hcju/isabelleps huggingface dataset:
set_mset_mset: fixes xs :: "'d list" shows "set_mset (mset xs) = set xs"
insert_messages_def: fixes M :: "('c, 'd) message set" shows "insert_messages M = {x. Insert x \<in> M}"
filter_paths_conv_free_restr: fixes S :: "'b set"   and t :: "'b ttree" shows "filter (\<lambda>x'. x' \<in> S) ` paths t = paths (ttree_restr S t)"
not_authKeys_not_AKcryptSK: fixes K :: "nat"   and evs :: "event list" assumes "K \<notin> authKeys evs"   and "K \<notin> range shrK"   and "evs \<in> kerbV" shows "\<forall>SK. \<not> AKcryptSK K SK evs"
partn_lst_VWF_imp_partn_lst: fixes \<beta> :: "V"   and \<alpha> :: "V list"   and n :: "nat"   and U :: "'a set"   and r :: "('a \<times> 'a) set" assumes "partn_lst_VWF \<beta> \<alpha> n"   and "\<beta> \<le> ordertype U r"   and "small U"   and "wf r"   and "trans r"   and "total_on U r" shows "Partitions.partn_lst r U \<alpha> n"
vcomp_eq: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "seq \<nu> \<mu>" shows "\<nu> \<bullet> \<mu> = \<lparr>Chn = Chn \<nu> \<cdot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr>"
hyps_1: shows "is_lub,P,E \<turnstile> e1 \<leadsto> e1'"
set_remdups: fixes xs :: "'a list" shows "set (remdups xs) = set xs"
ListAif2: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> 'b) list"   and C1 :: "acom"   and C2 :: "acom"   and l :: "'a \<Rightarrow> nat" assumes "\<not> bval b s" shows "preList upds (IF b THEN C1 ELSE C2) l s = preList upds C2 l s"
H_QM: shows "Hermite_reduce_above (mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))) (mod_type_class.to_nat i) i (LEAST n. A $ i $ n \<noteq> (0::'a)) res = Q ** mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))"
list_Previous_nth: fixes xs :: "'d list"   and init :: "'d"   and n :: "nat" shows "xs\<^bsup>\<leftarrow>' init\<^esup> n = (case n of 0 \<Rightarrow> init | Suc n' \<Rightarrow> xs ! n')"
apfst_def: fixes f :: "'c \<Rightarrow> 'e" shows "apfst f = map_prod f id"
hyps: assumes "map_of xs k = Some v" shows "\<exists>ys zs. xs = ys @ (k, v) # zs \<and> map_of ys k = None"
irreducible\<^sub>d_multD: fixes a :: "'a poly"   and b :: "'a poly" assumes "irreducible\<^sub>d (a * b)" shows "degree a = 0 \<and> a \<noteq> 0 \<and> irreducible\<^sub>d b \<or> degree b = 0 \<and> b \<noteq> 0 \<and> irreducible\<^sub>d a"
inv_in_hom: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and a :: "'d"   and b :: "'d" assumes "category C"   and "category.iso C f"   and "partial_magma.in_hom C f a b" shows "partial_magma.in_hom C (category.inv C f) b a"
lm003: fixes x :: "'c"   and y :: "'d" shows "{(x, {y})} \<in> allocationsUniverse"
FAss\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and fs' :: "Objects.fields"   and h\<^sub>2' :: "heap" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,NonStatic:t in D"   and "fs' = fs((F, D) \<mapsto> v)"   and "h\<^sub>2' = h\<^sub>2(a \<mapsto> (C, fs'))" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>unit,(h\<^sub>2', l\<^sub>2, sh\<^sub>2)\<rangle>"
all_in_listSubset: fixes b :: "('a, 'b) Combinators list"   and l :: "'a set list"   and a :: "('a, 'b) Combinators list" assumes "all_in_list b l"   and "singleCombinators a"   and "set a \<subseteq> set b" shows "all_in_list a l"
less_eq_multiset_empty_right: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "\<not> M \<le> {#}"
Upper_memI: fixes A :: "'c set"   and L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<in> carrier L" shows "x \<in> Upper L A"
elementLevelLtLevelImpliesMemberPrefixToLevel_aux: fixes e :: "'b"   and M :: "('b \<times> bool) list"   and i :: "nat"   and level :: "nat" assumes "e \<in> set (elements M)"   and "elementLevel e M + i \<le> level"   and "i \<le> level" shows "e \<in> set (elements (prefixToLevel_aux M level i))"
distinct_1: fixes x1 :: "agent"   and x2 :: "nat" shows "cAgent x1 \<noteq> cNumber x2"
is_snd_sylow: shows "snd_sylow (G\<lparr>carrier := P\<rparr>) p a k"
l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_disconnected_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs get_disconnected_document get_disconnected_document_locs ShadowRootClass.type_wf"
SD_inefficientE: fixes p :: "'alt pmf" assumes "\<not> SD_efficient R p" obtains q :: "'alt pmf" and i :: "'agent" where "q \<in> lotteries_on alts" and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q" and "i \<in> agents" and "p \<prec>[SD (R i)] q"
B: shows "w \<in> sinks_aux I D U (ipurge_tr_aux I' D' U' (xs @ [x]))"
choose_int_def: fixes m :: "nat"   and n :: "int" shows "choose_int m n = (if n < 0 then 0 else m choose nat n)"
span_mono: fixes A :: "('a, 'b) vec set"   and B :: "('a, 'b) vec set" assumes "A \<subseteq> B" shows "vec.span A \<subseteq> vec.span B"
Cod_dom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Cod (local.dom f) = Dom f"
int: fixes w :: "complex" assumes "w \<in> S - path_image \<gamma>" shows "((\<lambda>u. f' u / (u - w) ^ k) has_contour_integral f w) \<gamma>"
Int_Inter_eq_1: fixes A :: "'a set"   and \<B> :: "'a set set" shows "A \<inter> \<Inter> \<B> = (if \<B> = {} then A else \<Inter> ((\<inter>) A ` \<B>))"
mult_root_eq_root: fixes n :: "nat"   and y :: "real"   and x :: "real" assumes "0 < n"   and "0 \<le> y" shows "root n x * y = root n (x * y ^ n)"
fresh_bij: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'b" shows "pi \<bullet> a \<sharp> pi \<bullet> x = a \<sharp> x"
subsumeseq_termE: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "s \<le>\<cdot> t" obtains \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" where "t = s \<cdot> \<sigma>"
assms: fixes Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'" shows "\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel"
has_field_fun: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list"   and b' :: "staticb"   and T' :: "ty" assumes "P \<turnstile> C has F,b:T in D"   and "P \<turnstile> C has F,b':T' in D" shows "b = b' \<and> T' = T"
meet_related: fixes x :: "'b"   and y :: "'b" assumes "x \<sqsubseteq> y" shows "x \<sqinter> y = x"
disc_eq_case_2: fixes option :: "'c option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
r_inv_ex: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<exists>y\<in>carrier G. x \<oplus> y = \<zero>"
prems_1: shows "subtract (Node l x d r) t\<^sub>2 = Some t"
inf_r_def: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = a * (a r\<rightarrow> b)"
sv01_limit_V01: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> V\<^sub>0\<^sub>1" shows "\<exists>i n\<^sub>0. \<forall>n\<ge>n\<^sub>0. s\<^bsub>01\<^esub> (f \<triangleright> n) \<down>= i"
ideD_1: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<tau>.A.arr a"
smap_simps_2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b stream" shows "stl (smap f a) = smap f (stl a)"
codomains_char: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.codomains f = (if A1xA2.arr f then {A1xA2.cod f} else {})"
init: shows "nfa.init \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr> \<subseteq> nfa.states \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>"
that: fixes t :: "'a" assumes "rs = {|([], t)|}" shows "thesis"
ipresFreshAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'varSort, 'k, 'l, 'var, 'm, 'h, 'n) model_scheme" shows "ipresFreshAbs hA MOD \<equiv> \<forall>ys y us s A. wlsAbs (us, s) A \<longrightarrow> freshAbs ys y A \<longrightarrow> igFreshAbs MOD ys y (hA A)"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'f, 'g, 'j) SLin_state_scheme" assumes "\<not> P7 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'f, 'g, 'j) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (\<forall>c. status s1 c = Aborted \<and> \<not> initialized s2 \<longrightarrow> pending s2 c = pending s1 c \<and> status s2 c \<in> {Pending, Aborted})"
wt_subst_subst_upd: fixes \<theta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and x :: "('fun, 'atom) Term.term \<times> nat"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"   and "\<Gamma> (Var x) = \<Gamma> t" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<theta>(x := t))"
nat_pow_eq: fixes x :: "nat"   and k :: "nat" shows "x [^]\<^bsub>R\<^esub> k = x ^ k mod n"
union_commute: fixes M :: "'c multiset"   and N :: "'c multiset" shows "M + N = N + M"
ran_def: fixes m :: "'d \<Rightarrow> 'c option" shows "ran m = {b. \<exists>a. m a = \<lfloor>b\<rfloor>}"
simps: fixes a1 :: "'f LDCFG_node"   and a2 :: "'g"   and Def :: "'f \<Rightarrow> 'g set"   and E :: "'f"   and X :: "'f"   and H :: "'g set"   and L :: "'g set" shows "((a1, a2) \<in> lift_Def_set Def E X H L) = ((\<exists>V n. a1 = Node n \<and> a2 = V \<and> V \<in> Def n) \<or> (\<exists>V. a1 = Node E \<and> a2 = V \<and> V \<in> H))"
S_def: shows "S = {w \<in> space M. stake n w = stake n x}"
freshChainSym: fixes xvec :: "name list"   and yvec :: "name list" shows "xvec \<sharp>* yvec = yvec \<sharp>* xvec"
get_elements_by_tag_name_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and results :: "(_) element_ptr list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_elements_by_tag_name ptr' idd \<rightarrow>\<^sub>r results"   and "result \<in> set results" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set c) = (ptr' \<in> set c)"
NEQ_DEP_IMP_IN_DOM_i: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'f) fmap) set"   and v :: "'d" assumes "a \<in> PROB"   and "v \<in> fmdom' (fst a)" shows "v \<in> prob_dom PROB"
lebesgue_closedin: fixes S :: "'a set"   and T :: "'a set" assumes "closedin (top_of_set S) T"   and "S \<in> sets lebesgue" shows "T \<in> sets lebesgue"
leg1_simps_6: shows "D.cod (F g) = F g"
except_Int_keys: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and U :: "'b set" shows "except p (U \<inter> keys p) = except p U"
proofTreeD: fixes A :: "(nat \<times> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list) set" assumes "proofTree A" shows "bounded A \<and> founded subs (SATAxiom \<circ> sequent) A"
nccD_2: fixes X :: "'c set" assumes "ncc X" shows "closed X"
valid_pre: fixes ds :: "'d set"   and P' :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool"   and Q :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool"   and P :: "'d set \<Rightarrow> 'x set \<Rightarrow> bool" assumes "valid ds P' Q"   and "\<And>fp. P ds fp \<Longrightarrow> P' ds fp" shows "valid ds P Q"
VectorSpaceHom: shows "VectorSpaceHom (\<sharp>\<cdot>) V (\<sharp>\<star>) T"
simps: fixes p :: "nat"   and x0 :: "real"   and y0 :: "real"   and x1 :: "real"   and y1 :: "real"   and xl :: "real" shows "intersect_segment_xline' p ((x0, y0), x1, y1) xl = (if x0 \<le> xl \<and> xl \<le> x1 then if x0 = x1 then Some (min y0 y1, max y0 y1) else let yl = truncate_down p (truncate_down p (real_divl p (y1 - y0) (x1 - x0) * (xl - x0)) + y0); yr = truncate_up p (truncate_up p (real_divr p (y1 - y0) (x1 - x0) * (xl - x0)) + y0) in Some (yl, yr) else None)"
asInp_pickInp: fixes inp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm set option" assumes "goodInp inp" shows "asInp (pickInp inp) = inp"
sup_continuous_apply: fixes x :: "'a" shows "sup_continuous (\<lambda>f. f x)"
divide_le_0_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide one a) zero = less_eq a zero"
subword_nth1: fixes i :: "nat"   and j :: "nat"   and w :: "'c list"   and k :: "nat" assumes "i + j \<le> length w"   and "k < j" shows "subword w i j ! k = w ! (i + k)"
IH: assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> coprime (f x) (f y)" shows "order c (prod f A) = (\<Sum>x\<in>A. order c (f x))"
1: fixes i :: "'a" assumes "i \<in> B" shows "norm i = 1"
intros: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" assumes "r a b" shows "r\<inverse>\<inverse> b a"
le_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and o\<^sub>1 :: "'b option"   and o\<^sub>2 :: "'b option" shows "Opt.le r o\<^sub>1 o\<^sub>2 = (case o\<^sub>2 of None \<Rightarrow> o\<^sub>1 = None | \<lfloor>y\<rfloor> \<Rightarrow> case o\<^sub>1 of None \<Rightarrow> True | \<lfloor>x\<rfloor> \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
get_heap_E: fixes h :: "'a"   and x :: "'a" assumes "h \<turnstile> get_heap \<rightarrow>\<^sub>r x" shows "x = h"
ideal_1: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "ideal I R addition multiplication zero unit"   and "a \<in> R"   and "b \<in> I" shows "multiplication a b \<in> I"
ts: shows "listrelp (\<rightarrow>\<^sub>\<beta>\<^sup>*) ts ts'"
rev_singleton_conv: fixes xs :: "'g list"   and x :: "'g" shows "(rev xs = [x]) = (xs = [x])"
inj_on_the_inv_into: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "inj_on (the_inv_into A f) (f ` A)"
T_det: fixes P :: "'a process"   and Q :: "'a process" shows "T (P [+] Q) = T P \<union> T Q"
map_F_rel_cong: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L3 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co3 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra3 :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and xi :: "('n, 'p, 'r, 't, 'v, 'x, 'z, 'ab, 'ad, 'af) F"   and yi :: "('o, 'q, 's, 'u, 'w, 'y, 'aa, 'ac, 'ae, 'af) F"   and L1' :: "'ag \<Rightarrow> 'ah \<Rightarrow> bool"   and x :: "'n \<Rightarrow> 'ag"   and y :: "'o \<Rightarrow> 'ah"   and L2' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and xa :: "'p \<Rightarrow> 'ai"   and ya :: "'q \<Rightarrow> 'aj"   and L3' :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and xb :: "'r \<Rightarrow> 'ak"   and yb :: "'s \<Rightarrow> 'al"   and Co1' :: "'am \<Rightarrow> 'an \<Rightarrow> bool"   and xc :: "'t \<Rightarrow> 'am"   and yc :: "'u \<Rightarrow> 'an"   and Co2' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and xd :: "'v \<Rightarrow> 'ao"   and yd :: "'w \<Rightarrow> 'ap"   and Co3' :: "'aq \<Rightarrow> 'ar \<Rightarrow> bool"   and xe :: "'x \<Rightarrow> 'aq"   and ye :: "'y \<Rightarrow> 'ar"   and Contra1' :: "'as \<Rightarrow> 'at \<Rightarrow> bool"   and xf :: "'as \<Rightarrow> 'z"   and yf :: "'at \<Rightarrow> 'aa"   and Contra2' :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and xg :: "'au \<Rightarrow> 'ab"   and yg :: "'av \<Rightarrow> 'ac"   and Contra3' :: "'aw \<Rightarrow> 'ax \<Rightarrow> bool"   and xh :: "'aw \<Rightarrow> 'ad"   and yh :: "'ax \<Rightarrow> 'ae" assumes "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 xi yi"   and "\<And>x y. L1 x y \<Longrightarrow> L1' (x x) (y y)"   and "\<And>x y. L2 x y \<Longrightarrow> L2' (xa x) (ya y)"   and "\<And>x y. L3 x y \<Longrightarrow> L3' (xb x) (yb y)"   and "\<And>x y. Co1 x y \<Longrightarrow> Co1' (xc x) (yc y)"   and "\<And>x y. Co2 x y \<Longrightarrow> Co2' (xd x) (yd y)"   and "\<And>x y. Co3 x y \<Longrightarrow> Co3' (xe x) (ye y)"   and "\<And>x y. Contra1' x y \<Longrightarrow> Contra1 (xf x) (yf y)"   and "\<And>x y. Contra2' x y \<Longrightarrow> Contra2 (xg x) (yg y)"   and "\<And>x y. Contra3' x y \<Longrightarrow> Contra3 (xh x) (yh y)" shows "rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3' (map_F x xa xb xc xd xe xf xg xh xi) (map_F y ya yb yc yd ye yf yg yh yi)"
entry0_dominates_all: shows "\<forall>p\<in>set (g_V G). dominate 0 p"
superadditive_converges_bounded'': fixes u :: "nat \<Rightarrow> real"   and a :: "real" assumes "superadditive u"   and "\<And>n. 0 < n \<Longrightarrow> u n \<le> real n * a" shows "\<exists>l. (\<lambda>n. u n / real n) \<longlonglongrightarrow> l \<and> (\<forall>n>0. u n \<le> real n * l)"
assms_3: shows "(Q + M, Q + N) \<in> mul_eq r"
arr_char: fixes f :: "'b \<times> 'b" shows "D.VV.arr f = (D.arr (fst f) \<and> D.arr (snd f) \<and> src\<^sub>D (fst f) = trg\<^sub>D (snd f))"
scale_minus_right: fixes a :: "'a"   and x :: "('a, 'b) vec" shows "a *s - x = - (a *s x)"
mapl_F_comp: fixes l1 :: "'aa \<Rightarrow> 'x"   and l2 :: "'ab \<Rightarrow> 'y"   and l3 :: "'ac \<Rightarrow> 'z"   and l1' :: "'n \<Rightarrow> 'aa"   and l2' :: "'o \<Rightarrow> 'ab"   and l3' :: "'p \<Rightarrow> 'ac" shows "mapl_F l1 l2 l3 \<circ> mapl_F l1' l2' l3' = mapl_F (l1 \<circ> l1') (l2 \<circ> l2') (l3 \<circ> l3')"
qfN_mult2: fixes a :: "int"   and N :: "int"   and b :: "int"   and c :: "int"   and d :: "int" shows "(a\<^sup>2 + N * b\<^sup>2) * (c\<^sup>2 + N * d\<^sup>2) = (a * c - N * b * d)\<^sup>2 + N * (a * d + b * c)\<^sup>2"
ccSup_upper: 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"   and x :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "x \<in> A" shows "less_eq x (Sup A)"
rep_number_unfold_set: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int"   and x :: "'b" assumes "constant_rep_design point_set block_collection design_rep_number"   and "x \<in> point_set" shows "int (size (filter_mset ((\<in>) x) block_collection)) = design_rep_number"
r_search_enum_def: shows "r_search_enum \<equiv> Cn 2 r_le [Cn 2 r_length [recf.Id 2 1], Cn 2 r_enumdom [Cn 2 r_s [recf.Id 2 1], recf.Id 2 0]]"
asubg_subset1: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set"   and x :: "'e" assumes "aGroup A"   and "A +> H"   and "x \<in> H" shows "x \<in> carrier A"
less_induct1: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "class.well_founded less"   and "\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
fps_inverse_fps_X_plus1: shows "inverse (1 + fps_X) = Abs_fps ((^) (- (1::'b)))"
assms_2: shows "finite {i. X i \<noteq> UNIV}"
simps_3: fixes carrier' :: "'a set \<Rightarrow> 'a set"   and carrier :: "'a set"   and more :: "'b" shows "carrier_update carrier' \<lparr>carrier = carrier, \<dots> = more\<rparr> = \<lparr>carrier = carrier' carrier, \<dots> = more\<rparr>"
open_mapping_thm3: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "open S"   and "inj_on f S" shows "open (f ` S)"
Join_SKIP_right: fixes F :: "'c program" shows "F \<squnion> \<bottom> = F"
clo1: shows "closedin (top_of_set (S \<union> \<Union> G \<union> (S \<union> UF))) (S \<union> \<Union> G)"
WTrt_elim_cases_15: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> INIT C (Cs,b) \<leftarrow> e : T" obtains "P,E,h,sh \<turnstile> e : T" and "\<not> sub_RI e" and "\<forall>C'\<in>set (tl Cs). \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>" and "b \<longrightarrow> (\<forall>C'\<in>set Cs. \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>)" and "distinct Cs" and "supercls_lst P Cs" and "is_class P C" and "Ball (set Cs) (is_class P)"
distrib_left: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a * (b \<squnion> c) = a * b \<squnion> a * c"
has_extend_witness': fixes n :: "nat"   and S :: "'b set"   and xs :: "'b list" assumes "has n S"   and "hasw xs S"   and "length xs < n" shows "\<exists>x. hasw (x # xs) S"
DynT_conf: fixes mode :: "inv_mode"   and s :: "st"   and a' :: "val"   and statT :: "ref_ty"   and declC :: "qtname"   and G :: "prog" assumes "G\<turnstile>invocation_class mode s a' statT\<preceq>\<^sub>C declC"   and "wf_prog G"   and "isrtype G statT"   and "G,s\<turnstile>a'\<Colon>\<preceq>RefT statT"   and "mode = IntVir \<longrightarrow> a' \<noteq> Null"   and "mode \<noteq> IntVir \<longrightarrow> (\<exists>statC. statT = ClassT statC \<and> G\<turnstile>statC\<preceq>\<^sub>C declC) \<or> (\<forall>statC. statT \<noteq> ClassT statC \<and> declC = Object)" shows "G,s\<turnstile>a'\<Colon>\<preceq>Class declC"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "(a1 >\<^sub>t a2) = ((\<exists>t s. a1 = t \<and> a2 = s \<and> is_App t \<and> chop t \<ge>\<^sub>t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t >\<^sub>h\<^sub>d head s \<and> is_Sym (head s) \<and> chkchop (>\<^sub>t) t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> chkchop_same (>\<^sub>t) t s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s))))"