hcju/isabelleps huggingface dataset:
J_rel_coind_coind1: fixes R2 :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> bool"   and f :: "'d JF1 \<Rightarrow> 'e JF1 \<Rightarrow> ('d \<times> 'e, 'd JF1 \<times> 'e JF1, 'd JF2 \<times> 'e JF2) F1"   and R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R3 :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> bool"   and g :: "'d JF2 \<Rightarrow> 'e JF2 \<Rightarrow> ('d \<times> 'e, 'd JF1 \<times> 'e JF1, 'd JF2 \<times> 'e JF2) F2"   and x1 :: "'d JF1"   and x1' :: "'d JF1"   and x2 :: "'d JF2"   and x2' :: "'d JF2" assumes "\<forall>x y. R2 x y \<longrightarrow> f x y \<in> F1in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F1map fst fst fst (f x y) = dtor1 x \<and> F1map snd snd snd (f x y) = dtor1 y"   and "\<forall>x y. R3 x y \<longrightarrow> g x y \<in> F2in {(x, y). R1 x y} {(x, y). R2 x y} {(x, y). R3 x y} \<and> F2map fst fst fst (g x y) = dtor2 x \<and> F2map snd snd snd (g x y) = dtor2 y" shows "((\<exists>y. R2 x1 y \<and> x1' = JF1map fst (unfold1 (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y) (x1, y))) \<longrightarrow> x1' = x1) \<and> ((\<exists>y. R3 x2 y \<and> x2' = JF2map fst (unfold2 (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y) (x2, y))) \<longrightarrow> x2' = x2)"
zip_map_map: fixes f :: "'i \<Rightarrow> 'g"   and xs :: "'i list"   and g :: "'j \<Rightarrow> 'h"   and ys :: "'j list" shows "zip (map f xs) (map g ys) = map2 (\<lambda>x y. (f x, g y)) xs ys"
d_aux_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "d_aux a b c = a \<sqinter> b \<squnion> b \<sqinter> c \<squnion> c \<sqinter> a"
assms_5: shows "C \<in> \<Union> (\<Union>(k, v)\<in>{0..<length \<pi>} \<times> set (\<Pi>\<^sub>\<V>). {{{(State k (index (\<Pi>\<^sub>\<V>) v))\<^sup>+, (State (Suc k) (index (\<Pi>\<^sub>\<V>) v))\<inverse>} \<union> {(Operator k (index (\<Pi>\<^sub>\<O>) op))\<^sup>+ |op. op \<in> set (\<Pi>\<^sub>\<O>) \<and> v \<in> set (add_effects_of op)}}})"
is_poincare_line_trough_zero_trough_infty: fixes l :: "circline" assumes "is_poincare_line l"   and "0\<^sub>h \<in> circline_set l" shows "\<infinity>\<^sub>h \<in> circline_set l"
top_left_mult_increasing: fixes x :: "'a" shows "x \<le> top * x"
simps_2: fixes y :: "'b"   and v :: "'a"   and ls :: "('b \<times> 'a) list"   and x :: "'b" shows "lookup ((y, v) # ls) x = (if x = y then Some v else lookup ls x)"
convex_contains_open_segment: fixes s :: "'b set" shows "convex s = (\<forall>a\<in>s. \<forall>b\<in>s. {a<--<b} \<subseteq> s)"
psi_l_Suc_l_derive: fixes l :: "nat" shows "2 * (alpha_l l * complex_of_real (cos (\<theta> / 2)) - beta_l l * complex_of_real (sin (\<theta> / 2))) \<cdot>\<^sub>v \<psi> - psi'_l l = psi_l (l + 1)"
i_ma: shows "mod_type_class.from_nat i \<le> ma"
Rep_ivl_inject: fixes x :: "'b ivl"   and y :: "'b ivl" shows "(Rep_ivl x = Rep_ivl y) = (x = y)"
distinct_card: fixes xs :: "'a list" assumes "distinct xs" shows "card (set xs) = |xs|"
le_fin_prefix_fininf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
side: shows "left_forkee \<notin> RID\<^sub>G s\<^sub>1"
eventually_compose_filterlim: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and f :: "'b \<Rightarrow> 'a"   and G :: "'b filter" assumes "eventually P F"   and "filterlim f F G" shows "\<forall>\<^sub>F x in G. P (f x)"
vreal_inverse_inverse: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" shows "x = (x\<inverse>\<^sub>\<real>)\<inverse>\<^sub>\<real>"
ih: fixes a :: "'a"   and b :: "'a" assumes "a \<in> H"   and "b \<in> H"   and "a \<noteq> b" shows "(\<otimes>) a ` J \<inter> (\<otimes>) b ` J = {}"
assms_2: fixes n :: "nat" assumes "n < length cl"   and "n < length dl" shows "cl ! n \<approx>s dl ! n"
simps_3: fixes "is" :: "'b set"   and t :: "'b"   and b :: "bool" shows "redT_updI is (IsInterrupted t b) = is"
Collect_empty_eq_bot: fixes P :: "'a \<Rightarrow> bool" shows "(Collect P = {}) = (P = \<bottom>)"
SDom_def: fixes \<sigma> :: "('a, 'b, 'c) subst" shows "SDom \<sigma> = {Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}"
inv_char: fixes f :: "'a" assumes "hom.iso f" shows "hom.inv f = local.inv f"
reachable_refl: fixes G :: "('e, 'f) pre_digraph"   and v :: "'e" assumes "wf_digraph G"   and "v \<in> verts G" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
intro: fixes X :: "'b set set"   and C0 :: "'b set"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "SimplicialComplex.maxsimp X C0"   and "C0 \<sim> g ` C0"   and "C0 \<noteq> g ` C0"   and "f ` g ` C0 = C0" shows "OpposedThinChamberComplexFoldings_axioms X f g C0"
vwalk_consI: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph"   and a :: "'a" assumes "vwalk p G"   and "a \<in> verts G"   and "a \<rightarrow>\<^bsub>G\<^esub> hd p" shows "vwalk (a # p) G"
zipRT_Done_Done: fixes f :: "'h \<rightarrow> 'g \<rightarrow> 'e"   and x :: "'h"   and y :: "'g" shows "zipRT\<cdot>f\<cdot>(Done\<cdot>x)\<cdot>(Done\<cdot>y) = Done\<cdot>(f\<cdot>x\<cdot>y)"
power_strict_increasing_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less (power.power one times b x) (power.power one times b y) = (x < y)"
approx_add_minus: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star"   and d :: "'a star" assumes "a \<approx> b"   and "c \<approx> d" shows "a + - c \<approx> b + - d"
star_inductr_eq: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y \<cdot> x = y" shows "z \<cdot> x\<^sup>\<star> \<le> y"
eventually_nhds: fixes "open" :: "'c set \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "class.topological_space open" shows "eventually P (topological_space.nhds open a) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
lconj_relator_freeword_R: fixes rs :: "'g list"   and xs :: "('g \<times> bool) list" assumes "rs \<in> R"   and "proper_signed_list xs"   and "order.greater_eq S (fst ` set xs)" shows "lconjby (Abs_freeword xs) (relator_freeword rs) \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>"
true_suntil: fixes P :: "'b stream \<Rightarrow> bool" shows "(\<lambda>_. True) suntil P = ev P"
hom_closed: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "group_hom G H h"   and "x \<in> carrier G" shows "h x \<in> carrier H"
ennreal_spmf_map_conv_nn_integral: fixes f :: "'d \<Rightarrow> 'b"   and p :: "'d spmf"   and x :: "'b" shows "ennreal (spmf (map_spmf f p) x) = integral\<^sup>N (measure_spmf p) (indicator (f -` {x}))"
list_succ_simps_1: fixes xs :: "'b list"   and x :: "'b" assumes "index xs x + 1 = length xs" shows "list_succ xs x = xs ! 0"
vifunion_vrange_VLambda: fixes I :: "V"   and f :: "V \<Rightarrow> V" shows "\<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) = \<Union>\<^sub>\<circ> (\<R>\<^sub>\<circ> (VLambda I f))"
continuous_on_zeta: fixes A :: "complex set" assumes "1 \<notin> A" shows "continuous_on A zeta"
while_def: fixes p :: "'a"   and x :: "'a" shows "while p do x od = (tp \<cdot> x)\<^sup>\<star> \<cdot> at p"
simps_3: fixes h1 :: "'b hp"   and h2 :: "'b hp"   and hs :: "'b hp list" shows "pass\<^sub>1 (h1 # h2 # hs) = link h1 h2 # pass\<^sub>1 hs"
PP_def: shows "PP \<equiv> subst_fm (PfP (Var xx)) xx"
Sup_atLeast: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and x :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (ord.atLeast less_eq x) = top"
Reals_add: fixes a :: "'c"   and b :: "'c" assumes "a \<in> \<real>"   and "b \<in> \<real>" shows "a + b \<in> \<real>"
impE: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and \<gamma>'' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" assumes "(\<gamma> \<longrightarrow>\<^sup>c \<gamma>') t n"   and "\<gamma> t n"   and "\<gamma>' t n \<Longrightarrow> \<gamma>'' t n" shows "\<gamma>'' t n"
le_neq_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less"   and "less_eq a b"   and "a \<noteq> b" shows "less a b"
leftderives_rule: fixes N :: "symbol"   and w :: "symbol list" assumes "(N, w) \<in> \<RR>" shows "leftderives [N] w"
simps_1: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" shows "convergent_subseq_fun s a 0 = 0"
sgn_rat: fixes a :: "int"   and b :: "int" shows "sgn (Fract a b) = rat_of_int (sgn a * sgn b)"
lookup_filter: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('c, 'b) mapping"   and k :: "'c" shows "lookup (Mapping.filter P m) k = (case lookup m k of None \<Rightarrow> None | Some v \<Rightarrow> if P k v then Some v else None)"
ord_spmf_return_spmf: fixes x :: "'a"   and p :: "'a spmf" shows "ord_spmf (=) (return_spmf x) p = (p = return_spmf x)"
distinct_replace: fixes fs :: "'c list"   and newFs :: "'c list"   and oldF :: "'c" assumes "distinct fs"   and "distinct newFs"   and "set fs \<inter> set newFs \<subseteq> {oldF}" shows "distinct (replace oldF newFs fs)"
IH_1: fixes T :: "real tensor" assumes "T \<in> set\<^sub>v (tensors_from_net m1)" shows "dims T = input_sizes m1"
finite_final: fixes M :: "'b dfa" assumes "dfa M" shows "finite (dfa.final M)"
fst_Gauss_Jordan_upt_k_PA: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" shows "fst (Gauss_Jordan_upt_k_PA A k) ** A = snd (Gauss_Jordan_upt_k_PA A k)"
npZero_sub: fixes n :: "nat" shows "0 < n \<longrightarrow> \<zero>^\<^bsup>R n\<^esup> = \<zero>"
extending_weaken: fixes C :: "'d program \<Rightarrow> 'd set"   and h :: "'e \<times> 'f \<Rightarrow> 'd"   and F :: "'e program"   and Y' :: "'d program set"   and Y :: "'e program set"   and V' :: "'d program set"   and V :: "'e program set" assumes "extending C h F Y' Y"   and "Y' \<subseteq> V'"   and "V \<subseteq> Y" shows "extending C h F V' V"
paths_ne: fixes u :: "'v"   and v :: "'v" shows "paths u v \<noteq> {}"
hbase_singletonI: fixes n :: "C"   and b :: "nat" assumes "n \<in> hbase (Suc (Suc b))" shows "C [n] \<in> hbase (Suc (Suc b))"
ldropn_ltl: fixes n :: "nat"   and xs :: "'a llist" shows "ldropn n (ltl xs) = ldropn (Suc n) xs"
uminus_finite_4_def: fixes x :: "finite_4" shows "- x = (case x of a\<^sub>1 \<Rightarrow> a\<^sub>4 | a\<^sub>2 \<Rightarrow> a\<^sub>3 | a\<^sub>3 \<Rightarrow> a\<^sub>2 | a\<^sub>4 \<Rightarrow> a\<^sub>1)"
lambda: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and T1 :: "Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T2 :: "Syntax.type" assumes "\<Gamma>\<langle>0:T1\<rangle> , \<Delta> \<turnstile>\<^sub>T t : T2" shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>T \<lambda>T1:t : T1 \<rightarrow> T2"
assms: shows "\<not> fin (M[a\<turnstile>c>b]) y"
add_stalk_def: shows "stk2.add_stalk \<equiv> stk2.add_rel"
contra2: fixes s :: "real \<times> real"   and p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "distinct4 s p q r" shows "\<not> (Counterclockwise_2D_Arbitrary.lex s p \<and> Counterclockwise_2D_Arbitrary.lex p q \<and> Counterclockwise_2D_Arbitrary.lex q r \<and> indelta s p q r)"
eq: fixes z :: "complex" assumes "z \<in> S - {a}" shows "g z = (z - a) * f z"
polyn_expr_short: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and k :: "nat" assumes "pol_coeff S c"   and "k \<le> fst c" shows "polyn_expr R X k c = polyn_expr R X k (k, snd c)"
sym: fixes x :: "real \<times> real"   and y :: "real \<times> real" shows "invert_order (compare x y) = compare y x"
single_list_def: fixes a :: "'a" shows "single_list a = [a]"
is_RELABEL'_def: fixes pr_operation' :: "pr_operation'" shows "is_RELABEL' pr_operation' = (pr_operation' = RELABEL')"
CauchyD: fixes X :: "nat \<Rightarrow> 'b"   and e :: "real" assumes "Cauchy X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
simps_4: fixes a :: "('a, 'b) Combinators"   and v :: "'a"   and va :: "'a"   and vb :: "'b" shows "member a (AllowPortFromTo v va vb) = (a = AllowPortFromTo v va vb)"
BT_lemma_1_14: fixes F :: "'b poly"   and B :: "'b poly"   and G :: "'b poly"   and H :: "'b poly"   and J :: "nat" assumes "F + B * G = H"   and "degree G \<le> degree F"   and "degree H < degree G \<or> H = 0 \<and> F \<noteq> 0 \<and> G \<noteq> 0"   and "degree H < J"   and "J < degree G - 1" shows "subresultant J F G = 0"
lt_plus_lessI: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "p + q \<noteq> 0"   and "punit.lt q = punit.lt p"   and "punit.lc q = - punit.lc p" shows "punit.lt (p + q) \<prec> punit.lt p"
g_le_OUT: fixes x :: "'v" shows "g (vertex.SINK, \<langle>x\<rangle>) \<le> d_OUT g \<langle>x\<rangle>"
i_shrink_assoc: fixes f :: "nat \<Rightarrow> 'b message_af"   and a :: "nat"   and b :: "nat" shows "f \<div> a \<div> b = f \<div> (a * b)"
random_bst_altdef: fixes A :: "'a set" assumes "finite A" shows "random_bst A = map_pmf bst_of_list (pmf_of_set (permutations_of_set A))"
defs'_uses'_disjoint: fixes n :: "'a"   and g :: "('a, 'b, 'c) gen_cfg_wf" assumes "n \<in> set (gen_wf_\<alpha>n g)" shows "gen_wf_defs' g n \<inter> gen_wf.uses' g n = {}"
metric_LIM_imp_LIM: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a"   and g :: "'a \<Rightarrow> 'c"   and m :: "'c" assumes "f \<midarrow>a\<rightarrow> l"   and "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l" shows "g \<midarrow>a\<rightarrow> m"
pivot_eq_def: fixes e :: "nat \<times> linear_poly"   and y :: "nat" shows "pivot_eq e y \<equiv> let cy = coeff (rhs e) y in (y, (- 1 / cy) *R (rhs e - cy *R Abstract_Linear_Poly.Var y) + (1 / cy) *R Abstract_Linear_Poly.Var (lhs e))"
normal_series_snd_to_last: shows "\<GG> ! (length \<GG> - 2) \<lhd> G"
\<W>_elem_in_TokensAt: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and P :: "(symbol \<times> character list) list set"   and Doc :: "character list"   and u :: "symbol \<times> character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "P \<subseteq> LocalLexing.\<PP> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc"   and "u \<in> LocalLexing.\<W> \<NN> \<TT> \<RR> \<SS> Lex Doc P k" shows "u \<in> LocalLexing.TokensAt \<TT> Lex Doc k (LocalLexing.Gen \<NN> \<TT> \<RR> \<SS> Doc P)"
eq_Nil_null: fixes xs :: "'b list" shows "(xs = \<epsilon>) = List.null xs"
simps_2: fixes n :: "nat"   and A :: "'a multiset" shows "repeat_mset (Suc n) A = A + repeat_mset n A"
exhaust: fixes y :: "'b literal" obtains x1 :: "'b equation" where "y = Pos x1"   | x2 :: "'b equation" where "y = Neg x2"
oth_class_taut_1_b: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>(\<phi> \<^bold>\<equiv> \<^bold>\<not>\<phi>) in v]"
xt1_8: fixes U :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and z :: "'a" assumes "order_ow U le ls"   and "y \<in> U"   and "x \<in> U"   and "z \<in> U"   and "le y x"   and "ls z y" shows "ls z x"
A: shows "A \<in> sets (N \<Otimes>\<^sub>M L)"
monom_inv_def: fixes m :: "('c \<times> nat) list" shows "monom_inv m \<equiv> (\<forall>(x, n)\<in>set m. 1 \<le> n) \<and> distinct (map fst m) \<and> sorted (map fst m)"
w'_def: shows "w' \<equiv> rsd_winners (R(i := Ri')) alts"
vors_3: shows "between (vertices f) ram2 ram1 = []"
card_face_cycle_sets_da: shows "card G.face_cycle_sets = Suc (Suc (card H.face_cycle_sets))"
gcd_lcm_distrib: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm" shows "gcd x (lcm y z) = lcm (gcd x y) (gcd x z)"
\<sigma>1_3: shows "(\<sigma>', coll) \<in> csmall_nstep P \<sigma>1 n1"
arrE: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "arr (fst f) \<and> VV.arr (snd f) \<and> src (fst f) = trg (fst (snd f))"
exhaust: fixes y :: "('a, 'b) stateT" obtains x :: "'a \<Rightarrow> 'b" where "y = StateT x"
scaleR_right_diff_distrib: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
LEQ: shows "\<forall>j\<in>J. |A j| \<le>o |B j|"
exec_move_BlockNone: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and V :: "nat"   and T :: "ty"   and e :: "'addr expr1" shows "exec_move ci P t {V:T=None; e} = exec_move ci P t e"
vectorspace_def: fixes V :: "'b set" shows "vectorspace V \<equiv> ((V \<noteq> {} \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x + y \<in> V) \<and> (\<forall>x a. x \<in> V \<longrightarrow> a \<cdot> x \<in> V)) \<and> (\<forall>x y z. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> z \<in> V \<longrightarrow> x + y + z = x + (y + z)) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x + y = y + x) \<and> (\<forall>x. x \<in> V \<longrightarrow> x - x = (0::'b))) \<and> ((\<forall>x. x \<in> V \<longrightarrow> (0::'b) + x = x) \<and> (\<forall>x y a. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y) \<and> (\<forall>x a b. x \<in> V \<longrightarrow> (a + b) \<cdot> x = a \<cdot> x + b \<cdot> x)) \<and> ((\<forall>x a b. x \<in> V \<longrightarrow> (a * b) \<cdot> x = a \<cdot> b \<cdot> x) \<and> (\<forall>x. x \<in> V \<longrightarrow> 1 \<cdot> x = x)) \<and> (\<forall>x. x \<in> V \<longrightarrow> - x = - 1 \<cdot> x) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x - y = x + - y)"
src_prism_lens: fixes X :: "'d \<Longrightarrow>\<^sub>\<triangle> 'c" shows "\<S>\<^bsub>prism_lens X\<^esub> = range build\<^bsub>X\<^esub>"
map_Vals_no_step: fixes vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "((map Val vs, s, b), es', s', b') \<notin> reds P"
non_inf_image: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'c" assumes "non_inf r"   and "\<And>a b. (a, b) \<in> s \<Longrightarrow> (f a, f b) \<in> r" shows "non_inf s"
less_qsend_iff_not_discrCf: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and bT :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream"   and n :: "nat" assumes "enabled cf bT" shows "(enat n < qsend (cf ## bT)) = (\<not> discrCf ((cf ## bT) !! n))"
cases: fixes x :: "const" obtains n :: "int" where "x = Const n"   | "x = Any"
ind_cpa_def: fixes key_gen :: "'b spmf"   and encrypt :: "'b \<Rightarrow> 'c \<Rightarrow> 'd spmf"   and valid_plain :: "'c \<Rightarrow> bool"   and \<A> :: "(('c \<times> 'c) \<times> 'e, 'c, 'd) gpv \<times> ('d \<Rightarrow> 'e \<Rightarrow> (bool, 'c, 'd) gpv)" shows "IND_CPA.ind_cpa.ind_cpa key_gen encrypt valid_plain \<A> = (let (\<A>1, \<A>2) = \<A> in key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (IND_CPA.ind_cpa.encrypt_oracle encrypt key) (\<A>1 \<bind> (\<lambda>((m0, m1), \<sigma>). if valid_plain m0 \<and> valid_plain m1 then lift_spmf (encrypt key (if b then m0 else m1)) \<bind> (\<lambda>cipher. \<A>2 cipher \<sigma>) else lift_spmf coin_spmf)) () \<bind> (\<lambda>(guess, uu_). return_spmf (guess = b)))))"
F_Inf_have_prems: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and mgu :: "'b set set \<Rightarrow> 'c option"   and less_atm :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<iota> :: "'b literal multiset inference" assumes "FO_resolution_prover S subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm"   and "\<iota> \<in> FO_resolution_prover.F_Inf S subst_atm renamings_apart mgu less_atm" shows "prems_of \<iota> \<noteq> []"
isTrue_iff_bprv_PPf: fixes n :: "'trm"   and \<phi> :: "'fmla" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue (PPf n \<langle>\<phi>\<rangle>) = bprv (PPf n \<langle>\<phi>\<rangle>)"
inverse_inverse_transformation: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_isomorphism A B F G \<tau>" shows "inverse_transformation.map A B F (inverse_transformation.map A B G \<tau>) = \<tau>"
find_sub_True_sound: fixes xs :: "'b set list"   and ys :: "'b set list"   and w :: "'b set"   and ws :: "'b set list"   and z :: "'b set"   and zs :: "'b set list" assumes "find_sub_True xs = Some (ys, w, ws, z, zs)" shows "xs = ys @ w # ws @ z # zs \<and> (z \<subseteq> w \<or> w \<subseteq> z)"
simps_14: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (Plus e1 e2) (Minus v va) = True"
similar_mat_wit_four_block: fixes A1 :: "'c mat"   and B1 :: "'c mat"   and P1 :: "'c mat"   and Q1 :: "'c mat"   and A2 :: "'c mat"   and B2 :: "'c mat"   and P2 :: "'c mat"   and Q2 :: "'c mat"   and URA :: "'c mat"   and UR :: "'c mat"   and LLA :: "'c mat"   and LL :: "'c mat"   and n :: "nat"   and m :: "nat" assumes "similar_mat_wit A1 B1 P1 Q1"   and "similar_mat_wit A2 B2 P2 Q2"   and "URA = P1 * UR * Q2"   and "LLA = P2 * LL * Q1"   and "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat m m"   and "LL \<in> carrier_mat m n"   and "UR \<in> carrier_mat n m" shows "similar_mat_wit (four_block_mat A1 URA LLA A2) (four_block_mat B1 UR LL B2) (four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2) (four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2)"
ffold_finsert_fremove: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and x :: "'b"   and A :: "'b fset" assumes "comp_fun_commute f" shows "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
assms_2: shows "m2 \<in> carrier_mat d1 d1"
trancl_into_rtrancl: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> r\<^sup>+" shows "(a, b) \<in> r\<^sup>*"