hcju/isabelleps huggingface dataset:
exec_gpv_oracle_bisim_bad_plossless: fixes bad2 :: "'a \<Rightarrow> bool"   and s2 :: "'a"   and X_bad :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s1 :: "'b"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and bad1 :: "'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and oracle1 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and oracle2 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf"   and gpv :: "('e, 'c, 'd) gpv" assumes "if bad2 s2 then X_bad s1 s2 else X s1 s2"   and "bad1 s1 = bad2 s2"   and "\<And>s1 s2 x. \<lbrakk>X s1 s2; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>s2. bad2 s2 \<Longrightarrow> callee_invariant_on oracle1 (\<lambda>s1. bad1 s1 \<and> X_bad s1 s2) \<I>"   and "\<And>s1. bad1 s1 \<Longrightarrow> callee_invariant_on oracle2 (\<lambda>s2. bad2 s2 \<and> X_bad s1 s2) \<I>"   and "\<And>s1 x. \<lbrakk>bad1 s1; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> lossless_spmf (oracle1 s1 x)"   and "\<And>s2 x. \<lbrakk>bad2 s2; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> lossless_spmf (oracle2 s2 x)"   and "plossless_gpv \<I> gpv"   and "\<And>s1. \<I> \<turnstile>c oracle1 s1 \<surd>"   and "\<And>s2. \<I> \<turnstile>c oracle2 s2 \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "rel_spmf (\<lambda>(a, s1') (b, s2'). bad1 s1' = bad2 s2' \<and> (if bad2 s2' then X_bad s1' s2' else a = b \<and> X s1' s2')) (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
foldseq_almostzero: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and j :: "nat"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<forall>x. f (0::'b) x = x"   and "\<forall>x. f x (0::'b) = x"   and "\<forall>i. i \<noteq> j \<longrightarrow> s i = (0::'b)" shows "foldseq f s n = (if j \<le> n then s j else (0::'b))"
suppd: shows "supp \<delta> = supp (Neg (PfP (Var i))) - {atom i}"
lex_prod_def: fixes ra :: "('b \<times> 'b) set"   and rb :: "('c \<times> 'c) set" shows "ra <*lex*> rb = {((a, b), a', b'). (a, a') \<in> ra \<or> a = a' \<and> (b, b') \<in> rb}"
convergent_prod_def: fixes f :: "nat \<Rightarrow> 'b" shows "convergent_prod f \<equiv> \<exists>M p. raw_has_prod f M p"
mult_rel_well_defined: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b"   and z :: "'a set"   and z' :: "'a set" assumes "x \<in> Sigma I \<FF>"   and "y \<in> Sigma I \<FF>"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y"   and "z' \<in> I"   and "z' \<subseteq> fst x"   and "z' \<subseteq> fst y" shows "\<lfloor>z, \<cdot>\<^bsub>z\<^esub> (\<rho> (fst x) z (snd x)) (\<rho> (fst y) z (snd y))\<rfloor> = \<lfloor>z', \<cdot>\<^bsub>z'\<^esub> (\<rho> (fst x) z' (snd x)) (\<rho> (fst y) z' (snd y))\<rfloor>"
is_extensional: fixes f :: "J.Arr" assumes "\<not> J.arr f" shows "A.map f = C.null"
alw_holds: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a"   and t :: "'a stream" shows "alw (holds P) (h ## t) = (P h \<and> alw (holds P) t)"
cauchy_rel: fixes u :: "nat \<Rightarrow> 'a Bonk_Schramm_extension_unfolded"   and n :: "nat" shows "(map_aux (u n), map_aux (would_be_Cauchy u)) \<in> wo_aux - Id"
assms_2: shows "q \<prec>\<^sub>p monomial c v"
WatchInvariantsAfterApplyLearn: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantUniqC (getC state)" shows "let state' = applyLearn state in InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchesDiffer (getF state') (getWatch1 state') (getWatch2 state') \<and> InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state') \<and> InvariantWatchListsContainOnlyClausesFromF (getWatchList state') (getF state') \<and> InvariantWatchListsUniq (getWatchList state') \<and> InvariantWatchListsCharacterization (getWatchList state') (getWatch1 state') (getWatch2 state')"
mul_eq_singleton: fixes M :: "'a multiset"   and \<alpha> :: "'a"   and r :: "('a \<times> 'a) set" assumes "(M, {#\<alpha>#}) \<in> mul_eq r" shows "M = {#\<alpha>#} \<or> set_mset M \<subseteq> r \<down>m {#\<alpha>#}"
ide_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>f : f \<Rightarrow> f\<guillemotright>"
subset_\<I>Suc: fixes k :: "nat" shows "\<I> k \<subseteq> \<I> (Suc k)"
sngl_adjust: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and lv :: "nat"   and r :: "('b \<times> nat) tree" assumes "pre_adjust \<langle>l, (a, lv), r\<rangle>"   and "sngl \<langle>l, (a, lv), r\<rangle>"   and "lv = lvl (adjust \<langle>l, (a, lv), r\<rangle>)" shows "sngl (adjust \<langle>l, (a, lv), r\<rangle>)"
wf_disc_sel_common_primitive_4: shows "wf_disc_sel (is_Dst, dst_sel) Dst"
suffix_zero: fixes s :: "nat \<Rightarrow> 'b" shows "s |\<^sub>s 0 = s"
induct: fixes P :: "dynamic list \<Rightarrow> bool"   and a0 :: "dynamic list" assumes "\<And>x y. P [DNum x, DNum y]"   and "\<And>x y. P [DBool x, DBool y]"   and "\<And>uv_. P [DNil, uv_]"   and "\<And>v. P [DBool v, DNil]"   and "\<And>v va. P [DBool v, DNum va]"   and "\<And>uu_. P [uu_, DNil]"   and "\<And>v. P [DNil, DBool v]"   and "\<And>va v. P [DNum va, DBool v]"   and "P []"   and "P [DNil]"   and "\<And>v vc vd. P (DNil # v # vc # vd)"   and "\<And>vb. P [DBool vb]"   and "\<And>vb v va. P (DBool vb # DNil # v # va)"   and "\<And>vb vd v va. P (DBool vb # DNum vd # v # va)"   and "\<And>vb v vd ve. P (DBool vb # v # vd # ve)"   and "\<And>v. P [v]"   and "\<And>v va vb. P (v # DNil # va # vb)"   and "\<And>vd v va. P (DNil # DBool vd # v # va)"   and "\<And>va vd v vb. P (DNum va # DBool vd # v # vb)"   and "\<And>v vd va vb. P (v # DBool vd # va # vb)"   and "\<And>v vb vd ve. P (v # vb # vd # ve)" shows "P a0"
weakCongEE_1: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"
Ifun_star_of: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "star_of f \<star> star_of x = star_of (f x)"
rep_eq: fixes x :: "('c, 'd) sq_matrix" shows "Square_Matrix.det x = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. to_fun x i (p i)))"
simps_1: fixes x :: "ref"   and h :: "ref \<Rightarrow> ref"   and y :: "ref" shows "Path x h y [] = (x = y)"
deleteAround_prevVertex: fixes g :: "graph"   and a :: "nat"   and f :: "face"   and b :: "nat"   and ps :: "(nat \<times> nat) list" assumes "minGraphProps g"   and "a \<in> \<V> g"   and "f \<in> set (facesAt g a)" shows "(f\<^bsup>-1\<^esup> \<bullet> a, b) \<notin> set (deleteAround g a ps)"
Gamma_one_half_real: shows "Gamma (1 / 2) = sqrt pi"
compact_eq_totally_bounded: fixes s :: "'b set" shows "compact s = (complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)))"
DataSpace_EmptySet: shows "[UNIV] \<in> {L |L. DataSpace L}"
gfinprod_distrib: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'a"   and g :: "'f \<Rightarrow> 'a" assumes "finite {x \<in> A. f x \<noteq> \<one>}"   and "finite {x \<in> A. g x \<noteq> \<one>}"   and "f \<in> A \<rightarrow> carrier G"   and "g \<in> A \<rightarrow> carrier G" shows "gfinprod G (\<lambda>i. f i \<otimes> g i) A = gfinprod G f A \<otimes> gfinprod G g A"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and r\<^sub>0 :: "'b"   and r\<^sub>1 :: "'b"   and s :: "'b"   and \<sigma> :: "'b"   and s\<^sub>0 :: "'b"   and s\<^sub>1 :: "'b"   and t :: "'b"   and \<tau> :: "'b"   and t\<^sub>0 :: "'b"   and t\<^sub>1 :: "'b"   and \<mu> :: "'b"   and \<pi> :: "'b" assumes "bicategory_of_spans V H \<a> \<i> src trg"   and "tabulation_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1"   and "tabulation_in_maps V H \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1"   and "tabulation_in_maps V H \<a> \<i> src trg t \<tau> t\<^sub>0 t\<^sub>1"   and "arrow_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 \<mu>"   and "arrow_of_tabulations_in_maps V H \<a> \<i> src trg s \<sigma> s\<^sub>0 s\<^sub>1 t \<tau> t\<^sub>0 t\<^sub>1 \<pi>" shows "vertical_composite_of_arrows_of_tabulations_in_maps V H \<a> \<i> src trg r \<rho> r\<^sub>0 r\<^sub>1 s \<sigma> s\<^sub>0 s\<^sub>1 t \<tau> t\<^sub>0 t\<^sub>1 \<mu> \<pi>"
weakly_bisimilar_weak_simulation_step: fixes P :: "'state"   and Q :: "'state"   and \<alpha> :: "'act"   and P' :: "'state" assumes "P \<approx>\<cdot> Q"   and "bn \<alpha> \<sharp>* Q"   and "P \<Rightarrow>\<langle>\<alpha>\<rangle> P'" obtains Q' :: "'state" where "Q \<Rightarrow>\<langle>\<alpha>\<rangle> Q'" and "P' \<approx>\<cdot> Q'"
frgv_impl_rel_ext_internal_def: fixes Rm :: "('f \<times> 'h) set"   and Rv :: "('e \<times> 'g) set" shows "frgv_impl_rel_ext Rm Rv \<equiv> \<langle>Rm, \<langle>Rv\<rangle>list_set_rel, \<langle>Rv\<rangle>slg_rel, \<langle>Rv\<rangle>list_set_rel\<rangle>gen_g_impl_rel_ext"
expectation_gpv_def: fixes fail :: "ennreal"   and \<I> :: "('h, 'i) \<I>"   and f :: "'g \<Rightarrow> ennreal" shows "expectation_gpv fail \<I> f \<equiv> lfp.fixp_fun (\<lambda>expectation_gpv gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv (c r)) \<partial>measure_spmf (the_gpv gpv) + fail * ennreal (pmf (the_gpv gpv) None))"
nsum_cmp_eq_transpos: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "\<forall>j\<le>Suc n. f j \<in> carrier A"   and "i \<le> n"   and "i \<noteq> n" shows "\<Sigma>\<^sub>e A cmp f (cmp (transpos i n) (cmp (transpos n (Suc n)) (transpos i n))) Suc n = \<Sigma>\<^sub>e A cmp f (transpos i (Suc n)) Suc n"
ub: fixes x :: "float" assumes "0 < x" shows "sqrt (real_of_float x) \<le> real_of_float (ub_sqrt prec x)"
finite_fold_rbt_fold_eq: fixes f :: "'b \<times> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d"   and t :: "('b, 'c) RBT.rbt" assumes "comp_fun_commute f" shows "Finite_Set.fold f A (set (RBT.entries t)) = RBT.fold (curry f) t A"
Map_cod: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Map (cod f) = EVAL (Cod f)"
convolution_delta_delta: fixes b1 :: "'b"   and a1 :: "'a"   and b2 :: "'b"   and a2 :: "'a" shows "convolution (b1 \<delta> a1) (b2 \<delta> a2) = (b1 * b2) \<delta> (a1 + a2)"
IH_2: fixes n :: "nat"   and R :: "'a list set"   and \<phi>' :: "'a MFOTL.formula" assumes "wf_mformula \<sigma> j n R \<psi> \<phi>'" shows "case meval n (\<tau> \<sigma> j) (\<Gamma> \<sigma> j) \<psi> of (xs, \<phi>\<^sub>n) \<Rightarrow> wf_mformula \<sigma> (Suc j) n R \<phi>\<^sub>n \<phi>' \<and> list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>')) [Monitor.progress \<sigma> \<phi>' j..<Monitor.progress \<sigma> \<phi>' (Suc j)] xs"
finite_extension_mem: fixes K :: "'a list set"   and xs :: "'a list list" assumes "subring K (A [X])"   and "set xs \<subseteq> carrier (A [X])" shows "set xs \<subseteq> UP_A.finite_extension K xs"
elims_1: fixes x :: "'fsym trm"   and y :: "bool" assumes "Ik.wt x = y" obtains xa :: "Preliminaries.var" where "x = Var xa" and "y"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "y = (wtFsym f \<and> list_all Ik.wt Tl \<and> arOf f = map Ik.tpOf Tl)"
hmem_Sup_ne: fixes x :: "hf"   and y :: "hf" assumes "x \<^bold>\<in> y" shows "\<Squnion>x \<noteq> y"
conjI_realizer: fixes P :: "'a \<Rightarrow> bool"   and p :: "'a"   and Q :: "'b \<Rightarrow> bool"   and q :: "'b" assumes "P p"   and "Q q" shows "P (fst (p, q)) \<and> Q (snd (p, q))"
intro: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec" assumes "ES_valid ES"   and "isViewOn \<V> E\<^bsub>ES\<^esub>" shows "BSPTaxonomyDifferentCorrections ES \<V>"
neg_fun_distr3: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R' :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and S' :: "'g \<Rightarrow> 'i \<Rightarrow> bool" assumes "left_unique R"   and "right_total R"   and "right_unique S"   and "left_total S" shows "R OO R' ===> S OO S' \<le> (R ===> S) OO (R' ===> S')"
equivalence_star_closed: fixes x :: "'a" assumes "equivalence x" shows "equivalence (x\<^sup>\<star>)"
wU': shows "w \<in> local.span (insert w (set us))"
exhaust_sel: fixes flow_entry_match :: "('a, 'b) flow_entry_match" obtains "flow_entry_match = OFEntry (ofe_prio flow_entry_match) (ofe_fields flow_entry_match) (ofe_action flow_entry_match)"
either_both_distr2: fixes t' :: "'b ttree"   and t :: "'b ttree"   and t'' :: "'b ttree" shows "t' \<otimes>\<otimes> t \<oplus>\<oplus> t'' \<otimes>\<otimes> t = (t' \<oplus>\<oplus> t'') \<otimes>\<otimes> t"
set_of_graph_rules_def: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set" shows "set_of_graph_rules Rs \<equiv> \<forall>R\<in>Rs. graph_rule R"
pure_I_conv: fixes x :: "('a, 'b) monoid_ap" shows "Applicative.ap (pure (\<lambda>x. x)) x = x"
connected_impl_maximally_connected: fixes G :: "('c, 'd) graph"   and H :: "('c, 'd) graph" assumes "valid_graph G"   and "connected_graph H"   and "subgraph H G" shows "maximally_connected H G"
sup_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and o\<^sub>1 :: "'b option"   and o\<^sub>2 :: "'b option" shows "Opt.sup f o\<^sub>1 o\<^sub>2 = (case o\<^sub>1 of None \<Rightarrow> OK o\<^sub>2 | Some x \<Rightarrow> case o\<^sub>2 of None \<Rightarrow> OK o\<^sub>1 | Some y \<Rightarrow> case f x y of Err \<Rightarrow> Err | OK z \<Rightarrow> OK (Some z))"
norm_bound_complex_to_real: fixes A :: "real mat"   and n :: "nat"   and d :: "nat" assumes "A \<in> carrier_mat n n"   and "\<exists>c1 c2. \<forall>k. norm_bound (of_real_hom.mat_hom A ^\<^sub>m k) (c1 + c2 * real k ^ d)" shows "\<exists>c1 c2. \<forall>k a. a \<in> elements_mat (A ^\<^sub>m k) \<longrightarrow> \<bar>a\<bar> \<le> c1 + c2 * real k ^ d"
set_times_def: fixes A :: "'c set"   and B :: "'c set" shows "A * B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a * b}"
src_tab_eq: shows "(\<a>\<^sup>-\<^sup>1[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (TfTgh.tab \<star> TTfgh_TfTgh.chine)) \<cdot> TTfgh_TfTgh.the_\<nu> = TTfgh.tab"
logical_weak_head_closure': fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and T :: "ty"   and s' :: "trm" assumes "\<Gamma> \<turnstile> s is t : T"   and "s' \<leadsto> s" shows "\<Gamma> \<turnstile> s' is t : T"
assms: shows "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<circ>\<^sub>\<circ> s"
det3_nonneg_scaleR_segment2: fixes x :: "real \<times> real"   and y :: "real \<times> real"   and z :: "real \<times> real"   and a :: "real" assumes "0 \<le> det3 x y z"   and "0 < a" shows "0 \<le> det3 x ((1 - a) *\<^sub>R x + a *\<^sub>R y) z"
execn_elim_cases_7: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Basic f,s\<rangle> =n\<Rightarrow> t" obtains f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | s :: "'e" where "sb__ = Normal sc__" and "t = Normal (f__ sc__)"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
lnull_lprefix: fixes xs :: "'f llist"   and ys :: "'f llist" assumes "lnull xs" shows "lprefix xs ys"
sums_of_real_iff: fixes f :: "nat \<Rightarrow> real"   and c :: "real" shows "(\<lambda>n. of_real (f n)) sums of_real c = f sums c"
diamond_fin_word_inf_word: fixes v :: "'action list"   and w :: "'action stream"   and p :: "'state" assumes "Ind (set v) (sset w)"   and "path v p"   and "run w p" shows "run w (target v p)"
reds: shows "P,E(V \<mapsto> T) \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2,(h\<^sub>2, l\<^sub>2)\<rangle>"
inject_2: fixes x21 :: "'a"   and x22 :: "'b"   and x23 :: "'c"   and y21 :: "'a"   and y22 :: "'b"   and y23 :: "'c" shows "(State x21 x22 x23 = State y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
fold_congs_2: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "nat"   and f :: "nat \<Rightarrow> nat"   and f' :: "nat \<Rightarrow> nat" assumes "r = r'"   and "l r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "l_update f r = l_update f' r'"
group_add_hom_axioms: fixes i :: "nat" shows "group_add_hom (\<lambda>p. coeff p i)"
eq: fixes zs :: "'b llist" assumes "llength zs \<le> enat n" shows "the_enat (eSuc (llength zs)) = Suc (the_enat (llength zs))"
orbits_unfold: fixes f :: "'b perm" shows "orbits f = orbit f ` affected f"
deriv_body: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(I \<inter> b) c I,A"
cccFix_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "cccFix \<Gamma> = (\<Lambda> i. (Afix \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), CCfix \<Gamma>\<cdot>(Afix \<Gamma>\<cdot>(fst i \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>), snd i)))"
zero_less_mult_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero (times a b)"   and "less zero a" shows "less zero b"
mod_mult_self2_is_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a b) b = zero"
nemp_concat_hull_2: fixes us :: "'b list list"   and G :: "'b list set" assumes "us \<noteq> \<epsilon>"   and "us \<in> lists G\<^sub>+" shows "concat us \<noteq> \<epsilon>"
shiftr_div_2n_w: fixes n :: "nat"   and w :: "'b word" assumes "n < size w" shows "w >> n = w div 2 ^ n"
M_3: shows "PS.events = sets (density M (\<lambda>x. 1 + h x))"
intro: fixes p :: "'b \<rightarrow> 'a"   and e :: "'a \<rightarrow> 'b" assumes "\<And>x. p\<cdot>(e\<cdot>x) = x"   and "\<And>y. e\<cdot>(p\<cdot>y) \<sqsubseteq> y" shows "ep_pair e p"
Nom_3: shows "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A"
runit'_simps_1: fixes f :: "'a" assumes "ide f" shows "arr \<r>\<^sup>-\<^sup>1[f]"
intro: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and create_document :: "((_) heap, exception, (_) document_ptr) prog" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document" shows "l_get_dom_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr heap_is_wellformed parent_child_rel type_wf known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name create_document"
unique_poincare_distance_on_ray: fixes d :: "real"   and u :: "complex_homo"   and v :: "complex_homo"   and y :: "complex_homo"   and z :: "complex_homo" assumes "0 \<le> d"   and "u \<noteq> v"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "y \<in> unit_disc"   and "poincare_distance u y = d"   and "poincare_between u v y"   and "z \<in> unit_disc"   and "poincare_distance u z = d"   and "poincare_between u v z" shows "y = z"
neq_qe_inf_helper: fixes neq :: "(real \<times> real \<times> real) list"   and q :: "real" assumes "\<forall>(d, e, f)\<in>set neq. \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0" shows "\<exists>y'>q. \<forall>(d, e, f)\<in>set neq. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0"
cblinfun_eq_0_on_span: fixes x :: "'a"   and S :: "'a set"   and F :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "x \<in> cspan S"   and "\<And>s. s \<in> S \<Longrightarrow> F *\<^sub>V s = (0::'b)" shows "F *\<^sub>V x = (0::'b)"
farkas_coefficients_atoms_tableau: fixes t :: "(nat \<times> linear_poly) list"   and as :: "'a atom set" assumes "\<triangle> t"   and "finite as" shows "(\<exists>C. farkas_coefficients_atoms_tableau as t C) = (\<nexists>v. v \<Turnstile>\<^sub>t t \<and> v \<Turnstile>\<^sub>a\<^sub>s as)"
eq_cbox: fixes a :: "'c"   and b :: "'c"   and c :: "'c"   and d :: "'c" shows "(cbox a b = cbox c d) = (cbox a b = {} \<and> cbox c d = {} \<or> a = c \<and> b = d)"