hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xa :: "'e uprod"   and xb :: "'f uprod" shows "rel_uprod x xa xb = (case rep_uprod xa of (a, b) \<Rightarrow> \<lambda>(c, d). x a c \<and> x b d \<or> x a d \<and> x b c) (rep_uprod xb)"
END_def: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and i :: "'b" assumes "class.arelations meets \<I> e m b ov d s f" shows "arelations.END e m ov d s f i = {j |j. (j, i) \<in> e \<union> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse> \<union> f \<union> f\<inverse> \<union> m\<inverse>}"
closed_affine_hull: fixes S :: "'c set" shows "closed (affine hull S)"
continuous_on_swap: fixes A :: "('c \<times> 'd) set" shows "continuous_on A prod.swap"
not_in_csupportD: fixes x :: "'c"   and carrier :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'd" assumes "x \<notin> csupport_on carrier \<phi>"   and "x \<in> carrier" shows "\<phi> x = (0::'d)"
filter_Q_J_neg_same: fixes V :: "nat set"   and I :: "nat set"   and J :: "nat set"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set"   and Q_I_neg :: "(nat set \<times> 'a option list set) set"   and n :: "nat" assumes "2 \<le> card V"   and "(I, J) = getIJ Q Qn V"   and "Q_I_neg = filterQueryNeg I Qn"   and "rwf_query n V Q Qn" shows "filterQuery J (Qn - Q_I_neg) = Qn - Q_I_neg"
sum_mset_if_eq_constant: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "(\<Sum>x\<in>#M. if a = x then f x else (0::'b)) = ((+) (f a) ^^ count M a) (0::'b)"
CompFun_def: shows "CompFun = snd \<circ> snd \<circ> snd \<circ> Rep_hierauto"
pCons_eq_iff: fixes a :: "'b"   and p :: "'b poly"   and b :: "'b"   and q :: "'b poly" shows "(pCons a p = pCons b q) = (a = b \<and> p = q)"
D: shows "msgs cJ1 <pJ, inM False> = 0"
connexE: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "connex A less_eq"   and "x \<in> A"   and "y \<in> A" obtains "less_eq x y"   | "less_eq y x"
mu_bound_Gramian_determinant: fixes l :: "nat"   and k :: "nat" assumes "l < k"   and "k < length fs" shows "(\<mu> k l)\<^sup>2 \<le> d l * \<parallel>fs ! k\<parallel>\<^sup>2"
scaleC_add_left: fixes a :: "complex"   and b :: "complex"   and x :: "'b" shows "(a + b) *\<^sub>C x = a *\<^sub>C x + b *\<^sub>C x"
mkSPEC_def: fixes ABI :: "((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" shows "mkSPEC ABI Anno = (let (A, B, I) = ABI in (\<lambda>s0. B s0 (mkState s0), \<lambda>s0. I s0 (mkState s0), Anno))"
linorder_not_le: fixes x :: "'g"   and y :: "'g" shows "(\<not> x \<le> y) = (y < x)"
ev_at_shift: fixes X :: "'a set"   and i :: "nat"   and \<omega> :: "'a stream"   and \<omega>' :: "'a stream" shows "ev_at (HLD X) i (stake (Suc i) \<omega> @- \<omega>') = ev_at (HLD X) i \<omega>"
implicit_pres_def: fixes effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list" shows "implicit_pres effs \<equiv> map (\<lambda>(uu_, v, vpre, uu_). (v, the vpre)) (filter (\<lambda>(uu_, uu_, vpre, uu_). vpre \<noteq> None) effs)"
card_subset_eq: fixes B :: "'a set"   and A :: "'a set" assumes "finite B"   and "A \<subseteq> B"   and "card A = card B" shows "A = B"
congruentD: fixes r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'b"   and z :: "'b" assumes "f respects r"   and "(y, z) \<in> r" shows "f y = f z"
E31: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> \<box>F) \<and> \<diamond>F \<longrightarrow> \<diamond>\<box>F"
alw_HLD_iff_streams: fixes X :: "'c set"   and \<omega> :: "'c stream" shows "alw (HLD X) \<omega> = (\<omega> \<in> streams X)"
carrier_matD_1: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_row A = nr"
cong_mat: fixes nr :: "nat"   and nr' :: "nat"   and nc :: "nat"   and nc' :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b"   and f' :: "nat \<times> nat \<Rightarrow> 'b" assumes "nr = nr'"   and "nc = nc'"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> f (i, j) = f' (i, j)" shows "mat nr nc f = mat nr' nc' f'"
in_front: shows "ego.q \<delta> < other.s \<delta>"
last_rev: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "last (rev xs) = hd xs"
finite_fun_args: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and f0 :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<forall>a\<in>A. finite (B a)" shows "finite {f. \<forall>a. if a \<in> A then f a \<in> B a else f a = f0 a}"
rel_gpv''D: fixes A :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and R :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and gpv :: "('j, 'l, 'n) gpv"   and gpv' :: "('k, 'm, 'o) gpv" assumes "rel_gpv'' A C R gpv gpv'" shows "rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv')"
abs_eq: fixes xa :: "'a \<Rightarrow> real"   and x :: "'a \<Rightarrow> real" shows "Abs_euclidean_space xa \<bullet> Abs_euclidean_space x = (\<Sum>xb\<in>UNIV. xa xb * x xb)"
vsubst_ident: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X" shows "X #[z // z]_zs = X"
to_cnf_not_0: fixes x :: "ordinal" assumes "0 < x" shows "to_cnf x = oLog \<omega> x # to_cnf (x - \<omega> ** oLog \<omega> x)"
rep_eq: fixes x :: "('e list, 'f) generator" shows "generator (concat_trans x) = concat_raw (generator x)"
conf_def: fixes typeof_addr :: "'a \<Rightarrow> htype option"   and P :: "'c prog" shows "heap_base'.conf TYPE('b) typeof_addr P \<equiv> heap_base.conf (\<lambda>_. typeof_addr) P undefined"
swapAll_psubstAll: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> (\<forall>rho z1 z2. rho \<in> envsOf P \<and> {z1, z2} \<subseteq> varsOf P \<longrightarrow> X #[rho] #[z1 \<and> z2]_zs = X #[z1 \<and> z2]_zs #[rho &[z1 \<and> z2]_zs])) \<and> (goodAbs A \<longrightarrow> (\<forall>rho z1 z2. rho \<in> envsOf P \<and> {z1, z2} \<subseteq> varsOf P \<longrightarrow> A $[rho] $[z1 \<and> z2]_zs = A $[z1 \<and> z2]_zs $[rho &[z1 \<and> z2]_zs]))"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_new_ptr: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr|}"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F"
simps_8: fixes c :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "merge_guards (DynCom c) = DynCom (\<lambda>s. merge_guards (c s))"
induct: fixes P :: "'a vec \<Rightarrow> ('a vec \<times> 'a) list \<Rightarrow> 'a list \<Rightarrow> 'a vec \<Rightarrow> bool"   and a0 :: "'a vec"   and a1 :: "('a vec \<times> 'a) list"   and a2 :: "'a list"   and a3 :: "'a vec" assumes "\<And>f mus g'. P f [] mus g'"   and "\<And>f g ng n_gs mus g'. (\<And>x. x = f \<bullet> g / ng \<Longrightarrow> P f n_gs (x # mus) (- x \<cdot>\<^sub>v g + g')) \<Longrightarrow> P f ((g, ng) # n_gs) mus g'" shows "P a0 a1 a2 a3"
at_infinity_eq_at_top_bot: shows "at_infinity = sup at_top at_bot"
assms: shows "<\<nu>x>\<zero> \<longmapsto> Rs"
inorder_n1: fixes t :: "'c bro" shows "inorder (n1 t) = inorder t"
countable': shows "countable (support_flow (\<Squnion>g\<in>Y. (\<lambda>e. f e - g e)))"
ics_SDG_path_realizable: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n ics-ns\<rightarrow>\<^sub>d* n'" obtains ns' :: "'node SDG_node list" where "realizable n ns' n'" and "set ns \<subseteq> set ns'"
is_digit_simps_2: fixes n :: "nat" assumes "\<not> n < 10" shows "\<not> is_digit n"
const_on_edom_disj: fixes f :: "'a \<Rightarrow> 'b ttree"   and S :: "'a set" shows "const_on f S TTree.empty = (edom f \<inter> S = {})"
phis_addN_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and n :: "'node \<times> 'val" shows "uninst_code.ssa.phis_addN p g n = fold (\<lambda>v. Mapping.map_default v {} (insert n)) (case Mapping.lookup (p g) n of None \<Rightarrow> [] | Some x \<Rightarrow> id x)"
intros: fixes R :: "('b \<times> 'b) set" assumes "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R" shows "asym R"
Q3_def: shows "Q3 \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0>)"
filterlim_uminus_at_bot: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter" shows "filterlim f at_bot F = (LIM x F. - f x :> at_top)"
integrable_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set" assumes "finite I"   and "\<And>a. a \<in> I \<Longrightarrow> f a integrable_on S" shows "(\<lambda>x. \<Sum>a\<in>I. f a x) integrable_on S"
SF_Box: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> (\<box>SF(A)_v) = SF(A)_v"
le_infI: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a"   and "x \<le> b" shows "x \<le> inf a b"
coeff_mpoly_of_poly: fixes i :: "nat"   and p :: "'b poly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (mpoly_of_poly i p) m = (poly.coeff p (lookup m i) when keys m \<subseteq> {i})"
div_gcd_coprime: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "a \<noteq> zero \<or> b \<noteq> zero" shows "algebraic_semidom.coprime times one (divide a (gcd a b)) (divide b (gcd a b))"
bit_sint_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (sint w) n = (LENGTH('b) \<le> n \<and> bit w (LENGTH('b) - 1) \<or> bit w n)"
iso_class_def: fixes f :: "'a" shows "\<lbrakk>f\<rbrakk>\<^sub>B \<equiv> {f'. f \<cong>\<^sub>B f'}"
compl_structD1: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> nat \<times> 'h \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> 'g) list \<times> 'h"   and d :: "'a \<Rightarrow> nat"   and sps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" assumes "compl_struct compl"   and "dickson_grading d"   and "sps \<noteq> []"   and "set sps \<subseteq> set ps" shows "dgrad_p_set_le d (fst ` set (fst (compl gs bs (ps -- sps) sps data))) (args_to_set (gs, bs, ps))"
Un_subset_iff: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "(A \<union> B \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
fdomain_mono: fixes r :: "V"   and s :: "V" assumes "r \<subseteq>\<^sub>\<circ> s" shows "\<D>\<^sub>\<bullet> r \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<bullet> s"
assms_2: shows "fdisjnt (fmdom \<Gamma>') (fmdom \<Gamma>)"
simps: fixes a :: "'c"   and uu :: "'d"   and uv :: "('c, 'd) bintree list" shows "priority (Node a uu uv) = a"
is_nth_power_def: fixes n :: "nat"   and x :: "'a" shows "is_nth_power n x = (\<exists>y. x = y ^ n)"
elem_cong_l: fixes a :: "'a"   and a' :: "'a"   and A :: "'a set" assumes "a \<in> carrier S"   and "a' \<in> carrier S"   and "A \<subseteq> carrier S"   and "a' .= a"   and "a .\<in> A" shows "a' .\<in> A"
insertList_xlate: fixes q :: "('c, 'd) BsSkewBinomialTree list"   and q' :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (insertList q q') = SkewBinomialHeapStruc.insertList (map bsmapt q) (map bsmapt q')"
winding_number_zero_const: fixes a :: "complex"   and z :: "complex" assumes "a \<noteq> z" shows "winding_number (\<lambda>t. a) z = 0"
inconsist_consistent: fixes i :: "nat"   and e :: "nat" assumes "inconsist i e \<down>" shows "(inconsist i e \<down>= e_length e) = (\<forall>x<e_length e. \<phi> i x \<down>= e_nth e x)"
pullback_has_bottom: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_eq' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "ordering.has_bottom less_eq' (f ` P)" shows "ordering.has_bottom less_eq P"
history: shows "\<exists>hs. rev qs @ h = [x, y] @ hs"
R2: fixes xps :: "('e \<times> 'f) list"   and Y :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and X :: "('e \<times> 'f) set" assumes "(xps, Y) \<in> c_failures step out s\<^sub>0"   and "X \<subseteq> Y" shows "(xps, X) \<in> c_failures step out s\<^sub>0"
ChiRel_SA_OneAncestor: fixes S :: "'d"   and T :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and SA :: "('d, 'e, 'f) seqauto"   and U :: "'d" assumes "(S, T) \<in> ChiRel A"   and "T \<in> States SA"   and "U \<in> States SA"   and "SA \<in> SAs A" shows "(S, U) \<in> ChiRel A"
map_le_def: fixes m\<^sub>1 :: "'b \<Rightarrow> 'c option"   and m\<^sub>2 :: "'b \<Rightarrow> 'c option" shows "(m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2) = (\<forall>a\<in>dom m\<^sub>1. m\<^sub>1 a = m\<^sub>2 a)"
set_integrable_UN: fixes I :: "'b set"   and M :: "'c measure"   and A :: "'b \<Rightarrow> 'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> set_integrable M (A i) f"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets M" shows "set_integrable M (\<Union> (A ` I)) f"
simps_2: fixes x :: "'b list"   and xs :: "'b list list" shows "concat (x # xs) = x \<cdot> concat xs"
Lt_lt_conv: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(key_compare ko x y = Lt) = ko.lt ko x y"
subgroup_finite: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := conj.stabilizer P\<rparr>)"   and "finite (carrier (G\<lparr>carrier := conj.stabilizer P\<rparr>))" shows "finite H"
distinct_1: fixes x1 :: "'b rexp"   and x21 :: "'b list set"   and x22 :: "'b rexp" shows "Lam x1 \<noteq> Trn x21 x22"
induct: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and x1 :: "'b set"   and x2 :: "'c"   and P :: "'b set \<Rightarrow> 'c \<Rightarrow> bool" assumes "fold_graph f z x1 x2"   and "P {} z"   and "\<And>x A y. \<lbrakk>x \<notin> A; fold_graph f z A y; P A y\<rbrakk> \<Longrightarrow> P (insert x A) (f x y)" shows "P x1 x2"
digraphD_9: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<R>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) \<subseteq>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
cases: fixes a1 :: "('d, 'e, 'f) gpv"   and a2 :: "('d, 'e, 'f) gpv" assumes "ord_gpv a1 a2" obtains f :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" and g :: "('d, 'e, ('d, 'e, 'f) rpv) generat spmf" where "a1 = GPV f" and "a2 = GPV g" and "ord_spmf (rel_generat (=) (=) (rel_fun (=) ord_gpv)) f g"
Basis_prod_def: shows "Basis = (\<lambda>u. (u, 0::'d)) ` Basis \<union> Pair (0::'c) ` Basis"
Inf_from_q_def: fixes q :: "'a literal multiset set" shows "F.ground.Inf_from_q q = G.Inf_from q"
def_hashmap_size: shows "1 < def_hashmap_size TYPE('b)"
nop_axioms: shows "nop A (2\<^sub>\<nat>) f"
trb: shows "a' \<otimes> b' \<sim> a' \<otimes> b"
cut_le_Int: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>\<le> t = A \<down>\<le> t \<inter> B \<down>\<le> t"
proj_Cons_2: fixes n :: "'a"   and a :: "'b"   and A :: "('a strand_label \<times> 'b) list" shows "proj n ((\<star>, a) # A) = (\<star>, a) # proj n A"
adj_reachable_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<rightarrow>\<^bsub>G \<restriction> S\<^esub> b"   and "b \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> c" shows "a \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> c"
enumerator_length2: fixes ls :: "nat list"   and m :: "nat"   and n :: "nat" assumes "ls \<in> set (enumerator m n)"   and "2 < m" shows "|ls| = m"
similar_implies_exist_matrix': fixes A :: "(('a, 'b) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'b) vec" assumes "similar_matrices A B" shows "\<exists>X Y f. linear (*s) (*s) f \<and> matrix' X X f = A \<and> matrix' Y Y f = B \<and> is_basis (set_of_vector X) \<and> is_basis (set_of_vector Y)"
ipset_from_cidr_def: fixes addr :: "'a word"   and pflength :: "nat" shows "ipset_from_cidr addr pflength \<equiv> ipset_from_netmask addr (mask pflength << LENGTH('a) - pflength)"
moebius_dilatation_comp: fixes a1 :: "real"   and a2 :: "real" assumes "0 < a1"   and "0 < a2" shows "moebius_comp (moebius_dilatation a1) (moebius_dilatation a2) = moebius_dilatation (a1 * a2)"
Int_insert_left_if1: fixes a :: "'b"   and C :: "'b set"   and B :: "'b set" assumes "a \<in> C" shows "insert a B \<inter> C = insert a (B \<inter> C)"
prv_imp_neg_imp_cnjR: fixes \<phi> :: "'fmla"   and \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "bprv (imp \<phi> (neg \<phi>2))" shows "bprv (imp \<phi> (neg (cnj \<phi>1 \<phi>2)))"
resolvable_necessary_cond_v: shows "\<k> dvd \<v>"
distinct_5: fixes x11 :: "trm"   and x12 :: "trm"   and x41 :: "name"   and x42 :: "fmla" shows "x11 EQ x12 \<noteq> exi x41 x42"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" assumes "prebicategory V H \<a>"   and "weak_composition_with_homs V H src trg" shows "prebicategory_with_homs V H \<a> src trg"
fsyms_satC_cong: fixes intT :: "'tp \<Rightarrow> 'b \<Rightarrow> bool"   and intF1 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and intP :: "'psym \<Rightarrow> 'b list \<Rightarrow> bool"   and intF2 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and c :: "('fsym, 'psym) lit list"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'b" assumes "Struct wtFsym wtPsym arOf resOf intT intF1 intP"   and "Struct wtFsym wtPsym arOf resOf intT intF2 intP"   and "\<And>f. f \<in> fsymsC c \<Longrightarrow> intF1 f = intF2 f" shows "Struct.satC intF1 intP \<xi> c = Struct.satC intF2 intP \<xi> c"
F: shows "(\<Union> (range (relations M)))\<^sup>* `` {t} = (\<Union> (range (relations M')))\<^sup>* `` {t}"
forest_components_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "top \<sqinter> top \<le> top \<sqinter> top"
restr_eqE: fixes F :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and F' :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "restrict F A = restrict F' A"   and "x \<in> A" shows "F x = F' x"
simps: fixes v :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "luckyFind' v L F = (case find_lucky_eq v L of None \<Rightarrow> And (list_conj (map fm.Atom L)) (list_conj F) | Some p \<Rightarrow> elimVar v L F (Eq p))"
qSwapped_preserves_qGood: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm"   and Y :: "('b, 'c, 'varSort, 'var, 'd) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qGood Y = qGood X"
mahler_measure_ge_0: fixes h :: "int poly" shows "0 \<le> mahler_measure h"
j_def: shows "j = (SOME j. j < n \<and> P $$ (j, i - (m - 1)) \<noteq> 0)"
mergeable_4: shows "\<forall>x\<in>B. x < n \<and> v2 ! x = f x"
is_constraint_future_bounded: fixes \<phi> :: "Formula.formula" assumes "is_constraint \<phi>" shows "Formula.future_bounded \<phi>"
the_pmf_def: fixes p :: "'c pmf" shows "option_to_spmf.the_pmf p = (THE x. p = return_pmf x)"
init_def: fixes T :: "Javatype" shows "init T = (case T of BoolT \<Rightarrow> boolV False | IntgT \<Rightarrow> intgV 0 | ShortT \<Rightarrow> shortV 0 | ByteT \<Rightarrow> byteV 0 | _ \<Rightarrow> nullV)"
mono_constant: fixes f :: "nat \<Rightarrow> 'a event list"   and a :: "'a event list" assumes "mono f"   and "\<forall>i. f i \<le> a" shows "\<exists>i. \<forall>j\<ge>i. f j = f i"
invh_baldL_Black: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invh l"   and "invh r"   and "bheight l + 1 = bheight r"   and "color r = Black" shows "invh (baldL l a r) \<and> bheight (baldL l a r) = bheight r"
Nil: fixes y :: "'a"   and ys :: "'a list" shows "ord_class.lexordp [] (y # ys)"
map_sel_1: fixes a :: "'a msllist"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> MSLNil" shows "un_MSLCons1 (mslmap f a) = f (un_MSLCons1 a)"
eq_fold: fixes A :: "'b set" shows "card A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
var_Monotone: fixes b :: "BExpr" shows "Monotone (\<lambda>\<Phi> (s, t, \<beta>). (b, \<Phi>, \<beta>, s, t) \<in> var)"
terminal_path_is_hyps_free: fixes v :: "'d"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "terminal_path v v' pth" shows "hyps_free pth"
disjnt_subset1: fixes X :: "'a set"   and Y :: "'a set"   and Z :: "'a set" assumes "disjnt X Y"   and "Z \<subseteq> X" shows "disjnt Z Y"