hcju/isabelleps huggingface dataset:
Vitali_covering_theorem_cballs: fixes K :: "'b set"   and r :: "'b \<Rightarrow> real"   and S :: "'c set"   and a :: "'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> K \<Longrightarrow> 0 < r i"   and "\<And>x d. \<lbrakk>x \<in> S; 0 < d\<rbrakk> \<Longrightarrow> \<exists>i. i \<in> K \<and> x \<in> cball (a i) (r i) \<and> r i < d" obtains C :: "'b set" where "countable C" and "C \<subseteq> K" and "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C" and "negligible (S - (\<Union>i\<in>C. cball (a i) (r i)))"
dehomogenize_homogenize: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "dehomogenize x (homogenize x p) = dehomogenize x p"
functor_axioms: shows "functor C.CC.comp (\<cdot>\<^sub>D) FoT\<^sub>C.map"
Hash_Pair: fixes X :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Hash X \<noteq> \<langle>X', Y'\<rangle>"
emeasure_eq_0: fixes N :: "'b set"   and M :: "'b measure"   and K :: "'b set" assumes "N \<in> sets M"   and "emeasure M N = 0"   and "K \<subseteq> N" shows "emeasure M K = 0"
Red_I_def: fixes N :: "'f set" shows "no_labels.Red_I_\<G> N = (\<Inter>q\<in>Q. no_labels.Red_I_\<G>_q q N)"
list_in_find_eq: fixes var :: "nat"   and L :: "atom list"   and A :: "real mpoly list"   and L' :: "atom list" assumes "find_eq var L = (A, L')" shows "set (map Eq A @ L') = set L"
B'_def: shows "B' = filter (\<lambda>v. dim_vec v = d) B"
transitive_acyclic_asymmetric: fixes x :: "'a" assumes "transitive x" shows "pd_kleene_allegory_class.acyclic x = asymmetric x"
word_rev_tf: fixes bl :: "bool list" shows "to_bl (of_bl bl) = rev (takefill False LENGTH('b) (rev bl))"
ChiRel_HAStates_subseteq: fixes A :: "('a, 'b, 'c) hierauto" shows "ChiRel A \<subseteq> HAStates A \<times> HAStates A"
fds_integral_def: fixes c :: "'b"   and f :: "'b fds" shows "fds_integral c f = fds (\<lambda>n. if n = 1 then c else - fds_nth f n /\<^sub>R ln (real n))"
wf_dom_len_eq: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w" shows "length ss = n"
card_set: fixes xs :: "'a list" shows "card (set xs) = length (remdups xs)"
assms_4: shows "set qs \<subseteq> {a, b}"
map: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option cdecl list" shows "map_prog f (Program x) = Program (map (map_prod id (map_prod id (map_prod id (map (map_prod id (map_prod id (map_prod id (map_option f)))))))) x)"
eq_fold: fixes A :: "'b set" shows "Gcd\<^sub>f\<^sub>i\<^sub>n A = (if finite A then Finite_Set.fold gcd (0::'b) A else (1::'b))"
wtL: fixes l :: "('a, 'b) lit" assumes "M.wtL l" shows "local.wtL l"
distinct_279: fixes x131 :: "floatarith"   and x132 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Powr x131 x132 \<noteq> Power x151 x152"
natToVertexList_nths: fixes f :: "face"   and v :: "nat"   and es :: "nat list" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "incrIndexList es |es| |vertices f|" shows "nths (verticesFrom f v) (set es) = removeNones (natToVertexList v f es)"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'c \<Rightarrow> 'f" shows "g \<circ> (f \<circ> v) = g \<circ> f \<circ> v"
classical_PP_prv: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv (neg (neg (PP \<langle>\<phi>\<rangle>)))" shows "prv (PP \<langle>\<phi>\<rangle>)"
resolveFalseClauses: fixes literal :: "Literal"   and clause1 :: "Literal list"   and valuation :: "Literal list"   and clause2 :: "Literal list" assumes "clauseFalse (removeAll literal clause1) valuation"   and "clauseFalse (removeAll (opposite literal) clause2) valuation" shows "clauseFalse (resolve clause1 clause2 literal) valuation"
equivalence_def: fixes S :: "'c set"   and E :: "('c \<times> 'c) set" shows "equivalence S E \<equiv> (E \<subseteq> S \<times> S \<and> (\<forall>a. a \<in> S \<longrightarrow> (a, a) \<in> E)) \<and> (\<forall>a b. (a, b) \<in> E \<longrightarrow> (b, a) \<in> E) \<and> (\<forall>a b c. (a, b) \<in> E \<longrightarrow> (b, c) \<in> E \<longrightarrow> (a, c) \<in> E)"
measurable_finmap_of: fixes N :: "'e measure"   and J :: "'e \<Rightarrow> 'd set"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'f"   and M :: "'d \<Rightarrow> 'f measure"   and I :: "'d set set" assumes "\<And>i. \<exists>x\<in>space N. i \<in> J x \<Longrightarrow> (\<lambda>x. f x i) \<in> N \<rightarrow>\<^sub>M M i"   and "\<And>x. x \<in> space N \<Longrightarrow> J x \<in> I"   and "\<And>x. x \<in> space N \<Longrightarrow> finite (J x)"   and "\<And>S. {x. J x = S} \<inter> space N \<in> sets N" shows "(\<lambda>x. finmap_of (J x) (f x)) \<in> N \<rightarrow>\<^sub>M Pi\<^sub>F I M"
funion_fempty_left: fixes B :: "'d fset" shows "{||} |\<union>| B = B"
quotient_disj: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and Y :: "'a set" assumes "equiv A r"   and "X \<in> A // r"   and "Y \<in> A // r" shows "X = Y \<or> X \<inter> Y = {}"
that: assumes "nop_bij A (2\<^sub>\<nat>) f" shows "thesis"
left_total_alt_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "left_total R = ((=) \<le> R OO R\<inverse>\<inverse>)"
1: shows "\<i> * z + csqrt (1 - z\<^sup>2) \<notin> \<real>\<^sub>\<le>\<^sub>0"
a_mult_add: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> (y + x) = ad x \<cdot> y"
nth_append: fixes xs :: "'f list"   and ys :: "'f list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
main_2: assumes "euler_liar x n" shows "[x ^ ((n - 1) div 2) = b] (mod int k)"
simps: fixes twist :: "bool"   and s :: "'b tree"   and x :: "'b"   and t :: "'b tree" shows "node twist s x t = (if twist then \<langle>t, x, s\<rangle> else \<langle>s, x, t\<rangle>)"
Fract_cases_nonzero: fixes q :: "'b fract" obtains a :: "'b" and b :: "'b" where "q = Fract a b" and "b \<noteq> (0::'b)" and "a \<noteq> (0::'b)"   | "q = 0"
islimpt_in_closure: fixes x :: "'a"   and S :: "'a set" shows "x islimpt S = (x \<in> closure (S - {x}))"
one_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "one = Abs 1"
is_faithful: fixes f :: "'c"   and f' :: "'c" assumes "C.par f f'"   and "C.L f = C.L f'" shows "f = f'"
rbt_lookup_rbt_unionwk: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a" assumes "rbt_sorted t1"   and "rbt_sorted t2" shows "rbt_lookup (rbt_union_with_key f t1 t2) k = (case rbt_lookup t1 k of None \<Rightarrow> rbt_lookup t2 k | Some v \<Rightarrow> case rbt_lookup t2 k of None \<Rightarrow> Some v | Some w \<Rightarrow> Some (f k v w))"
split_int_eq_real: fixes a :: "int"   and b :: "real" shows "(real_of_int a = b) = (a = \<lfloor>b\<rfloor> \<and> b = real_of_int \<lfloor>b\<rfloor>)"
leadsETo_weaken_L: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B :: "'d set" assumes "F \<in> A leadsTo[CC] A'"   and "B \<subseteq> A" shows "F \<in> B leadsTo[CC] A'"
lhr_fps_numerator_def: fixes m :: "nat"   and cs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "lhr_fps_numerator m cs f = (let N = length cs - 1 in Poly (map (\<lambda>k. \<Sum>i\<le>min N k. cs ! (N - i) * f (k - i)) [0..<N + m]))"
map_gpv_conv_map_gpv': fixes f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e" shows "map_gpv f g = map_gpv' f g id"
SAT_def: fixes \<Phi> :: "('fsym, 'psym) lit list set" shows "Ik.SAT \<Phi> \<equiv> \<forall>\<xi>. Ik.wtE \<xi> \<longrightarrow> Ik.satPB \<xi> \<Phi>"
simps_6: fixes v :: "char list"   and va :: "aexp" shows "varacom (v ::= va) = {}"
word_set_bit_def: fixes a :: "'a word"   and n :: "nat"   and x :: "bool" shows "set_bit_class.set_bit a n x = word_of_int (bin_sc n x (uint a))"
assms_4: shows "R1 \<subseteq> upRules \<and> R = Ax \<union> R1"
3: shows "NR\<^sub>N n u l \<noteq> \<top>"
t: shows "wf\<^sub>t\<^sub>r\<^sub>m t"
clop_Inf_closed: fixes f :: "'b \<Rightarrow> 'b" assumes "clop f" shows "Inf_closed_set (Fix f)"
primeness_condition: fixes p :: "'a list" assumes "p \<in> carrier (K [X])" shows "pirreducible K p = pprime K p"
list_length_hint2a: fixes l :: "'d list" assumes "length l = Suc 0" shows "tl l = []"
NS_least_least: fixes w :: "'a \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and pr_weak :: "'a \<times> nat \<Rightarrow> 'a \<times> nat \<Rightarrow> bool"   and least :: "'a \<Rightarrow> bool"   and scf :: "'a \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'a"   and t :: "('a, 'b) Term.term" assumes "admissible_kbo w w0 pr_strict pr_weak least scf"   and "least f"   and "snd (kbo.kbo w w0 scf least pr_strict pr_weak (Fun f []) t)" shows "\<exists>g. t = Fun g [] \<and> least g"
policy_refinement_def: fixes p :: "'g \<Rightarrow> 'h decision option"   and abs\<^sub>a :: "'i \<Rightarrow> 'g"   and abs\<^sub>b :: "'j \<Rightarrow> 'h"   and q :: "'i \<Rightarrow> 'j decision option" shows "p \<sqsubseteq>\<^bsub>abs\<^sub>a\<^esub>\<^sub>,\<^bsub>abs\<^sub>b\<^esub> q \<equiv> \<forall>a. case p a of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> \<forall>a'\<in>{x. abs\<^sub>a x = a}. \<exists>b'. q a' = \<lfloor>allow b'\<rfloor> \<and> abs\<^sub>b b' = y | \<lfloor>deny y\<rfloor> \<Rightarrow> \<forall>a'\<in>{x. abs\<^sub>a x = a}. \<exists>b'. q a' = \<lfloor>deny b'\<rfloor> \<and> abs\<^sub>b b' = y"
sel_1: fixes x1 :: "float poly"   and x2 :: "float interval" shows "tm_poly (TaylorModel x1 x2) = x1"
match_map_regex: fixes t :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and r :: "'b Regex.regex" shows "Regex.match t (regex.map_regex f r) = Regex.match (\<lambda>k z. t k (f z)) r"
groupParts_same_fw_wi1: fixes V :: "'b wordinterval list"   and c :: "parts_connection"   and rs :: "'b simple_rule list" assumes "V \<in> set (groupWIs c rs)" shows "\<forall>w1\<in>set V. \<forall>w2\<in>set V. \<forall>a1\<in>wordinterval_to_set w1. \<forall>a2\<in>wordinterval_to_set w2. same_fw_behaviour_one a1 a2 c rs"
stl_def: fixes stream :: "'a stream" shows "stl stream = (case stream of x1 ## x2 \<Rightarrow> x2)"
assms_1: shows "P o\<^sub>C\<^sub>L P = P"
not_less_zero: fixes n :: "'c multiset" shows "\<not> n \<subset># {#}"
UNpart1_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list"   and P :: "nat \<Rightarrow> nat set set" shows "UNpart1 cl dl P \<equiv> \<Union>n\<in>theNFTBoth cl dl. local.shift cl n ` P n"
good_mapI: fixes h :: "'d \<times> 'e \<Rightarrow> 'c" assumes "surj h"   and "\<And>x x' y y'. h (x, y) = h (x', y') \<Longrightarrow> x = x'" shows "good_map h"
Suc: fixes a :: "'a"   and b :: "'a" shows "pochhammer (a + b) n = (\<Sum>k\<le>n. of_nat (n choose k) * pochhammer a k * pochhammer b (n - k))"
add_diff_le: fixes k :: "nat"   and n :: "nat"   and m :: "nat" assumes "k \<le> n" shows "m + k - n \<le> m"
fold_mset_def: fixes f :: "'l \<Rightarrow> 'k \<Rightarrow> 'k"   and s :: "'k"   and M :: "'l multiset" shows "fold_mset f s M = Finite_Set.fold (\<lambda>x. f x ^^ count M x) s (set_mset M)"
A_def: shows "A = [a, a', a'']\<^sub>\<circ>"
total: shows "\<forall>x y. le x y \<or> le y x"
pfa_last_def: shows "pfa_last \<equiv> arl_last"
finterE: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "c |\<in>| A |\<inter>| B" obtains "c |\<in>| A" and "c |\<in>| B"
of_nat_minus_hmset: fixes m :: "nat"   and n :: "nat" shows "of_nat (m - n) = of_nat m - of_nat n"
CCexp_def: fixes \<Gamma> :: "exp" shows "CCexp \<Gamma> = (\<Lambda> a. snd (cccExp \<Gamma>\<cdot>a))"
x': shows "complex_of_real x \<noteq> complex_of_real (- 1)"
ComposeC: fixes T :: "('fun, 'var) Term.term list"   and f :: "'fun"   and M :: "('fun, 'var) Term.term set" assumes "length T = arity f"   and "public f"   and "\<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t" shows "M \<turnstile>\<^sub>c Fun f T"
irreflp_on_converse_simp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "irreflp_on P\<inverse>\<inverse> A = irreflp_on P A"
prfOf_iff_PPf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "prf \<in> proof"   and "\<phi> \<in> fmla" shows "Fvars \<phi> = {} \<longrightarrow> prfOf prf \<phi> = bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
OF_same_priority_match2_def: fixes \<gamma> :: "'e set \<Rightarrow> 'f \<Rightarrow> bool"   and flow_entries :: "('e, 'd) flow_entry_match list"   and packet :: "'f" shows "OF_same_priority_match2 \<gamma> flow_entries packet \<equiv> let s = {ofe_action f |f. f \<in> set flow_entries \<and> \<gamma> (ofe_fields f) packet \<and> (\<forall>fo\<in>set flow_entries. ofe_prio f < ofe_prio fo \<longrightarrow> \<not> \<gamma> (ofe_fields fo) packet)} in case card s of 0 \<Rightarrow> NoAction | Suc 0 \<Rightarrow> Action (the_elem s) | Suc (Suc nata) \<Rightarrow> Undefined"
NatTransMapsTo: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and G :: "('g, 'h, 'i, 'j, 'k, 'l) Functor"   and X :: "'g" assumes "NT \<eta> : F \<Longrightarrow> G"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "\<eta> $$ X maps\<^bsub>CatCod G\<^esub> F @@ X to G @@ X"
inj1_hom: fixes R :: "('g, 'h) ring_scheme"   and M1 :: "('g, 'i) module"   and M2 :: "('g, 'j) module" assumes "module R M1"   and "module R M2" shows "mod_hom R M1 (direct_sum M1 M2) (inj1 M1 M2)"
empty_not_insert: fixes a :: "'b"   and A :: "'b set" shows "{} \<noteq> insert a A"
filter_sup_left_upper_bound: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and G :: "'b set"   and F :: "'b set" assumes "class.semilattice_inf inf less_eq less"   and "ord.filter less_eq G" shows "F \<subseteq> semilattice_inf.filter_sup inf less_eq F G"
steps'_steps: fixes as :: "'c list"   and qs :: "(nat \<times> nat list) list"   and s :: "'c list" assumes "length as = length qs" shows "steps' s as qs (length as) = Partial_Cost_Model.steps s as qs"
iso_char: fixes f :: "'a \<times> 'a" shows "B.VV.iso f = (B.VxV.iso f \<and> B.VV.arr f \<and> B.VV.arr (B.VxV.inv f))"
bidel_arc_not_biconnected_axioms: shows "bidel_arc_not_biconnected G M a"
lookup_pair_update_by_fun_gr_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k1 :: "'a"   and f :: "'c \<Rightarrow> 'c"   and k2 :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (update_by_fun_gr_pair x k1 f xs) k2 = (if k1 = k2 then f else id) (lookup_pair x xs k2)"
assms_3: shows "\<exists>n'. latestAct_cond tid t n n'"
proj_set_thickening: fixes p :: "'b"   and x :: "'b"   and Z :: "'b set"   and D :: "real"   and G :: "'b set" assumes "p \<in> proj_set x Z"   and "0 \<le> D"   and "D \<le> dist p x"   and "geodesic_segment_between G p x" shows "geodesic_segment_param G p D \<in> proj_set x (\<Union>z\<in>Z. cball z D)"
mono_lookup: fixes F :: "'b Abs_State.st"   and F' :: "'b Abs_State.st"   and x :: "char list" assumes "F \<sqsubseteq> F'" shows "lookup F x \<sqsubseteq> lookup F' x"
fmap_ErrorT: fixes f :: "'h \<rightarrow> 'e"   and m :: "'h\<cdot>'g error\<cdot>'f" shows "fmap\<cdot>f\<cdot>(ErrorT\<cdot>m) = ErrorT\<cdot>(fmap\<cdot>(fmap\<cdot>f)\<cdot>m)"
indicator_def: fixes S :: "'c set"   and x :: "'c" shows "indicator S x = (if x \<in> S then 1::'b else (0::'b))"
x: shows "x = cis (real i * 2 * pi / real n)"
slice1_ics_SDG_path: fixes n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "n \<in> sum_SDG_slice1 n'"   and "n \<noteq> n'" obtains ns :: "'node SDG_node list" where "CFG_node (_Entry_) ics-ns\<rightarrow>\<^sub>d* n'" and "n \<in> set ns"
raw_subst_frees: fixes t :: "'a" shows "abs_pred (\<lambda>t. \<forall>env. closed_env env \<longrightarrow> frees (subst t env) = frees t |-| fmdom env) t"
map_topology_def: fixes p :: "'e \<Rightarrow> 'd"   and X :: "'e topology" shows "map_topology p X = final_topology (p ` topspace X) (\<lambda>_. X) (\<lambda>_. p)"
frac_le: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and y :: "'d"   and x :: "'d"   and w :: "'d"   and z :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero y"   and "less_eq x y"   and "less zero w"   and "less_eq w z" shows "less_eq (divide x z) (divide y w)"
Gcd_Lcm: fixes A :: "'b set" shows "Gcd A = Lcm {b. \<forall>a\<in>A. b dvd a}"
distinct_7: shows "a\<^sub>2 \<noteq> a\<^sub>3"
induct: fixes P1 :: "(nat \<Rightarrow> nat) \<Rightarrow> bool"   and P2 :: "(nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> bool"   and P3 :: "(nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> bool"   and x1 :: "nat \<Rightarrow> nat"   and x2 :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x3 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "P1 (\<lambda>x. 0)"   and "P1 Suc"   and "P1 (\<lambda>x. x)"   and "P2 (\<lambda>x y. x)"   and "P2 (\<lambda>x y. y)"   and "P3 (\<lambda>x y z. x)"   and "P3 (\<lambda>x y z. y)"   and "P3 (\<lambda>x y z. z)"   and "\<And>f g. \<lbrakk>f \<in> PrimRec1; P1 f; g \<in> PrimRec1; P1 g\<rbrakk> \<Longrightarrow> P1 (\<lambda>x. f (g x))"   and "\<And>f g. \<lbrakk>f \<in> PrimRec1; P1 f; g \<in> PrimRec2; P2 g\<rbrakk> \<Longrightarrow> P2 (\<lambda>x y. f (g x y))"   and "\<And>f g. \<lbrakk>f \<in> PrimRec1; P1 f; g \<in> PrimRec3; P3 g\<rbrakk> \<Longrightarrow> P3 (\<lambda>x y z. f (g x y z))"   and "\<And>f g h. \<lbrakk>f \<in> PrimRec2; P2 f; g \<in> PrimRec1; P1 g; h \<in> PrimRec1; P1 h\<rbrakk> \<Longrightarrow> P1 (\<lambda>x. f (g x) (h x))"   and "\<And>f g h k. \<lbrakk>f \<in> PrimRec3; P3 f; g \<in> PrimRec1; P1 g; h \<in> PrimRec1; P1 h; k \<in> PrimRec1; P1 k\<rbrakk> \<Longrightarrow> P1 (\<lambda>x. f (g x) (h x) (k x))"   and "\<And>f g h. \<lbrakk>f \<in> PrimRec2; P2 f; g \<in> PrimRec2; P2 g; h \<in> PrimRec2; P2 h\<rbrakk> \<Longrightarrow> P2 (\<lambda>x y. f (g x y) (h x y))"   and "\<And>f g h. \<lbrakk>f \<in> PrimRec2; P2 f; g \<in> PrimRec3; P3 g; h \<in> PrimRec3; P3 h\<rbrakk> \<Longrightarrow> P3 (\<lambda>x y z. f (g x y z) (h x y z))"   and "\<And>f g h k. \<lbrakk>f \<in> PrimRec3; P3 f; g \<in> PrimRec2; P2 g; h \<in> PrimRec2; P2 h; k \<in> PrimRec2; P2 k\<rbrakk> \<Longrightarrow> P2 (\<lambda>x y. f (g x y) (h x y) (k x y))"   and "\<And>f g h k. \<lbrakk>f \<in> PrimRec3; P3 f; g \<in> PrimRec3; P3 g; h \<in> PrimRec3; P3 h; k \<in> PrimRec3; P3 k\<rbrakk> \<Longrightarrow> P3 (\<lambda>x y z. f (g x y z) (h x y z) (k x y z))"   and "\<And>g h. \<lbrakk>g \<in> PrimRec1; P1 g; h \<in> PrimRec3; P3 h\<rbrakk> \<Longrightarrow> P2 (PrimRecOp g h)" shows "(x1 \<in> PrimRec1 \<longrightarrow> P1 x1) \<and> (x2 \<in> PrimRec2 \<longrightarrow> P2 x2) \<and> (x3 \<in> PrimRec3 \<longrightarrow> P3 x3)"
filter_tab_def: fixes c :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "'a \<Rightarrow> 'b option" shows "filter_tab c t = (\<lambda>k. case t k of None \<Rightarrow> None | Some x \<Rightarrow> if c k x then Some x else None)"
reachable_in_vertsE: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" obtains "u \<in> verts G" and "v \<in> verts G"
PiE_fun_upd: fixes y :: "'a"   and T :: "'b \<Rightarrow> 'a set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "y \<in> T x"   and "f \<in> Pi\<^sub>E S T" shows "f(x := y) \<in> Pi\<^sub>E (insert x S) T"
induct: fixes x1 :: "L"   and x2 :: "Var \<Rightarrow> L"   and x3 :: "IMP"   and x4 :: "Var \<Rightarrow> L"   and P :: "L \<Rightarrow> (Var \<Rightarrow> L) \<Rightarrow> IMP \<Rightarrow> (Var \<Rightarrow> L) \<Rightarrow> bool" assumes "(x1, x2, x3, x4) \<in> HS"   and "\<And>p G. P p G Skip G"   and "\<And>G e t p x. (G, e, t) \<in> HS_E \<Longrightarrow> P p G (Assign x e) (upd G x (LUB p t))"   and "\<And>p G c K d H. \<lbrakk>(p, G, c, K) \<in> HS; P p G c K; (p, K, d, H) \<in> HS; P p K d H\<rbrakk> \<Longrightarrow> P p G (Comp c d) H"   and "\<And>G b t p c H d. \<lbrakk>(G, b, t) \<in> HS_B; (LUB p t, G, c, H) \<in> HS; P (LUB p t) G c H; (LUB p t, G, d, H) \<in> HS; P (LUB p t) G d H\<rbrakk> \<Longrightarrow> P p G (Iff b c d) H"   and "\<And>G b p c H d. \<lbrakk>(G, b, p) \<in> HS_B; (p, G, c, H) \<in> HS; P p G c H; (p, G, d, H) \<in> HS; P p G d H\<rbrakk> \<Longrightarrow> P p G (Iff b c d) H"   and "\<And>G b t p c H. \<lbrakk>(G, b, t) \<in> HS_B; (LUB p t, G, c, H) \<in> HS; P (LUB p t) G c H; H = G\<rbrakk> \<Longrightarrow> P p G (While b c) H"   and "\<And>pp GG c HH p G H. \<lbrakk>(pp, GG, c, HH) \<in> HS; P pp GG c HH; LEQ p pp; \<forall>x. LEQ (G x) (GG x); \<forall>x. LEQ (HH x) (H x)\<rbrakk> \<Longrightarrow> P p G c H" shows "P x1 x2 x3 x4"
prems_5: shows "((\<forall>nm\<in>incoming n'. nm < name n') \<and> name n' \<le> name (fst x) \<and> (incoming n' \<inter> incoming (fst x) \<noteq> {} \<longrightarrow> new n' \<subseteq> old (fst x) \<union> new (fst x))) \<and> (\<forall>nd\<in>snd x. \<forall>nm\<in>incoming n'. nm \<in> incoming nd \<longrightarrow> new n' \<subseteq> old nd)"
group_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b" shows "group f z inverse \<equiv> semigroup f \<and> group_axioms f z inverse"
terminalD: fixes subs :: "'b \<Rightarrow> 'b set"   and Gamma :: "'b"   and x :: "'b" assumes "terminal subs Gamma" shows "x \<notin> subs Gamma"
ent_true: fixes P :: "assn" shows "P \<Longrightarrow>\<^sub>A true"
coreflexive_mult_closed: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.non_associative_left_semiring one times sup less_eq less bot"   and "less_eq x one"   and "less_eq y one" shows "less_eq (times x y) one"
vwx: shows "v ^ 3 + w ^ 3 = x ^ 3 \<and> v * w * x \<noteq> 0 \<and> even x \<and> coprime v w"
trans: shows "\<forall>x\<in>{{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}. \<forall>y\<in>{{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}. \<forall>z\<in>{{y \<in> X. (y, x) \<in> r \<and> (x, y) \<in> r} |x. x \<in> X}. x \<noteq> y \<and> (SOME y. y \<in> x, SOME ya. ya \<in> y) \<in> r \<longrightarrow> y \<noteq> z \<and> (SOME ya. ya \<in> y, SOME y. y \<in> z) \<in> r \<longrightarrow> x \<noteq> z \<and> (SOME y. y \<in> x, SOME y. y \<in> z) \<in> r"
power_is_relpow: fixes X :: "('b \<times> 'b) set"   and n :: "nat" shows "rel_dioid.power X n = X ^^ n"
axioms_2: fixes C :: "('a, 'b, 'c) Category_scheme" assumes "Category C" shows "Category_axioms C"
not_None_eq: fixes x :: "'d option" shows "(x \<noteq> \<bottom>) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
abs_homogenous: fixes x :: "'a"   and a :: "real" assumes "x \<in> E" shows "\<parallel>a \<cdot> x\<parallel> = \<bar>a\<bar> * \<parallel>x\<parallel>"
open_imp_subset: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and U :: "'d set" assumes "topological_space S is_open"   and "is_open U" shows "U \<subseteq> S"