hcju/isabelleps huggingface dataset:
sublist_valid_plan: fixes as' :: "'c list"   and as :: "'c list"   and PROB :: "'c set" assumes "subseq as' as"   and "as \<in> valid_plans PROB" shows "as' \<in> valid_plans PROB"
dehomogenize_def: fixes x :: "'c" shows "dehomogenize x = poly_subst (dehomo_subst x)"
gState_progress_rel_def: fixes p :: "program" shows "gState_progress_rel p = {(g, g'). gState_inv p g \<and> gState_inv p g' \<and> length (gState.channels g) \<le> length (gState.channels g') \<and> dom (lm.\<alpha> (gState.vars g)) = dom (lm.\<alpha> (gState.vars g'))}"
new_exn: fixes \<sigma> :: "heap option"   and c :: "'b Heap"   and r :: "'b" assumes "\<not> is_exn \<sigma>"   and "execute c (the_state \<sigma>) = None" shows "run c \<sigma> None r"
ab_rationality_imp: assumes "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) \<in> \<rat>" shows "\<exists>B>0. \<exists>c. (\<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2) \<and> (\<lambda>n. real_of_int (c (Suc n)) / real_of_int (a n)) \<longlonglongrightarrow> 0"
seqI: fixes f :: "'a"   and g :: "'a" assumes "S.arr f"   and "S.arr g"   and "S.dom g = S.cod f" shows "S.seq g f"
Q_min: fixes Q' :: "'a list set" assumes "DisjointPaths H_x v0 v1 Q' \<and> card Q' = sep_size" shows "Q_weight Q \<le> Q_weight Q'"
simps: fixes a :: "'outPort"   and n :: "'node"   and p :: "'inPort" shows "(a \<in> hyps_for' n p) = (\<exists>h. a = h \<and> hyps n h = Some p)"
igAbsIPresIGWls_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igAbsIPresIGWls MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X)"
hom_uminus: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "group_add_hom hom" shows "hom (- x) = - hom x"
insert_Id_image_Acts: fixes f :: "('d \<times> 'd) set \<Rightarrow> ('c \<times> 'c) set"   and F :: "'d program" assumes "f Id = Id" shows "insert Id (f ` Acts F) = f ` Acts F"
assms: shows "match_term_list P \<sigma> = Some \<tau>"
minus_set_def: fixes A :: "'c set"   and B :: "'c set" shows "A - B = Collect ((\<lambda>x. x \<in> A) - (\<lambda>x. x \<in> B))"
g_def: shows "g \<equiv> \<lambda>x. 1 - sin (pi * x) / (pi * x)"
antisym: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "antisymmetric A less_eq"   and "less_eq x y"   and "less_eq y x"   and "x \<in> A"   and "y \<in> A" shows "x = y"
concat_same_lengths_nth: fixes XS :: "'a list list"   and N :: "nat"   and i :: "nat" assumes "\<And>xs. xs \<in> set XS \<Longrightarrow> length xs = N"   and "i < length XS * N"   and "0 < N" shows "concat XS ! i = XS ! (i div N) ! (i mod N)"
weakly_bisimilar_def: fixes P :: "'state"   and Q :: "'state" shows "P \<approx>\<cdot> Q \<equiv> \<exists>R. is_weak_bisimulation R \<and> R P Q"
accept\<^sub>G\<^sub>R_I: fixes \<phi> :: "'a ltl" assumes "accept\<^sub>G\<^sub>R (\<A>\<^sub>\<UU> \<Sigma> \<phi>) w" obtains \<pi> :: "'a ltl \<Rightarrow> nat option" where "dom \<pi> \<subseteq> \<^bold>G \<phi>" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>" and "accepting_pair\<^sub>R (\<delta>\<^sub>\<UU> \<Sigma>) (\<iota>\<^sub>\<UU> \<phi>) (M\<^sub>\<UU>_fin \<pi>, UNIV) w" and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (\<delta>\<^sub>\<UU> \<Sigma>) (\<iota>\<^sub>\<UU> \<phi>) (Acc\<^sub>\<UU> \<Sigma> \<pi> \<chi>) w"
ae_filter_def: fixes M :: "'b measure" shows "ae_filter M = (\<Sqinter>N\<in>null_sets M. principal (space M - N))"
s_Let_inv: fixes \<pi> :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and x :: "Syntax.var"   and e\<^sub>2 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Syntax.Let e\<^sub>1 x e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   and "atom x \<sharp> (e\<^sub>1, \<pi>)" obtains "e' = e\<^sub>2[e\<^sub>1 / x] \<and> value e\<^sub>1 \<and> \<pi> = \<pi>'"   | e\<^sub>1' :: "Syntax.term" where "\<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater> \<and> e' = Syntax.Let e\<^sub>1' x e\<^sub>2 \<and> \<not> value e\<^sub>1"
sorted_dual_rev_iff: fixes ws :: "'a list list" shows "linorder.sorted (\<lambda>x y. y \<le>lex x) ws = linorder.sorted (\<le>lex) (rev ws)"
assms: shows "ys \<in> listset (map ((`) f) xs)"
simps_2: fixes d :: "int"   and i :: "int"   and ks :: "int list" shows "hd_coeff (Dvd d i ks) = (case ks of [] \<Rightarrow> 0 | k # x \<Rightarrow> k)"
galleryD_adj: fixes xs :: "'a set list" assumes "folding_g.gallery xs" shows "adjacentchain xs"
monad_state_prob_stateT: fixes return :: "'b \<times> 'c \<Rightarrow> 'd"   and bind :: "'d \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'd) \<Rightarrow> 'd"   and sample :: "'e pmf \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd" assumes "monad_prob return bind sample" shows "monad_state_prob (return_state return) (bind_state bind) get_state put_state (sample_state sample)"
Linear_order_wf_diff_Id: fixes r :: "('d \<times> 'd) set" assumes "Linear_order r" shows "wf (r - Id) = (\<forall>A\<subseteq>Field r. A \<noteq> {} \<longrightarrow> (\<exists>a\<in>A. \<forall>a'\<in>A. (a, a') \<in> r))"
indRelLSTPO_impl_SRel_and_TRel_preserve_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelLSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
lowner_le_matrix_sum: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and d :: "nat"   and g :: "nat \<Rightarrow> complex mat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> f k \<le>\<^sub>L g k" shows "matrix_sum d f n \<le>\<^sub>L matrix_sum d g n"
transaction_dual_subst_unfold: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and \<theta> :: "(('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat \<Rightarrow> (('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term" shows "unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_strand T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)) = unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)) @ unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)) @ unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)) @ unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_updates T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>)) @ unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<theta>))"
\<alpha>_def: shows "\<alpha> \<in> set noteq \<and> m < \<alpha>"
link_size: fixes hp :: "'b tree" shows "size (link hp) = size hp"
ide_hcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.ide \<nu>"   and "D.ide \<mu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "D.ide (\<nu> \<star>\<^sub>D \<mu>)"
is_LTLGlobal_def: fixes ltl :: "'c ltl" shows "is_LTLGlobal ltl = (\<exists>x8. ltl = G x8)"
simClosedAux: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "eqvt Rel"   and "\<Psi> \<rhd> P \<leadsto>[Rel] Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<leadsto>[Rel] p \<bullet> Q"
SN: shows "SN_on (E\<^sup>* O R O E\<^sup>*) {s}"
totatives_eq: shows "totatives n = {k \<in> {1..<n}. coprime k n}"
simps_20: fixes v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a acom"   and ve :: "'a acom" shows "({v} WHILE va DO vb {vc} \<le> vd;; ve) = False"
inv_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and f :: "'e"   and S :: "'e set" assumes "Group.group G"   and "f \<in> carrier (subgroup_generated G S)" shows "inv\<^bsub>subgroup_generated G S\<^esub> f = inv\<^bsub>G\<^esub> f"
four_composable_identity_arrows_of_spans_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data"   and \<pi> :: "'b arrow_of_spans_data"   and \<rho> :: "'b arrow_of_spans_data" shows "four_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<rho> \<equiv> ((four_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<rho> \<and> identity_arrow_of_spans C \<mu>) \<and> identity_arrow_of_spans C \<nu> \<and> identity_arrow_of_spans C \<pi> \<and> identity_arrow_of_spans C \<rho>) \<and> (two_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<and> two_composable_identity_arrows_of_spans C prj0 prj1 \<nu> \<pi>) \<and> two_composable_identity_arrows_of_spans C prj0 prj1 \<pi> \<rho> \<and> three_composable_identity_arrows_of_spans C prj0 prj1 \<mu> \<nu> \<pi> \<and> three_composable_identity_arrows_of_spans C prj0 prj1 \<nu> \<pi> \<rho>"
ncf3: fixes y :: "real" assumes "0 \<le> y"   and "y \<le> x" shows "0 < numer_cf3 (- y)"
preserves_cod: fixes f :: "'a" assumes "S.arr f" shows "S.cod (R f) = R (S.cod f)"
length_n_lists_elem: fixes ys :: "'g list"   and n :: "nat"   and xs :: "'g list" assumes "ys \<in> set (List.n_lists n xs)" shows "length ys = n"
s\<^sub>2: shows "s\<^sub>2 = \<epsilon>"
step_z_norm_induct: fixes x1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and x2 :: "'d"   and x3 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and x7 :: "'d"   and x8 :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and P :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint) \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> 'd \<Rightarrow> (nat \<Rightarrow> nat \<Rightarrow> real DBMEntry) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>\<^sub>\<N> \<langle>x7, x8\<rangle>"   and "\<And>A l D l' D'. A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle> \<Longrightarrow> P A l D l' (DBM_Normalization.norm (FW D' n) (\<lambda>x. real ((k \<circ> v') x)) n)" shows "P x1 x2 x3 x7 x8"
cmp_swapI: fixes m :: "'a"   and n :: "'a"   and x :: "'a"   and y :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and arr :: "'a \<Rightarrow> 'a" assumes "m < n \<and> x < y"   and "\<lbrakk>m < n \<and> x < y; x = m; y = n\<rbrakk> \<Longrightarrow> P (arr n) (arr m)"   and "\<lbrakk>m < n \<and> x < y; x \<noteq> m; x \<noteq> n; y \<noteq> m; y \<noteq> n\<rbrakk> \<Longrightarrow> P (arr m) (arr n)"   and "\<lbrakk>m < n \<and> x < y; x = m; y \<noteq> m; y \<noteq> n\<rbrakk> \<Longrightarrow> P (arr y) (arr n)"   and "\<lbrakk>m < n \<and> x < y; x = n; y \<noteq> m; y \<noteq> n\<rbrakk> \<Longrightarrow> P (arr m) (arr y)"   and "\<lbrakk>m < n \<and> x < y; x \<noteq> m; x \<noteq> n; y = n\<rbrakk> \<Longrightarrow> P (arr m) (arr x)"   and "\<lbrakk>m < n \<and> x < y; x \<noteq> m; x \<noteq> n; y = m\<rbrakk> \<Longrightarrow> P (arr x) (arr n)" shows "P (swap arr x y m) (swap arr x y n)"
set_iterator_genord_image_filter_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and g :: "'e \<Rightarrow> 'f option"   and R' :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_genord it S R"   and "inj_on g (S \<inter> dom g)"   and "\<And>x y x' y'. \<lbrakk>x \<in> S; g x = Some x'; y \<in> S; g y = Some y'; R x y\<rbrakk> \<Longrightarrow> R' x' y'" shows "set_iterator_genord (set_iterator_image_filter g it) {y. \<exists>x. x \<in> S \<and> g x = Some y} R'"
arctan_0_1_bounds: fixes y :: "float"   and prec :: "nat"   and n :: "nat" assumes "0 \<le> real_of_float y"   and "real_of_float y \<le> 1" shows "arctan (sqrt (real_of_float y)) \<in> {sqrt (real_of_float y) * real_of_float (lb_arctan_horner prec (get_even n) 1 y)..sqrt (real_of_float y) * real_of_float (ub_arctan_horner prec (get_odd n) 1 y)}"
LAT3: fixes p :: "L"   and q :: "L" shows "LEQ p (LUB p q)"
hyps: fixes stack :: "'a list" shows "exec (xs @ ys) stack env = exec ys (exec xs stack env) env"
simps_1: fixes l :: "'b" shows "[]\<down>l = None"
r_dovetail: fixes i :: "nat"   and z :: "nat" shows "eval r_dovetail [i, z] \<down>= (if 0 < the (eval r_result1 [pdec2 z, i, pdec1 z]) then 0 else 1)"
unroll_def: fixes G :: "prog"   and l :: "label"   and e :: "expr"   and c :: "stmt" shows "unroll G l e c = {(s, t). \<exists>v s1 s2. G\<turnstile>s \<midarrow>e-\<succ>v\<rightarrow> s1 \<and> the_Bool v \<and> normal s1 \<and> G\<turnstile>s1 \<midarrow>c\<rightarrow> s2 \<and> t = abupd (absorb (Cont l)) s2}"
substitutes_unitE': fixes y :: "'a"   and M :: "'a trm"   and X :: "'a trm" assumes "substitutes Unit y M X" obtains "X = Unit"   | "Unit = Var ya__" and "X = M"   | x :: "'a" where "Unit = Var x" and "X = Unit" and "x \<noteq> ya__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" and B' :: "'a trm" where "Unit = App A__ B__" and "X = App A'__ B'__" and "substitutes A__ ya__ M A'__" and "substitutes B__ ya__ M B'__"   | y :: "'a" and A :: "'a trm" and A' :: "'a trm" and T :: "PreSimplyTyped.type" where "Unit = Fn y T Aa__" and "X = Fn y T A'a__" and "ya__ \<noteq> y" and "y \<notin> fvs M" and "substitutes Aa__ ya__ M A'a__"   | A :: "'a trm" and A' :: "'a trm" and B :: "'a trm" and B' :: "'a trm" where "Unit = SimplyTyped.Pair A B" and "X = SimplyTyped.Pair A' B'" and "substitutes A ya__ M A'" and "substitutes B ya__ M B'"   | P :: "'a trm" and P' :: "'a trm" where "Unit = Fst P__" and "X = Fst P'__" and "substitutes P__ ya__ M P'__"   | P :: "'a trm" and P' :: "'a trm" where "Unit = Snd P" and "X = Snd P'" and "substitutes P ya__ M P'"
subst_Abs: fixes t :: "dB"   and s :: "dB"   and k :: "nat" shows "Abs t[s/k] = Abs (t[lift s 0/k + 1])"
l2_step4_def: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and gnx :: "msg" shows "l2_step4 Rb A B gnx \<equiv> {(s, s'). guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress s Rb = Some {xny, xgnx, xgny, xsk} \<and> guessed_frame Rb xgnx = Some gnx \<and> Auth A B \<langle>Number 1, gnx, Exp Gen (NonceF (Rb $ ny))\<rangle> \<in> chan s \<and> s' = s\<lparr>progress := progress s(Rb \<mapsto> {xny, xgnx, xgny, xsk, xEnd}), secret := {x. x = Exp gnx (NonceF (Rb $ ny)) \<and> Rb = test} \<union> secret s, signalsResp := if can_signal s A B then addSignal (signalsResp s) (Commit A B (Exp gnx (NonceF (Rb $ ny)))) else signalsResp s\<rparr>}"
adds_lcs: fixes s :: "'a"   and t :: "'a" shows "s adds lcs s t"
rep_eq: fixes x :: "uint64" shows "lsb x = lsb (Rep_uint64 x)"
free_category_axioms: shows "free_category {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True)"
all: shows "\<forall>u v. u \<noteq> \<epsilon> \<and> u \<in> \<langle>G\<rangle> \<and> v \<noteq> \<epsilon> \<and> v \<in> \<langle>G\<rangle> \<longrightarrow> u \<cdot> v \<noteq> b"
eq_\<I>_attach_on': fixes \<I>' :: "('i, 'j) \<I>"   and res :: "('i, 'j) resource"   and \<I> :: "('k, 'l) \<I>"   and conv :: "('k, 'l, 'i, 'j) converter"   and conv' :: "('k, 'l, 'i, 'j) converter"   and A :: "'k set" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"   and "A \<subseteq> outs_\<I> \<I>" shows "A \<turnstile>\<^sub>R conv \<rhd> res \<sim> conv' \<rhd> res"
multiple_block_in_original: fixes b :: "'a set"   and n :: "nat" assumes "b \<in># fis.multiple_blocks n" shows "b \<in># multiple_blocks n"
cont: fixes x :: "'b" assumes "x \<in> X" shows "continuous_on T (\<lambda>t. f t x)"
same_orbit_rel_def: shows "same_orbit_rel = {p \<in> M \<times> M. \<exists>g\<in>carrier G. \<phi> g (snd p) = fst p}"
small_imp_big: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> o[F](g)" shows "f \<in> O[F](g)"
viewL_def: fixes t :: "('c, 'd) FingerTree" shows "viewL t \<equiv> extract_viewres (viewL_aux t)"
h1: shows "prod_list (retrieve_polys qs (a # I)) = qs ! a * prod_list (retrieve_polys qs I)"
simps_9: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "Cod \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = Cod t \<^bold>\<otimes> Cod u \<^bold>\<otimes> Cod v"
c_snd_def: fixes u :: "nat" shows "c_snd u = c_sum u - c_fst u"
\<epsilon>_nonneg: shows "0 \<le> \<epsilon>"
simps: fixes f :: "'a \<Rightarrow> 'b"   and a1 :: "'a set"   and a2 :: "'a formula"   and a3 :: "'b" shows "hom_graph f a1 a2 a3 = (a1 = {} \<and> a2 = \<bottom> \<and> a3 = \<bottom> \<or> a1 = {} \<and> a2 = \<top> \<and> a3 = \<top> \<or> (\<exists>i S x a y b. a1 = insert i S \<and> a2 = ifte (var i) x y \<and> a3 = ifte (f i) a b \<and> i \<notin> S \<and> hom_graph f S x a \<and> hom_graph f S y b))"
is_ring: fixes R :: "('e, 'f) ring_scheme" assumes "field R" shows "ring R"
mem_adjoin: fixes H :: "'a set"   and x :: "'a"   and a :: "'a" assumes "subgroup H G"   and "x \<in> H"   and "a \<in> carrier G" shows "x \<in> adjoin G H a"
last_drop: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "last (drop n xs) = last xs"
sep_conj_sep_true': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "P h" shows "sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) P h"
analz_Un_partition: fixes S :: "msg set"   and H :: "msg set" assumes "analz S \<subseteq> synth (analz (S \<inter> payload \<union> S \<inter> valid \<union> S \<inter> range LtK \<union> Tags))"   and "H \<subseteq> payload \<union> valid \<union> range LtK" shows "analz (H \<union> S) \<subseteq> synth (analz ((H \<union> S) \<inter> payload \<union> (H \<union> S) \<inter> valid \<union> (H \<union> S) \<inter> range LtK \<union> Tags))"
poly_add_hom: fixes p :: "int poly"   and q :: "int poly" assumes "of_int (lead_coeff p) \<noteq> (0::'a)"   and "of_int (lead_coeff q) \<noteq> (0::'a)" shows "of_int_poly (poly_add p q) = poly_add (of_int_poly p) (of_int_poly q)"
conc_UNION_distrib_2: fixes M :: "'d \<Rightarrow> 'c list set"   and I :: "'d set"   and A :: "'c list set" shows "\<Union> (M ` I) @@ A = (\<Union>i\<in>I. M i @@ A)"
finite_UNIV: assumes "class.finite TYPE('b)" shows "finite UNIV"
free_tv_gen_cons: fixes S :: "nat \<Rightarrow> typ"   and A :: "type_scheme list"   and t :: "typ" shows "free_tv (gen ($ S A) t # $ S A) = free_tv ($ S A)"
axioms_2: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" assumes "constant_functor A B b" shows "category B"
dvd: shows "[:- a, 1::'a:] ^ k dvd p"
step_3_a_dims_2: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_col (??.Jordan_Normal_Form_Existence.step_3_a i j A) = dim_col A"
assms_2: fixes \<xi> :: "'a"   and \<xi>' :: "'a" assumes "S \<xi> \<xi>'" shows "U \<xi> \<xi>'"
keys_mult_scalar_monomial_right_subset: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c" shows "keys (p \<odot> monomial c v) \<subseteq> (\<lambda>t. t \<oplus> v) ` keys p"
m2_lt_n: shows "c_snd (c_snd n) < n"
infs_sset: fixes w :: "'a stream" shows "infs (sset w) w"
strategies_continue: fixes v :: "'a"   and w :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> winning_region p"   and "v \<rightarrow> w"   and "v \<in> VV p \<Longrightarrow> \<sigma> v = w"   and "\<sigma> \<in> good v" shows "\<sigma> \<in> good w"
connected_component_refl_eq: fixes S :: "'b set"   and x :: "'b" shows "connected_component S x x = (x \<in> S)"
continuous_on_mult_right: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. f x * c)"
f: shows "(fi, f) \<in> A \<rightarrow> \<langle>B\<rangle>nres_rel"
distinct_15: fixes x31 :: "'d regex"   and x32 :: "'d regex"   and x41 :: "'d regex"   and x42 :: "'d regex" shows "Plus x31 x32 \<noteq> Times x41 x42"
less_depth_nmset_imp_less_nmultiset: fixes X :: "'b nmultiset"   and Y :: "'b nmultiset" assumes "|X| < |Y|" shows "X < Y"
connect_eq_resource_cong: fixes \<I> :: "('h, 'i) \<I>"   and distinguisher :: "('h, 'i) distinguisher"   and res :: "('h, 'i) resource"   and res' :: "('h, 'i) resource" assumes "\<I> \<turnstile>g distinguisher \<surd>"   and "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"   and "\<I> \<turnstile>res res \<surd>" shows "connect distinguisher res = connect distinguisher res'"
g_def: shows "g = (\<lambda>r. part_circlepath z0 r 0 pi)"
fin_cut_same_tl: fixes s :: "('b \<times> 'c list) stream"   and x :: "'b \<times> 'c list" assumes "\<exists>n. sdrop n s = sconst x" shows "fin_cut_same (\<pi> x) (map \<pi> (cut_same x s)) = cut_same (\<pi> x) (smap \<pi> s)"
sel_2: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "stl (siterate f x) = siterate f (f x)"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "aggregation.sum_0 g A = aggregation.sum_0 h B"
eq: fixes X :: "'a quot"   and Y :: "'b quot" assumes "P X Y" shows "f X Y \<equiv> g (pick X) (pick Y)"
g_1: shows "g = (\<lambda>x. ennreal (spmf q x))"
assms_3: assumes "x \<in> set (local.filter_min xs)" shows "thesis"
prm_pi_def: fixes pi :: "('a \<times> 'a) list"   and x :: "pi" shows "pi \<bullet> x \<equiv> Abs_pi (pi \<bullet> Rep_pi x)"
line_ext_def: fixes l :: "Lines" shows "line_ext l \<equiv> {P. incid P l}"
name_append_less: fixes xs :: "name"   and ys :: "name" assumes "xs \<noteq> Name []" shows "ys < Name.append ys xs"
ide_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.ide (trg\<^sub>C \<mu>)"
part_equiv_trans_sym_2: fixes R :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "part_equiv R"   and "(a, b) \<in> R"   and "(a, c) \<in> R" shows "(b, c) \<in> R"
count_greater_zero_iff: fixes M :: "'b multiset"   and x :: "'b" shows "(0 < count M x) = (x \<in># M)"
Int_Collect_mono: fixes A :: "'c set"   and B :: "'c set"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x" shows "A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
kbo_std_def: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat" shows "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym \<equiv> kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"
firstPosLeLength: fixes a :: "'b"   and l :: "'b list" assumes "a \<in> set l" shows "firstPos a l < length l"
VcompNml_Nml_Dom: fixes t :: "'c local.term" assumes "Nml t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Dom t = t"
spray_def: fixes x :: "'a" shows "spray x \<equiv> {y. \<exists>R\<in>SPRAY x. y \<in> R}"
simps_5: fixes val :: "val" shows "constVal (Lit val) = Some val"
less_sb: shows "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"
op_map_pick_remove_def: fixes m :: "'d \<Rightarrow> 'e option" shows "op_map_pick_remove m \<equiv> SPEC (\<lambda>((k, v), m'). m k = Some v \<and> m' = m |` (- {k}))"