hcju/isabelleps huggingface dataset:
le_iff_sup: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "less_eq x y = (sup x y = y)"
BasicProperties: shows "\<forall>n. nonUniform (last (fe n)) \<and> prefixList (fe n) (fe (n + 1)) \<and> prefixList (ft n) (ft (n + 1)) \<and> execution trans sends start (fe n) (ft n) \<and> initial (hd (fe (n + 1)))"
NatTransFtorCod: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTransP \<eta>" shows "NTCatCod \<eta> = CatCod (NTDom \<eta>)"
fps_to_fls_eq_iff: fixes f :: "'b fps"   and g :: "'b fps" shows "(fps_to_fls f = fps_to_fls g) = (f = g)"
unit_factor_prime: fixes x :: "'a" assumes "prime x" shows "unit_factor x = (1::'a)"
P_subset: shows "P x \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V"
ends_del_arc: fixes u :: "'b" shows "arc_to_ends (del_arc u) = arc_to_ends G"
head_head: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "head (head p) = head p"
pgallery_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "pgallery (xs @ ys)" shows "pgallery xs"
p2: fixes x :: "'a" shows "isCont (\<lambda>x. x - P *\<^sub>V x) x"
map_poly_pCons_hom: fixes a :: "'b"   and p :: "'b poly" shows "map_poly to_fract (pCons a p) = pCons (to_fract a) (map_poly to_fract p)"
rel_funI: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "(A ===> B) f g"
affine_parallel_def: fixes S :: "'a set"   and T :: "'a set" shows "affine_parallel S T = (\<exists>a. T = (+) a ` S)"
negate_def: fixes g :: "'a gexp list" shows "negate g = gNot (fold gAnd g (Bc True))"
tObsC_tStep_eq_inv: fixes t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" assumes "tObsC t' = tObsC (t \<leadsto> s)" shows "\<exists>t'' s'. t' = t'' \<leadsto> s'"
tag_Times_def: fixes A :: "'a list set"   and B :: "'a list set" shows "tag_Times A B \<equiv> \<lambda>x. (\<approx>A `` {x}, {uu_. \<exists>x\<^sub>p x\<^sub>s. uu_ = \<approx>B `` {x\<^sub>s} \<and> x\<^sub>p \<in> A \<and> (x\<^sub>p, x\<^sub>s) \<in> Partitions x})"
eq: shows "\<Squnion> (value_flow \<Delta>'' ` {g. flow \<Delta>'' g}) = ennreal \<alpha>'"
continuous_blinfun_vec': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow>\<^sub>L real" assumes "\<And>i. continuous_on UNIV (f i)" shows "continuous_on UNIV (\<lambda>x. blinfun_vec (\<lambda>i. f i x))"
ring: shows "ring (RDirProd_list (map (\<lambda>i. R Quot I i) [0..<Suc n]))"
unapp_app: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unapp (app u\<^sub>1 u\<^sub>2) = Some (u\<^sub>1, u\<^sub>2)"
int_of_uint64_mod: fixes x :: "uint64"   and y :: "uint64" shows "int_of_uint64 (x mod y) = int_of_uint64 x mod int_of_uint64 y"
simps_1: fixes x :: "nat" shows "f4 x x = 1"
tllist_all2_cases: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys" obtains b :: "'c" and b' :: "'d" where "xs = TNil b" and "ys = TNil b'" and "Q b b'"   | x :: "'a" and xs' :: "('a, 'c) tllist" and y :: "'b" and ys' :: "('b, 'd) tllist" where "xs = TCons x xs'" and "ys = TCons y ys'" and "P x y" and "tllist_all2 P Q xs' ys'"
scale: fixes r :: "'a"   and b :: "'b" shows "f (r *a b) = r *b f b"
ideal_quotient_idealTr: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and x :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "ideal R A"   and "ideal R B"   and "ideal R C"   and "x \<in> carrier R"   and "\<forall>c\<in>C. x \<cdot>\<^sub>r c \<in> A \<dagger>\<^sub>R B" shows "f \<in> {j. j \<le> n} \<rightarrow> set_mult R B C \<longrightarrow> x \<cdot>\<^sub>r \<Sigma>\<^sub>e R f n \<in> A"
eq_general: fixes B :: "'a set"   and A :: "'b set"   and h :: "'b \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> 'c"   and \<phi> :: "'b \<Rightarrow> 'c" assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma> (h x) = \<phi> x" shows "sum \<phi> A = sum \<gamma> B"
distinct_27: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x31@[x32] \<noteq> x41 ;; x42"
tObsC_ii_abs_conv: fixes v :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace" shows "(v \<in> tObsC_ii_abs t) = (\<exists>t'. t' \<in> SPR.jkbpC \<and> tObsC t' = tObsC t \<and> v = tLast t')"
Inf_greatest: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'g set"   and z :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq z x" shows "less_eq z (Inf A)"
concat_inf_truncate_nonempty: fixes z :: "nat \<Rightarrow> 'd list"   and i :: "nat" assumes "z i \<noteq> []" shows "concat (inf_truncate z i) \<noteq> []"
qfp_as_extreme_bound: fixes c :: "'a" assumes "extreme_bound A (\<sqsubseteq>) C c"   and "\<forall>x\<in>A. x \<sqsubseteq> f x \<or> (\<forall>y\<in>A. y \<sqsubseteq> x \<longrightarrow> f y \<sqsubseteq> f x)" shows "f c \<sim> c"
C_extension_options: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat"   and x :: "'e" assumes "vs @ xs \<in> C M2 M1 \<Omega> V m i"   and "mcp (vs @ xs @ [x]) V vs"   and "x \<in> inputs M2"   and "0 < i" shows "vs @ xs @ [x] \<in> C M2 M1 \<Omega> V m (Suc i) \<or> vs @ xs \<in> RM M2 M1 \<Omega> V m i"
integral_spike: fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "integral T f = integral T g"
assms_7: shows "(\<phi>\<^sub>S \<Psi> s) (v, a') = Some True"
Left_Coset_memE: fixes a :: "'a"   and x :: "'a"   and H :: "'a set" assumes "a \<in> x \<cdot>| H" obtains h :: "'a" where "h \<in> H" and "a = x \<cdot> h"
hd_take: fixes j :: "nat"   and xs :: "'b list" assumes "0 < j" shows "hd (take j xs) = hd xs"
derivable_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and P :: "'b eclause set"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> k C' = ((\<exists>P1 P2. P1 \<in> S \<and> P2 \<in> S \<and> P = {P1, P2} \<and> basic_superposition.superposition trm_ord sel pos_ord filter_trms P1 P2 C \<sigma> k C') \<or> (\<exists>P1. P1 \<in> S \<and> P = {P1} \<and> basic_superposition.factorization trm_ord sel filter_trms P1 C \<sigma> k C') \<or> (\<exists>P1. P1 \<in> S \<and> P = {P1} \<and> basic_superposition.reflexion trm_ord sel filter_trms P1 C \<sigma> k C'))"
f_map: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and r :: "'a"   and m :: "'b" assumes "VectorSpaceHom smult V smult' T"   and "r \<in> UNIV"   and "m \<in> V" shows "T (smult r m) = smult' r (T m)"
le_epred: fixes l :: "enat"   and k :: "enat" shows "(l < epred k) = (eSuc l < k)"
betw4_weak: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]] \<and> [[a c d]] \<or> [[a b c d]] \<or> [[a b d]] \<and> [[b c d]] \<or> [[a b d]] \<and> [[b c d]]" shows "[[a b c d]]"
Cd_simps_1: fixes d :: "'b" shows "Cd d {} = {}"
is_expansion_lift: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and C :: "'a" assumes "basis_wf (b # basis)"   and "is_expansion C basis" shows "is_expansion (lift_ms C) (b # basis)"
rel_conversep: fixes R1 :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'c \<Rightarrow> bool" shows "rel_sp\<^sub>\<mu> R1\<inverse>\<inverse> R2\<inverse>\<inverse> = (rel_sp\<^sub>\<mu> R1 R2)\<inverse>\<inverse>"
lg_bin: fixes c :: "cell list" assumes "\<forall>rs n. c \<noteq> Oc \<up> Suc rs @ Bk \<up> n"   and "bl2wc c = 2 ^ lg (Suc (bl2wc c)) 2 - Suc 0" shows "bl2wc c = 0"
add: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
sums_Suc_imp: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a" assumes "f 0 = (0::'a)"   and "(\<lambda>n. f (Suc n)) sums s" shows "f sums s"
has_domain_iff_has_codomain: fixes f :: "'b" shows "(D.domains f \<noteq> {}) = (D.codomains f \<noteq> {})"
trans_from_irrefl: fixes A :: "'a set"   and zs :: "'a list"   and ys :: "'a list"   and xs :: "'a list"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite A"   and "zs \<in> lists A"   and "ys \<in> lists A"   and "xs \<in> lists A"   and "\<forall>x\<in>A. \<not> gt x x"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "ext gt zs ys"   and "ext gt ys xs" shows "ext gt zs xs"
assms_1: shows "\<forall>t. subalgebra M (F t)"
simps_12: fixes v :: "String.literal"   and va :: "typ" shows "eta_reducible (Ct v va) = False"
prebicategory_with_units_axioms_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b" shows "prebicategory_with_units_axioms V H \<i> \<equiv> (\<forall>a. weak_composition.weak_unit V H a \<longrightarrow> partial_magma.in_hom V (\<i> a) (H a a) a) \<and> (\<forall>a. weak_composition.weak_unit V H a \<longrightarrow> category.iso V (\<i> a))"
isOK_update_error: fixes m :: "'c + 'b"   and f :: "'c \<Rightarrow> 'a" shows "isOK (try m catch (\<lambda>x. Inl (f x))) = isOK m"
toSet_ODList: fixes xs :: "'c list" assumes "distinct xs"   and "sorted xs" shows "toSet (ODList xs) = set xs"
steps_epsilon: fixes p :: "bool list"   and q :: "bool list"   and w :: "'b list" shows "((p, q) \<in> NAe.steps epsilon w) = (w = [] \<and> p = q)"
assms_3: shows "(G ===> (=)) \<alpha>n \<alpha>n2"
distinct_41: fixes x31 :: "('a, 'b, 'c, 'd) seqp"   and x32 :: "('a, 'b, 'c, 'd) seqp"   and x71 :: "'d"   and x72 :: "'a \<Rightarrow> 'b"   and x73 :: "('a, 'b, 'c, 'd) seqp" shows "x31 \<oplus> x32 \<noteq> {x71}send(x72) . x73"
map_append: fixes f :: "'h \<Rightarrow> 'g"   and xs :: "'h list"   and ys :: "'h list" shows "map f (xs @ ys) = map f xs @ map f ys"
imirror_bounds_iff: fixes n :: "nat"   and I :: "nat set"   and l :: "nat"   and r :: "nat" shows "(n \<in> imirror_bounds I l r) = (\<exists>x\<in>I. n = l + r - x)"
sets_UNIV: fixes A :: "'b set" shows "A \<in> sets (count_space UNIV)"
exhaust: fixes y :: "('b, 'a) optionT" obtains x :: "'a" where "y = OptionT x"
assms_1: shows "subst_domain \<delta> \<union> range_vars \<delta> \<subseteq> fv\<^sub>s\<^sub>t S"
simps_3: fixes c :: "'b" shows "nullable (Atom c) = False"
axioms_1: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "ab_semigroup_add_ow S pls" shows "semigroup_add_ow S pls"
primary_decomposition_Zn: obtains ns :: "nat list" where "DirProds (\<lambda>n. Z (ns ! n)) {..<length ns} \<cong> G" and "\<forall>n\<in>set ns. n = 0 \<or> (\<exists>p k. normalization_semidom_class.prime p \<and> 0 < k \<and> n = p ^ k)"
quantity_updated_borel: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and Mkt :: "('a, 'c) discrete_market" assumes "\<forall>n. \<forall>asset\<in>support_set p. p asset (Suc n) \<in> borel_measurable (F n)"   and "\<forall>n. \<forall>asset\<in>support_set p. prices Mkt asset n \<in> borel_measurable (F n)" shows "\<forall>n. cls_val_process Mkt p n \<in> borel_measurable (F n)"
col_def: fixes A :: "'d mat"   and j :: "nat" shows "col A j = Matrix.vec (dim_row A) (\<lambda>i. A $$ (i, j))"
\<pi>: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "(x, y) \<in> r" shows "(\<pi> x, \<pi> y) \<in> s"
inv_p: shows "\<forall>x\<in>A. f' x = f (inv p x)"
price_structure_borel_measurable: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and pyf :: "'c \<Rightarrow> real"   and T :: "nat"   and \<pi> :: "real"   and pr :: "nat \<Rightarrow> 'c \<Rightarrow> real" assumes "disc_filtr_prob_space M F"   and "disc_filtr_prob_space.price_structure M F pyf T \<pi> pr" shows "pr T \<in> borel_measurable (F T)"
simps: fixes fs :: "'b list"   and rs :: "'b list" shows "list (fs, rs) = fs @ rev rs"
PPs_closed_single: fixes x :: "'a"   and X :: "'a set"   and e :: "nat" assumes "x \<in> X" shows "monomial e x \<in> .[X]"
rel_prod_inject: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'e"   and b :: "'g"   and c :: "'f"   and d :: "'h" shows "rel_prod R1 R2 (a, b) (c, d) = (R1 a c \<and> R2 b d)"
change_on_0_hyp_change: fixes i :: "nat"   and t :: "nat" assumes "change_on_0 (block i t) (gap i t)" shows "\<sigma> i (\<tau> i \<triangleright> (t + 1)) \<noteq> \<sigma> i (\<tau> i \<triangleright> (gap i t - 1))"
image_eqvt: fixes p :: "perm"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "p \<bullet> f ` A = (p \<bullet> f) ` (p \<bullet> A)"
fold_premise: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "healthy t"   and "\<guillemotleft> P \<guillemotright> \<tturnstile> t \<guillemotleft> Q \<guillemotright>" shows "\<forall>s. P s \<longrightarrow> 1 \<le> t \<guillemotleft> Q \<guillemotright> s"
prime_multiplicity_gt_zero_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "x \<noteq> zero" shows "(0 < factorial_semiring.multiplicity times one p x) = dvd.dvd times p x"
ObjMap_def: shows "ObjMap = []\<^sub>\<circ>"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "a \<cong> a'" obtains f :: "'a" where "\<guillemotleft>f : a \<rightarrow> a'\<guillemotright> \<and> local.iso f"
simps: fixes E :: "('b \<times> 'b) set"   and a1 :: "'b"   and a2 :: "'b list"   and a3 :: "'b" shows "path E a1 a2 a3 = ((\<exists>u. a1 = u \<and> a2 = [] \<and> a3 = u) \<or> (\<exists>u v l w. a1 = u \<and> a2 = u # l \<and> a3 = w \<and> (u, v) \<in> E \<and> path E v l w))"
op_elem_def: fixes oper :: "('d, 'c) operation" shows "op_elem oper \<equiv> case oper of Create i e \<Rightarrow> e | Delete is e \<Rightarrow> e | Append i e \<Rightarrow> e | Expunge e mo i \<Rightarrow> e | Store e mo i \<Rightarrow> e"
uniformly_continuous_on_sequentially: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "uniformly_continuous_on s f = (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and> (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0 \<longrightarrow> (\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0)"
induct: fixes P :: "('a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> ('a \<times> 'c) list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "('a \<times> 'c) list"   and a2 :: "('a \<times> 'c) list" assumes "\<And>f k v as k' v' bs. \<lbrakk>k' < k \<Longrightarrow> P f ((k, v) # as) bs; \<lbrakk>\<not> k' < k; k < k'\<rbrakk> \<Longrightarrow> P f as ((k', v') # bs); \<lbrakk>\<not> k' < k; \<not> k < k'\<rbrakk> \<Longrightarrow> P f as bs\<rbrakk> \<Longrightarrow> P f ((k, v) # as) ((k', v') # bs)"   and "\<And>f bs. P f [] bs"   and "\<And>f as. P f as []" shows "P a0 a1 a2"
polytope_lowdim_imp_simplex: fixes P :: "'b set" assumes "polytope P"   and "aff_dim P \<le> 1" obtains n :: "int" where "n simplex P"
List_rec_unfold: fixes a :: "('c, unit) Old_Datatype.node set"   and c :: "'b"   and d :: "('c, unit) Old_Datatype.node set \<Rightarrow> ('c, unit) Old_Datatype.node set \<Rightarrow> 'b \<Rightarrow> 'b" shows "List_rec a c d = List_case c (\<lambda>x y. d x y (cut (\<lambda>M. List_rec M c d) (pred_sexp\<^sup>+) a y)) a"
induct: fixes P :: "'c ltl set \<Rightarrow> 'c ltl \<Rightarrow> bool"   and a0 :: "'c ltl set"   and a1 :: "'c ltl" assumes "\<And>\<A>. P \<A> true"   and "\<And>\<A>. P \<A> false"   and "\<And>\<A> \<phi> \<psi>. \<lbrakk>P \<A> \<phi>; P \<A> \<psi>\<rbrakk> \<Longrightarrow> P \<A> (\<phi> and \<psi>)"   and "\<And>\<A> \<phi> \<psi>. \<lbrakk>P \<A> \<phi>; P \<A> \<psi>\<rbrakk> \<Longrightarrow> P \<A> (\<phi> or \<psi>)"   and "\<And>\<A> v. P \<A> p(v)"   and "\<And>\<A> v. P \<A> (np(v))"   and "\<And>\<A> v. P \<A> (X v)"   and "\<And>\<A> v. P \<A> (G v)"   and "\<And>\<A> v. P \<A> (F v)"   and "\<And>\<A> v va. P \<A> (v U va)" shows "P a0 a1"
map_invar_impl_les: fixes m :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and s :: "bdd"   and s' :: "bdd" assumes "mi.map_invar_impl m s"   and "mi.les s s'" shows "mi.map_invar_impl m s'"
horizontal_chop_own_trans: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<longrightarrow> own u = own w"
regular_languages_def: fixes A :: "'a \<Rightarrow> nat option" shows "regular_languages A \<equiv> {ta_lang TA |TA. ranked_tree_automaton TA A}"
fg_local_transition_from_round: fixes A :: "('f, 'g, 'h, 'i) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> ('g, 'f, 'h) config"   and HOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and SHOs :: "nat \<Rightarrow> 'f \<Rightarrow> 'f set"   and coords :: "nat \<Rightarrow> 'f \<Rightarrow> 'f"   and p :: "'f"   and r :: "nat" assumes "fg_run A rho HOs SHOs coords" obtains n :: "nat" where "round (rho n) p = r" and "fg_start_round rho p (Suc r) = Suc n" and "fg_local A p (HOs r p) (coords (Suc r) p) (rho n) (rho (Suc n))"
Times_mono: fixes r1 :: "'a language"   and s1 :: "'a language"   and r2 :: "'a language"   and s2 :: "'a language" assumes "r1 \<le> s1"   and "r2 \<le> s2" shows "Times r1 r2 \<le> Times s1 s2"
mustT_If_L: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and tst :: "'a"   and s :: "'b"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com" assumes "tval tst s"   and "PL.mustT tval aval (if tst then c1 else c2) s" shows "PL.mustT tval aval c1 s"
poly_induct4: fixes p :: "nat \<Rightarrow> 'a"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "p \<in> carrier P"   and "\<And>p q. \<lbrakk>q \<in> carrier P; p \<in> carrier P; Q p; Q q\<rbrakk> \<Longrightarrow> Q (p \<oplus>\<^bsub>P\<^esub> q)"   and "\<And>a. a \<in> carrier R \<Longrightarrow> Q (monom P a 0)"   and "\<And>a n. a \<in> carrier R \<Longrightarrow> Q (monom P a (Suc n))" shows "Q p"
simps_2: fixes uv :: "'b"   and uw :: "real"   and l :: "'b kdt"   and r :: "'b kdt" shows "height (Node uv uw l r) = max (height l) (height r) + 1"
simps: fixes R :: "real interval"   and I :: "real interval"   and R' :: "real interval"   and I' :: "real interval" shows "Complex_Interval R I - Complex_Interval R' I' = Complex_Interval (R - R') (I - I')"
Gauss_Jordan_in_ij_PA_def': fixes A' :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec"   and i :: "'b"   and j :: "'c" shows "Gauss_Jordan_in_ij_PA A' i j = (let P = fst A'; A = snd A'; n = LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n; interchange_A = interchange_rows A i n; A' = mult_row interchange_A i ((1::'a) / interchange_A $ i $ j); interchange_P = interchange_rows P i n; P' = mult_row interchange_P i ((1::'a) / interchange_A $ i $ j) in (Gauss_Jordan_wrapper i j P' interchange_A, Gauss_Jordan_wrapper i j A' interchange_A))"
quality_increases_rt_fresherD: fixes \<xi> :: "state"   and \<xi>' :: "state"   and ip :: "nat" assumes "quality_increases \<xi> \<xi>'"   and "ip \<in> kD (rt \<xi>)" shows "rt \<xi> \<sqsubseteq>\<^bsub>ip\<^esub> rt \<xi>'"
a: shows "length (remzero (maskedDB_zero (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt)))))))) (length (nat_to_bv (p * q)) - Suc 0) @ sha1 (generate_M' (sha1 m) salt) @ BC)) \<le> length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))))) @ sha1 (generate_M' (sha1 m) salt) @ BC) - (8 - (length (nat_to_bv (p * q)) - Suc 0) mod 8)"
tmsubst0_nb: fixes t :: "tm"   and a :: "tm" assumes "tmbound0 t" shows "tmbound0 (tmsubst0 t a)"
trading_strategy_adapted: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and Mkt :: "(bool stream, 'b) discrete_market" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "borel_adapt_stoch_proc G (val_process Mkt p)"
widen_Class: fixes P :: "'c prog"   and T :: "ty"   and C :: "String.literal" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C) \<or> C = Object \<and> (\<exists>A. T = A\<lfloor>\<rceil>))"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRel Enc"
simps_8: shows "pratt2_max \<infinity>\<^sub>2 \<infinity>\<^sub>2 = \<infinity>\<^sub>2"
UNlift01_BrnFT: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" shows "UNlift01 cl dl P F (BrnFT cl dl) = BrnFT dl cl"
module_hom_sub: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "module_hom s1 s2 f"   and "module_hom s1 s2 g" shows "module_hom s1 s2 (\<lambda>x. f x - g x)"
unit_div_1_div_1: fixes a :: "'a" assumes "is_unit a" shows "(1::'a) div ((1::'a) div a) = a"