hcju/isabelleps huggingface dataset:
mono: shows "mono (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> mk_less ord x y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> mk_less ord x y \<and> \<not> mk_less ord y x \<and> p xs ys))"
that: fixes a :: "nat \<Rightarrow> real"   and b :: "nat \<Rightarrow> real"   and n :: "nat" assumes "\<And>x. x \<in> {- pi..pi} \<Longrightarrow> \<bar>f x - (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))\<bar> < e" shows "thesis"
oppOrdering_simps_3: shows "oppOrdering\<cdot>GT = LT"
sel_rews_4: fixes a :: "'a"   and Stream :: "'a Stream" shows "sttail\<cdot>(a && Stream) = Stream"
vintersection_commutativity: fixes A :: "V"   and B :: "V" shows "A \<inter>\<^sub>\<circ> B = B \<inter>\<^sub>\<circ> A"
Derives1_unique_src: fixes u :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and w :: "symbol list"   and v :: "symbol list" assumes "Derives1 u i r w"   and "Derives1 v i r w" shows "u = v"
wp_def: fixes pr :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wp pr \<equiv> pr True"
free_group_is_group: fixes gens :: "'d set" shows "Group.group \<F>\<^bsub>gens\<^esub>"
simps_4: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "min_dnf (\<phi> or\<^sub>n \<psi>) = min_dnf \<phi> \<union>\<^sub>m min_dnf \<psi>"
ucast_short_ucast_long_ingoreLeadingZero: fixes w :: "'a word" assumes "length (dropWhile Not (to_bl w)) \<le> LENGTH('b)"   and "LENGTH('b) \<le> LENGTH('a)" shows "UCAST('b \<rightarrow> 'a) (UCAST('a \<rightarrow> 'b) w) = w"
Limit_vifunion_in_VsetI: fixes \<alpha> :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "Limit \<alpha>"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> A i \<in>\<^sub>\<circ> Vset \<alpha>"   and "vfinite I" shows "\<Union>\<^sub>\<circ> (VLambda I A `\<^sub>\<circ> I) \<in>\<^sub>\<circ> Vset \<alpha>"
R'_def: shows "R' = map_prod (to_nat_on A) (to_nat_on B) ` R"
simps_1: fixes v :: "'d option"   and m :: "('e \<times> ('e, 'd) trie) list" shows "lookup_trie (Trie v m) [] = v"
non_unique_continuous_unfolding: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf"   and q :: "'a pmf"   and r :: "'a pmf" assumes "continuous_vnm (\<P> outcomes) \<R>"   and "rational_preference (\<P> outcomes) \<R>"   and "p \<succeq>[\<R>] q"   and "q \<succeq>[\<R>] r"   and "p \<succeq>[\<R>] r \<and> (r, p) \<notin> \<R>" shows "\<exists>\<alpha>\<in>{0..1}. q \<approx>[\<R>] mix_pmf \<alpha> p r"
exhaust: fixes y :: "X3" obtains "y = X3.Xd1"   | "y = X3.Xd1'"   | "y = X3.Xd2"
fmdrop_set_fmdom: fixes A :: "('a, 'b) fmap" shows "fmdrop_set (fmdom' A) A = fmempty"
trad_foundation: fixes z :: "V" assumes "z \<noteq> 0" shows "\<exists>w. w \<in>\<^sub>\<circ> z \<and> vdisjnt w z"
exhaust: fixes y :: "('a, 'b) rbt" obtains "y = Empty"   | x21 :: "color" and x22 :: "('a, 'b) rbt" and x23 :: "'a" and x24 :: "'b" and x25 :: "('a, 'b) rbt" where "y = Branch x21 x22 x23 x24 x25"
Stable_Int: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set" assumes "F \<in> Stable A"   and "F \<in> Stable A'" shows "F \<in> Stable (A \<inter> A')"
characteristic_weak_formula_is_characteristic': fixes Q :: "'state"   and P :: "'state" shows "Q \<Turnstile> characteristic_weak_formula P = (P \<equiv>\<cdot> Q)"
assms_2: shows "(I \<Rightarrow> Post) \<sqsubseteq> P"
assms_6: shows "roots = (\<lambda>(a, b, c). if a = 0 \<and> b \<noteq> 0 then {- c / b} else if a \<noteq> 0 \<and> 0 \<le> b\<^sup>2 - 4 * a * c then {(- b + sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} \<union> {(- b - sqrt (b\<^sup>2 - 4 * a * c)) / (2 * a)} else {})"
zero_closed: shows "\<zero>\<^bsub>S\<^esub> \<in> carrier S"
dres_cases: fixes x :: "'a dres" obtains "x = dSUCCEED"   | r :: "'a" where "x = dRETURN r"   | "x = dFAIL"
intro: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>a a' b. prod (a + a') b = prod a b + prod a' b"   and "\<And>a b b'. prod a (b + b') = prod a b + prod a b'"   and "\<And>r a b. prod (r *\<^sub>R a) b = r *\<^sub>R prod a b"   and "\<And>a r b. prod a (r *\<^sub>R b) = r *\<^sub>R prod a b"   and "\<exists>K. \<forall>a b. \<parallel>prod a b\<parallel> \<le> (\<parallel>a\<parallel>) * (\<parallel>b\<parallel>) * K" shows "bounded_bilinear prod"
OclAsReal\<^sub>I\<^sub>n\<^sub>t_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "X->oclAsType\<^sub>I\<^sub>n\<^sub>t(Real) = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>real_of_int \<lceil>\<lceil>X \<tau>\<rceil>\<rceil>\<rfloor>\<rfloor> else invalid \<tau>)"
map_seq_def: fixes f :: "'b \<Rightarrow> natural \<Rightarrow> bool \<Rightarrow> 'a lazy_sequence option"   and xq :: "'b lazy_sequence"   and i :: "natural"   and pol :: "bool" shows "map_seq f xq i pol = map_option Lazy_Sequence.flat (Lazy_Sequence.those (Lazy_Sequence.map (\<lambda>x. f x i pol) xq))"
mset_subset_eq_insertD: fixes x :: "'k"   and A :: "'k multiset"   and B :: "'k multiset" assumes "add_mset x A \<subseteq># B" shows "x \<in># B \<and> A \<subset># B"
s_def: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "distributed_system channel" shows "distributed_system.s channel trans send recv c t i = (THE c'. distributed_system.trace channel trans send recv c (take i t) c')"
l3_defs_7: shows "l3_trans \<equiv> \<Union>m M KE Rb Ra A B K. (l3_step1 Ra A B \<union> l3_step2 Rb A B KE \<union> l3_step3 Ra A B m \<union> l3_dy M \<union> l3_lkr_others A \<union> l3_lkr_after A \<union> l3_skr Ra K)\<^sup>="
orbit_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> E"   and "y \<in> E"   and "z \<in> E"   and "y \<in> orbit G \<phi> x"   and "z \<in> orbit G \<phi> y" shows "z \<in> orbit G \<phi> x"
conv_imp_valley: fixes r :: "('a \<times> 'a) set"   and s :: "'b"   and \<alpha>_step :: "'a \<times> 'b"   and \<rho>_step :: "('a \<times> 'b) list"   and \<beta>_step :: "'a \<times> 'b"   and \<upsilon>_step :: "('a \<times> 'b) list"   and ars :: "('b \<times> 'a \<times> 'b) set"   and \<delta>1 :: "'b \<times> (bool \<times> 'a \<times> 'b) list"   and M :: "'a multiset" assumes "trans r"   and "\<And>y. \<lbrakk>(y, (s, [\<alpha>_step] @ \<rho>_step), s, [\<beta>_step] @ \<upsilon>_step) \<in> pex r; peak ars y\<rbrakk> \<Longrightarrow> \<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')"   and "\<delta>1 \<in> conv ars"   and "set_mset (measure_conv r \<delta>1) \<subseteq> r \<down>m M"   and "(M, {#fst \<alpha>_step, fst \<beta>_step#}) \<in> mul_eq r" shows "\<exists>\<sigma> \<tau>. {\<sigma>, \<tau>} \<subseteq> seq ars \<and> fst \<sigma> = fst \<delta>1 \<and> fst \<tau> = lst_conv \<delta>1 \<and> lst \<sigma> = lst \<tau> \<and> set_mset (Decreasing_Diagrams.measure r (\<sigma>, \<tau>)) \<subseteq> r \<down>m M"
admissible_def: fixes lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" shows "ccpo.admissible lub ord P = (\<forall>A. Complete_Partial_Order.chain ord A \<longrightarrow> A \<noteq> {} \<longrightarrow> (\<forall>x\<in>A. P x) \<longrightarrow> P (lub A))"
e_def: shows "e = (case ms_aux_hd_exp xs of None \<Rightarrow> 0 | Some e \<Rightarrow> e + 1)"
rational_root_test_main_2: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and p :: "rat poly" assumes "divisors_fun df"   and "divisors_pos_fun dp"   and "rational_root_test_main df dp p = None" shows "\<nexists>x. poly p x = 0"
qSwapAbs_preserves_qGoodAbs1: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "qGoodAbs A" shows "qGoodAbs (A $[[x \<and> y]]_zs)"
contlub_def: fixes f :: "'b \<Rightarrow> 'c" shows "contlub f = (\<forall>Y. chain Y \<longrightarrow> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i)))"
subterms_refl: fixes p :: "('a, 'b, 'c, 'd) seqp" shows "p \<in> subterms p"
s_learn_prenum: fixes b :: "nat" shows "learn_bc \<phi> {prenum b} s"
g: fixes z :: "complex"   and e :: "real" assumes "z \<in> S"   and "0 < e" shows "\<exists>N. \<forall>n\<ge>N. cmod (\<F> (k n) z - g z) < e"
open_greaterThan: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.order_topology less_eq less open" shows "open (ord.greaterThan less a)"
nonempty': fixes \<rho> :: "nat \<Rightarrow> val" assumes "\<rho> \<in> space (dens_ctxt_measure \<Y> \<rho>)" shows "space (M \<rho>) \<noteq> {}"
hcomp_simps_1: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "hseq \<nu> \<mu>" shows "src (\<nu> \<star> \<mu>) = src \<mu>"
Lcm_eucl: shows "normalization_euclidean_semiring_class.Lcm = Lcm"
icdj: shows "m icd\<^bsup>\<pi>\<^esup>\<rightarrow> j"
assms_2: shows "Counterclockwise_2D_Arbitrary.lex q r"
intro: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" assumes "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "locally_ringed_space_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str" shows "locally_ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
ln_diff_le: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "ln x - ln y \<le> (x - y) / y"
lt_monom_mult: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "lt (monom_mult c t p) = t \<oplus> lt p"
\<beta>_boundedness_eq: fixes m :: "int"   and x :: "'a" assumes "m \<le> int (k x)"   and "x \<in> X" shows "\<exists>U\<subseteq>\<R>. \<Union> U = {u \<in> V. u x = real_of_int m}"
ext: fixes g :: "'c"   and f :: "'c" assumes "g \<cdot> f \<noteq> C.null" shows "C.seq g f"
nsubst_sterm': fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<theta> :: "('sf + 'b, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and a :: "(real, 'sz) vec"   and b :: "(real, 'sz) vec" assumes "TadmitFFO \<sigma> \<theta>"   and "\<And>i. dsafe (\<sigma> i)" shows "sterm_sem I (TsubstFO \<theta> \<sigma>) a = sterm_sem (adjointFO I \<sigma> (a, b)) \<theta> a"
f1: shows "(s |\<^bsub>rel\<^esub> \<one>) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> (\<one> |\<^bsub>rel\<^esub> s) = (\<one> |\<^bsub>rel\<^esub> \<one>)"
supp_2: fixes x1 :: "tyvrs" shows "supp (Tvar x1) = supp x1"
exec_step_ind_Putfield: fixes v :: "val"   and stk :: "val list"   and r :: "val"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and h :: "heap"   and D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and C :: "char list"   and F :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "v = hd stk"   and "r = hd (tl stk)"   and "a = the_Addr r"   and "(D, fs) = the (h a)"   and "(D', b, t) = field P C F"   and "r \<noteq> Null"   and "P \<turnstile> D has F,NonStatic:t in C" shows "exec_step_ind (StepI (Putfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h(a \<mapsto> (D, fs((F, C) \<mapsto> v))), (tl (tl stk), loc, C\<^sub>0, M\<^sub>0, pc + 1, ics) # frs, sh)"
transitive_rule_def: fixes t :: "'b" shows "transitive_rule t = transl_rule (A_Cmp (A_Lbl t) (A_Lbl t) \<sqsubseteq> A_Lbl t)"
e2stack_def: fixes e :: "nat" shows "e2stack e \<equiv> pdec1 e"
d_Bisimulation_Up_To_USdB_def: fixes d :: "'d"   and R :: "('com list \<times> 'com list) set" shows "d_Bisimulation_Up_To_USdB d R \<equiv> sym R \<and> (\<forall>(V, V')\<in>R. length V = length V') \<and> (\<forall>(V, V')\<in>R. \<forall>i<length V. \<forall>m1 m1' W m2. \<langle>V ! i,m1\<rangle> \<rightarrow> \<langle>W,m2\<rangle> \<and> m1 =\<^bsub>d\<^esub> m1' \<longrightarrow> (\<exists>W' m2'. \<langle>V' ! i,m1'\<rangle> \<rightarrow> \<langle>W',m2'\<rangle> \<and> (W, W') \<in> R \<union> (\<approx>\<^bsub>d\<^esub>) \<and> m2 =\<^bsub>d\<^esub> m2'))"
map_sel_1: fixes a :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "a \<noteq> []" shows "hd (map f a) = f (hd a)"
restrict_bind_pmf: fixes p :: "'d pmf"   and g :: "'d \<Rightarrow> 'c spmf"   and A :: "'c set" shows "(p \<bind> g) \<upharpoonleft> A = p \<bind> (\<lambda>x. g x \<upharpoonleft> A)"
assms_4: fixes x :: "'alt" assumes "x \<in> alts" shows "measure_pmf.prob q (preferred_alts (R i) x) \<le> measure_pmf.prob p (preferred_alts (R i) x)"
d_OUT_lift_Inner: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" shows "d_OUT (lift f) (Inner x) = d_OUT f x"
is_singletonE: fixes A :: "'e set" assumes "is_singleton A" obtains x :: "'e" where "A = {x}"
cupcake_list_preserve0: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list" assumes "list_all2_shortcircuit (\<lambda>e r. cupcake_evaluate_single env e r \<and> (is_cupcake_all_env env \<longrightarrow> is_cupcake_exp e \<longrightarrow> if_rval is_cupcake_value r)) es rs"   and "is_cupcake_all_env env"   and "list_all is_cupcake_exp es"   and "sequence_result rs = Rval vs" shows "list_all is_cupcake_value vs"
norm_eq: fixes x :: "'a"   and y :: "'b" shows "(norm x = norm y) = (inner x x = inner y y)"
case2: assumes "\<not> x \<le> y" shows "t_ins_aux \<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr> \<in> t_ins_set \<lparr>t_type.folding = True, item = x, subtrees = Branch y yl xt # ts\<rparr>"
Cignorabimus_empty: fixes \<alpha> :: "game"   and V :: "variable set" shows "{} \<in> Cignorabimus \<alpha> V"
is_weak_bisimulation_def: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "weak_nominal_ts satisfies transition \<tau>" shows "weak_nominal_ts.is_weak_bisimulation satisfies transition \<tau> R \<equiv> symp R \<and> (\<forall>P Q \<phi>. R P Q \<and> satisfies P \<phi> \<longrightarrow> (\<exists>Q'. weak_nominal_ts.tau_transition transition \<tau> Q Q' \<and> R P Q' \<and> satisfies Q' \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> transition P \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. weak_nominal_ts.weak_transition transition \<tau> Q \<alpha> Q' \<and> R P' Q')))"
exists_least_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m. less m n \<longrightarrow> \<not> P m))"
evaln_elim_cases_19: fixes G :: "prog"   and s :: "st"   and e0 :: "expr"   and e1 :: "expr"   and e2 :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (e0 ? e1 : e2)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains b :: "val" and s1 :: "state" and va :: "val" where "v = In1 va" and "G\<turnstile>Norm s \<midarrow>e0-\<succ>b\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>(if the_Bool b then e1 else e2)-\<succ>va\<midarrow>n\<rightarrow> s'"
indep_sets_mono_sets: fixes F :: "'d \<Rightarrow> 'a set set"   and I :: "'d set"   and G :: "'d \<Rightarrow> 'a set set" assumes "indep_sets F I"   and "\<And>i. i \<in> I \<Longrightarrow> G i \<subseteq> F i" shows "indep_sets G I"
assms_2: shows "z \<notin> path_image \<gamma>1"
power_both_mono: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and m :: "nat"   and n :: "nat"   and b :: "'a" assumes "class.semiring_1_real_line one times plus zero minus less_eq less"   and "less_eq one a"   and "m \<le> n"   and "less_eq a b" shows "less_eq (power.power one times a m) (power.power one times b n)"
GF_advice_sync_and: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set"   and \<psi> :: "'b ltln" assumes "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu>"   and "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<psi> (prefix i w)[X]\<^sub>\<nu>" shows "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[X]\<^sub>\<nu> \<and> suffix i w \<Turnstile>\<^sub>n af \<psi> (prefix i w)[X]\<^sub>\<nu>"
error_msg_notin_dom_def: fixes i :: "nat" shows "error_msg_notin_dom i = show i @ error_msg_notin_dom_err"
has_bochner_integral_minus: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" assumes "has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. - f x) (- x)"
\<O>_sb_def: fixes t :: "'d \<times> 'e \<times> 'f \<times> 'c \<times> 'g \<times> 'b \<times> 'h" shows "\<O>_sb t = (let (p, instrs, \<theta>, sb, \<D>, \<O>, \<R>) = t in (\<O>, sb))"
SN_on_iff_wf_below: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "SN_on r A = wf_below (r\<inverse>) A"
to_metric_completion_dense': shows "closure (range to_metric_completion) = UNIV"
assms_4: fixes x :: "'alt" assumes "x \<in> alts" shows "lottery_prob (sds (R(i := Ri'))) (preferred_alts (R i) x) \<le> lottery_prob (sds R) (preferred_alts (R i) x)"
rBangTauI: fixes S :: "('a, 'b, 'c) psi"   and S' :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b" assumes "\<Psi>' \<rhd> S \<parallel> S \<Longrightarrow>\<^sup>^\<^sub>\<tau> S'"   and "guarded S" shows "\<exists>T. \<Psi>' \<rhd> !S \<Longrightarrow>\<^sup>^\<^sub>\<tau> T \<and> (\<Psi>', T, S' \<parallel> !S) \<in> Rel'"
Gauss_Jordan_wrapper_def: fixes i :: "'f"   and j :: "'g"   and A :: "(('d, 'e) vec, 'f) vec"   and B :: "(('d, 'g) vec, 'f) vec" shows "Gauss_Jordan_wrapper i j A B = (\<chi>s. if s = i then A $ s else row_add A s i (- B $ s $ j) $ s)"
fact_num_eq_if: fixes m :: "nat" shows "fact m = (if m = 0 then 1::'b else of_nat m * fact (m - 1))"
differentiable_chain_within: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and S :: "'c set"   and g :: "'d \<Rightarrow> 'e" assumes "f differentiable at x within S"   and "g differentiable at (f x) within f ` S" shows "g \<circ> f differentiable at x within S"
has_codomain_iff_arr: fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" shows "(S.VV.codomains f \<noteq> {}) = S.VV.arr f"
zero_mat_def: fixes nr :: "nat"   and nc :: "nat" shows "0\<^sub>m nr nc \<equiv> mat nr nc (\<lambda>ij. 0::'b)"
iter_induct_isolate: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice) \<sqinter> c\<^sup>\<infinity> = lfp (\<lambda>x::'a::refinement_lattice. d \<sqinter> c ; x)"
Cons_7: shows "c2 \<sqsubseteq> c1"
abs_eq: fixes x :: "('a \<times> 'b) set['c]" shows "var_of_lts_emb x = abs_var (BPS x)"
simps_2: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b list"   and xs :: "'b list list" assumes "Matrix_Tensor.mult id f" shows "mult.reduct (x # xs) = xs"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'w" assumes "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
valid_abs_init_1: fixes v :: "nat" shows "(abs_ast_initial_state v \<noteq> None) = (v \<in> set abs_ast_variable_section)"
suffix_butlast: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys" shows "suffix (butlast xs) (butlast ys)"
map_comp: fixes g :: "'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d multiset" shows "image_mset g (image_mset f v) = image_mset (g \<circ> f) v"
one_eq_prime_power_iff: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "(one = power.power one times p n) = (n = 0)"
locals_def: shows "locals = case_st (\<lambda>g l. l)"
a_rcos_module: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme"   and x :: "'a"   and x' :: "'a" assumes "abelian_subgroup H G"   and "x \<in> carrier G"   and "x' \<in> carrier G" shows "(x' \<in> H +>\<^bsub>G\<^esub> x) = (x' \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x \<in> H)"
P3: shows "Bet PO a'' A \<and> Cong Po a PO a''"
merge_apply_2: fixes I :: "'d set"   and J :: "'d set"   and i :: "'d"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" assumes "I \<inter> J = {}"   and "i \<in> J" shows "merge I J (x, y) i = y i"
\<section>: fixes xs :: "'a list"   and ys :: "'a list" assumes "(xs, ys) \<in> lenlex {(u, v). u < v}"   and "(ys, xs) \<in> lenlex {(u, v). u < v}" shows "False"
take_subseq_def: fixes p :: "int"   and s :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> nat" assumes "padic_integers p" shows "take_subseq s f = (\<lambda>k. s (f k))"
delta: fixes S :: "'a set"   and a :: "'a"   and b :: "'a \<Rightarrow> 'b" assumes "finite S" shows "(\<Prod>k\<in>S. if k = a then b k else (1::'b)) = (if a \<in> S then b a else (1::'b))"
iso_iff_section_and_epi: fixes f :: "'b" shows "D.iso f = (D.section f \<and> D.epi f)"
hyps: assumes "p \<noteq> 0" shows "card (proots_within p B) = card (proots_within (p \<circ>\<^sub>p q) A)"
char_less_iff_nat_of_char: fixes x :: "char"   and y :: "char" shows "(x < y) = (of_char x < of_char y)"
lnull_lfilter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lnull (lfilter P xs) = (\<forall>x\<in>lset xs. \<not> P x)"
h3: shows "last (fin_make_untimed (inf_truncate z i)) = l i"
cod_src_eq_dom_src: shows "assoc_fgh.cod.src = assoc_fgh.dsrc"
liminf_upper_bound: fixes u :: "nat \<Rightarrow> ereal"   and l :: "ereal"   and k :: "nat" assumes "liminf u < l" shows "\<exists>N>k. u N < l"
fun_cong: fixes f :: "'n \<Rightarrow> 'o"   and g :: "'n \<Rightarrow> 'o"   and x :: "'n" assumes "f = g" shows "f x = g x"
roofed_circ_essential: fixes \<Gamma> :: "('a, 'b) web_scheme"   and S :: "'a set" shows "RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (\<E>\<^bsub>\<Gamma>\<^esub> S) = RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S"
circle_arcs_neq: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real"   and k :: "real"   and n :: "real"   and m :: "real"   and q :: "real" assumes "circle i j d"   and "0 \<le> k"   and "k \<le> 1"   and "0 \<le> n"   and "n \<le> 1"   and "n < k"   and "k + n < 1" shows "subcube k m (1, circle.circle_polar d) \<noteq> subcube n q (1, circle.circle_polar d)"
x0: fixes n :: "nat" shows "x0 \<in> cbox (A n) (B n)"