\<^sub>m y"
final: shows "dfa.final Canon.DFA \<subseteq> dfa.states Canon.DFA"
spmf_bind: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf"   and y :: "'a" shows "spmf (p \<bind> f) y = LINT x|measure_spmf p. spmf (f x) y"
of_nat_1: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 1 = one"
arr_dom_iff_arr: fixes f :: "'a" shows "A.arr (A.dom f) = A.arr f"
ipresIGWls_def: fixes h :: "'q \<Rightarrow> 'r"   and MOD :: "('s, 't, 'u, 'v, 'w, 'x, 'q, 'y, 'z) model_scheme"   and MOD' :: "('aa, 'ab, 'ac, 'v, 'ad, 'ae, 'r, 'af, 'ag) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
iso_in_hom_def: fixes f :: "'b"   and a :: "'b"   and b :: "'b" shows "B.iso_in_hom f a b \<equiv> \<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright> \<and> B.iso f"
in_hom_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and b :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] b\<guillemotright> = (A_B.arr f \<and> A_B.ide a \<and> A_B.ide b \<and> A_B.Dom f = A_B.Dom a \<and> A_B.Cod f = A_B.Dom b)"
simple_list_index_equality: fixes a :: "'b list"   and n :: "nat"   and b :: "'b list" assumes "length a = n"   and "length b = n"   and "\<forall>i<n. a ! i = b ! i" shows "a = b"
pdevs_val_cmul: fixes u :: "real"   and e :: "nat \<Rightarrow> real"   and xs :: "'b pdevs" shows "pdevs_val (\<lambda>i. u * e i) xs = u *\<^sub>R pdevs_val e xs"
int_to_ipv6preferred_def: fixes i :: "128 word" shows "int_to_ipv6preferred i = IPv6AddrPreferred (UCAST(128 \<rightarrow> 16) (i && 340277174624079928635746076935438991360 >> 16 * 7)) (UCAST(128 \<rightarrow> 16) (i && 5192217630372313364192902785269760 >> 16 * 6)) (UCAST(128 \<rightarrow> 16) (i && 79226953588444722964369244160 >> 16 * 5)) (UCAST(128 \<rightarrow> 16) (i && 1208907372870555465154560 >> 16 * 4)) (UCAST(128 \<rightarrow> 16) (i && 18446462598732840960 >> 16 * 3)) (UCAST(128 \<rightarrow> 16) (i && 281470681743360 >> 16 * 2)) (UCAST(128 \<rightarrow> 16) (i && 4294901760 >> 16 * 1)) (UCAST(128 \<rightarrow> 16) (i && 65535))"
count_simps_2: fixes x :: "'b"   and n :: "nat"   and xs :: "('b \<times> nat) list" shows "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"
of_complex_one_dim_iso: fixes \<psi> :: "'b" shows "of_complex (one_dim_iso \<psi>) = one_dim_iso \<psi>"
overlapping_sym: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'b"   and u :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.overlapping subst t u" shows "term.overlapping subst u t"
fresh_left: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'d" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
comp_char: fixes g :: "'a"   and f :: "'a" shows "S.comp g f = (if S.arr f \<and> S.arr g \<and> B.seq g f then g \<cdot>\<^sub>B f else B.null)"
cexp_czero: fixes r :: "('d \<times> 'd) set" shows "r ^c czero =o cone"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "\<And>C. Prop C \<Psi> P \<alpha> P'"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "Prop C \<Psi> (Cases Cs) \<alpha> P'"
assms_2: shows "ipresIGWlsAll h hA MOD MOD'"
concl_cex: shows "\<not> I_of N \<Turnstile> {concl_of \<iota>}"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "\<^bold>\<lfloor>\<^bold>\<a>\<^bold>[t, u, v\<^bold>]\<^bold>\<rfloor> = (\<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor>) \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<lfloor>v\<^bold>\<rfloor>"
simps_1: fixes cert :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and B :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and pc :: "nat"   and s :: "'c" shows "wtl_inst_list [] cert f r T B step pc s = s"
subset: shows "V \<subseteq> U\<^sub>V\<^sub>S"
prognosis_reorder: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and u :: "Arity"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> = map_of \<Delta>" shows "prognosis ae as u (\<Gamma>, e, S) = prognosis ae as u (\<Delta>, e, S)"
conv_self_conjugate_var: shows "is_conjugation relation_algebra_class.converse relation_algebra_class.converse"
valid_before: shows "valid_mmuaux' args cur nt (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length) auxlist"
rep_eq: fixes x :: "('form, 'rule) graph_node"   and xa :: "('form, 'var) antecedent" shows "fset (hyps_for x xa) = hyps_for' x xa"
redT_has_lock_inv: fixes ls :: "('l, 't) locks"   and ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and m :: "'m"   and ws :: "'t \<Rightarrow> 'w wait_set_status option"   and "is" :: "'t set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and ls' :: "('l, 't) locks"   and ts' :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and m' :: "'m"   and ws' :: "'t \<Rightarrow> 'w wait_set_status option"   and is' :: "'t set"   and t' :: "'t"   and l :: "'l" assumes "\<langle>ls, (ts, m), ws, is\<rangle> -t\<triangleright>ta\<rightarrow> \<langle>ls', (ts', m'), ws', is'\<rangle>"   and "t \<noteq> t'" shows "has_lock (ls' $ l) t' = has_lock (ls $ l) t'"
prems_4: shows "set (ah # as) \<subseteq> carrier G"
degenerate_has_max_preferred: shows "{x \<in> local.degenerate_lotteries. \<forall>y\<in>local.degenerate_lotteries. x \<succeq>[\<R>] y} \<noteq> {}"
strict_trans2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<prec> b"   and "b \<preceq> c" shows "a \<prec> c"
of_fun_def: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "IArray.of_fun f n = IArray (map f [0..<n])"
simps_18: fixes x1 :: "'a"   and x6 :: "'a local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[x6\<^bold>]"
ivl_disj_un_two_touch_4: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m..u} = {l..u}"
BSI_def: fixes \<V> :: "'b V_rec"   and Tr :: "'b list set" shows "BSI \<V> Tr \<equiv> \<forall>\<alpha> \<beta>. \<forall>c\<in>C\<^bsub>\<V>\<^esub>. \<beta> @ \<alpha> \<in> Tr \<and> \<alpha> \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [] \<longrightarrow> (\<exists>\<alpha>'. \<beta> @ [c] @ \<alpha>' \<in> Tr \<and> \<alpha>' \<upharpoonleft> V\<^bsub>\<V>\<^esub> = \<alpha> \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> \<alpha>' \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
irreduccible_recurrent_class: fixes x :: "'s" assumes "recurrent x" shows "acc `` {x} \<in> UNIV // communicating"
VDMNew: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and x :: "Var"   and C :: "Class" shows "G \<rhd> New x C : \<lambda>s t. \<exists>l. l \<notin> Dom (snd s) \<and> t = (update (fst s) x (RVal (Loc l)), (l, C, []) # snd s)"
semigroup_ow_axioms: shows "semigroup_ow U (*\<^sub>o\<^sub>w)"
assms_1: shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
\<psi>_im: shows "\<psi> ` GRepHomSet (\<star>) W \<subseteq> HRepHomSet (\<star>) W"
crel_vs_update: fixes param :: "'k" shows "crel_vs (=) () (update param (dp param))"
abs_eq: fixes y :: "64 word" shows "Abs_uint64' y = Abs_uint64 y"
simps_28: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> x41 ;; x42"
induced_signed_eq_lconjpermutation: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "(\<rightarrow>) (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) signed_lconjpermutation (the_inv_into (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) id) (sum_list ss))) = signed_list_lconjaction ss"
comp_functor_identity: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "functor A B F" shows "F \<circ> identity_functor.map A = F"
Ig_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant"   and I :: "'c set" shows "Ig K v I = (SOME x. x \<in> I \<and> v x = LI K v I)"
similar_mat_refl: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat A A"
collapse: fixes x :: "x" shows "X (a x) (b x) = x"
strategy_attracts_extends_VVpstar: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and W :: "'a set"   and v0 :: "'a" assumes "strategy_attracts p \<sigma> S W"   and "v0 \<notin> VV p"   and "v0 \<in> directly_attracted p S" shows "strategy_attracts_via p \<sigma> v0 (insert v0 S) W"
ipset_prefix_match_nm: fixes pfx :: "'b prefix_match"   and rg :: "'b word set" shows "snd (ipset_prefix_match pfx rg) = rg - prefix_to_wordset pfx"
leg1_uniquely_isomorphic_1: shows "r\<^sub>1 \<cong> s\<^sub>1 \<star> w"
Lp_Norm_1: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> borel_measurable M" shows "Norm (\<LL> (ennreal p) M) f = (LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p)"
simps_1: fixes at :: "('a, 'c) atm" shows "gL (Pos at) = Pos (gA at)"
f_def: shows "f \<equiv> \<lambda>x. poly q' x / poly p' x"
Domainp_conversep: fixes R :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp R\<inverse>\<inverse> = Rangep R"
getChoices_p: fixes a :: "edge"   and b :: "pState"   and g :: "gState\<^sub>I"   and p :: "pState"   and es :: "edge list" assumes "(a, b) \<in> set (getChoices g p es)" shows "b = p"
set_pmf_iff: fixes x :: "'b"   and M :: "'b pmf" shows "(x \<in> set_pmf M) = (pmf M x \<noteq> 0)"
r1: shows "ls ! (Suc z mod |ls|) = rotate1 ls ! (z mod |ls|)"
assms_1: shows "a \<in> {x. P x} \<and> a' \<in> {x. P' x}"
d_def: shows "d = gcd (nat \<bar>t\<bar>) u"
fps_deriv_numeral: fixes n :: "num" shows "fps_deriv (numeral n) = 0"
hseqE: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "C.hseq \<nu> \<mu>" obtains "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>"
sorted_wrt_Cons: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and ys :: "'c list" shows "sorted_wrt P (x # ys) = (Ball (set ys) (P x) \<and> sorted_wrt P ys)"
infsetsum_cmult_left: fixes c :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "c \<noteq> (0::'a) \<Longrightarrow> f abs_summable_on A" shows "(\<Sum>\<^sub>ax\<in>A. f x * c) = infsetsum f A * c"
c_sum_eq_arg: fixes x :: "nat" assumes "c_sum x = x" shows "x \<le> 1"
a2_omega: shows "(\<lambda>x. real_of_int ((a x)\<^sup>2)) \<in> \<omega>(\<lambda>x. real x * ln (real x))"
comp_right_one: fixes x :: "'a" shows "x * (1::'a) = x"
cases: fixes r :: "s" obtains inst :: "inst list" and funcs :: "cl list" and tab :: "cl option list list" and mem :: "mem list" and globs :: "global list" where "r = \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs\<rparr>"
discI_4: fixes event :: "('a, 'b, 'c) event"   and x4 :: "'a" assumes "event = Snapshot x4" shows "isSnapshot event"
exec_step_ind_Getstatic_Called: fixes D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and D :: "char list"   and F :: "char list"   and C :: "char list"   and v :: "val"   and sh :: "sheap"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and frs :: "frame list" assumes "(D', b, t) = field P D F"   and "P \<turnstile> C has F,Static:t in D"   and "v = the (fst (the (sh D')) F)" shows "exec_step_ind (StepI (Getstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc (Called Cs) frs sh (None, h, (v # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, No_ics) # frs, sh)"
lex_ext_unbounded_map_S: fixes ss :: "'a list"   and ts :: "'a list"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and f :: "'a \<Rightarrow> 'a" assumes "\<And>i. \<lbrakk>i < length ss; i < length ts; fst (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> fst (r (map f ss ! i) (map f ts ! i))"   and "\<And>i. \<lbrakk>i < length ss; i < length ts; snd (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> snd (r (map f ss ! i) (map f ts ! i))"   and "fst (lex_ext_unbounded r ss ts)" shows "fst (lex_ext_unbounded r (map f ss) (map f ts))"
rHom_npow: fixes A :: "('e, 'f) Ring_scheme"   and R :: "('g, 'h) Ring_scheme"   and x :: "'e"   and f :: "'e \<Rightarrow> 'g"   and n :: "nat" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "f \<in> rHom A R" shows "f (x^\<^bsup>A n\<^esup>) = f x^\<^bsup>R n\<^esup>"
assms_1: fixes x :: "'a" shows "g (f x) = g x"
wt_ge_0: fixes c :: "('test, 'atom, 'choice) cmd"   and i :: "nat"   and s :: "'state" assumes "proper c"   and "i < brn c" shows "0 \<le> wt c s i"
IH: fixes s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(s, t) \<in> D c" shows "(s, t) \<in> Big_step c"
write_annul_def: fixes b :: "bool"   and v :: "sparc_state_var" shows "write_annul b v \<equiv> v\<lparr>annul := b\<rparr>"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'd"   and p :: "'b"   and Tl :: "'a trm list" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP" shows "Struct.satA intF intP \<xi> (Pr p Tl) = intP p (map (Struct.int intF \<xi>) Tl)"
restrict_variables_subset: fixes b :: "'b ifex"   and var :: "'b"   and val :: "bool" shows "ifex_var_set (restrict b var val) \<subseteq> ifex_var_set b"
next_recvcapD_4: fixes c0 :: "('b, 'loc \<times> 't) Exchange.configuration"   and c1 :: "('b, 'loc \<times> 't) Exchange.configuration"   and p :: "'b"   and t :: "'loc \<times> 't" assumes "cri.next_recvcap' c0 c1 p t" shows "c_glob c1 = c_glob c0"
well_shaped_lchild: fixes t :: "'b tree" assumes "well_shaped t" shows "well_shaped (lchild t)"
comp_fun_commute_axioms: shows "comp_fun_commute ((+) \<circ> pair_size)"
bisim1NewArrayFail: fixes P :: "'a prog"   and h :: "'heap"   and T :: "ty"   and e :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val" shows "P,newA T\<lfloor>e\<rceil>,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v], xs, length (compE2 e), \<lfloor>a\<rfloor>)"
map_eq_conv: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and g :: "'c \<Rightarrow> 'b" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
union_mset_add_mset_left: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "add_mset a A + B = add_mset a (A + B)"
assms_1: shows "rb_aux_inv2 (bs, ss, ps)"
construct_partition_on_def: fixes B :: "'a set"   and C :: "'a set" shows "construct_partition_on B C = {0..card B} \<bind> (\<lambda>k. {0..card C} \<bind> (\<lambda>j. {P. partition_on C P \<and> card P = j} \<bind> (\<lambda>P. {B'. B' \<subseteq> B \<and> card B' = k} \<bind> (\<lambda>B'. {Q. partition_on B' Q} \<bind> (\<lambda>Q. B - B' \<rightarrow>\<^sub>E P \<bind> (\<lambda>f. {(\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P} \<bind> (\<lambda>P'. {P' \<union> Q})))))))"
univ_poly_a_inv_def': fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (poly_ring R)"   and "p \<in> carrier (K [X]\<^bsub>poly_ring R\<^esub>)" shows "\<ominus>\<^bsub>K [X]\<^bsub>poly_ring R\<^esub>\<^esub> p = map (a_inv (poly_ring R)) p"
is_bdd_impl_def: fixes bdd :: "bdd"   and bddi :: "bddi" shows "is_bdd_impl bdd bddi = is_pointermap_impl (dpm bdd) (dpmi bddi) * is_hashmap (dcl bdd) (dcli bddi)"
wn_eq: shows "winding_number p z = winding_number \<gamma> z"
invar_not_found: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and src :: "nat"   and dst :: "nat"   and PRED :: "nat \<Rightarrow> nat option"   and C :: "nat set"   and N :: "nat set"   and d :: "nat" assumes "nf_invar c src dst PRED C N d"   and "C = {}" shows "Graph.bfs_spec c src dst None"
valid_weaken_pre: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and F :: "'c set"   and P :: "'b set"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set"   and P' :: "'b set" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P c Q, A"   and "P' \<subseteq> P" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P' c Q, A"
IUSpec_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals" shows "IUSpec ch mm rs \<equiv> (\<forall>p. UPSpec ch mm rs p) \<and> (\<forall>l. #l \<in> #MemLoc \<longrightarrow> MSpec ch mm rs l)"
max_sup: fixes x :: "'a"   and y :: "'a" shows "max x y = x \<squnion> y"
i_set0_cont_def: shows "i_set0_cont \<equiv> insert {} i_set_cont"
permutes_drop_cycle_size_two: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "p permutes S"   and "p (p x) = x" shows "Fun.swap x (p x) p permutes S - {x, p x}"
UN_E: fixes b :: "'g"   and B :: "'h \<Rightarrow> 'g set"   and A :: "'h set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'h" where "x \<in> A" and "b \<in> B x"
not_in_hyp: shows "\<forall>n<dim_vec C. C $ n = 0 \<longrightarrow> B ! n \<notin> A"
I_def: shows "I = (\<lambda>n. complex_set_lebesgue_integral lborel {r..R n} (g s))"
compact_sphere: fixes a :: "'b"   and r :: "real" shows "compact (sphere a r)"
same_orbit_rel_def: fixes G :: "('c, 'd) monoid_scheme"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and M :: "'e set" assumes "group_action G \<phi> M" shows "group_action.same_orbit_rel G \<phi> M = {p \<in> M \<times> M. \<exists>g\<in>carrier G. \<phi> g (snd p) = fst p}"
FSwSbImorph_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b, 'c) model_scheme" shows "FSwSbImorph h hA termMOD MOD = termFSwSbImorph h hA MOD"
e2ennreal'_inverse: fixes y :: "ereal" assumes "y \<in> {x. 0 \<le> x}" shows "enn2ereal (e2ennreal' y) = y"
pc: fixes k :: "real" shows "piecewise_continuous_on a k (I k) f"
g: shows "(2 ^ 4 * (3 * 2 ^ (4 - 1) - 1) * (3 * 2 ^ 4 - 1)) Amic (2 ^ 4 * (9 * 2 ^ (2 * 4 - 1) - 1))"
tanh_conv_tan: fixes z :: "complex" shows "tanh z = - \<i> * tan (\<i> * z)"
distinct_17: fixes x21 :: "char list"   and x22 :: "aexp"   and x61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and x62 :: "bexp"   and x63 :: "acom" shows "x21 ::= x22 \<noteq> {x61} WHILE x62 DO x63"
simps_10: fixes x41 :: "('a, 'b) process_term"   and x42 :: "('a, 'b) process_term"   and x1 :: "'b" shows "CH x41 x42 \<noteq> VAR x1"
abs_eq: shows "0 = Abs_fpxs (\<lambda>r. 0::'b)"
simps: fixes \<pi> :: "'b ltl \<Rightarrow> nat option" shows "M\<^sub>\<UU>_fin \<pi> = {((\<phi>', m), \<nu>, p). \<not> (\<forall>S. (\<forall>\<chi>\<in>dom \<pi>. S \<up>\<Turnstile>\<^sub>P Abs \<chi> \<and> S \<up>\<Turnstile>\<^sub>P \<up>eval\<^sub>G (dom \<pi>) (Abs (theG \<chi>)) \<and> (\<forall>q. (\<exists>j\<ge>the (\<pi> \<chi>). the (m \<chi>) q = Some j) \<longrightarrow> S \<up>\<Turnstile>\<^sub>P \<up>eval\<^sub>G (dom \<pi>) (\<up>step q \<nu>))) \<longrightarrow> S \<up>\<Turnstile>\<^sub>P \<up>step \<phi>' \<nu>)}"
fun_upd_eqD: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b"   and g :: "'a \<Rightarrow> 'b"   and z :: "'b" assumes "f(x := y) = g(x := z)" shows "y = z"
strict_implies_order: fixes a :: "'b list"   and b :: "'b list" assumes "strict_prefix a b" shows "prefix a b"
verit_comp_simplify_17: fixes n :: "num" shows "((1::'a) < numeral n) = (num.One < n)"
inner_commute: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x y = inner y x"
simple_ruleset_def: fixes rs :: "'b rule list" shows "simple_ruleset rs \<equiv> \<forall>r\<in>set rs. get_action r = Accept \<or> get_action r = Drop"
wf: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "wf {(x, y). less x y}"