hcju/isabelleps huggingface dataset:
CD_on_def: fixes ds :: "D2 set"   and X :: "X3 set" shows "Theorem_9.CD_on ds X = (\<Union>d\<in>ds. Theorem_9.Cd d X)"
sctn_rel_internal: fixes A :: "('c \<times> 'd) set" shows "sctn_rel A = {(a, b). rel_sctn (in_rel A) a b}"
right_def: fixes \<sigma> :: "'a" shows "right \<sigma> \<equiv> \<lambda>\<mu>. \<mu> \<star> \<sigma> \<noteq> null"
eval_embed_false: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "\<not> P s" shows "\<guillemotleft> P \<guillemotright> s = 0"
extreme_point_of_convex_hull_affine_independent: fixes S :: "'b set"   and x :: "'b" assumes "\<not> affine_dependent S" shows "(x extreme_point_of convex hull S) = (x \<in> S)"
counit_simps_5: shows "trg \<epsilon> = trg f"
Sem_eval_cases_2: fixes s :: "Var \<Rightarrow> Val"   and x :: "Var"   and e :: "Expr"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , Assign x e \<rightarrow>\<^sub>n  t " obtains "n = Suc 0" and "t = update s x (evalE e s)"
Live2_def: shows "Live2 \<equiv> SF(n1)_vars \<and> SF(n2)_vars"
powr_less_eq_inverse_iff: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x"   and "0 < y"   and "0 < z" shows "(x powr y \<le> z) = (x \<le> z powr inverse y)"
indRelSTEQ_impl_SRel_and_TRel_reflect_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_reflects_binary_pred (indRelSTEQ SRel TRel) (\<lambda>b. hasBarb b (STCalWB SWB TWB))" shows "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)"
gto_outer_impl_def: fixes brk :: "('q set \<times> 'q set) option"   and s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gto_outer_impl brk s \<equiv> let (A, S, B, I, P) = s in (brk, I)"
poly_different_sign_imp_root': fixes p :: "real poly"   and a :: "real"   and b :: "real" assumes "sgn (poly p a) \<noteq> sgn (poly p b)" shows "\<exists>x. poly p x = 0"
assms_2: fixes i :: "nat" shows "f i \<noteq> (0::'a)"
type_less_eq_OclVoid_x_intro: fixes \<tau> :: "'a basic_type" shows "OclVoid \<le> \<tau>"
wellformed1_sorted: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "wellformed_policy1 p" shows "wellformed_policy1 (FWNormalisationCore.sort p l)"
llist_abs_def: shows "llist_abs \<equiv> PRJ('a\<cdot>llist) oo EMB(one \<oplus> 'a\<^sub>\<bottom> \<otimes> ('a\<cdot>llist)\<^sub>\<bottom>)"
unboxedval_axioms: shows "unboxedval is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2"
verts3_subdivide: fixes e :: "'a \<times> 'a"   and w :: "'a" assumes "e \<in> parcs G'"   and "w \<notin> pverts G'" shows "verts3 (with_proj (subdivide G' e w)) = verts3 (with_proj G')"
pfsubset_eq: fixes A :: "'b fset"   and B :: "'b fset" shows "(A |\<subset>| B) = (A |\<subseteq>| B \<and> A \<noteq> B)"
0: fixes X :: "'var list"   and \<gamma> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> set X = {}"   and "subst_domain \<gamma> = set X"   and "ground (subst_range \<gamma>)" shows "\<gamma> \<circ>\<^sub>s \<delta> = \<delta> \<circ>\<^sub>s \<gamma>"
isUCont_def: fixes f :: "'c \<Rightarrow> 'd" shows "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
equalityCE: fixes A :: "'ag set"   and B :: "'ag set"   and c :: "'ag" assumes "A = B" obtains "c \<in> A" and "c \<in> B"   | "c \<notin> A" and "c \<notin> B"
Sup_V_insert: fixes A :: "V set"   and x :: "V" assumes "small A" shows "\<Squnion> (insert x A) = x \<union>\<^sub>\<circ> \<Squnion> A"
update: fixes xs :: "'a option list set list"   and ys :: "'a option list set list"   and aux' :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and buf' :: "'a option list set list \<times> 'a option list set list"   and nts' :: "nat list"   and zs :: "'a option list set list"   and aux'' :: "(nat \<times> 'a option list set \<times> 'a option list set) list" assumes "fst (meval n (\<tau> \<sigma> j) (\<Gamma> \<sigma> j) \<phi>) = xs"   and "fst (meval n (\<tau> \<sigma> j) (\<Gamma> \<sigma> j) \<psi>) = ys"   and "mbuf2t_take (update_until I pos) aux (mbuf2_add xs ys buf) (nts @ [\<tau> \<sigma> j]) = (aux', buf', nts')"   and "eval_until I (case nts' of [] \<Rightarrow> \<tau> \<sigma> j | nt # x \<Rightarrow> nt) aux' = (zs, aux'')" shows "wf_until_aux \<sigma> n R pos \<phi>'' I \<psi>'' aux'' (Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') (Suc j)) \<and> Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') j + length zs = Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') (Suc j) \<and> Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') j + length zs + length aux'' = min (Monitor.progress \<sigma> \<phi>''' (Suc j)) (Monitor.progress \<sigma> \<psi>'' (Suc j)) \<and> list_all2 (\<lambda>i. qtable n (fv \<psi>'') (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i (formula.Until (if pos then \<phi>'' else formula.Neg \<phi>'') I \<psi>''))) [Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') j..<Monitor.progress \<sigma> (formula.Until \<phi>''' I \<psi>'') j + length zs] zs"
uint64_divmod_def: fixes x :: "uint64"   and y :: "uint64" shows "uint64_divmod x y = (if y = 0 then (undefined (div) x 0, undefined (mod) x 0) else (x div y, x mod y))"
IH: assumes "local.inv_upto local.jb B j"   and "local.inv_from local.uppert B j"   and "local.inv_from_bot (\<lambda>A i. local.one_zero A i j) B i"   and "local.ev_block n B" shows "local.inv_from local.uppert (step_3_a i j B) j \<and> local.inv_upto local.jb (step_3_a i j B) j \<and> local.inv_at local.one_zero (step_3_a i j B) j \<and> local.same_diag B (step_3_a i j B)"
ZObis_transT: fixes indis :: "('a \<times> 'a) set"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a"   and s' :: "'a" assumes "PL_Indis indis"   and "(c, d) \<in> PL_Indis.ZObis tval aval indis"   and "(s, t) \<in> indis"   and "PL.transT tval aval (c, s) s'" obtains t' :: "'a" where "PL.transT tval aval (d, t) t'a__" and "(s', t'a__) \<in> indis"   | d' :: "('b, 'c) com" and t' :: "'a" where "PL.transC tval aval (d, t) (d', t')" and "(s', t') \<in> indis" and "PL_Indis.discr tval aval indis d'"   | "(s', t) \<in> indis" and "PL_Indis.discr tval aval indis d"
distinct_1: fixes x1 :: "nat"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat" shows "Prime x1 \<noteq> Triple x21 x22 x23"
assms_7: shows "w * e\<^sup>T * top = bot"
case_sum_if: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a"   and p :: "bool"   and x :: "'b"   and y :: "'c" shows "(case if p then Inl x else Inr y of Inl x \<Rightarrow> f x | Inr x \<Rightarrow> g x) = (if p then f x else g y)"
gcd_mult_right: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "gcd (a * c) (b * c) = normalize (gcd b a * c)"
unique_x_axis_poincare_distance_positive: fixes d :: "real" assumes "0 \<le> d" shows "\<exists>!z. is_real z \<and> 0 \<le> Re z \<and> Re z < 1 \<and> poincare_distance 0\<^sub>h (of_complex z) = d"
equivalence_functor_G: shows "equivalence_functor hom.comp hom'.comp (\<lambda>\<mu>'. trg f \<star> \<mu>' \<star> e)"
\<L>\<^sub>P_def: shows "\<L>\<^sub>P = {uu_. \<exists>u v. uu_ = u \<and> is_word u \<and> is_derivation (u @ v)}"
assms: fixes i :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<phi> i : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA> i"
axioms_1: 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"   and I :: "'c set set"   and x :: "'c" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"
prime_imp_prime_elem: 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" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "comm_semiring_1.prime_elem times one zero p"
preallocatedD: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and C :: "char list" assumes "preallocated h"   and "C \<in> sys_xcpts" shows "\<exists>S. h (addr_of_sys_xcpt C) = \<lfloor>(C, S)\<rfloor>"
simps_2: fixes tst :: "test"   and s :: "Concrete.var \<Rightarrow> nat" shows "cval (Inr tst) s = (if tval tst s then 1::'e else (0::'e))"
extends_def: fixes S :: "'b list"   and S' :: "'b list" shows "(S \<lesssim> S') = (\<exists>S''. S' = S'' @ S)"
ac_redu: fixes M :: "trm"   and M' :: "trm" assumes "M \<longrightarrow>\<^sub>c M'" shows "M \<longrightarrow>\<^sub>a M'"
limit_inter_INF: fixes w :: "nat \<Rightarrow> 'e"   and S :: "'e set" assumes "limit w \<inter> S \<noteq> {}" shows "\<exists>\<^sub>\<infinity>n. w n \<in> S"
induct: fixes x1 :: "'a OCL_Types.type"   and x2 :: "'a OCL_Types.type"   and P :: "'a OCL_Types.type \<Rightarrow> 'a OCL_Types.type \<Rightarrow> bool" assumes "to_unique_collection x1 x2"   and "\<And>\<tau>. P (Collection \<tau>) (Set \<tau>)"   and "\<And>\<tau>. P (Set \<tau>) (Set \<tau>)"   and "\<And>\<tau>. P (OrderedSet \<tau>) (OrderedSet \<tau>)"   and "\<And>\<tau>. P (Bag \<tau>) (Set \<tau>)"   and "\<And>\<tau>. P (Sequence \<tau>) (OrderedSet \<tau>)" shows "P x1 x2"
integrable_on_indicator: fixes S :: "'c set"   and T :: "'c set" shows "(indicat_real S integrable_on T) = (S \<inter> T \<in> lmeasurable)"
assms: shows "adjunction_data_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
exhaust: fixes y :: "WS1S_Prelim.order" obtains "y = FO"   | "y = SO"
lt_monom_mult_zero: fixes c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'c)" shows "lt (monom_mult c (0::'a) p) = lt p"
append_bv_chop_id: fixes w :: "bit list"   and l :: "nat" shows "fst (bv_chop w l) @ snd (bv_chop w l) = w"
tr_var_def: fixes T :: "x \<Rightarrow> x option"   and var :: "char list" shows "tr_var T var = (case tr_x T (x_var var) of x_var var' \<Rightarrow> var' | x_this \<Rightarrow> var)"
analz_subset: fixes H :: "msg set" shows "H \<subseteq> analz H"
le: fixes e :: "'a \<times> 'a" shows "f e \<le> g e"
floor_less_zero: fixes x :: "'a" shows "(\<lfloor>x\<rfloor> < 0) = (x < (0::'a))"
bounded_iff: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" shows "(a \<subseteq># b \<inter># c) = (a \<subseteq># b \<and> a \<subseteq># c)"
wf_trm_Var: fixes x :: "'c" shows "wf\<^sub>t\<^sub>r\<^sub>m (Var x)"
CD_on_def: fixes ds :: "'d set"   and X :: "'x set" shows "CD_on ds X = (\<Union>d\<in>ds. Cd d X)"
adjoined_in_adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := H\<rparr>)" shows "a \<in> adjoin (G\<lparr>carrier := H\<rparr>) H a"
rel_pmf_reflI: fixes p :: "'d pmf"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" shows "rel_pmf P p p"
NatTransCompDefCod: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and f :: "'i"   and X :: "'g"   and Y :: "'g" assumes "NatTrans \<eta>"   and "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y" shows "\<eta> $$ X \<approx>>\<^bsub>NTCatCod \<eta>\<^esub> NTCod \<eta> ## f"
octo_of_nat_add: fixes a :: "nat"   and b :: "nat" shows "octo_of_nat (a + b) = octo_of_nat a + octo_of_nat b"
reachable_refl: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" assumes "wf_digraph G"   and "v \<in> verts G" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
hyps: assumes "y \<sharp> As" shows "[y;x]As = [(y, x)] \<bullet> As"
crename_lredu: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and M' :: "trm" assumes "M[a\<turnstile>c>b] \<longrightarrow>\<^sub>l M'" shows "\<exists>M0. M0[a\<turnstile>c>b] = M' \<and> M \<longrightarrow>\<^sub>l M0"
simps_1: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'a list" assumes "Matrix_Tensor.mult id f" shows "mult.vec_mat_Tensor f xs [] = []"
infsetsum_swap: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "countable A"   and "countable B"   and "(\<lambda>(x, y). f x y) abs_summable_on A \<times> B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) B) = (\<Sum>\<^sub>ay\<in>B. \<Sum>\<^sub>ax\<in>A. f x y)"
runit_char_2: fixes f :: "'a" assumes "B.ide f" shows "B.R (B.runit f) = (f \<star>\<^sub>B \<i>\<^sub>B[src\<^sub>B f]) \<cdot>\<^sub>B \<a>\<^sub>B[f, src\<^sub>B f, src\<^sub>B f]"
simps_2: fixes Vs :: "char list list"   and e :: "char list exp"   and es :: "char list exp list" shows "compEs\<^sub>1 Vs (e # es) = compE\<^sub>1 Vs e # compEs\<^sub>1 Vs es"
commute: fixes a :: "'g"   and b :: "'g" shows "a + b = b + a"
Abs'_def: fixes x :: "int" shows "Abs' x = Abs (x mod int CARD('b))"
image_mset_cong: fixes M :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in># M \<Longrightarrow> f x = g x" shows "f `# M = g `# M"
fold_cong: fixes a :: "'g"   and b :: "'g"   and xs :: "'h list"   and ys :: "'h list"   and f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'g \<Rightarrow> 'g" assumes "a = b"   and "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "fold f xs a = fold g ys b"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "a + b \<in> R"
cosh_real_nonneg: fixes x :: "real" shows "0 \<le> cosh x"
uminus_conjugate_vec: fixes v :: "'a vec" shows "- conjugate v = conjugate (- v)"
ran: shows "\<R>\<^sub>\<circ> [a, b]\<^sub>\<circ> \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (VLambda (2\<^sub>\<nat>) f `\<^sub>\<circ> 2\<^sub>\<nat>)"
hypsubst: fixes A :: "'d set"   and B :: "'d set"   and x :: "'d" assumes "A = B"   and "x \<in> B" obtains "x \<in> A"
sup_top_left: fixes 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 top :: "'b"   and x :: "'b" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup top x = top"
min_number_of_1: fixes u :: "num"   and v :: "num" shows "min (numeral u) (numeral v) = (if numeral u \<le> numeral v then numeral u else numeral v)"
span_li_not_depend_1: fixes S :: "'a vec set"   and N :: "'a vec set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule class_ring N V" shows "LinearCombinations.module.span class_ring (vs N) S = local.span S"
Union_mono: fixes A :: "'a set set"   and B :: "'a set set" assumes "A \<subseteq> B" shows "\<Union> A \<subseteq> \<Union> B"
cinner_sum_left: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and y :: "'a" shows "cinner (sum f A) y = (\<Sum>x\<in>A. cinner (f x) y)"
axioms: fixes X :: "'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"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" assumes "morphism_sheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"
simps_123: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x2 :: "48 word" shows "(case EtherSrc x2 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f2 x2"
spr_sim_def: shows "spr_sim \<equiv> \<lambda>t. \<lparr>sprFst = tFirst t, sprLst = tLast t, sprCRel = tObsC_abs t\<rparr>"
fd: shows "f ` ball x d \<subseteq> ball (f x) e'"
negative_cycle_dest_diag': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<not> cycle_free m n" shows "\<exists>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<and> len m i i xs < (0::'a)"
d_cp0: fixes d :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" assumes "profile_single d" shows "d X \<tau> = d (\<lambda>_. X \<tau>) \<tau>"
Gr_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "BNF_Def.Gr A f = {(a, f a) |a. a \<in> A}"
fds_perzeta_frac: fixes q :: "real" shows "fds_perzeta (frac q) = fds_perzeta q"
Abs_def: shows "Abs \<equiv> quot_type.abs (\<equiv>\<^sub>P) Abs_ltl_prop_equiv_quotient"
fcomp_apply: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'b"   and x :: "'c" shows "fcomp f g x = g (f x)"
expandFrame: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and C :: "'d"   and C' :: "'e" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>P \<sharp>* C'"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* C'" obtains p :: "(name \<times> name) list" and \<Psi>' :: "'b" and A\<^sub>P' :: "name list" and \<Psi>\<^sub>P' :: "'b" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn (p \<bullet> \<alpha>))" and "(p \<bullet> \<Psi>\<^sub>P) \<otimes> \<Psi>' \<simeq> \<Psi>\<^sub>P'" and "distinctPerm p" and "extractFrame P' = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>" and "A\<^sub>P' \<sharp>* P'" and "A\<^sub>P' \<sharp>* \<alpha>" and "A\<^sub>P' \<sharp>* (p \<bullet> \<alpha>)" and "A\<^sub>P' \<sharp>* C" and "bn (p \<bullet> \<alpha>) \<sharp>* C'" and "bn (p \<bullet> \<alpha>) \<sharp>* \<alpha>" and "bn (p \<bullet> \<alpha>) \<sharp>* P'" and "distinct A\<^sub>P'"
fold_graph_closed: fixes z :: "'a"   and B :: "'a set"   and y :: "'a" assumes "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "y \<in> A"
rminusinf_\<Upsilon>: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and x :: "real" assumes "isrlfm p"   and "\<not> Ifm (a # bs) (minusinf p)"   and "Ifm (x # bs) p" shows "\<exists>(s, m)\<in>set (\<Upsilon> p). Inum (a # bs) s / real_of_int m \<le> x"
a2: shows "j < dim_col (\<langle>|v\<rangle>| * A\<^sup>\<dagger>)"
lin_indpt_list_def: fixes fs :: "rat vec list" shows "gs.lin_indpt_list fs = (set fs \<subseteq> Rn \<and> distinct fs \<and> gs.lin_indpt (set fs))"
Seq_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e;; e') h s s'"
elims: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and x :: "'c"   and xa :: "nat"   and y :: "nat option" assumes "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w x xa = y" obtains q :: "'c" and n :: "nat" where "x = q" and "xa = n" and "y = (if semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n \<noteq> {} then Some (Min (semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n)) else None)"
lspasl_u: fixes Gamma :: "bool"   and h :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (h,0::'a\<triangleright>h) \<longrightarrow> Delta" shows "Gamma \<longrightarrow> Delta"
leg_simps_4: shows "C.dom f.cod.leg1 = f.cod.apex"
foldr_max_length: fixes lss :: "'a list list"   and n :: "nat" shows "foldr (\<lambda>xs. max (length xs)) lss n = fold max (map length lss) n"
num_params_tm_pow_le: fixes t :: "taylor_model"   and z :: "nat"   and prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and n :: "nat" assumes "num_params (tm_poly t) \<le> z" shows "num_params (tm_poly (tm_pow prec ord I a t n)) \<le> z"
tensor_preserves_iso: fixes f :: "'c"   and g :: "'c" assumes "C.iso f"   and "C.iso g" shows "C.iso (f \<otimes>\<^sub>C g)"
sb: shows "sb = filter is_Write\<^sub>s\<^sub>b rws"
that: fixes p1 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "red_single p p1 g (t + (lp f - lp g))" shows "thesis"
conjugate_id_iff: fixes a :: "complex_homo" shows "(conjugate a = a) = (is_real (to_complex a) \<or> a = \<infinity>\<^sub>h)"
simps_2: fixes b :: "nat"   and t1 :: "'b trm"   and t2 :: "'b trm" shows "fvi b (Eq t1 t2) = fvi_trm b t1 \<union> fvi_trm b t2"
trss_er_path_proc_const: fixes p :: "'b"   and c :: "'a list multiset"   and w :: "('b, 'c) label list"   and q :: "'b"   and c' :: "'a list multiset" assumes "(([entry fg p], c), w, [return fg q], c') \<in> trcl (trss fg)" shows "p = q"
hyps: fixes i :: "nat"   and S :: "nat \<Rightarrow> bool option" assumes "i < length ls"   and "ls ! i = (p, Read volatile a t # is, \<theta>, sb, \<D>, \<O>)" shows "share_all_until_volatile_write (ls[i := (p, is, \<theta>(t \<mapsto> v), sb @ [Read\<^sub>s\<^sub>b volatile a t v], \<D>', \<O>)]) S = share_all_until_volatile_write ls S"
d1_def: shows "ps2_P.d1 = prod_list ps2_P.dims1"