hcju/isabelleps huggingface dataset:
eval_Neg_And: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and L :: "real list" shows "eval (Neg (And \<phi> \<psi>)) L = eval (Or (Neg \<phi>) (Neg \<psi>)) L"
FunctorComp: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c"   and g :: "'c" assumes "PreFunctor F"   and "f \<approx>>\<^bsub>CatDom F\<^esub> g" shows "F ## f ;;\<^bsub>CatDom F\<^esub> g = F ## f ;;\<^bsub>CatCod F\<^esub> (F ## g)"
distinct_6: fixes x3 :: "nat"   and x2 :: "nat" shows "Regions.intv.Greater x3 \<noteq> Regions.intv.Intv x2"
t1: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "name"   and \<tau> :: "ty" assumes "valid \<Gamma>"   and "(a, \<tau>) \<in> set \<Gamma>" shows "\<Gamma> \<turnstile> Var a : \<tau>"
card_proots_within_rsquarefree: fixes p :: "'a poly"   and s :: "'a set" assumes "rsquarefree p" shows "proots_count p s = card (proots_within p s)"
simE2_1: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel> S"   and "eqvt Rel"   and "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
lprefix_lfilterI: fixes xs :: "'b llist"   and ys :: "'b llist"   and P :: "'b \<Rightarrow> bool" assumes "lprefix xs ys" shows "lprefix (lfilter P xs) (lfilter P ys)"
incl_in_img_cod: fixes f :: "'s" assumes "arr f" shows "incl_in (img f) (cod f)"
list_emb_refl: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x x" shows "list_emb P xs xs"
extend_in_C_no_delta: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat" assumes "extend S C f n \<union> {f n} \<in> C"   and "\<forall>p. f n \<noteq> Exists p"   and "\<forall>p. f n \<noteq> Neg (Forall p)" shows "extend S C f (Suc n) \<in> C"
powr_0: fixes z :: "'a" shows "(0::'a) powr z = (0::'a)"
Sbis_prefix: shows "Sbis \<subseteq> Sretr Sbis"
simps: fixes M :: "('d, 'e, 'f) FSM" shows "well_formed M = (finite_FSM M \<and> (\<forall>s1 x y. x \<notin> inputs M \<or> y \<notin> outputs M \<longrightarrow> succ M (x, y) s1 = {}) \<and> inputs M \<noteq> {} \<and> outputs M \<noteq> {})"
RelM_def: shows "RelM \<equiv> {(N1, N2). N1 \<in> calM \<and> N2 \<in> calM \<and> (\<exists>g\<in>carrier G. N1 = N2 #> g)}"
lists_image_witness: fixes x :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and Q :: "'b set" assumes "x \<in> lists (f ` Q)" obtains xo :: "'b list" where "xo \<in> lists Q" and "x = map f xo"
whilefalse: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and c :: "('b, 'c) MWLfCom" assumes "BMap (E b m) = False" shows "((while b do c od, m), [], m) \<in> MWLf_semantics.MWLfSteps_det E BMap"
replicate_iff2: fixes n :: "nat"   and l :: "color list" assumes "n < length l" shows "(\<forall>i<n. l ! i = R) = (\<exists>l'. l = replicate n R @ l')"
simps_1: fixes P :: "'e \<Rightarrow> bool" shows "filter P [] = []"
adjust_steps_eq: fixes st :: "nat"   and l :: "cell list"   and r :: "cell list"   and ap :: "(action \<times> nat) list"   and stp :: "nat"   and st' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (st, l, r) ap stp = (st', l', r')"   and "tm_wf0 ap"   and "0 < st'" shows "steps0 (st, l, r) (adjust0 ap) stp = (st', l', r')"
icard_insert_if: fixes x :: "'a"   and A :: "'a set" shows "icard (insert x A) = (if x \<in> A then icard A else eSuc (icard A))"
assms_3: shows "cross_ratio (apply_cltn2 p C) (apply_cltn2 q C) (apply_cltn2 r C) t = cross_ratio p q r s"
outs: shows "outs_\<I> \<I> = outs_\<I> \<I>'"
ih: fixes x :: "hf" assumes "\<And>y. y \<^bold>\<in> x \<Longrightarrow> P y" shows "P x"
assms_1: shows "b0 \<otimes> inv b1 = (h0 \<otimes> inv h1) [^] r"
funpow_commute: fixes k :: "nat"   and f :: "'c \<Rightarrow> 'b"   and c :: "'c \<Rightarrow> 'c"   and s :: "'c"   and c' :: "'b \<Rightarrow> 'b" assumes "\<forall>k'<k. f (c ((c ^^ k') s)) = c' (f ((c ^^ k') s))" shows "f ((c ^^ k) s) = (c' ^^ k) (f s)"
check_def: fixes M :: "'a kripke"   and f :: "'a set" shows "M \<turnstile> f \<equiv> init M \<subseteq> {s \<in> states M. s \<in> f}"
finite': fixes P :: "nat \<Rightarrow> bool" assumes "finite {x. \<not> P x}" shows "eventually P \<U>"
assoc_closed: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "Arr f \<and> C.ide f"   and "Arr g \<and> C.ide g"   and "Arr h \<and> C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "Arr \<a>\<^sub>C[f, g, h]"
SCallInitThrow\<^sub>1: fixes P :: "nat exp prog"   and ps :: "nat exp list"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and h\<^sub>1 :: "heap"   and ls\<^sub>1 :: "val list"   and sh\<^sub>1 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list"   and a :: "nat exp"   and s' :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = body in D"   and "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit"   and "P \<turnstile>\<^sub>1 \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>1, ls\<^sub>1, sh\<^sub>1)\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>"
abelian_groupE_4: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
H: fixes x :: "(char list list \<times> int) list" assumes "mset p = mset x"   and "sorted_wrt (rel2p (Id \<union> term_order_rel)) (map fst x)" shows "(x, p') \<in> \<langle>term_poly_list_rel \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel"
get_dom_component_separates_tree_order_general: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr'' :: "(_) object_ptr"   and to'' :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and c' :: "(_) object_ptr list" 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> to_tree_order ptr'' \<rightarrow>\<^sub>r to''"   and "ptr'' \<in> set c"   and "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"   and "ptr' \<notin> set c" shows "set to'' \<inter> set c' = {}"
tvs'_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [a] : tvs' _> tvs"
phiArgs_def_distinct: fixes g :: "'g"   and p :: "'val"   and q :: "'val"   and r :: "'val" assumes "phiArg g p q"   and "phiArg g p r"   and "q \<noteq> r"   and "p \<in> allVars g" shows "defNode g q \<noteq> defNode g r"
impl_of_cases: fixes y :: "('a, 'b) RBT_Impl.rbt" assumes "y \<in> {t. is_rbt t}" obtains x :: "('a, 'b) RBT.rbt" where "y = impl_of x"
equal_llist_def: fixes x :: "'a llist"   and y :: "'a llist" shows "equal_class.equal x y = (x = y)"
intros_1: fixes xs :: "'c llist" shows "LNil \<sqsubseteq> xs"
distinct_58: fixes x14 :: "Syntax.term"   and x2 :: "Syntax.var" shows "Unroll x14 \<noteq> Var x2"
Borsuk_maps_homotopic_in_path_component: fixes s :: "'a set"   and a :: "'a"   and b :: "'a" assumes "path_component (- s) a b" shows "homotopic_with_canon (\<lambda>x. True) s (sphere (0::'a) 1) (\<lambda>x. (x - a) /\<^sub>R norm (x - a)) (\<lambda>x. (x - b) /\<^sub>R norm (x - b))"
A1: shows "A1 = \<Union> (B1 ` N)"
red_Diag: fixes a :: "'c local.term" assumes "Diag a" shows "a\<^bold>\<down> = a"
normalize_quot_def: shows "normalize_quot = (\<lambda>(a, b). if b = (0::'a) then (0::'a, 1::'a) else let d = gcd a b * unit_factor b in (a div d, b div d))"
pair_map_pmf1: fixes f :: "'f \<Rightarrow> 'd"   and A :: "'f pmf"   and B :: "'e pmf" shows "pair_pmf (map_pmf f A) B = map_pmf (apfst f) (pair_pmf A B)"
takefill_same: fixes fill :: "'a"   and xs :: "'a list" shows "takefill fill (length xs) xs = xs"
le_neq_trans: fixes b :: "('c, 'd) Term.term"   and a :: "('c, 'd) Term.term" assumes "b \<unrhd> a"   and "a \<noteq> b" shows "b \<rhd> a"
take_Cons': fixes n :: "nat"   and x :: "'d"   and xs :: "'d list" shows "take n (x # xs) = (if n = 0 then [] else x # take (n - 1) xs)"
cis_minus_pi2: shows "cis (- pi / 2) = - \<i>"
restrict_current_IN_not_RF: fixes \<Gamma> :: "('a, 'b) web_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> g"   and "x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_IN (g \<upharpoonleft> \<Gamma> / f) x = d_IN g x"
listset_map_imageI: fixes ys' :: "'b list"   and xs :: "'b set list"   and ys :: "'c list"   and f :: "'b \<Rightarrow> 'c" assumes "ys' \<in> listset xs"   and "ys = map f ys'" shows "ys \<in> listset (map ((`) f) xs)"
Red_in_Sup: fixes Bot :: "'b set"   and Inf :: "'b inference set"   and entails :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I :: "'b set \<Rightarrow> 'b inference set"   and Red_F :: "'b set \<Rightarrow> 'b set"   and Ns :: "'b set llist" assumes "calculus Bot Inf entails Red_I Red_F"   and "chain (calculus.derive Red_F) Ns" shows "Sup_llist Ns - Liminf_llist Ns \<subseteq> Red_F (Sup_llist Ns)"
mcont_Pair: fixes lub :: "'a set \<Rightarrow> 'a"   and ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c" assumes "mcont lub ord luba orda f"   and "mcont lub ord lubb ordb g" shows "mcont lub ord (prod_lub luba lubb) (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
h2: shows "\<exists>F'. convert_fm var (convertDerivative var p) xs = Some F'"
intros_11: fixes Env :: "env"   and B :: "lname set"   and C :: "qtname" shows "Env\<turnstile> B \<guillemotright>\<langle>Init C\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
card_psubset: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subseteq> B"   and "card A < card B" shows "A \<subset> B"
cases: fixes x :: "'a Word.word list" obtains "x = []"   | xa :: "'a Word.word" and ys :: "'a Word.word list" where "x = xa # ys"
toplevel_inters_Full: fixes r :: "'b rexp"   and n :: "nat" assumes "toplevel_inters r = {Full}"   and "local.wf n r" shows "lang n r = lists (\<Sigma> n)"
C13: fixes x :: "'b" shows "(x\<^sup>\<star>)\<^sup>\<star> = x\<^sup>\<star>"
tab\<^sub>0_simps_3: fixes r :: "'a" assumes "ide r" shows "trg (tab\<^sub>0 r) = src r"
add_closed: fixes U :: "'b set"   and V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "Subspace.subspace U V"   and "x \<in> U"   and "y \<in> U" shows "x + y \<in> U"
Cons: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "validTrans trn"   and "tgtOf trn = srcOf (hd tr)"   and "valid tr" shows "valid (trn # tr)"
a_comm: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
G: shows "G = \<lparr>nodes = nodes G1 \<union> nodes G2, edges = edges G1 \<union> edges G2 \<union> {(v1', w, v2), (v2, w, v1')}\<rparr>"
degree_reflect_poly_le: fixes p :: "'b poly" shows "degree (reflect_poly p) \<le> degree p"
zcount_ps_apply_cm: fixes c :: "('c, 't, 'd) Propagate.configuration_scheme"   and loc :: "'c"   and t :: "'t"   and n :: "int"   and loc' :: "'c"   and t' :: "'t" shows "zcount (c_pts (apply_cm c loc t n) loc') t' = zcount (c_pts c loc') t' + (if loc = loc' \<and> t = t' then n else 0)"
not_real_square_gt_zero: fixes x :: "real" shows "(\<not> 0 < x * x) = (x = 0)"
A9_DAcc_level0: shows "DAcc level0 sA9 = {sA8}"
length_rk2_slp_le: shows "2 * D \<le> length rk2_slp"
replete_concrete_set_category_def: fixes S :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and U :: "'ah set"   and \<iota> :: "'ah \<Rightarrow> 'ag" shows "replete_concrete_set_category S U \<iota> \<equiv> replete_set_category S \<and> concrete_set_category S (cardSuc (cmax |UNIV| natLeq)) U \<iota>"
assms: shows "\<R>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> \<R>\<^sub>\<bullet> s = []\<^sub>\<circ>"
remove_state_def: fixes s :: "'a"   and e :: "(('a \<times> 'a) \<times> 'b) fset" shows "remove_state s e = ffilter (\<lambda>((from, to), t). from \<noteq> s \<and> to \<noteq> s) e"
attract: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "y \<sqsubseteq> z"   and "x \<in> B"   and "y \<in> B"   and "z \<in> B" shows "x \<sqsubseteq> z"
homogenize_in_Polys: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and x :: "'c" assumes "p \<in> P[X]" shows "homogenize x p \<in> P[insert x X]"
chamber_retraction1: fixes C :: "'a set" assumes "C \<in> f \<turnstile> \<C>" shows "fixespointwise f C"
simps_4: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and va :: "'a"   and v :: "'b" shows "sum_le P Q (Inl va) (Inr v) = False"
fresh_star_restict_stack: fixes a :: "atom set"   and S :: "stack_elem list"   and V :: "Vars.var set" assumes "a \<sharp>* S" shows "a \<sharp>* restr_stack V S"
ins3_ins2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and t :: "'b tree"   and d :: "nat"   and x :: "'b" assumes "RBTi2 TYPE('b) bal_i e"   and "bal_i n (height t + d)" shows "RBTi2.ins3 bal_i n d x t = RBTi2.ins2 bal_i n d x t"
IH: fixes s :: "'a \<Rightarrow> bool option" assumes "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "foldl (++) s (map (map_of \<circ> effect_to_assignments) ops) v = s v"
group_closure_minus_iff: fixes s :: "'a"   and S :: "'a set" shows "(- s \<in> group_closure S) = (s \<in> group_closure S)"
sgn_poly_def: fixes x :: "'b poly" shows "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
vars\<^sub>s\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>s\<^sub>t_bvars\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('f, 'e) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t S = fv\<^sub>s\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>s\<^sub>t S"
size_multiset_empty: fixes f :: "'k \<Rightarrow> nat" shows "size_multiset f {#} = 0"
mpoly_base_conv_3: fixes x :: "'b" shows "x = poly [:0::'b, 1::'b:] x"
ceiling_log2_div2: fixes n :: "nat" assumes "2 \<le> n" shows "\<lceil>log 2 (real n)\<rceil> = \<lceil>log 2 (real ((n - 1) div 2 + 1))\<rceil> + 1"
open_state_space: fixes t0 :: "real" shows "open (SIGMA x0:X. (+) t0 ` existence_ivl0 x0)"
cSUP_const: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and c :: "'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}" shows "Sup ((\<lambda>x. c) ` A) = c"
concurrent_def: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s1 :: "'c"   and s2 :: "'c" assumes "happens_before hb_weak hb" shows "happens_before.concurrent hb s1 s2 \<equiv> \<not> hb s1 s2 \<and> \<not> hb s2 s1"
abs_summable_on_finite: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A" shows "f abs_summable_on A"
wp_cond_aux1: fixes p :: "'a"   and x :: "'a"   and y :: "'a"   and q :: "'a" shows "d p \<cdot> wp (if p then x else y fi) q = d p \<cdot> wp x q"
conc: shows "List.member (map (\<lambda>x. map (map_fun Rep_fin_string Abs_fin_string (string_cons x)) (fin_string_upto n)) enum_class.enum) LL"
j0_notin: shows "prems_of \<iota> ! j0 \<notin> active_subset (lnth Ns n)"
inject: fixes x :: "'b"   and ya :: "'b" shows "(\<partial> x = \<partial> ya) = (x = ya)"
null_setsD2: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> null_sets M" shows "A \<in> sets M"
assms_4: shows "rs \<bullet>\<cdot> ms = (0::'m)"
distinct_formal_ins: fixes p :: "'pname"   and ins :: "'var list"   and outs :: "'var list" assumes "(p, ins, outs) \<in> set procs" shows "distinct ins"
distinct_144: fixes x151 :: "floatarith"   and x152 :: "nat"   and x5 :: "floatarith" shows "x151 ^\<^sub>e x152 \<noteq> Cos x5"
convex_hull_eq_empty: fixes S :: "'a set" shows "(convex hull S = {}) = (S = {})"
bot_lub: fixes S :: "'b set" shows "S <<| \<bottom> = (S \<subseteq> {\<bottom>})"
abs_abs: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (abs a) = abs a"
assms_1: shows "\<Theta>,\<Gamma> \<turnstile> B"
bconf: shows "P,shp s \<turnstile>\<^sub>b (es,b) \<surd>"
xor_list_commute: fixes xs :: "'a list"   and ys :: "'a list" shows "xs [\<oplus>] ys = ys [\<oplus>] xs"
simps_7: fixes bs :: "'a list"   and t :: "poly"   and n :: "nat" shows "\<lparr>Pw t n\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>t\<rparr>\<^sub>p\<^bsup>bs\<^esup> ^ n"
proj_def: fixes n :: "'a"   and S :: "('a strand_label \<times> 'b) list" shows "proj n S \<equiv> filter (\<lambda>s. is_LabelN n s \<or> is_LabelS s) S"
univ_poly_consistent: fixes K :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) set" assumes "subring K L" shows "univ_poly (L\<lparr>carrier := K\<rparr>) = univ_poly L"
rHom_inv_inv: fixes A :: "('e, 'f) Ring_scheme"   and R :: "('g, 'h) Ring_scheme"   and x :: "'e"   and f :: "'e \<Rightarrow> 'g" assumes "Ring A"   and "Ring R"   and "x \<in> carrier A"   and "f \<in> rHom A R" shows "f (-\<^sub>a\<^bsub>A\<^esub> x) = -\<^sub>a\<^bsub>R\<^esub> f x"
represents_imp_degree: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" shows "Polynomial.degree p \<noteq> 0"
simps_1: fixes T :: "'s Trace set" shows "jkbpT\<^bsub>0\<^esub> T = {tInit s |s. s \<in> set envInit}"
min_stepsize_def: fixes n :: "nat" shows "min_stepsize n = Min (stepsize ` {0..n})"
id: shows "neville_aitken_interpolation_poly xs_ys = neville_aitken (\<lambda>i. fst (xs_ys ! i)) (\<lambda>i. snd (xs_ys ! i)) (length xs_ys - 1)"
splitAt_distinct_snd_aux: fixes vs :: "'c list"   and ram :: "'c" assumes "distinct vs" shows "distinct (snd (splitAt ram vs))"
Order_fs_def: fixes A :: "'c set"   and B :: "'d set" shows "Order_fs A B = \<lparr>carrier = {Z. \<exists>A1 f. A1 \<in> Pow A \<and> f \<in> A1 \<rightarrow> B \<and> f \<in> extensional A1 \<and> Z = (A1, f)}, rel = {Y \<in> {Z. \<exists>A1 f. A1 \<in> Pow A \<and> f \<in> A1 \<rightarrow> B \<and> f \<in> extensional A1 \<and> Z = (A1, f)} \<times> {Z. \<exists>A1 f. A1 \<in> Pow A \<and> f \<in> A1 \<rightarrow> B \<and> f \<in> extensional A1 \<and> Z = (A1, f)}. fst (fst Y) \<subseteq> fst (snd Y) \<and> (\<forall>a\<in>fst (fst Y). snd (fst Y) a = snd (snd Y) a)}\<rparr>"
filterlim_INF: fixes f :: "'b \<Rightarrow> 'c"   and G :: "'d \<Rightarrow> 'c filter"   and B :: "'d set"   and F :: "'b filter" shows "filterlim f (Inf (G ` B)) F = (\<forall>b\<in>B. filterlim f (G b) F)"
remove_vertex_E': fixes v :: "'a"   and w :: "'a"   and x :: "'a" assumes "v \<rightarrow>\<^bsub>H.remove_vertex x\<^esub> w" shows "v \<rightarrow>\<^bsub>remove_vertex x\<^esub> w"
element_exists: fixes A :: "'a set" assumes "A \<in> Partition" shows "\<exists>a\<in>S. a \<in> A"
T_on_rand'_append: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf"   and xs :: "'a list"   and ys :: "'a list" shows "Move_to_Front.T_on_rand' A s (xs @ ys) = Move_to_Front.T_on_rand' A s xs + Move_to_Front.T_on_rand' A (Partial_Cost_Model.config'_rand A s xs) ys"
Smith_1xn_two_matrices_all: fixes n :: "nat"   and Smith_1xn :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat" assumes "\<forall>A\<in>carrier_mat 1 n. is_SNF A (Smith_1xn A)" shows "\<exists>Smith_1xn'. \<forall>A\<in>carrier_mat 1 n. is_SNF A (1\<^sub>m 1, Smith_1xn' A)"
pair_of_term_injective: fixes u :: "'t"   and v :: "'t" assumes "pair_of_term u = pair_of_term v" shows "u = v"
cnt_X_pres: fixes xa :: "'a" assumes "count (mset (y # ys)) xa < count (mset (x # xs)) xa"   and "xa \<noteq> x" shows "count (mset ys) xa < count (mset xs) xa"
disjuncts: fixes p :: "fm"   and bbs :: "bool list"   and bs :: "int list" shows "(\<exists>q\<in>set (disjuncts p). Ifm bbs bs q) = Ifm bbs bs p"
subgraph_node: fixes H :: "('c, 'd) graph"   and G :: "('c, 'd) graph"   and v :: "'c" assumes "subgraph H G" shows "(v \<in> nodes G) = (v \<in> nodes H)"
dom_m_fmdrop_All: fixes C :: "'c"   and N :: "('c, 'd) fmap" shows "dom_m (fmdrop C N) = removeAll_mset C (dom_m N)"
False: shows "\<not> (r = 0 \<or> s = t)"
iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "iterates f x = LCons x (iterates f (f x))"
simps_5: fixes b :: "bexp"   and c :: "com" shows "lvars (WHILE b DO c) = lvars c"
c1: shows " App (Lam [a].t2) s2 \<longrightarrow>\<^sub>\<beta> t2[a::=s2]"
piecewise_C1_imp_differentiable: fixes f :: "real \<Rightarrow> 'a"   and i :: "real set" assumes "f piecewise_C1_differentiable_on i" shows "f piecewise_differentiable_on i"
abs_eq: shows "rbt_entries_generator = Generator (rbt_has_next, rbt_entries_next)"
WT_sim: fixes \<eta> :: "nat" shows "\<I>_real \<eta>, \<I>_ideal \<eta> \<turnstile>\<^sub>C sim \<eta> \<surd>"
less_TC_minimal: fixes P :: "V \<Rightarrow> bool"   and a :: "V" assumes "P a" obtains x :: "V" where "P x" and "x \<sqsubseteq> a" and "\<And>y. y \<sqsubset> x \<Longrightarrow> \<not> P y"
degree_poly_of_vec_less: fixes v :: "'b vec" assumes "0 < dim_vec v" shows "degree (poly_of_vec v) < dim_vec v"
\<epsilon>o_def: fixes x :: "'d" shows "Adj.\<epsilon>o x = \<psi> x (G x)"
ODnum_segmentTr: fixes D :: "'c Order"   and x :: "'c Order set"   and y :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D"   and "y \<in> ODnums"   and "y \<sqsubset> x" shows "\<exists>c. c \<in> carrier D \<and> (\<forall>Y\<in>y. ord_equiv Y (Iod D (segment D c)))"
rev_map: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "'g list" shows "rev (map f xs) = map f (rev xs)"
sfis_rv: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> sfis f M" shows "f \<in> rv M"
spoly_red_zero_imp_crit_pair_cbelow_on: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'e) set"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'e"   and q :: "'a \<Rightarrow>\<^sub>0 'e" assumes "dickson_grading d"   and "F \<subseteq> punit.dgrad_p_set d m"   and "p \<in> punit.dgrad_p_set d m"   and "q \<in> punit.dgrad_p_set d m"   and "p \<noteq> 0"   and "q \<noteq> 0"   and "(punit.red F)\<^sup>*\<^sup>* (punit.spoly p q) 0" shows "punit.crit_pair_cbelow_on d m F p q"
is_rat_real_alg_def: fixes x :: "real_alg" shows "is_rat_real_alg x = (case to_rat_real_alg_main x of None \<Rightarrow> False | Some q \<Rightarrow> True)"
inc_dot_origin: fixes d :: "nat"   and x :: "item" shows "item_origin (inc_dot d x) = item_origin x"
normalize_simps_7: fixes p :: "'e" shows "Language.normalize (Call p) = Call p"
eq_onp_top_eq_eq: shows "eq_onp top = (=)"