hcju/isabelleps huggingface dataset:
assms_2: shows "card {q. prime q \<and> q \<le> p} = Suc n"
Sem_eval_cases_1: fixes s :: "Var \<Rightarrow> Val"   and n :: "nat"   and t :: "Var \<Rightarrow> Val" assumes " s , Skip \<rightarrow>\<^sub>n  t " obtains "n = Suc 0" and "t = s"
nGt_le_conv: fixes x :: "'b"   and y :: "'b" shows "(c1 x y \<noteq> Gt) = c1.le x y"
V_disjoint_iff: fixes x :: "V"   and y :: "V" shows "vdisjnt x y = (elts x \<inter> elts y = {})"
Liminf_mono: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x \<le> g x" shows "Liminf F f \<le> Liminf F g"
comm_monoid_axioms_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" shows "comm_monoid_axioms f z \<equiv> \<forall>a. f a z = a"
conjugation_conj: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "((x \<lhd> y) \<sqinter> z = \<bottom>) = (y \<sqinter> (z \<rhd> x) = \<bottom>)"
INT_D: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set"   and a :: "'d" assumes "b \<in> \<Inter> (B ` A)"   and "a \<in> A" shows "b \<in> B a"
intros: fixes X\<^sub>0 :: "'d set"   and X :: "'d set"   and I :: "'d \<Rightarrow> intv"   and r :: "('d \<times> 'd) set"   and k :: "'d \<Rightarrow> nat" assumes "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}"   and "refl_on X\<^sub>0 r"   and "trans r"   and "total_on X\<^sub>0 r"   and "\<forall>x\<in>X. valid_intv (k x) (I x)" shows "valid_region X k I r"
that: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "op' = \<phi>\<^sub>O \<Psi> op"   and "is_valid_operator_sas_plus \<Psi> op" shows "thesis"
simps_1: fixes a :: "variable"   and s :: "String.literal set"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubstT (Tv a s) \<rho> = \<rho> a s"
segment_Expre: fixes a :: "'a"   and b :: "'a" assumes "a \<in> segment D b" shows "ExPre (Iod D (segment D b)) a = ExPre D a"
bigtheta_trans2: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)"   and "g \<in> \<Theta>[F](h)" shows "f \<in> \<Theta>[F](h)"
prim_invar_impl_finish: fixes Q :: "'q"   and \<pi> :: "'m" assumes "Q_is_empty Q"   and "prim_invar_impl Q \<pi>" shows "invar_MST \<pi> \<and> is_MST (G_\<alpha>w g) rg (\<alpha>_MST \<pi>)"
distinct_31: fixes x41 :: "('c, 'd) Term.term"   and x42 :: "('c, 'd) Term.term"   and x51 :: "('c, 'd) Term.term"   and x52 :: "('c, 'd) Term.term" shows "insert\<langle>x41,x42\<rangle> \<noteq> delete\<langle>x51,x52\<rangle>"
ipurge_tr_all: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list" shows "(ipurge_tr I D u xs = xs) = (\<not> (\<exists>v\<in>D ` set xs. (u, v) \<in> I))"
fermat_3: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "x ^ 3 + y ^ 3 = z ^ 3" shows "x * y * z = 0"
F2set2_bd': fixes x :: "('e, 'd, 'f) F2"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" shows "|F2set2 x| \<le>o dir_image (r1 +c bd_F2) Abs_bd_type_F"
S1: shows "pr_gr = PrimRecOp1 0 (\<lambda>x y. g_step y (c_fst x))"
MakeFtor: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" assumes "FunctorM F" shows "Functor (MakeFtor F)"
valid_path_edges': fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c"   and Ps :: "'c llist" assumes "Digraph G"   and "Digraph.valid_path G (LCons v (LCons w Ps))" shows "v \<rightarrow>\<^bsub>G\<^esub> w"
"2_5": fixes x :: "'a"   and y :: "'b" assumes "(x, y) \<in> Ra"   and "x \<in> set (x__ # xs)"   and "y \<in> set (y__ # ys)" shows "(f x, g y) \<in> Rs \<rightarrow> Rs"
hinv_l_bal: fixes l :: "'a tree"   and r :: "'a tree"   and n :: "'a" assumes "hinv l"   and "hinv r"   and "height (erase l) = height (erase r) + 2" shows "hinv (l_bal n l r)"
ask_env_def: fixes f :: "'d \<Rightarrow> ('d, 'e) envT" shows "ask_env f = EnvT (\<lambda>r. run_env (f r) r)"
x_grid: shows "x \<in> grid (start dm) ({0..<dm} - ds)"
simps_1: fixes uu :: "'i" shows "dest_IO (Pure uu) = None"
diameter_cball: fixes a :: "'b"   and r :: "real" shows "diameter (cball a r) = (if r < 0 then 0 else 2 * r)"
init_obj_predI: fixes x :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and xa :: "char list"   and xb :: "char list list \<times> (char list, val) mapping" assumes "init_obj' x xa xb" shows "pred.eval (init_obj_pred x xa) xb"
eq_Min_iff: fixes A :: "'t set"   and m :: "'t" assumes "finite A"   and "A \<noteq> {}" shows "(m = ord_term_lin.Min A) = (m \<in> A \<and> (\<forall>a\<in>A. m \<preceq>\<^sub>t a))"
strict_linear_order_on_diff_Id: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "linear_order_on A r" shows "strict_linear_order_on A (r - Id)"
mono_below: fixes iv1 :: "ivl"   and iv2 :: "ivl" assumes "iv1 \<le> iv2" shows "below iv1 \<le> below iv2"
igAbsRenSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igAbsRenSTR MOD \<equiv> \<forall>xs y x X. igFresh MOD xs y X \<longrightarrow> igAbs MOD xs y (igSubst MOD xs (igVar MOD xs y) x X) = igAbs MOD xs x X"
fixp_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) f" shows "fixp f = f (fixp f)"
M1_subset_G: shows "M1 \<subseteq> carrier G"
eqpoll_iff_bijections: fixes A :: "'a set"   and B :: "'b set" shows "(A \<approx> B) = (\<exists>f g. (\<forall>x\<in>A. f x \<in> B \<and> g (f x) = x) \<and> (\<forall>y\<in>B. g y \<in> A \<and> f (g y) = y))"
expand_map_alt_def: fixes f :: "'c \<Rightarrow> 'd set option" shows "expand_map f = {g. dom g = dom f \<and> (\<forall>x S y. f x = Some S \<longrightarrow> g x = Some y \<longrightarrow> y \<in> S)}"
semiring_normalization_rules_32: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "power.power one times x 0 = one"
basic_trans_rules_24: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
rn_rev_price_cond_expect: fixes N :: "bool stream measure"   and q :: "real"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "der \<in> borel_measurable (G matur)"   and "Suc n \<le> matur" shows "expl_cond_expect N (proj_stoch_proc geom_proc n) (rn_rev_price N der matur (matur - Suc n)) w = q * rn_rev_price N der matur (matur - Suc n) (pseudo_proj_True n w) + (1 - q) * rn_rev_price N der matur (matur - Suc n) (pseudo_proj_False n w)"
b: shows "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
FrG: shows "G = \<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle>"
Unbal: shows "ins (size t) 0 x t = Unbal t'"
adds_term_trans: fixes u :: "'t"   and v :: "'t"   and w :: "'t" assumes "u adds\<^sub>t v"   and "v adds\<^sub>t w" shows "u adds\<^sub>t w"
context_equiv_tyenv_eq: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and mem :: "'a \<Rightarrow> 'b"   and mem' :: "'a \<Rightarrow> 'b"   and P :: "'d set"   and \<Gamma>' :: "'a \<Rightarrow> 'd set option" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "sifum_types.tyenv_eq ev\<^sub>B dma_type \<Gamma> mem mem'"   and "sifum_types.context_equiv ev\<^sub>B \<Gamma> P \<Gamma>'"   and "sifum_lang_no_dma.pred ev\<^sub>B P mem" shows "sifum_types.tyenv_eq ev\<^sub>B dma_type \<Gamma>' mem mem'"
zX: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A" shows "perron_frobenius.z A \<in> perron_frobenius.X"
sets_Collect_countable_All: fixes M :: "'d measure"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>i. {x \<in> space M. P i x} \<in> sets M" shows "{x \<in> space M. \<forall>i. P i x} \<in> sets M"
f_def: shows "f = (\<lambda>n. card {p. p \<in># prime_factorization n \<and> ln (real n) < real p})"
jozsa_transform_coeff_1: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row U\<^sub>f \<and> j < dim_col U\<^sub>f" shows "i = j \<longrightarrow> U\<^sub>f $$ (i, j) = of_nat (1 - f (i div 2))"
dterms_psimps_1: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and g :: "'b \<Rightarrow> 'b set"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "dterms \<Gamma> ({l}\<langle>g\<rangle> p) = sterms \<Gamma> p"
assms_4: fixes x :: "'a"   and y :: "'b" assumes "a = Inl x"   and "b = Inr y" shows "thesis"
OrdE: fixes v :: "i"   and x :: "\<kappa>" assumes "[\<lparr>O!,x\<rparr> in v]" shows "\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y"
param_RETURN: fixes R :: "('b \<times> 'c) set" shows "(RETURN, RETURN) \<in> R \<rightarrow> \<langle>R\<rangle>nres_rel"
het_def: fixes l :: "real list" shows "het l = length (list_neq l (mean l))"
one_normh: fixes n :: "nat" shows "isnpolyh (1::'b)\<^sub>p n"
AbstrLevels_S2_A11: fixes i :: "AbstrLevelsID" assumes "sS2 \<in> AbstrLevel i" shows "sA11 \<notin> AbstrLevel i"
reflexivity_rule_def: fixes t :: "'c" shows "reflexivity_rule t = (LG {} {0}, LG {(t, 0, 0)} {0})"
union_tsI2: fixes c :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and k :: "'b"   and A :: "'b \<Rightarrow> 'a set" assumes "c \<in> B k" shows "c \<in> (A \<Rightarrow>\<union> B) k"
invariant_reaches: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Graph_Invariant_Strong E P"   and "E\<^sup>*\<^sup>* a b"   and "P a" shows "P b"
strategyproof_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "strategyproof_sds agents alts sds \<equiv> social_decision_scheme agents alts sds \<and> strategyproof_sds_axioms agents alts sds"
A: fixes ys :: "'a list"   and zs :: "'a list"   and ys' :: "'a list" shows "xs \<simeq> {ys, zs, P} \<longrightarrow> xs \<simeq> {ys', zs, P} \<longrightarrow> ys = ys'"
dg_digraph_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "digraph \<beta> \<CC>"
gen_iapath_def: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "H.gen_iapath V u p v \<equiv> u \<in> V \<and> v \<in> V \<and> H.apath u p v \<and> set (H.inner_verts p) \<inter> V = {} \<and> p \<noteq> []"
rt_graph_vD: fixes ip :: "nat"   and ip' :: "nat"   and \<sigma> :: "nat \<Rightarrow> state"   and dip :: "nat" assumes "(ip, ip') \<in> rt_graph \<sigma> dip" shows "dip \<in> vD (rt (\<sigma> ip))"
not_mod2_eq_Suc_0_eq_0: fixes n :: "nat" shows "(n mod 2 \<noteq> Suc 0) = (n mod 2 = 0)"
not_in_set_insert: fixes x :: "'b"   and xs :: "'b list" assumes "x \<notin> set xs" shows "List.insert x xs = x # xs"
set_comp: fixes U :: "'b set"   and g :: "'b set_arrow"   and f :: "'b set_arrow" shows "g \<bullet>\<^bsub>set_cat U\<^esub> f = g \<odot> f"
invert_side_of_triangle_2: fixes h :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.arr h"   and "f \<cdot>\<^sub>S g = h"   and "S.iso g" shows "S.seq h (S.inv g) \<and> f = h \<cdot>\<^sub>S S.inv g"
rel_pmf_restrict_relpE: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c pmf"   and y :: "'d pmf" assumes "rel_pmf (R \<upharpoonleft> P \<otimes> Q) x y" obtains "rel_pmf R x y" and "pred_pmf P x" and "pred_pmf Q y"
solve_construct: shows "w_vec_R p qs signs = solve_for_lhs_R p qs subsets (M_mat_R signs subsets)"
abs_eq: fixes xa :: "'b multiset \<times> 'b multiset"   and x :: "'b multiset \<times> 'b multiset" shows "abs_zmultiset xa + abs_zmultiset x = abs_zmultiset ((case xa of (Mp, Mn) \<Rightarrow> \<lambda>(Np, Nn). (Mp + Np, Mn + Nn)) x)"
closeSubstUnfold: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<sigma> :: "(name list \<times> 'a list) list" assumes "(\<Psi>, P, Q) \<in> closeSubst Rel"   and "wellFormedSubst \<sigma>" shows "(\<Psi>, P[<\<sigma>>], Q[<\<sigma>>]) \<in> closeSubst Rel"
continuous_on_open_Collect_neq: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "continuous_on S g"   and "open S" shows "open {x \<in> S. f x \<noteq> g x}"
Sup_fun_def: fixes A :: "('e \<Rightarrow> 'f) set" shows "\<Squnion>A = (\<lambda>x. SUP f\<in>A. f x)"
continuous_map_snd: fixes X :: "'b topology"   and Y :: "'c topology" shows "continuous_map (prod_topology X Y) Y snd"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'j, 'k, 'l) SLin_state_scheme" assumes "\<not> P13 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'j, 'k, 'l) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (finite (initVals s2) \<and> finite (abortVals s1) \<and> finite (abortVals s2))"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Inv \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[Inv t, Inv u, Inv v\<^bold>]"
uv1: fixes X :: "'b program set"   and GG :: "'b program set" assumes "uv_prop X"   and "finite GG"   and "GG \<subseteq> X"   and "OK GG (\<lambda>G. G)" shows "(\<Squnion>G\<in>GG. G) \<in> X"
\<pi>_Faba_respect: shows "rel_fun Eab Ea \<pi>_Faba \<pi>_Faba"
Q_initQ_def: fixes G_invar :: "'a \<Rightarrow> bool"   and G_succ :: "'a \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and G_empty :: "'a"   and G_add :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> nat option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat"   and G_\<alpha> :: "'a \<Rightarrow> 'b \<times> 'b \<Rightarrow> enat"   and s :: "'b" assumes "Dijkstra_Impl_Defs G_invar G_succ G_empty G_add M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin G_\<alpha>" shows "Dijkstra_Impl_Defs.Q_initQ Q_empty Q_update s \<equiv> Q_update s 0 Q_empty"
simps: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state"   and a3 :: "state" shows "\<gamma>,p\<turnstile> \<langle>a1, a2\<rangle> \<Rightarrow>\<^sub>\<alpha> a3 = ((\<exists>t. a1 = [] \<and> a2 = t \<and> a3 = t) \<or> (\<exists>m. a1 = [Rule m action.Accept] \<and> a2 = Undecided \<and> a3 = Decision FinalAllow \<and> Matching_Ternary.matches \<gamma> m action.Accept p) \<or> (\<exists>m. a1 = [Rule m action.Drop] \<and> a2 = Undecided \<and> a3 = Decision FinalDeny \<and> Matching_Ternary.matches \<gamma> m action.Drop p) \<or> (\<exists>m. a1 = [Rule m Reject] \<and> a2 = Undecided \<and> a3 = Decision FinalDeny \<and> Matching_Ternary.matches \<gamma> m Reject p) \<or> (\<exists>m. a1 = [Rule m Log] \<and> a2 = Undecided \<and> a3 = Undecided \<and> Matching_Ternary.matches \<gamma> m Log p) \<or> (\<exists>m. a1 = [Rule m Empty] \<and> a2 = Undecided \<and> a3 = Undecided \<and> Matching_Ternary.matches \<gamma> m Empty p) \<or> (\<exists>m a. a1 = [Rule m a] \<and> a2 = Undecided \<and> a3 = Undecided \<and> \<not> Matching_Ternary.matches \<gamma> m a p) \<or> (\<exists>rs X. a1 = rs \<and> a2 = Decision X \<and> a3 = Decision X) \<or> (\<exists>rs\<^sub>1 t rs\<^sub>2 t'. a1 = rs\<^sub>1 @ rs\<^sub>2 \<and> a2 = Undecided \<and> a3 = t' \<and> \<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> t \<and> \<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow>\<^sub>\<alpha> t'))"
CR_for_One_star: fixes t :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes " t \<longrightarrow>\<^sub>1\<^sup>* t1"   and " t \<longrightarrow>\<^sub>1\<^sup>* t2" shows "\<exists>t3.  t2 \<longrightarrow>\<^sub>1\<^sup>* t3 \<and>  t1 \<longrightarrow>\<^sub>1\<^sup>* t3"
Iset_def: fixes A :: "'c set star" shows "Iset A = {x. (*p2* (\<in>)) x A}"
\<R>\<^sub>S_comp: fixes \<beta> :: "'d \<Rightarrow> 'd"   and \<beta>' :: "'d \<Rightarrow> 'd"   and \<alpha>' :: "'e \<Rightarrow> 'e"   and \<alpha> :: "'e \<Rightarrow> 'e"   and s :: "'d \<Rightarrow> ('e, 'd, 'f) val option" assumes "bij \<beta>"   and "bij \<beta>'" shows "\<R>\<^sub>S \<alpha>' \<beta>' (\<R>\<^sub>S \<alpha> \<beta> s) = \<R>\<^sub>S (\<alpha>' \<circ> \<alpha>) (\<beta>' \<circ> \<beta>) s"
finite_mins: fixes i :: "int"   and j :: "int" shows "finite (mins i j)"
discr_Sbis: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "proper c"   and "proper d"   and "discr c"   and "discr d" shows "c \<approx>s d"
xor_distr_L: fixes A :: "bool"   and B :: "bool"   and C :: "bool" shows "A \<oplus> (B \<oplus> C) = (A \<and> \<not> B \<and> \<not> C) \<or> A \<and> B \<and> C \<or> \<not> A \<and> B \<and> \<not> C \<or> \<not> A \<and> \<not> B \<and> C"
OSC_def: fixes L :: "'b \<Rightarrow> nat"   and E :: "('b \<times> 'b) set" shows "OSC L E = (\<forall>e\<in>E. L (fst e) = 1 \<or> L (snd e) = 1 \<or> L (fst e) = L (snd e) \<and> 1 < L (fst e))"
subspace_Collect_diff_fun: shows "subspace (Collect (diff_fun k charts_eucl))"
seq: shows "\<forall>n. f n \<in> PiC I X"
agiop_giop: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "iop (b_ag A) = mop A"
F_def: shows "F \<equiv> rec_nat M0 (\<lambda>n N. SOME N'. N' \<subseteq> N \<and> infinite N' \<and> decides \<F> (f (Suc n)) N')"
transpI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
function_comp_is_closed_seq: fixes s :: "nat \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "s \<in> carrier R\<^bsup>\<omega>\<^esup>"   and "f \<in> carrier (function_ring (carrier R) R)" shows "f \<circ> s \<in> carrier R\<^bsup>\<omega>\<^esup>"
has_distD: fixes G :: "graph"   and m :: "nat"   and n :: "nat" assumes "has_dist G m n" shows "\<exists>p. is_shortest_path G m n p"
IFind_2: fixes y :: "'a ifex" assumes "ro_ifex xb2"   and "ro_ifex y"   and "\<And>ass. val_ifex xb2 ass = val_ifex y ass" shows "xb2 = y"
rel_distinct_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'c"   and y22 :: "'c list" shows "\<not> list_all2 R (y21 # y22) []"
tm_dghm_ArrMap_in_Vset: shows "\<FF>\<lparr>ArrMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
index_p: fixes p :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "infinitely_many p" shows "p (infinitely_many.index p n)"
distinct_1: fixes x1 :: "'c"   and x2 :: "'c" shows "Le x1 \<noteq> Lt x2"
ideal_eq_left_ideal: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'b set" assumes "class.comm_ring plus zero minus uminus times" shows "ring.ideal plus zero uminus times I = ring.left_ideal plus zero uminus times I"
refl_on_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. x \<^bsub>r\<^esub>\<preceq> x))"
const_lt_component_subset: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "const_lt_component ` (B - {0}) - {None} \<subseteq> Some ` component_of_term ` Keys B"
image_subgroup: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d"   and G :: "'c set" assumes "monoid_isomorphism \<eta> M composition unit M' composition' unit'"   and "subgroup G M composition unit" shows "subgroup (\<eta> ` G) M' composition' unit'"
simps: fixes a :: "'c"   and g :: "'c \<Rightarrow> ('d, 'c) step" shows "(a \<in> productive_on g) = ((\<exists>s. a = s \<and> g s = Done) \<or> (\<exists>s s'. a = s \<and> g s = Skip s' \<and> s' \<in> productive_on g) \<or> (\<exists>s x s'. a = s \<and> g s = Yield x s'))"
IH: assumes "A \<subseteq> ON"   and "A \<noteq> {}" shows "\<Squnion> A \<in> A"
jozsa_transform_def: shows "U\<^sub>f \<equiv> Matrix.mat (2 ^ (n + 1)) (2 ^ (n + 1)) (\<lambda>x. of_nat (case x of (i, j) \<Rightarrow> if i = j then 1 - f (i div 2) else if i = j + 1 \<and> odd i then f (i div 2) else if i = j - 1 \<and> even i \<and> 1 \<le> j then f (i div 2) else 0))"
markedElementsToAppend: fixes e :: "'a"   and M1 :: "('a \<times> bool) list"   and M2 :: "('a \<times> bool) list" shows "markedElementsTo e (M1 @ M2) = (if e \<in> set (elements M1) then markedElementsTo e M1 else markedElements M1 @ markedElementsTo e M2)"
at_le: fixes "open" :: "'c set \<Rightarrow> bool"   and s :: "'c set"   and t :: "'c set"   and x :: "'c" assumes "class.topological_space open"   and "s \<subseteq> t" shows "topological_space.at_within open x s \<le> topological_space.at_within open x t"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Left_a.in_hom f a b \<equiv> a \<in> Left_a.domains f \<and> b \<in> Left_a.codomains f"
a: shows "0 < a * r\<^sup>2 + b * r + c"
natural_isomorphism_axioms: shows "natural_isomorphism C.CC.comp (\<cdot>\<^sub>D) \<phi> \<phi> \<phi>"
path_eq: fixes p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" assumes "path p"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t" shows "path q"
sin_product_formula_real': fixes x :: "real" assumes "x \<noteq> 0" shows "(\<lambda>n. \<Prod>k = 1..n. 1 - x\<^sup>2 / (real k)\<^sup>2) \<longlonglongrightarrow> sin (pi * x) / (pi * x)"
G_map: fixes g :: "'g"   and v :: "'v" assumes "g \<in> G"   and "v \<in> set vs" shows "T (g *\<cdot> v) = g *\<star> T v"
p_residue_ring_car_memE_1: fixes m :: "int"   and k :: "nat" assumes "m \<in> carrier (residue_ring (p ^ k))" shows "m < p ^ k"
the_inv_into_f_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'b" assumes "inj_on f A"   and "f x = y"   and "x \<in> A" shows "the_inv_into A f y = x"
span_base: fixes a :: "'c"   and S :: "'c set" assumes "a \<in> S" shows "a \<in> vs2.span S"
bvs_2: shows "set bvs \<subseteq> RSpan vs \<and> RSpan vs = RSpan bvs \<and> R_lin_independent bvs"
return_time_returns: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and P :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "auto_ll_on_open.returns_to f X P x"   and "closed P" shows "auto_ll_on_open.flow0 f X x (auto_ll_on_open.return_time f X P x) \<in> P"
Lang_eq2: fixes w :: "'c list"   and G :: "('d \<times> ('d, 'c) RHS) list"   and S :: "'d" shows "(w \<in> Lang G S \<and> 1 < length w) = (\<exists>A B. (S, Branch A B) \<in> set G \<and> (\<exists>l r. w = l \<cdot> r \<and> l \<in> Lang G A \<and> r \<in> Lang G B))"