hcju/isabelleps huggingface dataset:
ball_imageD: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>f ` A. P x" shows "\<forall>x\<in>A. P (f x)"
length_upt: fixes i :: "nat"   and j :: "nat" shows "\<^bold>|[i..<j]\<^bold>| = j - i"
assms: shows "n i-ns\<rightarrow>\<^sub>d* n'"
lhd_ldropn: fixes n :: "nat"   and xs :: "'c llist" assumes "enat n < llength xs" shows "lhd (ldropn n xs) = lnth xs n"
sets_into_space: fixes x :: "'a set" assumes "x \<in> sigma_sets \<Omega> E" shows "x \<subseteq> \<Omega>"
aless: fixes n :: "int"   and m :: "int" shows "(n < m) = (ant n < ant m)"
visible_def: shows "visible \<equiv> {a. \<exists>q. en a q \<and> int q \<noteq> int (ex a q)}"
hyps_1: shows "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"
ldrop_LNil: fixes n :: "enat" shows "ldrop n <> = <>"
measure_pmf_single: fixes M :: "'c pmf"   and x :: "'c" shows "measure_pmf.prob M {x} = pmf M x"
f1: shows "3 \<le> rk {A, C, A', P}"
sum_negf: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(\<Sum>x\<in>A. - f x) = - sum f A"
charpoly_transpose: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "charpoly (Finite_Cartesian_Product.transpose A) = charpoly A"
assms: shows "\<forall>l. l el elements M \<longrightarrow> formulaEntailsLiteral (F @ val2form (decisionsTo l M)) l"
rep_nat_term_inj: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat_term rep_nat_term splus"   and "rep_nat_term x = rep_nat_term y" shows "x = y"
cut3: fixes A :: "fmla"   and B :: "fmla"   and C :: "fmla"   and D :: "fmla"   and H :: "fmla set" assumes "{A, B, C} \<turnstile> D"   and "H \<turnstile> A"   and "H \<turnstile> B"   and "H \<turnstile> C" shows "H \<turnstile> D"
rank_senior_senior: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "rank (senior x n) n = rank x n"
assms: shows "(\<lambda>n. real (card (A \<inter> {..<n})) / real n) \<longlonglongrightarrow> l"
diff_closed: fixes V :: "'b set"   and x :: "'b"   and y :: "'b" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "x - y \<in> V"
Ds: shows "Ds = (Cs @\<^sub>p Cs'') @\<^sub>p Cs'"
f_non_negative: fixes e :: "nat \<times> nat" shows "(0::'capacity) \<le> f e"
chamber_system_into: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "order.greater_eq (ChamberComplex.chamber_system Y) (f \<turnstile> ChamberComplex.chamber_system X)"
prems_2: shows "s' e\<longrightarrow>\<^bsub>SES\<^esub> sn"
extends_subst_fun_upd_new: fixes \<sigma> :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and t :: "'a"   and \<tau> :: "'b \<Rightarrow> 'a option" assumes "\<sigma> x = None" shows "extends_subst (\<sigma>(x \<mapsto> t)) \<tau> = (extends_subst \<sigma> \<tau> \<and> \<tau> x = Some t)"
edges_of_walk_edge: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "Digraph G"   and "Digraph.walk G xs"   and "(v, w) \<in> Digraph.edges_of_walk xs" shows "v \<rightarrow>\<^bsub>G\<^esub> w"
eq_up_to_le: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and k :: "nat"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a" assumes "IFC prob"   and "k \<le> n"   and "IFC_def.eq_up_to \<pi> n \<pi>'" shows "IFC_def.eq_up_to \<pi> k \<pi>'"
pow_simps_1: fixes P :: "pol" shows "Commutative_Ring.pow 0 P = Pc 1"
i: shows "(f i \<longlongrightarrow> a i) F"
add_pos_neg_is_real: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "b < (0::'a)" shows "is_real (a + b)"
ceiling_def: fixes x :: "'a" shows "\<lceil>x\<rceil> = - \<lfloor>- x\<rfloor>"
r2_2: shows "R2 = stabilizer G \<phi> x #> r2"
eventually_def: fixes F :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool" shows "\<diamond>F \<equiv> \<not> \<box>\<not> F"
lnull_cons_def: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and s :: "'c" shows "lnull_cons g s = lnull (lunstream g s)"
sorted_distinct_ffold_ord: fixes l :: "'b list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c" assumes "sorted l"   and "distinct l" shows "ffold_ord f (fset_of_list l) b = fold f l b"
preserves_arr: fixes f :: "'a" assumes "hom.arr f" shows "hom'.arr (trg f \<star> f \<star> e)"
SC_mini_to_full: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "image_mset to_mini_formula \<Gamma> \<Rightarrow> image_mset to_mini_formula \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
lv_rel_def: shows "lv_rel = br eucl_of_list (\<lambda>xs. length xs = DIM('c))"
iMODb_modD: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "x \<in> [ r, mod m, c ]" shows "x mod m = r mod m"
dimcf: shows "\<forall>n. complex_of_real c \<cdot>\<^sub>m f n \<in> carrier_mat d d"
l_cancel: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<otimes> a = c \<otimes> b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a = b"
max_list_as_Greatest: fixes xs :: "'b list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> x \<le> y \<or> y \<le> x" shows "Missing_Lemmas.max_list xs = (GREATEST a. a \<in> set xs)"
domIff: fixes a :: "'c"   and m :: "'c \<Rightarrow> 'd option" shows "(a \<in> dom m) = (m a \<noteq> None)"
cases: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and a1 :: "'s"   and a2 :: "'a list"   and a3 :: "'a literal multiset" assumes "eligible S a1 a2 a3" obtains DA :: "'a literal multiset" and As :: "'a list" and \<sigma> :: "'s" where "a1 = \<sigma>" and "a2 = As" and "a3 = DA" and "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0 \<cdot>a \<sigma>) (DA \<cdot> \<sigma>)"
primitive_part_fract_def: fixes p :: "'a fract poly" shows "primitive_part_fract p = primitive_part (map_poly (fst \<circ> quot_of_fract) (smult (to_fract (Lcm (snd ` quot_of_fract ` set (coeffs p)))) p))"
interval_extension_powr: fixes p :: "nat" shows "interval_extension2 (powr_float_interval p) (powr)"
simps_1: fixes x :: "'b"   and l :: "nat"   and k :: "nat"   and A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3_c x l k [] A = A"
no_Empty_merge: fixes h1 :: "'b heap"   and h2 :: "'b heap" assumes "no_Empty h1"   and "no_Empty h2" shows "no_Empty (merge h1 h2)"
multiple_proper_design: fixes n :: "nat" assumes "0 < n" shows "proper_design \<V> (multiple_blocks n)"
that: fixes Z :: "V" assumes "X \<approx>\<^sub>\<circ> Z"   and "Z \<inter>\<^sub>\<circ> Y = []\<^sub>\<circ>"   and "Z \<subseteq>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)" shows "thesis"
preserves_arr: fixes f :: "'j" assumes "J.arr f" shows "C.arr (A'.map f)"
map_option_eq_Some: fixes f :: "'e \<Rightarrow> 'd"   and xo :: "'e option"   and y :: "'d" shows "(map_option f xo = Some y) = (\<exists>z. xo = Some z \<and> f z = y)"
assms_2: shows "(\<Union>io\<in>L\<^sub>i\<^sub>n M1 T. {io} \<times> B M1 io \<Omega>) \<subseteq> (\<Union>io\<in>L\<^sub>i\<^sub>n M2 T. {io} \<times> B M2 io \<Omega>)"
exhaust: fixes y :: "'a rose_tree" obtains x :: "'a \<times> 'a rose_tree list" where "y = Tree x"
elims: fixes x :: "'b set"   and xa :: "'b set list"   and y :: "'b set list" assumes "partList4 x xa = y" obtains s :: "'b set" where "x = sa__" and "xa = []" and "y = []"   | s :: "'b set" and t :: "'b set" and ts :: "'b set list" where "x = s" and "xa = t # ts" and "y = (if s = {} then t # ts else if s \<inter> t = {} then t # partList4 s ts else if t - s = {} then t # partList4 (s - t) ts else t \<inter> s # (t - s) # partList4 (s - t) ts)"
wf_abs_eff: fixes name :: "char list"   and pre :: "(nat \<times> nat) list"   and eff :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pre, eff, cost)" shows "wf_partial_state (map (\<lambda>(uu_, v, uu_, a). (v, a)) eff)"
blocks_def: fixes m :: "M"   and w :: "W"   and \<mu> :: "(M \<times> W) set" shows "Non_Strict.blocks m w \<mu> = (w \<in> Non_Strict.m_strictly_prefers m \<mu> \<and> m \<in> Non_Strict.w_strictly_prefers w \<mu>)"
span_zero: fixes S :: "'d set" shows "(0::'d) \<in> cspan S"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "Oldp x1 \<noteq> Guard x2"
subdiv_props_1: shows "chain_subdiv_chain one_chain1 subdiv"
run_scons_elim: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and a :: "nat \<times> 'b"   and r :: "(nat \<times> 'b) stream"   and p :: "nat \<times> 'b" assumes "grun A w V (a ## r) p" obtains "case p of (k, p) \<Rightarrow> w !! k \<in> alphabet A \<and> a \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V" and "grun A w V r a"
ACI_idem: fixes r :: "'b rexp" shows "ACI (Plus r r) r"
obj_simps_4: fixes a :: "'d" assumes "D.obj a" shows "D.dom a = a"
fr_in_3: shows "frontier (inside (path_image c1 \<union> path_image c)) = path_image c1 \<union> path_image c"
subgroup_generated_is_monoid: fixes H :: "'a set" shows "Group_Theory.monoid (additive.subgroup_generated H) (+) \<zero>"
all_chains_no_call_upd: fixes c :: "char list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and x :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" assumes "all_chains (no_call_to c) \<Gamma> rs" shows "\<Gamma>(c \<mapsto> x),\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t = \<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t"
that_2: fixes i :: "'x" assumes "i \<in> Basis" shows "k \<bullet> i \<le> d \<bullet> i"
matrix_component_le_onorm: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and i :: "'b"   and j :: "'a" shows "\<bar>A $h i $h j\<bar> \<le> onorm ((*v) A)"
proj_preserves: fixes x :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A" shows "Equiv_Relations.proj r x \<in> A // r"
has_derivative_locally_injective: fixes a :: "'a"   and S :: "'a set"   and g' :: "'b \<Rightarrow> 'a"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "a \<in> S"   and "open S"   and "bounded_linear g'"   and "g' \<circ> f' a = id"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x)"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e" obtains r :: "real" where "0 < r" and "ball a r \<subseteq> S" and "inj_on f (ball a r)"
assms_1: shows "vs @ xs \<notin> TS M2 M1 \<Omega> V m i"
simps: fixes V :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set" shows "filterQueryNeg V Q = Set.filter (\<lambda>(A, uu_). A \<subseteq> V) Q"
lesub_step_typeD: fixes A :: "(nat \<times> 'c) set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "(nat \<times> 'c) set"   and x :: "nat"   and y :: "'c" assumes "A {\<sqsubseteq>\<^bsub>r\<^esub>} B"   and "(x, y) \<in> A" shows "\<exists>y'. (x, y') \<in> B \<and> y \<sqsubseteq>\<^bsub>r\<^esub> y'"
tensor_relations_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and MN :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "TR\<^bsub>R\<^esub> M N MN = LSM\<^bsub>R\<^esub> MN (TR1 R M N MN \<union> TR2 R M N MN \<union> TR3 R M N MN \<union> TR4 R M N MN)"
one: shows "x ; 1 \<cdot> (- x \<cdot> 1') \<le> x ; x\<^sup>\<smile> ; (- x \<cdot> 1')"
H: shows "\<forall>x\<in>S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S"
u_def: shows "u = (\<lambda>y n. Busemann_function_at xi ((f ^^ n) y) y)"
satC_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c :: "('a, 'b) lit list" shows "U.satC \<xi> c = F.satC \<xi> c"
emeasure_bind_prob_algebra: fixes A :: "'c measure"   and N :: "'c measure"   and B :: "'c \<Rightarrow> 'd measure"   and L :: "'d measure"   and X :: "'d set" assumes "A \<in> space (prob_algebra N)"   and "B \<in> N \<rightarrow>\<^sub>M prob_algebra L"   and "X \<in> sets L" shows "emeasure (A \<bind> B) X = \<integral>\<^sup>+ x. emeasure (B x) X \<partial>A"
not_is_Tensor_Unity: shows "\<not> is_Tensor \<^bold>\<I>"
augment_cf_def: fixes edges :: "(nat \<times> nat) set"   and cap :: "'capacity" shows "f.cf.augment_cf edges cap \<equiv> \<lambda>e. if e \<in> edges then f.cf e - cap else if prod.swap e \<in> edges then f.cf e + cap else f.cf e"
abs_apply_terms_def: fixes M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and \<alpha> :: "nat \<times> bool \<Rightarrow> 'c set" shows "M \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t \<alpha> \<equiv> (\<lambda>t. t \<cdot>\<^sub>\<alpha> \<alpha>) ` M"
equiv_class_self: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and a :: "'b" assumes "equiv A r"   and "a \<in> A" shows "a \<in> r `` {a}"
absorb_iff1: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (a \<sqinter> b = a)"
cindex_def: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" shows "cindex a b f = (\<Sum>x | jump f x \<noteq> 0 \<and> a < x \<and> x < b. jump f x)"
is_conjugation_def: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" shows "is_conjugation f g \<equiv> \<forall>x y. (f x \<cdot> y = 0) = (x \<cdot> g y = 0)"
binomial_antimono: fixes k :: "nat"   and k' :: "nat"   and n :: "nat" assumes "k \<le> k'"   and "n div 2 \<le> k"   and "k' \<le> n" shows "n choose k' \<le> n choose k"
vector_mult_closed: fixes y :: "'a"   and x :: "'a" assumes "y \<sqinter> top = y" shows "x \<sqinter> y \<sqinter> top = x \<sqinter> y"
of_nat_catalan_Suc': fixes n :: "nat" shows "of_nat (catalan (Suc n)) = of_nat (2 * (2 * n + 1)) / of_nat (n + 2) * of_nat (catalan n)"
max_fun_diff_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" shows "max_fun_diff f g \<equiv> maxim {a \<in> Field r. f a \<noteq> g a}"
dlet2: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and p :: "pat"   and mn :: "char list list"   and locs :: "locn \<times> locn" assumes "evaluate ck env s1 e (s2, Rval v1) \<and> allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s2) p v1 [] = No_match" shows "evaluate_dec ck mn env s1 (Dlet locs p e) (s2, Rerr (Rraise Bindv))"
elims_3: fixes x :: "'b infty"   and xa :: "'b infty" assumes "\<not> x < xa" obtains uu_ :: "'b infty" where "x = Infty" and "xa = uu_"   | a :: "'b" and b :: "'b" where "x = Num a" and "xa = Num b" and "\<not> a < b"
Sigma_empty_iff: fixes I :: "'d set"   and X :: "'d \<Rightarrow> 'e set" shows "(Sigma I X = {}) = (\<forall>i\<in>I. X i = {})"
less_hd_imp_less: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs < [hd ys]" shows "xs < ys"
tail_in_verts: fixes G :: "('e, 'f) pre_digraph"   and e :: "'f" assumes "wf_digraph G"   and "e \<in> arcs G" shows "tail G e \<in> verts G"
mult_smult_distrib: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat"   and k :: "'c" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * (k \<cdot>\<^sub>m B) = k \<cdot>\<^sub>m (A * B)"
real_roots_of_int_poly_1: fixes p :: "int poly" assumes "p \<noteq> 0" shows "set (real_roots_of_int_poly p) = {x. poly (real_of_int_poly p) x = 0}"
tarski: fixes x :: "'a" shows "(x \<noteq> 0) = (1 ; x ; 1 = 1)"
a: shows "a \<sharp> (\<theta>_c, c, N)"
S1: shows "S1 = (\<lambda>s. if s ''a'' \<le> s ''z'' then s else s(''y'' := s ''a'' * s ''b'', ''z'' := s ''a'', ''g'' := s ''b''))"
ltl_inf_llist: fixes f :: "nat \<Rightarrow> 'c" shows "ltl (inf_llist f) = inf_llist (\<lambda>n. f (Suc n))"
bind_return_distr: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and N :: "'b measure" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "M \<bind> return N \<circ> f = distr M N f"
add: fixes x :: "complex"   and y :: "complex"   and xa :: "'d" shows "(x + y) *\<^sub>C xa = x *\<^sub>C xa + y *\<^sub>C xa"
det_def': fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A = (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>i = 0..<n. A $$ (i, p i)))"
lowEquivalence_def: fixes s :: "'state"   and s' :: "'state" shows "s \<approx>\<^sub>L s' \<equiv> \<forall>V\<in>L. state_val s V = state_val s' V"
poincare_on_ray_def: fixes p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo" shows "poincare_on_ray p a b = (poincare_between a p b \<or> poincare_between a b p)"
pure_universal_weak_until_simp: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" assumes "pure_universal \<phi>"   and "pure_universal \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<phi> or\<^sub>n \<psi>"
inject_2: fixes x31 :: "'c"   and x32 :: "'d"   and y31 :: "'c"   and y32 :: "'d" shows "(Yield x31 x32 = Yield y31 y32) = (x31 = y31 \<and> x32 = y32)"
induct: fixes P :: "'a negation_type list \<Rightarrow> bool"   and a0 :: "'a negation_type list" assumes "P []"   and "\<And>e es. P es \<Longrightarrow> P (Pos e # es)"   and "\<And>e es. P es \<Longrightarrow> P (Neg e # es)" shows "P a0"
wtl_not_top: fixes ls :: "'b list"   and c :: "'a list"   and pc :: "nat"   and s :: "'a" assumes "wtl ls c pc s \<noteq> \<top>" shows "s \<noteq> \<top>"
all_consts_def: shows "all_consts = fst |`| rs |\<union>| C"
x_c_lb2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> (y \<cdot> 1\<^sub>\<pi>) \<le> y \<cdot> 1\<^sub>\<pi>"
gen_model_def: fixes M :: "('d, 'e, 'f) KripkeStructure"   and w :: "'f" shows "gen_model M w \<equiv> let ws' = worlds M \<inter> (\<Union> (range (relations M)))\<^sup>* `` {w} in \<lparr>worlds = ws', relations = \<lambda>a. Restr (relations M a) ws', valuation = valuation M\<rparr>"
Dom_in_Obj: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Dom f \<in> {A. |A| <o \<AA>}"
bitstrings_finite: fixes m :: "nat" shows "finite {xs. length xs = m}"
cSim: shows "weak_reduction_correspondence_simulation (indRelRTPO TRel) (STCal Source Target)"
symetric: fixes P :: "pi"   and Q :: "pi" assumes "P \<sim>\<^sup>s Q" shows "Q \<sim>\<^sup>s P"
Cn_not: fixes A :: "'a set"   and p :: "'a" shows "(A \<turnstile> .\<not> p) = (Cn (A \<union> {p}) = UNIV)"
is_weak_order_B: shows "weak_partial_order \<Y>\<^bsub>G\<^esub>"
generate_minimal: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set" assumes "Group.group G"   and "H \<subseteq> carrier G" shows "generate G H = \<Inter> {H'. subgroup H' G \<and> H \<subseteq> H'}"
compl_sets: fixes a :: "'d set"   and M :: "'d measure" assumes "a \<in> sets M" shows "space M - a \<in> sets M"
composite_pseudofunctor_def: fixes V\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>B :: "'r \<Rightarrow> 'r"   and src\<^sub>B :: "'r \<Rightarrow> 'r"   and trg\<^sub>B :: "'r \<Rightarrow> 'r"   and V\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>C :: "'s \<Rightarrow> 's"   and src\<^sub>C :: "'s \<Rightarrow> 's"   and trg\<^sub>C :: "'s \<Rightarrow> 's"   and V\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and H\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and \<a>\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't \<Rightarrow> 't"   and \<i>\<^sub>D :: "'t \<Rightarrow> 't"   and src\<^sub>D :: "'t \<Rightarrow> 't"   and trg\<^sub>D :: "'t \<Rightarrow> 't"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'s \<Rightarrow> 't"   and \<Phi>\<^sub>G :: "'s \<times> 's \<Rightarrow> 't" shows "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<equiv> (bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<and> bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C) \<and> bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D \<and> pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F \<and> pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"
Fxz: shows "finite_entropy (S \<Otimes>\<^sub>M P) (\<lambda>x. (X x, Z x)) Pxz"
octo_divide_numeral: fixes x :: "octo"   and y :: "num" shows "x / numeral y = x /\<^sub>R numeral y"
not_None_eq: fixes x :: "'e option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
continuous_on_product_then_coordinatewise_UNIV: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and i :: "'d" assumes "continuous_on UNIV f" shows "continuous_on UNIV (\<lambda>x. f x i)"
nn_integral_cong: fixes M :: "'a measure"   and u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
const_measurable: fixes S :: "'b set set"   and c :: "real"   and X :: "real set set" assumes "sigma_algebra S" shows "(\<lambda>x. c) \<in> measurable S X"
gcs_comm: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs_powerprod.gcs plus minus lcs s t = lcs_powerprod.gcs plus minus lcs t s"
isSame_equi_dev: fixes n :: "nat"   and V :: "nat set"   and A :: "nat set"   and t1 :: "'a option list"   and B :: "nat set"   and t2 :: "'a option list"   and s :: "nat set" assumes "wf_set n V"   and "wf_tuple n A t1"   and "wf_tuple n B t2"   and "s \<subseteq> A"   and "s \<subseteq> B"   and "A \<subseteq> V"   and "B \<subseteq> V" shows "isSameIntersection t1 s t2 = (restrict s t1 = restrict s t2)"
subsetD: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "(A::'b set) \<subseteq> (B::'b set)"   and "(c::'b) \<in> (A::'b set)" shows "(c::'b) \<in> (B::'b set)"
rep_eq: fixes x :: "('a, 'b) mapping"   and xa :: "'a \<Rightarrow> 'b"   and xb :: "'a set" shows "mapping.rep (upd_set x xa xb) = (\<lambda>a. if a \<in> xb then Some (xa a) else mapping.rep x a)"
3: fixes b :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and x :: "String.literal"   and y :: "nat"   and a :: "(String.literal \<times> String.literal) list" assumes "fst ` set b = fst ` set tao"   and "(x, y) \<in> set tao"   and "sa = (a, b)" shows "\<exists>y. map_of (map (apsnd map_of) b) x = Some y"
properfactor_divides: fixes a :: "'a"   and b :: "'a" assumes "properfactor G a b" shows "a divides b"
ex_in_inf: fixes A :: "'a set" assumes "at TYPE('a)"   and "finite A" obtains c :: "'a" where "c \<notin> A"
eexp_special_values_3: shows "eexp (- \<infinity>) = 0"
hp_conv: fixes h :: "'e"   and l :: "'f" shows "hp (h, l) = h"
state_io_rel_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "('a, 'b) state" shows "state_io_rel P m = (\<forall>s. P s (snd (run_state m s)))"
cancel_to_append: fixes a :: "(bool \<times> 'c) list"   and b :: "(bool \<times> 'c) list"   and l :: "(bool \<times> 'c) list"   and l' :: "(bool \<times> 'c) list" assumes "cancels_to a b" shows "cancels_to (l @ a @ l') (l @ b @ l')"
set_intros_14: fixes yl :: "'b"   and x121 :: "'b ltlc"   and x122 :: "'b ltlc" assumes "yl \<in> atoms_ltlc x121" shows "yl \<in> atoms_ltlc (x121 R\<^sub>c x122)"