hcju/isabelleps huggingface dataset:
AssertionsImplyMethInvariants: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and C :: "Class"   and m :: "Method"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and s0 :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "G \<rhd> \<lbrace> A \<rbrace> C,m,l \<lbrace> B \<rbrace> I"   and "A s0 s" shows "I s0 s (heap s)"
simps_2: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and s :: "'a"   and r :: "'b"   and rs :: "'b list"   and a :: "'c"   and as :: "'c list" shows "On_Off.T step t s (r # rs) (a # as) = t s r a + On_Off.T step t (step s r a) rs as"
distinct_25: fixes x41 :: "'b rexp"   and x42 :: "'b rexp"   and x51 :: "'b rexp"   and x52 :: "'b rexp" shows "Plus x41 x42 \<noteq> Times x51 x52"
Polys_closed_monom_mult: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and c :: "'d" assumes "t \<in> .[X]"   and "p \<in> P[X]" shows "punit.monom_mult c t p \<in> P[X]"
dghm_up_ObjMap_vrange: fixes I :: "V"   and \<alpha> :: "V"   and \<CC> :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> \<phi> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i" shows "\<R>\<^sub>\<circ> (dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
convex_fx_plus_fy_ineq: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "convex_on {0..} f"   and "0 \<le> x"   and "0 \<le> y"   and "f 0 = 0" shows "f x + f y \<le> f (x + y)"
Var_castVar: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isVar s X" shows "Var (fst (castVar s X)) (snd (castVar s X)) = X"
decomps\<^sub>e\<^sub>s\<^sub>t_ik_subset: fixes D :: "('fun, 'var) extstrand_step list"   and M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>" shows "ik\<^sub>e\<^sub>s\<^sub>t D \<subseteq> subterms\<^sub>s\<^sub>e\<^sub>t (M \<union> N)"
many_strongly_connected_implies_11: fixes x :: "'a" assumes "many_strongly_connected x" shows "x\<^sup>\<star> ; x\<^sup>T ; x \<le> x\<^sup>+"
Sup_fun_def: fixes A :: "('b \<Rightarrow> 'c) set" shows "\<Squnion>A = (\<lambda>x. \<Squnion>f\<in>A. f x)"
bind_nonempty: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure" assumes "space M \<noteq> {}" shows "M \<bind> f = join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f)"
rconf_def: fixes G :: "prog"   and L :: "lenv"   and s :: "st"   and t :: "Term.term"   and v :: "vals"   and T :: "tys" shows "G,L,s\<turnstile>t\<succ>v\<Colon>\<preceq>T = (case T of Inl T \<Rightarrow> if \<exists>var. t = In2 var then (\<forall>n. the_In2 t = LVar n \<longrightarrow> fst (the_In2 v) = the (locals s n) \<and> (locals s n \<noteq> None \<longrightarrow> G,s\<turnstile>fst (the_In2 v)\<Colon>\<preceq>T)) \<and> ((\<nexists>n. the_In2 t = LVar n) \<longrightarrow> G,s\<turnstile>fst (the_In2 v)\<Colon>\<preceq>T) \<and> s\<le>|snd (the_In2 v)\<preceq>T\<Colon>\<preceq>(G, L) else G,s\<turnstile>the_In1 v\<Colon>\<preceq>T | Inr Ts \<Rightarrow> list_all2 (conf G s) (the_In3 v) Ts)"
simps_1: fixes f :: "'a" shows "Can \<^bold>\<langle>f\<^bold>\<rangle> = C.ide f"
euler_criterion_aux: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "[Legendre a (int p) = a ^ ((p - 1) div 2)] (mod int p)"
language_ltlc_def: fixes \<phi> :: "'a ltlc" shows "language_ltlc \<phi> \<equiv> {\<xi>. \<xi> \<Turnstile>\<^sub>c \<phi>}"
Reals_cases: fixes q :: "'b" assumes "q \<in> \<real>" obtains r :: "real" where "q = of_real r"
induct: fixes P :: "'c formula \<Rightarrow> bool"   and a0 :: "'c formula" assumes "\<And>F G. P (F \<^bold>\<rightarrow> G)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<and> G)"   and "\<And>F G. \<lbrakk>P F; P G\<rbrakk> \<Longrightarrow> P (F \<^bold>\<or> G)"   and "\<And>v. P (Atom v)"   and "P \<bottom>"   and "\<And>v. P (\<^bold>\<not> v)" shows "P a0"
freshTransition_2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>a[b] \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
trss_ret: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and p :: "'g"   and r :: "'f list"   and c :: "'f list multiset" shows "((return fg p # r, c), LRet, r, c) \<in> trss fg"
unfold_appendStep: fixes sb0 :: "'b"   and ha :: "'c \<rightarrow> ('d, 'c) Step"   and hb :: "'b \<rightarrow> ('d, 'b) Step" assumes "sb0 \<noteq> \<bottom>" shows "(\<forall>sa. sa \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(appendStep\<cdot>ha\<cdot>hb\<cdot>sb0)\<cdot>(Left\<cdot>sa) = appendL\<cdot>(unfold\<cdot>ha\<cdot>sa)\<cdot>(unfold\<cdot>hb\<cdot>sb0)) \<and> (\<forall>sb. sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>(appendStep\<cdot>ha\<cdot>hb\<cdot>sb0)\<cdot>(Right\<cdot>sb) = unfold\<cdot>hb\<cdot>sb)"
nterm_to_term_insert_dupl: fixes y :: "name"   and n :: "nat"   and \<Gamma> :: "name list"   and t :: "nterm" assumes "y \<in> set (take n \<Gamma>)"   and "n \<le> length \<Gamma>" shows "nterm_to_term \<Gamma> t = incr_bounds (- 1) (Suc n) (nterm_to_term (insert_nth n y \<Gamma>) t)"
hom_char: fixes f :: "'d"   and b :: "'d"   and a :: "'d" shows "Fop.Aop.in_hom f b a = Dop.in_hom f a b"
G_def: shows "G = {(i, j). A $h i $h j \<noteq> (0::'a)}"
abs_of_diff_def: shows "abs_of_diff = (\<lambda>x y. x - y + (y - x))"
intros_1: fixes M :: "lam" shows " M \<longrightarrow>\<^sub>1 M"
ofilter_def: fixes A :: "'c set" shows "t.ofilter A \<equiv> A \<subseteq> Field t \<and> (\<forall>a\<in>A. t.under a \<subseteq> A)"
countable_converseI: fixes A :: "('a \<times> 'b) set" assumes "countable A" shows "countable (A\<inverse>)"
map_simp: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'a \<times> 'c" assumes "symmetry_functor A1 A2"   and "partial_magma.arr (product_category.comp A1 A2) f" shows "symmetry_functor.map A1 A2 f = (snd f, fst f)"
acyclic_4_implies_2: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pd_kleene_allegory star one times sup less_eq less bot inf top conv uminus"   and "conv x = x"   and "pd_kleene_allegory.acyclic_4 star less_eq inf uminus x" shows "pd_kleene_allegory.acyclic_2 star one times less_eq bot inf conv uminus x"
set_foldr_Cons: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and as :: "'b list" shows "set (foldr (\<lambda>x xs. if P x xs then x # xs else xs) as []) \<subseteq> set as"
indep_sets_collect_sigma: fixes E :: "'e \<Rightarrow> 'a set set"   and I :: "'f \<Rightarrow> 'e set"   and J :: "'f set" assumes "indep_sets E (\<Union> (I ` J))"   and "\<And>i j. \<lbrakk>j \<in> J; i \<in> I j\<rbrakk> \<Longrightarrow> Int_stable (E i)"   and "disjoint_family_on I J" shows "indep_sets (\<lambda>j. sigma_sets (space M) (\<Union> (E ` I j))) J"
existence_ivl_undefined: fixes x0 :: "'a" assumes "x0 \<notin> X" shows "existence_ivl0 x0 = {}"
analz_image_freshK: fixes evs :: "event list"   and KK :: "nat set"   and K :: "nat" assumes "evs \<in> bankerberos"   and "KK \<subseteq> - range shrK" shows "(Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
nstut_eq: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b" assumes "NSTUTINV F"   and "NSTUTINV G" shows "NSTUTINV F = G"
pr1_in_hom: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr1 a b : local.prod a b \<rightarrow> a\<guillemotright>"
simps: fixes i :: "nat"   and t :: "nat"   and e :: "nat"   and s :: "bddi" shows "iteci_lu i t e s = hm_lookup (i, t, e) (dcli s) \<bind> (\<lambda>lu. case lu of None \<Rightarrow> param_optci i t e s \<bind> (\<lambda>(po, s). case po of None \<Rightarrow> lowest_topsci [i, t, e] s \<bind> (\<lambda>lt. case lt of None \<Rightarrow> raise STR ''Cannot happen'' | Some a \<Rightarrow> restrict_topci i a True s \<bind> (\<lambda>ti. restrict_topci t a True s \<bind> (\<lambda>tt. restrict_topci e a True s \<bind> (\<lambda>te. restrict_topci i a False s \<bind> (\<lambda>fi. restrict_topci t a False s \<bind> (\<lambda>ft. restrict_topci e a False s \<bind> (\<lambda>fe. iteci_lu ti tt te s \<bind> (\<lambda>(tb, s). iteci_lu fi ft fe s \<bind> (\<lambda>(fb, s). ifci a tb fb s \<bind> (\<lambda>(r, s). hm_update (i, t, e) r (dcli s) \<bind> (\<lambda>cl. return (r, dcli_update (const cl) s)))))))))))) | Some b \<Rightarrow> return (b, s)) | Some b \<Rightarrow> return (b, s))"
strict_implies_not_eq: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "ordering less_eq less"   and "less a b" shows "a \<noteq> b"
intros_18: fixes i :: "nat"   and \<C> :: "t_context"   and ts :: "t list"   and t1s :: "t list"   and t2s :: "t list" assumes "i < length (label \<C>)"   and "label \<C> ! i = ts" shows "\<C> \<turnstile> [Br i] : t1s @ ts _> t2s"
suminf_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'c" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'c)" shows "suminf f = sum f N"
LID\<^sub>GI: fixes s :: "'b \<Rightarrow> (('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr) option"   and r :: "'b"   and ls :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and l :: "'a" assumes "s r = Some ls"   and "l \<in> LID\<^sub>L ls" shows "l \<in> LID\<^sub>G s"
EmptyMap_Union_ran_override: fixes S :: "'d set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option" assumes "S \<noteq> {}"   and "S \<inter> dom G = {}" shows "\<Union> (ran (G ++ EmptyMap S)) = \<Union> (ran G)"
isPartition_wrt_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and lo :: "nat"   and hi :: "nat"   and p :: "nat" shows "isPartition_wrt R xs lo hi p \<equiv> (\<forall>i. lo \<le> i \<and> i < p \<longrightarrow> R (xs ! i) (xs ! p)) \<and> (\<forall>j. p < j \<and> j \<le> hi \<longrightarrow> R (xs ! p) (xs ! j))"
iwlsFSbSw_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "iwlsFSbSw MOD \<equiv> iwlsFSb MOD \<and> igSwapAllIPresIGWlsAll MOD \<and> igSwapCls MOD"
complete_real: assumes "\<exists>x\<in>elts \<real>\<^sub>\<circ>. x \<in>\<^sub>\<circ> A"   and "\<exists>z\<in>elts \<real>\<^sub>\<circ>. \<forall>x\<in>elts A. x \<le>\<^sub>\<real> z" shows "\<exists>y\<in>elts \<real>\<^sub>\<circ>. (\<forall>x\<in>elts A. x \<le>\<^sub>\<real> y) \<and> (\<forall>z\<in>elts \<real>\<^sub>\<circ>. (\<forall>x\<in>elts A. x \<le>\<^sub>\<real> z) \<longrightarrow> y \<le>\<^sub>\<real> z)"
prems_2: shows "f \<in> borel_measurable (F (Suc n))"
l3_init_def: shows "l3_init \<equiv> {\<lparr>ik = ik_init, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0, l3_state.bad = bad_init\<rparr>}"
dsr4: shows "d (zero_class.zero::'a) = (zero_class.zero::'a)"
mono_neutral_right: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = z" shows "abel_semigroup_set.F f z g T = abel_semigroup_set.F f z g S"
\<psi>_\<phi>: fixes y :: "'c"   and f :: "'b"   and x :: "'b" assumes "C.ide y"   and "\<guillemotleft>f : F' y \<rightarrow>\<^sub>B x\<guillemotright>" shows "\<psi>' x (\<phi>' y f) = f"
terminates_iff_terminates_merge_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" shows "\<Gamma>\<turnstile>c \<down> s = \<Gamma>\<turnstile>merge_guards c \<down> s"
A_ia_least_A: shows "A $ ia $ (LEAST n. A $ ia $ n \<noteq> (0::'a)) \<noteq> (0::'a)"
star_if_lang: fixes w :: "'a list"   and A :: "'a list set" assumes "w \<in> A" shows "w \<in> A\<star>"
product_nn_integral_prod: fixes I :: "'i set"   and f :: "'i \<Rightarrow> 'a \<Rightarrow> ennreal" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable (M i)" shows "\<integral>\<^sup>+ x. (\<Prod>i\<in>I. f i (x i)) \<partial>Pi\<^sub>M I M = (\<Prod>i\<in>I. integral\<^sup>N (M i) (f i))"
spmf_sample_uniform: fixes n :: "nat"   and x :: "nat" shows "spmf (sample_uniform n) x = indicat_real {..<n} x / real n"
inv_closed: fixes x :: "int" assumes "x \<in> carrier R" shows "\<ominus> x \<in> carrier R"
C_1: fixes z :: "complex" assumes "cmod z = r"   and "z \<notin> k" shows "cmod (f z) \<le> C"
ordering_sym: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>a b c. ord a b c \<Longrightarrow> ord c b a"   and "finite X"   and "TernaryOrdering.ordering f ord X" shows "TernaryOrdering.ordering (\<lambda>n. f (card X - 1 - n)) ord X"
cases: fixes a1 :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a2 :: "Vars.var \<Rightarrow> Value"   and a3 :: "Vars.var \<Rightarrow> Value" assumes "eq\<rho> a1 a2 a3" obtains ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" and \<rho>1 :: "Vars.var \<Rightarrow> Value" and \<rho>2 :: "Vars.var \<Rightarrow> Value" where "a1 = ae" and "a2 = \<rho>1" and "a3 = \<rho>2" and "\<And>x a. ae x = up\<cdot>a \<Longrightarrow> eq a (\<rho>1 x) (\<rho>2 x)"
simps: fixes u :: "int"   and p :: "int"   and \<Sigma> :: "int \<Rightarrow> (int \<Rightarrow> int set option) option" shows "hasLR u p \<Sigma> = (case \<Sigma> p of \<bottom> \<Rightarrow> False | \<lfloor>lrs\<rfloor> \<Rightarrow> \<exists>lr. lr \<in> ran lrs \<and> u \<in> lr)"
simps_2: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat" shows "TS M2 M1 \<Omega> V m (Suc 0) = V"
nb: shows "\<forall>(v, a)\<in>set (effect_of op'). \<forall>(v', a')\<in>set (effect_of op'). v \<noteq> v' \<or> a = a'"
simps_2: fixes s :: "String.literal"   and n :: "int" shows "(Str s < Num n) = False"
dg_length: shows "vcard \<BB> = 4\<^sub>\<nat>"
nonzero_eq_divide_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(a = divide b c) = (times a c = b)"
prj_simps_14: shows "C.dom Prj\<^sub>1\<^sub>0 = \<mu>\<nu>\<pi>.chine"
lt_Koszul_syz_init: fixes i :: "nat"   and j :: "nat" assumes "0 \<notin> set fs"   and "i < j"   and "j < length fs" shows "lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j)) - fs ! j \<odot> monomial (1::'b) (term_of_pair (0::'a, i))) = ord_term_lin.max (term_of_pair (punit.lt (fs ! i), j)) (term_of_pair (punit.lt (fs ! j), i))"
WT_parallel_oracle: fixes \<I>l :: "('a, 'b) \<I>"   and sl :: "'s1"   and \<I>r :: "('c, 'd) \<I>"   and sr :: "'s2" assumes "\<I>l \<turnstile>c left sl \<surd>"   and "\<I>r \<turnstile>c right sr \<surd>" shows "\<I>l \<oplus>\<^sub>\<I> \<I>r \<turnstile>c local.parallel_oracle (sl, sr) \<surd>"
simps_2: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c" shows "dfa.init \<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = init"
intros_2: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> \<^bold>\<not> F"   and "\<Gamma> \<turnstile> F" shows "\<Gamma> \<turnstile> \<bottom>"
gt_sus: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "\<And>x. local.ground_heads (Var x) = UNIV"   and "t >\<^sub>t s" shows "subst \<rho> t >\<^sub>t subst \<rho> s"
fail_bind: fixes f :: "'a \<times> 's \<Rightarrow> 'm" shows "bind fail f = fail"
packing: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int"   and ps :: "'b set" assumes "t_packing_design point_set block_collection u_block_size grouping min_index"   and "ps \<subseteq> point_set"   and "int (card ps) = grouping" shows "int (block_collection index ps) \<le> min_index"
Ind: fixes A :: "fm"   and H :: "fm set" assumes "A \<in> induction_axioms" shows "H \<turnstile> A"
left_cancel: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "group f z inverse" shows "(f a b = f a c) = (b = c)"
IH: fixes s' :: "'a" assumes "(E ^^ n) s s'" shows "\<exists>xs. local.steps xs \<and> hd xs = s \<and> last xs = s' \<and> length xs = n + 1"
bind_spmf_const: fixes p :: "'d spmf"   and q :: "'c spmf" shows "p \<bind> (\<lambda>x. q) = scale_spmf (weight_spmf p) q"
substitute_mono2': fixes t :: "'a ttree"   and t' :: "'a ttree"   and f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set" assumes "t \<sqsubseteq> t'" shows "substitute f T t \<sqsubseteq> substitute f T t'"
subdegree_decompose: fixes f :: "'a fps" shows "f = fps_shift (subdegree f) f * fps_X ^ subdegree f"
prems: shows "local.prefixes (Suc t) j \<down>= b"
fsmult_closed: fixes v :: "'v"   and a :: "'f" assumes "v \<in> V" shows "a \<sharp>\<cdot> v \<in> V"
chain: shows "Complete_Partial_Order.chain (fun_ord (\<le>)) Y"
simps_4: fixes M :: "('fun, 'var) Term.term set"   and uu :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" shows "\<lbrakk>M; \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S\<rbrakk>\<^sub>d' = (\<lambda>\<I>. Unifier \<I> t t' \<and> \<lbrakk>M; S\<rbrakk>\<^sub>d' \<I>)"
cases: fixes a1 :: "'node"   and a2 :: "'edge list"   and a3 :: "'node" assumes "a1 -a2\<rightarrow>* a3" obtains n :: "'node" where "a1 = nb__" and "a2 = []" and "a3 = nb__" and "valid_node nb__"   | n'' :: "'node" and as :: "'edge list" and n' :: "'node" and a :: "'edge" and n :: "'node" where "a1 = n" and "a2 = a # as" and "a3 = n'" and "n'' -as\<rightarrow>* n'" and "valid_edge a" and "sourcenode a = n" and "targetnode a = n''"
d: shows "d = p * A $ from_nat i $ from_nat i + A $ from_nat k $ from_nat k * q"
finite_extesion_mem_imp_algebraic: fixes K :: "'a set"   and xs :: "'a list"   and y :: "'a" assumes "subfield K R"   and "set xs \<subseteq> carrier R"   and "\<And>x. x \<in> set xs \<Longrightarrow> (algebraic over K) x"   and "y \<in> finite_extension K xs" shows "(algebraic over K) y"
cod_cod: fixes f :: "'a" assumes "B.arr f" shows "B.cod (B.cod f) = B.cod f"
induces_adjunction: shows "adjunction (\<cdot>\<^sub>B) (\<cdot>\<^sub>C) replete_setcat.comp F'G'.\<phi>C F'G'.\<phi>D F' G' \<phi>' \<psi>' F'G'.\<eta> F'G'.\<epsilon> F'G'.\<Phi> F'G'.\<Psi>"
Cons_2: shows "[a] \<preceq>\<^sub>F w"
vc: shows "x \<sharp> \<Gamma>\<^sub>2"
Collect_empty_eq: fixes P :: "'d \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
prefix_msg_in_history: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat"   and m :: "'a \<times> ('a, 'b) operation" assumes "es prefix of i"   and "m \<in> set (node_deliver_messages es)" shows "Deliver m \<in> set (history i)"
lookup_monom_mult: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and c :: "'f"   and t :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and v :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.monom_mult pair_of_term term_of_pair c t p) v = (if term_powerprod.adds_pp pair_of_term t v then c * lookup p (term_powerprod.sminus pair_of_term term_of_pair v t) else (0::'f))"
\<alpha>\<gamma>_defl: fixes x :: "'b" shows "\<alpha> (\<gamma> x) \<le> x"
le_oFix: fixes a :: "ordinal"   and F :: "ordinal \<Rightarrow> ordinal" shows "a \<le> oFix F a"
norm_psi: shows "inner_prod \<psi> \<psi> = 1"
intro: fixes rep_key_order :: "'b \<Rightarrow> 'c key_order"   and list_of_oalist :: "'a \<Rightarrow> ('c \<times> 'd) list \<times> 'b"   and oalist_of_list :: "('c \<times> 'd) list \<times> 'b \<Rightarrow> 'a" assumes "\<And>x. oalist_raw.oalist_inv rep_key_order (list_of_oalist x)"   and "\<And>xs. list_of_oalist (oalist_of_list xs) = oalist_raw.sort_oalist_raw rep_key_order xs"   and "\<And>x. oalist_of_list (list_of_oalist x) = x" shows "oalist_abstract rep_key_order list_of_oalist oalist_of_list"
size_ffilter: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b fset" shows "size (ffilter P f) \<le> size f"
has_label: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "\<exists>l. l \<in> labels \<Gamma> p"
term_powerprod_axioms: shows "term_powerprod (\<lambda>x. x) (\<lambda>x. x)"
comm_monoid_set_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "comm_monoid_set f z \<equiv> comm_monoid f z"
irreducible_prime: fixes a :: "'a" assumes "a \<in> carrier G"   and "irreducible G a" shows "prime G a"