hcju/isabelleps huggingface dataset:
sum_upto_sqrt_bound: fixes x :: "real" assumes "0 \<le> x" shows "norm ((sum_upto (\<lambda>_. 1) (sqrt x))\<^sup>2 - x) \<le> 2 * norm (sqrt x)"
poly_eval_map_indets: fixes a :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and p :: "('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (map_indets g p) = poly_eval (a \<circ> g) p"
semiring_simprules_14: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "semiring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
HomCHom: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and g :: "'e" assumes "LSCategory C"   and "f \<in> mor\<^bsub>C\<^esub>"   and "g \<in> mor\<^bsub>C\<^esub>" shows "(HomC\<^bsub>C\<^esub>[g,dom\<^bsub>C\<^esub> f]) ;;\<^bsub>SET\<^esub> (Hom\<^bsub>C\<^esub>[dom\<^bsub>C\<^esub> g,f]) = (Hom\<^bsub>C\<^esub>[cod\<^bsub>C\<^esub> g,f]) ;;\<^bsub>SET\<^esub> (HomC\<^bsub>C\<^esub>[g,cod\<^bsub>C\<^esub> f])"
onorm_sum_le: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f"   and z :: "real" assumes "finite S"   and "\<And>s. s \<in> S \<Longrightarrow> bounded_linear (f s)"   and "(\<Sum>s\<in>S. onorm (f s)) \<le> z" shows "onorm (\<lambda>x. \<Sum>s\<in>S. f s x) \<le> z"
wPlusU: fixes \<theta>\<^sub>1 :: "trm"   and \<theta>\<^sub>2 :: "trm"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([Plus \<theta>\<^sub>1 \<theta>\<^sub>2]<>\<nu>) = (let (l1, u1) = [\<theta>\<^sub>1]<>\<nu>; (l2, u2) = [\<theta>\<^sub>2]<>\<nu> in (pl l1 l2, pu u1 u2))"
someI: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e" assumes "P x" shows "P (Eps P)"
bernpoly_1: fixes n :: "nat" assumes "n \<noteq> 1" shows "bernpoly n (1::'a) = of_real (bernoulli n)"
div_nat_eqI: fixes n :: "nat"   and q :: "nat"   and m :: "nat" assumes "n * q \<le> m"   and "m < n * Suc q" shows "m div n = q"
x: shows "\<forall>i j. i < m \<and> j < n \<longrightarrow> x dvd A $$ (i, j)"
right_diff_distrib_numeral: fixes v :: "num"   and b :: "'a"   and c :: "'a" shows "numeral v \<cdot> (b - c) = numeral v \<cdot> b - numeral v \<cdot> c"
measurable_dist_dens: fixes f :: "'a \<Rightarrow> val"   and M :: "'a measure"   and dst :: "pdf_dist"   and g :: "'a \<Rightarrow> val" assumes "f \<in> M \<rightarrow>\<^sub>M stock_measure (dist_param_type dst)"   and "g \<in> M \<rightarrow>\<^sub>M stock_measure (dist_result_type dst)" shows "(\<lambda>x. dist_dens dst (f x) (g x)) \<in> borel_measurable M"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and nat :: "nat" shows "h (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = (case nat of 0 \<Rightarrow> h f1 | Suc x \<Rightarrow> h (f2 x))"
eq_refl: fixes less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'h"   and y :: "'h" assumes "class.preorder less_eq less"   and "x = y" shows "less_eq x y"
A: fixes X :: "('a, 'b) action"   and Y :: "('a, 'c) action" assumes "X \<preceq>S Y" shows "P X \<preceq>S Q Y"
simps_2: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
gt_0_lt_mult_gt_1_hmset: fixes m :: "hmultiset"   and n :: "hmultiset" assumes "0 < m"   and "1 < n" shows "m < m * n"
gh: fixes y :: "(real, 'n) vec" assumes "y \<in> T" shows "h y \<in> S \<and> g (h y) = y"
simps_1: fixes f :: "'c \<Rightarrow> 'b"   and P :: "'c" shows "ACom.map_acom f (SKIP {P}) = SKIP {f P}"
Bignorabimus_union: fixes M :: "variable set"   and \<alpha> :: "game"   and N :: "variable set" assumes "M \<in> Bignorabimus \<alpha>"   and "N \<in> Bignorabimus \<alpha>" shows "M \<union> N \<in> Bignorabimus \<alpha>"
connected_iff2: fixes x :: "'a" assumes "is_inj x"   and "is_p_fun x" shows "connected x = (x ; 1 ; x\<^sup>T \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star>)"
term_to_nterm_nterm_to_term0: fixes \<Gamma> :: "name list"   and t :: "Term_Class.term"   and s :: "name" assumes "wellformed' (length \<Gamma>) t"   and "fdisjnt (fset_of_list \<Gamma>) (frees t)"   and "distinct \<Gamma>"   and "fBall (frees t |\<union>| fset_of_list \<Gamma>) (\<lambda>x. x \<le> s)" shows "nterm_to_term \<Gamma> (fst (run_state (term_to_nterm \<Gamma> t) s)) = t"
alphaAbs_all_qAFresh_def: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" shows "alphaAbs_all_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<forall>y. qAFresh xs y X \<and> qAFresh xs y X' \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
is_natural_2: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.map (V.cod f) \<bullet> V.map f = V.map f"
induct: fixes P :: "('b \<times> color) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> color) tree \<Rightarrow> bool"   and a0 :: "('b \<times> color) tree"   and a1 :: "'b"   and a2 :: "('b \<times> color) tree" assumes "\<And>l x r. \<lbrakk>\<And>x21 x22 x23 xa y. \<lbrakk>\<not> bheight l \<le> bheight r; l = \<langle>x21, x22, x23\<rangle>; (xa, y) = x22; y = Red\<rbrakk> \<Longrightarrow> P x23 x r; \<And>x21 x22 x23 xa y. \<lbrakk>\<not> bheight l \<le> bheight r; l = \<langle>x21, x22, x23\<rangle>; (xa, y) = x22; y = Black\<rbrakk> \<Longrightarrow> P x23 x r\<rbrakk> \<Longrightarrow> P l x r" shows "P a0 a1 a2"
simps_16: fixes x41 :: "sterm"   and x42 :: "sterm"   and x3 :: "(Term_Class.term \<times> sterm) list" shows "x41 $\<^sub>s x42 \<noteq> Sabs x3"
simps_3: fixes u :: "'a local.term"   and v :: "'a local.term"   and w :: "'a local.term"   and x :: "'a local.term" shows "(u \<^bold>\<star> v) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (w \<^bold>\<star> x) = u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w \<^bold>\<star> v \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> x"
contfs: fixes y :: "'c" assumes "y \<in> U" shows "continuous_on ({0..1} \<times> V y) (fs y)"
mset_filter_mono: fixes A :: "'b multiset"   and B :: "'b multiset"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A \<subseteq># B"   and "\<And>x. \<lbrakk>x \<in># A; P x\<rbrakk> \<Longrightarrow> Q x" shows "filter_mset P A \<subseteq># filter_mset Q B"
image_def: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "f ` A = _Collect y (\<exists>x\<in>A. y = f x)"
convex_hull_mono: fixes Vs :: "'b vec set"   and Ws :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> Ws" shows "gram_schmidt.convex_hull n Vs \<subseteq> gram_schmidt.convex_hull n Ws"
silent_moves_Cons: fixes S :: "'node SDG_node set"   and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and ms :: "'node list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a :: "'edge"   and ms' :: "'node list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and as :: "'edge list"   and ms'' :: "'node list"   and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"   and "S,f \<turnstile> (ms',s') =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')" shows "S,f \<turnstile> (ms,s) =a # as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"
map_member_alt_def: fixes x :: "'c"   and y :: "'d"   and A :: "'c \<Rightarrow> 'd option" shows "((x, y) \<in>\<^sub>m A) = (x \<in> dom A \<and> A(x)\<^sub>m = y)"
power_series_and_derivative: fixes a :: "nat \<Rightarrow> complex"   and r :: "real"   and w :: "complex" assumes "summable (\<lambda>n. a n * complex_of_real (r ^ n))" obtains g :: "complex \<Rightarrow> complex" and g' :: "complex \<Rightarrow> complex" where "\<forall>z\<in>ball w r. (\<lambda>n. a n * (z - w) ^ n) sums g z \<and> (\<lambda>n. of_nat n * a n * (z - w) ^ (n - 1)) sums g' z \<and> (g has_field_derivative g' z) (at z)"
match_4: fixes g :: "'a"   and f :: "'a"   and h :: "'a" assumes "seq g f"   and "seq h g" shows "seq h (g \<cdot> f)"
cod_simps_2: shows "dom_\<nu>.cod.leg0 = doms.\<nu>.leg0"
WT_gpv_coinduct_bind: fixes X :: "('d, 'e, 'f) gpv \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and \<I> :: "('e, 'f) \<I>" assumes "X gpv"   and "\<And>gpv out c. \<lbrakk>X gpv; IO out c \<in> set_spmf (the_gpv gpv)\<rbrakk> \<Longrightarrow> out \<in> outs_\<I> \<I> \<and> (\<forall>input\<in>responses_\<I> \<I> out. X (c input) \<or> \<I> \<turnstile>g c input \<surd> \<or> (\<exists>gpv' f. c input = gpv' \<bind> f \<and> \<I> \<turnstile>g gpv' \<surd> \<and> (\<forall>x\<in>results_gpv \<I> gpv'. X (f x))))" shows "\<I> \<turnstile>g gpv \<surd>"
list_all_cons_def: fixes s :: "'s" shows "local.list_all_cons s = list_all P (unstream g s)"
Mp_prod_mset: fixes m :: "int"   and b :: "int poly multiset" shows "poly_mod.Mp m (\<Prod>\<^sub># (image_mset (poly_mod.Mp m) b)) = poly_mod.Mp m (\<Prod>\<^sub># b)"
simps_3: fixes x3 :: "nat"   and y3 :: "nat" shows "(Regions.intv.Greater x3 = Regions.intv.Greater y3) = (x3 = y3)"
assms_2: shows "\<turnstile>\<^sub>! (q \<^bold>\<longleftrightarrow>\<^sub>! q')"
in_carrier_atlasI: fixes c :: "('a, 'b) chart"   and x :: "'a" assumes "c \<in> atlas"   and "x \<in> domain c" shows "x \<in> carrier"
distinct_3: fixes x31 :: "'e"   and x32 :: "'f" shows "Done \<noteq> Yield x31 x32"
collapse: fixes tm :: "'a tm" shows "??.Time_Monad.tm.TM (tm.val tm) (un_TM2 tm) = tm"
ideduct_vars: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "M \<turnstile> t" shows "fv t \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
split_time_def: fixes s :: "nat \<Rightarrow> 'a list"   and n :: "nat"   and t :: "nat" shows "split_time s n t \<equiv> if t mod n = 0 then s (t div n) else []"
prod1: shows "mat_of_rows n fs_init = of_int_hom.mat_hom U1 * mat_of_rows n gs"
ipoly_reflect_poly: fixes x :: "'a"   and p :: "int poly" assumes "x \<noteq> (0::'a)" shows "ipoly (reflect_poly p) x = x ^ degree p * ipoly p (inverse x)"
rhs_def: shows "rhs \<equiv> List.maps (\<lambda>x. [Suc 0..<sum_list_cons (replicate_prod x) x]) [2..<n]"
winding_number_circlepath_centre: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "winding_number (circlepath z r) z = 1"
ltl_models_equiv_prop_entailment: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl" shows "w \<Turnstile> \<phi> = {\<chi>. w \<Turnstile> \<chi>} \<Turnstile>\<^sub>P \<phi>"
CFG_SSA_Transformed_notriv_linorder_def: fixes \<alpha>e :: "'a \<Rightarrow> ('b \<times> 'c \<times> 'b) set"   and \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b"   and oldDefs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and oldUses :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'e set"   and phis :: "'a \<Rightarrow> 'b \<times> 'e \<Rightarrow> 'e list option"   and var :: "'a \<Rightarrow> 'e \<Rightarrow> 'd"   and chooseNext_all :: "('b \<Rightarrow> 'e set) \<Rightarrow> ('b \<times> 'e \<Rightarrow> 'e list option) \<Rightarrow> 'a \<Rightarrow> 'b \<times> 'e" shows "CFG_SSA_Transformed_notriv_linorder \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var chooseNext_all \<equiv> CFG_SSA_Transformed_notriv \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs uses phis var chooseNext_all"
progress_le: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> Map.empty \<phi> j \<le> j"
contraposition_2: fixes q :: "('a, 'b) fm"   and p :: "('a, 'b) fm" shows "\<turnstile> (\<^bold>\<not> q \<^bold>\<longrightarrow> \<^bold>\<not> p) \<^bold>\<longrightarrow> p \<^bold>\<longrightarrow> q"
thin_triangles1: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and H :: "'a set"   and z :: "'a"   and t :: "real" assumes "geodesic_segment_between G x y"   and "geodesic_segment_between H x z"   and "t \<in> {0..Gromov_product_at x y z}" shows "dist (geodesic_segment_param G x t) (geodesic_segment_param H x t) \<le> 4 * deltaG TYPE('a)"
bij_betw_permutes_compose_left: fixes \<pi> :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "\<pi> permutes A" shows "bij_betw ((\<circ>) \<pi>) {\<sigma>. \<sigma> permutes A} {\<sigma>. \<sigma> permutes A}"
iiface_Some: fixes iface :: "iface" assumes "iface_conjunct iif1 iif2 = Some iface" shows "match_iface iface (p_iiface p) = (match_iface iif1 (p_iiface p) \<and> match_iface iif2 (p_iiface p))"
Hermite_Lindemann': fixes X :: "complex set"   and \<beta> :: "complex \<Rightarrow> complex" assumes "finite X"   and "\<And>x. x \<in> X \<Longrightarrow> algebraic x"   and "\<And>x. x \<in> X \<Longrightarrow> algebraic (\<beta> x)"   and "(\<Sum>x\<in>X. \<beta> x * exp x) = 0" shows "\<forall>x\<in>X. \<beta> x = 0"
hyps_1: shows "a \<in> sigma_sets (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j)) {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "U.satC \<xi> \<equiv> list_ex (U.satL \<xi>)"
ennreal_add_diff_cancel_right: fixes y :: "ennreal"   and x :: "ennreal" assumes "y \<noteq> \<top>" shows "x + y - y = x"
x_pow_n: fixes n :: "nat" shows "monom (1::'a) 1 ^ n = monom (1::'a) n"
seq_compact_eq_acc_point: fixes s :: "'b set" shows "seq_compact s = (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x\<in>s. \<forall>U. x \<in> U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
assms_7: shows "\<forall>x. insertion (nth_default 0 (L[var := x])) c = Cv"
notEqual: shows "\<not> (A = D \<and> distance B C = distance E F)"
Lemma_2: fixes xs :: "'b list"   and ys :: "'b list" assumes "length xs = length ys"   and "\<And>k. k < length xs \<Longrightarrow> xs ! k = ys ! k" shows "xs = ys"
dom_op_def: fixes x :: "'a" shows "d x = ad (ad x)"
is_bcv_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "'a"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and n :: "nat"   and A :: "'a set"   and bcv :: "'a list \<Rightarrow> 'a list" shows "is_bcv r T step n A bcv = (\<forall>\<tau>s\<^sub>0\<in>list n A. (\<forall>p<n. bcv \<tau>s\<^sub>0 ! p \<noteq> T) = (\<exists>\<tau>s\<in>list n A. \<tau>s\<^sub>0 [\<sqsubseteq>\<^bsub>r\<^esub>] \<tau>s \<and> wt_step r T step \<tau>s))"
\<T>o_e_induces_transformation: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and e :: "'b" assumes "yoneda_lemma C S \<phi> F a"   and "e \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)" shows "transformation_by_components (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) F (yoneda_lemma.\<T>o C S \<phi> F a e)"
preallocated_upd_obj: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and fs' :: "char list \<times> char list \<Rightarrow> val option" assumes "preallocated h"   and "h a = \<lfloor>(C, fs)\<rfloor>" shows "preallocated (h(a \<mapsto> (C, fs')))"
cfun_map_def: shows "cfun_map = (\<Lambda> a b f x. b\<cdot>(f\<cdot>(a\<cdot>x)))"
inheritedJoinI: fixes subs :: "'a \<Rightarrow> 'a set"   and P :: "(nat \<times> 'a) set \<Rightarrow> bool"   and Q :: "(nat \<times> 'a) set \<Rightarrow> bool"   and R :: "(nat \<times> 'a) set \<Rightarrow> bool" assumes "inherited subs P"   and "inherited subs Q"   and "R = (\<lambda>x. P x \<and> Q x)" shows "inherited subs R"
final_conc: fixes L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" shows "fin (RegExp2NAe.conc L R) p = (\<exists>s. p = False # s \<and> fin R s)"
a3: shows "p \<cdot> x = x \<cdot> p"
f_eq_2: fixes x :: "'a" shows "poly q (f x) = x"
cor_eq_0: fixes x :: "real"   and y :: "real" shows "(cor x + \<i> * cor y = 0) = (x = 0 \<and> y = 0)"
push_refine: fixes s :: "'v list \<times> nat list \<times> ('v \<Rightarrow> node_state option) \<times> (nat \<times> 'v set) list"   and p :: "'v set list"   and D :: "'v set"   and pE :: "('v \<times> 'v) set"   and v :: "'v"   and v' :: "'v" assumes "(s, p, D, pE) \<in> GS_rel"   and "(v, v') \<in> Id"   and "v \<notin> \<Union> (set p)"   and "v \<notin> D" shows "(push_impl v s, push v' (p, D, pE)) \<in> GS_rel"
iface_conjunct_ifaceAny: fixes i :: "iface" shows "iface_conjunct ifaceAny i = Some i"
bot_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times bot x = bot"
real_mon_conv_times: fixes x :: "nat \<Rightarrow> real"   and y :: "real"   and z :: "real" assumes "x\<up>y"   and "0 \<le> z" shows "(\<lambda>m. z * x m)\<up>z * y"
bounded_lattice_bot_ow_axioms: shows "bounded_lattice_bot_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 bot\<^sub>2"
enableds_def: fixes p :: "'state" shows "R.enableds p \<equiv> {a. ren a p}"
length_geq_zero: shows "\<forall>ts v. \<parallel>ext v\<parallel> = 0 \<or> 0 < \<parallel>ext v\<parallel>"
lm087: fixes P :: "'d \<Rightarrow> bool"   and l :: "'d list" assumes "P -` {True} \<inter> set l \<noteq> {}" shows "takeAll P l \<noteq> []"
filtermap_times_pos_at_right: fixes c :: "'b"   and p :: "'b" assumes "(0::'b) < c" shows "filtermap ((*) c) (at_right p) = at_right (c * p)"
\<phi>\<^sub>0_in_hom_1: fixes f :: "'a" assumes "\<guillemotleft>f : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>"   and "ide f" shows "\<guillemotleft>(e\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[d\<^sub>1, f \<star> e\<^sub>0, d\<^sub>0]) \<cdot> \<a>[e\<^sub>1, d\<^sub>1, (f \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, e\<^sub>0, d\<^sub>0]) \<cdot> (local.inv \<epsilon>\<^sub>1 \<star> f \<star> local.inv \<epsilon>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src d\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f] : src d\<^sub>0 \<rightarrow> src d\<^sub>1\<guillemotright>"
inject_3: fixes x51 :: "'b rexp"   and x52 :: "'b rexp"   and y51 :: "'b rexp"   and y52 :: "'b rexp" shows "(Times x51 x52 = Times y51 y52) = (x51 = y51 \<and> x52 = y52)"
ring_mat_simps_1: fixes ty :: "'a itself"   and n :: "nat"   and b :: "'b" shows "(\<otimes>\<^bsub>ring_mat ty n b\<^esub>) = (*)"
inverse_functors_axioms: shows "inverse_functors J.comp J' \<phi> \<psi>"
substs_preserve_finiteness: fixes C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "finite C" shows "finite (subst_cl C \<sigma>)"
stationary_distribution_implies_pmf_of_st_vec: fixes N :: "'i pmf" assumes "stationary_distribution N" shows "\<exists>x. N = pmf_of_st_vec x"
exact_seq_imp_exact_hom: fixes G3 :: "'b monoid"   and G2 :: "'b monoid"   and G1 :: "'b monoid"   and l :: "'b monoid list"   and q :: "('b \<Rightarrow> 'b) list"   and g1 :: "'b \<Rightarrow> 'b"   and g2 :: "'b \<Rightarrow> 'b" assumes "exact_seq (G1 # l, q)  \<longlongrightarrow>\<^bsub>g1\<^esub> G2  \<longlongrightarrow>\<^bsub>g2\<^esub> G3" shows "g1 ` carrier G1 = kernel G2 G3 g2"
wary_grounding_\<rho>: shows "wary_subst grounding_\<rho>"
valid_Cn_disj: fixes p :: "'a"   and q :: "'a" shows "\<tturnstile> (p .\<or>. q) = (Cn {q} \<subseteq> Cn {.\<not> p})"
ipoly_of_rat: fixes p :: "int poly"   and y :: "rat" shows "ipoly p (of_rat y) = of_rat (ipoly p y)"
IH: fixes \<sigma> :: "nat \<Rightarrow> val"   and x :: "nat"   and e :: "cexpr" shows "cexpr_sem \<sigma> (cexpr_subst x e e') = cexpr_sem (insert_var x \<sigma> (cexpr_sem \<sigma> e)) e'"
assms: fixes \<psi> :: "'a" shows "\<psi> \<bullet>\<^sub>C (A *\<^sub>V \<psi>) \<in> \<real>"
Suc_1: fixes r :: "'a list"   and q :: "'a list" assumes "n = 1 + length r - length d" shows "Poly (divide_poly_main_list lc q (rev r) (rev d) n) = divide_poly_main lc (monom (1::'a) n * Poly q) (Poly r) (Poly d) (length r - 1) n"
addsI: fixes a :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and k :: "'b" assumes "a = plus b k" shows "plus.adds plus b a"
preserves_reflects_arr: fixes f :: "'c" shows "S.arr (\<T> e f) = Cop.arr f"
append_eq_appendI: fixes xs :: "'e list"   and xs1 :: "'e list"   and zs :: "'e list"   and ys :: "'e list"   and us :: "'e list" assumes "xs @ xs1 = zs"   and "ys = xs1 @ us" shows "xs @ ys = zs @ us"
ALGxy_def: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list"   and init :: "'c list"   and x :: "'c"   and y :: "'c" shows "ALGxy A qs init x y = (\<Sum>i\<in>{..<length qs} \<inter> {i. qs ! i \<in> {y, x}}. ALG' A qs init i y + ALG' A qs init i x)"
assms: shows "(s, t) \<in> timpl_closure' TI"
arg_emb: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "s \<in> set (args t)" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
assms_1: shows "x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B"
sym_order_trans: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiattractive A less_eq"   and "sympartp less_eq x y"   and "less_eq y z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "less_eq x z"
uint_div_def: fixes x :: "uint"   and y :: "uint" shows "uint_div x y = fst (uint_divmod x y)"
constr_rep_2: fixes x2 :: "freeRes"   and x1 :: "pi" shows "Rep_residual (x2 \<prec> x1) = FreeR_Rep x2 x1"
thetaSeqZOT_sym: shows "sym thetaSeqZOT"
sum_f_int: shows "ennreal (infsetsum f A) = integral\<^sup>N (count_space A) fe"
dist_le_zero_iff: fixes x :: "'a"   and y :: "'a" shows "(dist x y \<le> 0) = (x = y)"
ideD_3: fixes a :: "(bool, unit) arr" assumes "ide a" shows "cod a = a"
implies_galois_swap: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<le> - y \<squnion> z) = (y \<le> - x \<squnion> z)"
assms_1: shows "\<forall>asset\<in>support_set p. p asset (Suc n) \<in> borel_measurable (F n)"
all_subset_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a set \<Rightarrow> bool" shows "(\<forall>B\<subseteq>f ` A. P B) = (\<forall>B\<subseteq>A. P (f ` B))"
scheme_list_substitutions_only_on_free_variables: fixes A :: "type_scheme list"   and S :: "nat \<Rightarrow> typ"   and S' :: "nat \<Rightarrow> typ" assumes "\<And>x. x \<in> free_tv A \<Longrightarrow> S x = S' x" shows "$ S A = $ S' A"
swapidseq_ext_endswap: fixes S :: "'a set"   and n :: "nat"   and p :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "swapidseq_ext S n p"   and "a \<noteq> b" shows "swapidseq_ext (insert a (insert b S)) (Suc n) (p \<circ> Fun.swap a b id)"
split_lemma: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool" shows "{((e, s), n, s'). P e s n s'} = {(v, x, y). (case v of (x, xa) \<Rightarrow> P x xa) x y}"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree" assumes "\<And>l a lv. P \<langle>l, (a, lv), \<langle>\<rangle>\<rangle>"   and "\<And>l a lv v va vb. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>l, (a, lv), \<langle>v, va, vb\<rangle>\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"