hcju/isabelleps huggingface dataset:
stake_sdrop: fixes n :: "nat"   and s :: "'b stream" shows "stake n s @- sdrop n s = s"
bisim: fixes s1 :: "'s"   and s2 :: "'s"   and x :: "'c" assumes "S s1 s2"   and "\<not> stop s2" shows "ord_spmf (\<lambda>(r1, s1') (r2, s2'). le_option r2 r1 \<and> S s1' s2' \<and> (r2 = None \<and> r1 \<noteq> None) = stop s2') (callee1 s1 x) (callee2 s2 x)"
enumerate_replicate_eq: fixes n :: "nat"   and m :: "nat"   and a :: "'a" shows "enumerate n (replicate m a) = map (\<lambda>q. (q, a)) [n..<n + m]"
init_adverse_eq_prefixes: fixes z :: "nat"   and i :: "nat"   and j :: "nat"   and n :: "nat" shows "local.adverse z i j \<triangleright> n = Partial_Recursive.list_encode (local.prefixes z i j n)"
covector_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "covector x"   and "covector y" shows "covector (x \<sqinter> y)"
fst_conv: fixes x1 :: "'g"   and x2 :: "'h" shows "fst (x1, x2) = x1"
mkeps_nullable: fixes r :: "'b rexp" assumes "nullable r" shows "\<turnstile> mkeps r : r"
isLubI2: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isUb R S x"   and "x <=* Collect (isUb R S)" shows "isLub R S x"
cnvalidD: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and n :: "nat"   and F :: "'g set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and s :: "'f"   and t :: "('f, 'g) xstate" assumes "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>(P, p, Q, A)\<in>\<Theta>. \<Gamma>\<Turnstile>n:\<^bsub>/F\<^esub> P Call p Q,A"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"   and "s \<in> P"   and "t \<notin> Fault ` F" shows "t \<in> Normal ` Q \<union> Abrupt ` A"
word_eq_iff_signed: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and v :: "'b word"   and w :: "'b word" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(v = w) = (ring_1.signed one plus zero minus v = ring_1.signed one plus zero minus w)"
interval_lebesgue_integrable_combine: fixes A :: "ereal"   and B :: "ereal"   and f :: "real \<Rightarrow> 'a"   and C :: "ereal" assumes "interval_lebesgue_integrable lborel A B f"   and "interval_lebesgue_integrable lborel B C f"   and "set_borel_measurable borel (einterval A C) f"   and "A \<le> B"   and "B \<le> C" shows "interval_lebesgue_integrable lborel A C f"
matchs_appI: 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 abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "term list"   and ys :: "'c list"   and env\<^sub>1 :: "(name, 'c) fmap"   and xs' :: "term list"   and ys' :: "'c list"   and env\<^sub>2 :: "(name, 'c) fmap" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.matchs unapp unconst xs ys = Some env\<^sub>1"   and "term.matchs unapp unconst xs' ys' = Some env\<^sub>2" shows "term.matchs unapp unconst (xs @ xs') (ys @ ys') = Some (env\<^sub>1 ++\<^sub>f env\<^sub>2)"
C1: shows "f \<ominus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f) \<in> carrier (UP R)"
Max_ge: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "x \<le> Max A"
neq_iff: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(x \<noteq> y) = (x <s y \<or> y <s x)"
eval_monom_mult: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and m :: "'d monom"   and n :: "'d monom" shows "eval_monom \<alpha> (m * n) = eval_monom \<alpha> m * eval_monom \<alpha> n"
pairs_inf_closed: fixes x :: "'c regular"   and y :: "'c dense Filters.filter"   and z :: "'c regular"   and w :: "'c dense Filters.filter" assumes "(x, y) \<in> stone_phi.pairs"   and "(z, w) \<in> stone_phi.pairs" shows "stone_phi.pairs_inf (x, y) (z, w) \<in> stone_phi.pairs"
arr_char: fixes f :: "'c \<times> 'c" shows "CopxC.arr f = (Cop.arr (fst f) \<and> C.arr (snd f))"
sum_UNIV_set_def: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "sum_UNIV_set f n = sum f {..<n}"
assms_2: shows "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>"
unit_prod: fixes S :: "'a set"   and U :: "(('b, 'a) vec, 'a) vec" assumes "finite S" shows "is_unit (\<Prod>i\<in>S. U $ i $ i) = (\<forall>i\<in>S. is_unit (U $ i $ i))"
NilTrans_1: fixes \<alpha> :: "act"   and P' :: "ccs" assumes "\<zero> \<longmapsto>\<alpha> \<prec> P'" shows "False"
inverse_eq_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(inverse x = one) = (x = one)"
indets_empty_imp_monomial: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "indets p = {}" shows "monomial (lookup p 0) 0 = p"
alphaInputDistinct: fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" assumes "inputChain xvec M P = inputChain yvec N Q"   and "distinct xvec"   and "\<And>x. x mem xvec \<Longrightarrow> x \<in> supp M"   and "xvec \<sharp>* yvec"   and "xvec \<sharp>* N"   and "xvec \<sharp>* Q" shows "distinct yvec"
weakenU: fixes \<sigma> :: "'e"   and \<sigma>' :: "'e" assumes "U \<sigma> \<sigma>'" shows "U' \<sigma> \<sigma>'"
inducts: fixes x1 :: "('procS, 'procT) Proc"   and x2 :: "('procS, 'procT) Proc"   and P :: "('procS, 'procT) Proc \<Rightarrow> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "indRelRp x1 x2"   and "\<And>S. P (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))" shows "P x1 x2"
map_add_def: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" shows "Map.map_add m1 m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
hyps_3: shows "\<turnstile> \<Gamma>,\<S>,P +\<^sub>\<S> e {c} \<Gamma>',\<S>',P'"
NextRecvUpd'_def: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and q :: "'b" shows "NextRecvUpd' c0 c1 p q = (init c0 q \<and> cri.next_recvupd' (exchange_config c0) (exchange_config c1) p q \<and> init c1 = init c0 \<and> (\<forall>p'. prop_config c1 p' = (if p' = q then cm_all (prop_config c0 q) (hd (c_msg (exchange_config c0) p q)) else prop_config c0 p')))"
intro: fixes point_set :: "'c set" assumes "finite point_set" shows "finite_incidence_system_axioms point_set"
MStar_1: fixes \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex" assumes "ok (length \<phi>s) mr"   and "fv_regex (from_mregex mr \<phi>s) \<subseteq> A"   and "list_all2 (\<lambda>\<phi>. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) j \<phi>)) \<phi>s rels"   and "ok_lctxt \<phi>s \<kappa> \<kappa>'"   and "\<forall>ms\<in>\<kappa> ` LPD mr. qtable n A (mem_restr R) (\<lambda>v. Q (map the v) (from_mregex ms \<phi>s)) (lookup rel ms)" shows "qtable n A (mem_restr R) (\<lambda>v. \<exists>a\<in>Regex.lpd\<kappa> \<kappa>' (Formula.sat \<sigma> V (map the v)) j (from_mregex mr \<phi>s). Q (map the v) a) (l\<delta> \<kappa> rel rels mr)"
cut_le_all_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<le> t = I) = (\<forall>x\<in>I. x \<le> t)"
ovlap: fixes C :: "'a set" assumes "C \<in> components (T - S)" shows "C \<inter> L \<noteq> {}"
least_contains: fixes k :: "'d"   and A :: "'d set" assumes "k \<in> A" shows "least A \<in> A"
matrix_to_iarray_nth: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c" shows "matrix_to_iarray A !! mod_type_class.to_nat i !! mod_type_class.to_nat j = A $ i $ j"
pre_mono: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and C :: "acom"   and s :: "(char list \<Rightarrow> int option) \<times> nat" assumes "\<forall>s. P s \<longrightarrow> Q s"   and "pre C P s" shows "pre C Q s"
bisimulation_upto_def: fixes R :: "'s \<Rightarrow> 's \<Rightarrow> bool"   and f :: "('s \<Rightarrow> 's \<Rightarrow> bool) \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> bool" shows "bisimulation_upto R f = R \<rightarrow> f R"
append_leq_symm: fixes t2 :: "(val \<times> val) list"   and t1 :: "(val \<times> val) list" shows "t2 @ t1 \<lesssim> t1 @ t2"
is_discovered_impl_def: fixes u :: "'f"   and s :: "('f, 'g) simple_state_scheme" shows "is_discovered_impl u s \<equiv> u \<in> visited s"
cases: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'e"   and a2 :: "'d" assumes "r\<inverse>\<inverse> a1 a2" obtains a :: "'d" and b :: "'e" where "a1 = b" and "a2 = a" and "r a b"
union_def: fixes u :: "nat"   and v :: "nat" shows "union u v = f v (c_pair u v)"
nz: shows "fst fctrs \<noteq> (0::'a)"
eq_game_trans: fixes a :: "game"   and b :: "game"   and c :: "game" assumes "eq_game a b"   and "eq_game b c" shows "eq_game a c"
intro: fixes M :: "'b set"   and \<eta> :: "'b \<Rightarrow> 'c"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'b"   and unit' :: "'c" assumes "\<And>x y. \<lbrakk>x \<in> M; y \<in> M\<rbrakk> \<Longrightarrow> \<eta> (composition x y) = composition' (\<eta> x) (\<eta> y)"   and "\<eta> unit = unit'" shows "monoid_homomorphism_axioms \<eta> M composition unit composition' unit'"
map_simp_ide: fixes a :: "'d" assumes "D.ide a" shows "\<epsilon>FoF\<eta>.map a = (\<epsilon> \<circ> F) a \<cdot>\<^sub>C (F \<circ> \<eta>) a"
induct: fixes P :: "('a, 'b) rbt \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"   and a0 :: "('a, 'b) rbt"   and a1 :: "('a, 'b) rbt" assumes "\<And>x. P Empty x"   and "\<And>v va vb vc vd. P (Branch v va vb vc vd) Empty"   and "\<And>a k x b c s y d. P b c \<Longrightarrow> P (Branch R a k x b) (Branch R c s y d)"   and "\<And>a k x b c s y d. P b c \<Longrightarrow> P (Branch B a k x b) (Branch B c s y d)"   and "\<And>va vb vc vd b k x c. P (Branch B va vb vc vd) b \<Longrightarrow> P (Branch B va vb vc vd) (Branch R b k x c)"   and "\<And>a k x b va vb vc vd. P b (Branch B va vb vc vd) \<Longrightarrow> P (Branch R a k x b) (Branch B va vb vc vd)" shows "P a0 a1"
map_fun_apply: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'f"   and h :: "'h \<Rightarrow> 'i"   and x :: "'g" shows "map_fun f g h x = g (h (f x))"
sim: shows "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"
IH: shows "Marker \<notin> set (msgs (S t n) cid)"
simps_2: fixes e :: "exp"   and v :: "Vars.var"   and y :: "Vars.var"   and z :: "Vars.var" shows "(App e v)[y::=z] = App e[y::=z] v[y::v=z]"
simps_1: fixes k :: "nat" shows "[] \<odot> k = []"
NSDERIV_iff2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" shows "NSDERIV f x :> D = (\<forall>w. w \<noteq> star_of x \<and> w \<approx> star_of x \<longrightarrow> (*f* (\<lambda>z. (f z - f x) / (z - x))) w \<approx> star_of D)"
rep_eq: fixes x :: "float" shows "is_float_zero x = (0 = real_of_float x)"
diff_add_mset_swap: fixes b :: "'c"   and A :: "'c multiset"   and M :: "'c multiset" assumes "b \<notin># A" shows "add_mset b M - A = add_mset b (M - A)"
blocks_subset: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" assumes "sub_set_system \<U> \<A> \<V> \<B>" shows "\<A> \<subseteq># \<B>"
scaleRat_less1: fixes a :: "'a"   and b :: "'a"   and k :: "rat" assumes "a < b"   and "0 < k" shows "k *R a < k *R b"
nth_length_exceeds: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "xs ! n = [] ! (n - length xs)"
dg_is_arrD_5: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
card_subset_le: fixes i :: "nat_int"   and i' :: "nat_int" shows "i \<le> i' \<longrightarrow> |i| \<le> |i'|"
dg_prod_3_Dom_vsv: shows "vsv ((\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Dom\<rparr>)"
OR4_imp_Gets: fixes B :: "agent"   and A :: "agent"   and M :: "nat"   and Na :: "nat"   and K :: "nat"   and evs :: "event list" assumes "Says B A \<lbrace>Nonce M, Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>\<rbrace> \<in> set evs"   and "B \<notin> bad"   and "evs \<in> orb" shows "\<exists>Nb. Gets B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>, Nonce Nb, Key K\<rbrace>\<rbrace> \<in> set evs"
gcd_diff1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and m :: "'b"   and n :: "'b" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd (minus m n) n = gcd m n"
addpreRT_welldefined: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:16..PRreq-:18} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRreq-:17 \<longrightarrow> oip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRrep-:4 \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>)) \<and> (l = PRrep-:5 \<longrightarrow> the (nhop (rt \<xi>) (dip \<xi>)) \<in> kD (rt \<xi>)))"
closedin_subtopology_refl: fixes U :: "'c topology"   and X :: "'c set" shows "closedin (subtopology U X) X = (X \<subseteq> topspace U)"
iterate_add_eq_add_iff1: fixes i :: "nat"   and j :: "nat"   and u :: "'a"   and m :: "'a"   and n :: "'a" assumes "i \<le> j" shows "(iterate_add j u + m = iterate_add i u + n) = (iterate_add (j - i) u + m = n)"
that: fixes M :: "(nat \<times> (nat, nat) Op) set" assumes "finite M"   and "deterministic M"   and "0 \<notin> fst ` M"   and "Suc 0 \<notin> fst ` M"   and "\<And>x vs. \<lbrakk>vs 0 = x; vs 1 = 0; x \<in> A \<union> B\<rbrakk> \<Longrightarrow> \<exists>vs'. ((2, vs), 0, vs') \<in> (step M)\<^sup>* \<or> ((2, vs), Suc 0, vs') \<in> (step M)\<^sup>*"   and "\<And>x vs vs'. \<lbrakk>vs 0 = x; vs 1 = 0; ((2, vs), 0, vs') \<in> (step M)\<^sup>*\<rbrakk> \<Longrightarrow> x \<in> A"   and "\<And>x vs vs'. \<lbrakk>vs 0 = x; vs 1 = 0; ((2, vs), Suc 0, vs') \<in> (step M)\<^sup>*\<rbrakk> \<Longrightarrow> x \<in> B" shows "thesis"
empty: fixes g :: "'c \<Rightarrow> 'a" shows "F g {} = \<^bold>1"
red_unionI2: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and F :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "punit.red G p q" shows "punit.red (F \<union> G) p q"
hyps_2: shows "w\<^sub>2 =\<^sub>S w\<^sub>3"
flv_round_refines: fixes r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and d_f :: "process \<Rightarrow> val option" shows "{flv_ref_rel \<inter> (Vinv1 \<inter> Vinv2) \<times> UNIV} v_round r v_f d_f, flv_round r v_f d_f {> flv_ref_rel}"
comp_def: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" shows "g \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]\<^sub>x\<^sub>A f = (if (A_B.arr (fst f) \<and> A.arr (snd f)) \<and> (A_B.arr (fst g) \<and> A.arr (snd g)) \<and> (if A_B.arr (fst f) \<and> A.arr (snd f) then (A_B.cod (fst f), A.cod (snd f)) else (A_B.null, A.null)) = (if A_B.arr (fst g) \<and> A.arr (snd g) then (A_B.dom (fst g), A.dom (snd g)) else (A_B.null, A.null)) then (fst g \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] fst f, snd g \<cdot>\<^sub>A snd f) else (A_B.null, A.null))"
intro: fixes smult :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M :: "'c set"   and smult' :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd" assumes "RModuleHom UNIV smult M smult' T" shows "ModuleHom smult M smult' T"
content_ball_eq_0_iff: fixes c :: "'b"   and r :: "real" shows "(content (ball c r) = 0) = (r \<le> 0)"
assms_3: shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n'"
theI': fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x) \<and> (\<forall>y. P y \<longrightarrow> y = (THE x. P x))"
h3: assumes "Av + Bv * sqrt Cv = 0" shows "Av * Bv \<le> 0"
def_hashmap_size_prod_def: shows "def_hashmap_size = (\<lambda>_. def_hashmap_size TYPE('a) + def_hashmap_size TYPE('b))"
RPDi_safe_fv_regex: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex"   and n :: "nat" assumes "safe_regex Past Strict (from_mregex mr \<phi>s)"   and "ms \<in> RPDi n mr" shows "fv_regex (from_mregex ms \<phi>s) = fv_regex (from_mregex mr \<phi>s)"
continuous_on_inv_into: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "compact s"   and "inj_on f s" shows "continuous_on (f ` s) (the_inv_into s f)"
\<sigma>3': shows "(fst \<sigma>3', snd \<sigma>3' @ snd \<sigma>3'') \<in> seq ars"
hypreal_of_real_not_eq_epsilon: fixes x :: "real" shows "hypreal_of_real x \<noteq> \<epsilon>"
smc_0_components_2: shows "smc_0\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
bind_eq_return_pmf_None: fixes p :: "'b spmf"   and f :: "'b \<Rightarrow> 'a spmf" shows "(p \<bind> f = return_pmf None) = (\<forall>x\<in>set_spmf p. f x = return_pmf None)"
rpo_eq_wpo: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" shows "rpo_pr s t = wpo s t"
trms_unlabel_proj: fixes l :: "'f"   and x :: "('d, 'e) strand_step" shows "trms\<^sub>s\<^sub>t\<^sub>p (snd (ln l, x)) \<subseteq> trms\<^sub>l\<^sub>s\<^sub>t (proj l [(ln l, x)])"
alpha_sym: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and X' :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "X #= X'" shows "X' #= X"
return_pmf_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun A (rel_pmf A) return_pmf return_pmf"
maximal_independent_subset: fixes U\<^sub>V\<^sub>S :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and V :: "'d set" assumes "vector_space_on U\<^sub>V\<^sub>S scale"   and "V \<subseteq> U\<^sub>V\<^sub>S" obtains B :: "'d set" where "B \<subseteq> U\<^sub>V\<^sub>S" and "B \<subseteq> V" and "\<not> module_on.dependent scale B" and "V \<subseteq> module_on.span scale B"
assms_2: shows "(l, m, r) = partition_by_median k (x # y # zs)"
row_add_code: fixes A :: "(('c, 'b) vec, 'd) vec"   and a :: "'d"   and b :: "'d"   and q :: "'c"   and i :: "'d" shows "($) (row_add_row A a b q i) = (\<lambda>j. if i = a then A $ a $ j + q * A $ b $ j else A $ i $ j)"
le_of_comp_def: fixes acomp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" shows "le_of_comp acomp x y = (case acomp x y of Gt \<Rightarrow> False | _ \<Rightarrow> True)"
Aexp_edom: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and a :: "Arity" assumes "EdomArityAnalysis Aexp" shows "edom (Aexp e\<cdot>a) \<subseteq> fv e"
check_bot: fixes \<C> :: "t_context"   and es :: "b_e list" shows "check \<C> es Bot = Bot"
nat_mirror_commute: fixes n :: "nat"   and l :: "nat"   and r :: "nat" shows "nat_mirror n l r = nat_mirror n r l"
cases: fixes a1 :: "'b"   and a2 :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a1, a2) \<in> r\<^sup>+" obtains a :: "'b" and b :: "'b" where "a1 = a__" and "a2 = b__" and "(a__, b__) \<in> r"   | a :: "'b" and b :: "'b" and c :: "'b" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>+" and "(b, c) \<in> r"
simps_3: fixes C\<^sub>1 :: "'b acom"   and C\<^sub>2 :: "'b acom" shows "strip (C\<^sub>1;; C\<^sub>2) = strip C\<^sub>1;; strip C\<^sub>2"
drop_append: fixes n :: "nat"   and xs :: "'k list"   and ys :: "'k list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
simps_1: fixes I :: "'g \<Rightarrow> 'h option"   and P :: "'h \<Rightarrow> 'g \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> bool" shows "upd_invs I P [] = I"
assms_3: fixes \<alpha> :: "V" assumes "Ord \<alpha>"   and "P \<alpha>" shows "P (ZFC_in_HOL.succ \<alpha>)"
StrongF_def: fixes F :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "SF(F)_v \<equiv> \<box>\<diamond>Enabled \<langle>F\<rangle>_v \<longrightarrow> \<box>\<diamond>\<langle>F\<rangle>_v"
hyperb_quad_ineq0: shows "Gromov_hyperbolic_subset (deltaG TYPE('a)) UNIV"
iso_cancel_left: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and g' :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.comp f g = VV.comp f g'"   and "VV.seq f g" shows "g = g'"
ket_one_to_mat_of_cols_list: shows "|Deutsch.one\<rangle> = Tensor.mat_of_cols_list 2 [[0, 1]]"
cos_x_cos_y: fixes x :: "'a"   and y :: "'a" shows "(\<lambda>p. \<Sum>n\<le>p. if even p \<and> even n then (real_of_int ((- 1) ^ (p div 2) * int (p choose n)) / fact p) *\<^sub>R x ^ n * y ^ (p - n) else (0::'a)) sums (cos x * cos y)"
IH_1: assumes "x \<in> ifex_var_set (ifex_ite (restrict_top i a True) (restrict_top t a True) (restrict_top e a True))" shows "x \<in> ifex_var_set (restrict_top i a True) \<or> x \<in> ifex_var_set (restrict_top t a True) \<or> x \<in> ifex_var_set (restrict_top e a True)"
ys_no_opts: shows "h \<turnstile> filter_M (\<lambda>x. return (x \<noteq> None)) ys_opts \<rightarrow>\<^sub>r ys_no_opts"
singleton_is_point: fixes p :: "'a" assumes "singleton p" shows "point (p ; 1)"
oinvariantI: fixes A :: "('a \<times> 'b, 'c) automaton"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>s. s \<in> init A \<Longrightarrow> P s"   and "\<And>g g' l. \<lbrakk>(g, l) \<in> oreachable A S U; P (g, l); U g g'\<rbrakk> \<Longrightarrow> P (g', l)"   and "\<And>s a s'. \<lbrakk>s \<in> oreachable A S U; P s; (s, a, s') \<in> automaton.trans A; S (fst s) (fst s') a\<rbrakk> \<Longrightarrow> P s'" shows "A \<Turnstile> (S, U \<rightarrow>) P"
ex_rotation_mapping_u_to_positive_x_axis: fixes u :: "complex_homo" assumes "u \<noteq> 0\<^sub>h"   and "u \<noteq> \<infinity>\<^sub>h" shows "\<exists>\<phi>. moebius_pt (moebius_rotation \<phi>) u \<in> positive_x_axis"
list_all2_dropI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and n :: "nat" assumes "list_all2 P xs ys" shows "list_all2 P (drop n xs) (drop n ys)"
less_floor_iff: fixes z :: "int"   and x :: "'a" shows "(z < \<lfloor>x\<rfloor>) = (of_int z + (1::'a) \<le> x)"
vdiff_vsingleton_in: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "set {a} -\<^sub>\<circ> A = 0"
UP_subring_eval: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S" shows "to_function R g a = to_function (R\<lparr>carrier := S\<rparr>) g a"
max_stepsize_ge_stepsize: fixes j :: "nat"   and n :: "nat" assumes "j \<le> n" shows "stepsize j \<le> max_stepsize n"
Inf_nres_def: fixes X :: "'b nres set" shows "Inf X \<equiv> if \<exists>x. RES x \<in> X then RES (\<Inter> {x. RES x \<in> X}) else FAILi"