hcju/isabelleps huggingface dataset:
filt_equiv_sets: fixes F :: "'d \<Rightarrow> 'e measure"   and M :: "'e measure"   and N :: "'e measure" assumes "filt_equiv F M N" shows "sets M = sets N"
in_homE: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and a :: "'a"   and b :: "'a" assumes "category C"   and "partial_magma.in_hom C f a b" obtains "partial_magma.arr C f" and "partial_magma.dom C f = a" and "partial_magma.cod C f = b"
filter: shows "set \<circ> (\<lambda>u. filter (\<lambda>x. fst x = u) (snd G_list)) = out_arcs (with_proj (list_digraph G_list))"
path2_unsnoc: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "2 \<le> length ns" obtains "g \<turnstile> n-butlast ns\<rightarrow>last (butlast ns)" and "last (butlast ns) \<in> set (predecessors g m)"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'b"   and bit :: "bit" shows "h (case bit of \<zero> \<Rightarrow> f1 | \<one> \<Rightarrow> f2) = (case bit of \<zero> \<Rightarrow> h f1 | \<one> \<Rightarrow> h f2)"
smallo_neg_powr_iff: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "p < 0"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x"   and "\<forall>\<^sub>F x in F. f x \<noteq> 0"   and "\<forall>\<^sub>F x in F. g x \<noteq> 0" shows "((\<lambda>x. f x powr p) \<in> o[F](\<lambda>x. g x powr p)) = (g \<in> o[F](f))"
ereal_open_mono_set: fixes S :: "ereal set" shows "(open S \<and> mono_set S) = (S = UNIV \<or> S = {Inf S<..})"
expectation_X_non_zero: shows "\<exists>i\<in>I. 0 < expectation (X i)"
has_integral_Beta2: fixes a :: "real" assumes "a < - 1 / 2" shows "((\<lambda>x. (1 + x\<^sup>2) powr a) has_integral Beta (- a - 1 / 2) (1 / 2) / 2) {0<..}"
cset_fin_simps_3: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
vector_smult_lzero: fixes x :: "('b, 'c) Finite_Cartesian_Product.vec" shows "(0::'b) *s x = 0"
hyps: fixes ys :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "dgrad_p_set_le d (set (comp_red_basis_aux xs ys)) (set xs \<union> set ys)"
width_V_empty: fixes G :: "('e, 'f) Graph_scheme"   and T :: "('g, 'h) Graph_scheme"   and bag :: "'g \<Rightarrow> 'e set" assumes "TreeDecomposition G T bag"   and "V\<^bsub>G\<^esub> = {}" shows "TreeDecomposition.width T bag = 0"
rep_upd: fixes \<omega> :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and x :: "fin_string"   and \<theta> :: "trm"   and \<nu>' :: "fin_string \<Rightarrow> real"   and r :: "real" assumes "\<omega> = \<nu>(Inr x := snd ([\<theta>]<>\<nu>), Inl x := fst ([\<theta>]<>\<nu>))"   and "\<nu> REP \<nu>'"   and "[\<theta>]\<nu>' \<down> r" shows "\<omega> REP \<nu>'(x := r)"
cancel: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point"   and p\<^sub>3 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "ell_field_class.add a p\<^sub>1 p\<^sub>2 = ell_field_class.add a p\<^sub>1 p\<^sub>3" shows "p\<^sub>2 = p\<^sub>3"
bij_betw_imp_surj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "bij_betw f A UNIV" shows "surj f"
sign_extend_bitwise_if: fixes i :: "nat"   and w :: "'b word"   and e :: "nat" assumes "i < size w" shows "sign_extend e w !! i = (if i < e then w !! i else w !! e)"
filter_cong: fixes xs :: "'e list"   and ys :: "'e list"   and P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
hd_let: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e2 :: "expr"   and g :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "({}, V \<union> V', \<Gamma>, \<lambda>_. 1) \<turnstile>\<^sub>d e1 \<Rightarrow> f"   and "(shift_var_set V, Suc ` V', case_nat (the (expr_type \<Gamma> e1)) \<Gamma>, insert_dens V V' f \<delta>) \<turnstile>\<^sub>d e2 \<Rightarrow> g" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d LET e1 IN e2 \<Rightarrow> (\<lambda>\<rho>. g (case_nat undefined \<rho>))"
gfp_upperbound: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and X :: "'b::type"   and f :: "'b::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (X::'b::type) ((f::'b::type \<Rightarrow> 'b::type) X)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (X::'b::type) (complete_lattice.gfp (Sup::'b::type set \<Rightarrow> 'b::type) less_eq (f::'b::type \<Rightarrow> 'b::type))"
keys_Mapping_1: fixes m :: "'b \<Rightarrow> 'c option" shows "Mapping.keys (Mapping m) = {k. m k \<noteq> None}"
mydiv_le_E: fixes k :: "nat"   and B :: "nat"   and A :: "nat" assumes "0 < k"   and "B \<le> k * A" shows "mydiv B k \<le> A"
WTrt_elim_cases_10: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'c J_prog"   and h :: "'b"   and E :: "String.literal \<Rightarrow> ty option"   and a :: "'c expr"   and i :: "'c expr"   and e :: "'c expr"   and T :: "ty" assumes "J_heap_base.WTrt typeof_addr P h E (a\<lfloor>i\<rceil> := e) T" obtains T :: "ty" and T' :: "ty" where "Ta__ = Void" and "J_heap_base.WTrt typeof_addr P h E a (T\<lfloor>\<rceil>)" and "J_heap_base.WTrt typeof_addr P h E i Integer" and "J_heap_base.WTrt typeof_addr P h E e T'__"   | T' :: "ty" where "Ta__ = Void" and "J_heap_base.WTrt typeof_addr P h E a NT" and "J_heap_base.WTrt typeof_addr P h E i Integer" and "J_heap_base.WTrt typeof_addr P h E e T'"
less_eq_Sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>v. v \<in> A \<Longrightarrow> less_eq u v"   and "A \<noteq> {}" shows "less_eq u (Sup A)"
commute: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "abel_semigroup f" shows "f a b = f b a"
distrib_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y + z) = x \<cdot> y + x \<cdot> z"
Pi_I: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i"   and B :: "'h \<Rightarrow> 'i set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
hom_prod_list: fixes xs :: "'a poly list" shows "map_poly hom (prod_list xs) = prod_list (map (map_poly hom) xs)"
extg_def: shows "extg \<equiv> \<lambda>((\<sigma>, l1), l2). (\<sigma>, l1, l2)"
type_definition_cars: shows "type_definition Rep_cars Abs_cars {n. True}"
enumerate_mono: fixes m :: "nat"   and n :: "nat"   and S :: "'b set" assumes "m < n"   and "infinite S" shows "wellorder_class.enumerate S m < wellorder_class.enumerate S n"
induct: fixes v :: "letter list"   and x :: "letter list list"   and P :: "letter list list \<Rightarrow> bool" assumes "L v x"   and "\<And>w ws. emb w v \<Longrightarrow> P (w # ws)"   and "\<And>ws w. \<lbrakk>L v ws; P ws\<rbrakk> \<Longrightarrow> P (w # ws)" shows "P x"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. (\<not> is_digit n \<Longrightarrow> P (n div 10)) \<Longrightarrow> P n" shows "P a0"
mod_Suc_conv: fixes a :: "nat"   and m :: "nat"   and b :: "nat" shows "(Suc a mod m = Suc b mod m) = (a mod m = b mod m)"
degree_prod_eq_sum_degree: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd poly" assumes "\<forall>i\<in>A. f i \<noteq> 0" shows "degree (prod f A) = (\<Sum>i\<in>A. degree (f i))"
1: shows "type_update (TopType cts) [TSome T_i32] (TopType [TAny]) = tm"
fold_insert_idem2: fixes A :: "'v set"   and X :: "'v \<Rightarrow> int pmf"   and x :: "'v"   and z :: "'m" assumes "finite A" shows "sample_vars X (insert x A) z = sample_vars X A (bind (memo (sample_var X) x) (\<lambda>_. z))"
casts: shows "P \<turnstile> Ts Casts vs to vs' "
finite_intersection_of_idempot: fixes P :: "'b set \<Rightarrow> bool" shows "finite intersection_of finite intersection_of P = finite intersection_of P"
inter: fixes x :: "'a set"   and y :: "'a set" assumes "x open"   and "y open" shows "x \<inter> y open"
real_cond_exp_charact: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "\<And>A. A \<in> sets F \<Longrightarrow> set_lebesgue_integral M A f = set_lebesgue_integral M A g"   and "integrable M f"   and "integrable M g"   and "g \<in> borel_measurable F" shows "AE x in M. real_cond_exp M F f x = g x"
iprev_fix_iff: fixes n :: "nat"   and I :: "nat set" shows "(n \<notin> I \<or> iMin I = n) = (iprev n I = n)"
wellformed_mojmir_opt: fixes \<G> :: "'b ltl set"   and \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "Only_G \<G>"   and "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "mojmir \<Sigma> \<up>af\<^sub>G\<^sub>\<UU> (Abs \<phi>) w {q. \<G> \<Turnstile>\<^sub>P Rep q}"
sup1CI: fixes B :: "'b \<Rightarrow> bool"   and x :: "'b"   and A :: "'b \<Rightarrow> bool" assumes "\<not> B x \<Longrightarrow> A x" shows "sup A B x"
filtration_measurable: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and f :: "'c \<Rightarrow> 'e"   and t :: "'d"   and N :: "'e measure" assumes "filtrated_prob_space M F"   and "f \<in> F t \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
simps_20: fixes x41 :: "('b, 'c) fm"   and x42 :: "('b, 'c) fm"   and x2 :: "'c" shows "(x41 \<^bold>\<or> x42) \<noteq> Nom x2"
closed_max: fixes m :: "nat"   and p :: "('a, 'b) form"   and n :: "nat"   and q :: "('c, 'd) form" assumes "closed m p"   and "closed n q" shows "closed (max m n) p \<and> closed (max m n) q"
sum_set_simps_4: fixes x :: "'a" shows "Basic_BNFs.setr (Inr x) = {x}"
E_generates: fixes i :: "'i" assumes "i \<in> I" shows "sets (M i) = sigma_sets (space (M i)) (E i)"
simps: fixes a1 :: "any_unop + boolean_unop + numeric_unop + string_unop + collection_unop"   and a2 :: "call_kind"   and a3 :: "'a OCL_Types.type"   and a4 :: "'a OCL_Types.type" shows "unop_type a1 a2 a3 a4 = ((\<exists>op \<tau> \<sigma>. a1 = Inl op \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> any_unop_type op \<tau> \<sigma>) \<or> (\<exists>op \<tau> \<sigma>. a1 = Inr (Inl op) \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> boolean_unop_type op \<tau> \<sigma>) \<or> (\<exists>op \<tau> \<sigma>. a1 = Inr (Inr (Inl op)) \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> numeric_unop_type op \<tau> \<sigma>) \<or> (\<exists>op \<tau> \<sigma>. a1 = Inr (Inr (Inr (Inl op))) \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> string_unop_type op \<tau> \<sigma>) \<or> (\<exists>op \<tau> \<sigma>. a1 = Inr (Inr (Inr (Inr op))) \<and> a2 = ArrowCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> collection_unop_type op \<tau> \<sigma>))"
lower_asymptotic_density_def: fixes A :: "nat set" shows "lower_asymptotic_density A = real_of_ereal (liminf (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x)))"
that_1: shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'"
fold_graph_finite: fixes z :: "'b fset set"   and A :: "'b fset set set"   and y :: "'b fset set" assumes "fold_graph (\<union>\<^sub>m) z A y" shows "finite A"
pred_equals_eq2: fixes R :: "('b \<times> 'd) set"   and S :: "('b \<times> 'd) set" shows "((\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> S)) = (R = S)"
c_manifolds_axioms: shows "c_manifolds k charts2 charts2"
iso_imp_finite: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme" assumes "finite_group G"   and "G \<cong> H"   and "Group.group H" shows "finite_group H"
atMost_subset_iff: fixes x :: "'b"   and y :: "'b" shows "({..x} \<subseteq> {..y}) = (x \<le> y)"
termMOD_igSwapIGVar: shows "igSwapIGVar termMOD"
insert_disjoint_1: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(insert a A \<inter> B = {}) = (a \<notin> B \<and> A \<inter> B = {})"
ys1_def: shows "ys1 = filter (P xs) ys"
sinits_strict: shows "sinits\<cdot>\<bottom> = \<bottom>"
l_zero: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_monoid G"   and "x \<in> carrier G" shows "\<zero>\<^bsub>G\<^esub> \<oplus>\<^bsub>G\<^esub> x = x"
underS_incr2: fixes r :: "('a \<times> 'a) set"   and r' :: "('a \<times> 'a) set"   and a :: "'a" assumes "r \<subseteq> r'" shows "underS r a \<subseteq> underS r' a"
Iff_wk: fixes p :: "'a pfm"   and q :: "'a pfm"   and r :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! q)" shows "\<turnstile>\<^sub>! ((r \<^bold>\<longrightarrow>\<^sub>! p) \<^bold>\<longleftrightarrow>\<^sub>! r \<^bold>\<longrightarrow>\<^sub>! q)"
eval_elim_cases_29: fixes G :: "prog"   and s :: "st"   and e1 :: "expr"   and e2 :: "expr"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In2 (e1.[e2])\<succ>\<rightarrow> (v, s')" obtains a :: "val" and s1 :: "state" and i :: "val" and s2 :: "state" and va :: "vvar" where "v = In2 va" and "G\<turnstile>Norm s \<midarrow>e1-\<succ>a\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>e2-\<succ>i\<rightarrow> s2" and "(va, s') = avar G i a s2"
ts_invI: fixes ts :: "'g \<Rightarrow> ('h \<times> 'i \<Rightarrow>f nat) option"   and I :: "'g \<Rightarrow> 'j option"   and P :: "'j \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'k \<Rightarrow> bool"   and m :: "'k" assumes "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> \<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m" shows "ts_inv P I ts m"
attractor_has_strategy: fixes W :: "'a set"   and p :: "Player" assumes "W \<subseteq> V\<^bsub>G'\<^esub>" shows "\<exists>\<sigma>. G'.strategy p \<sigma> \<and> G'.strategy_attracts p \<sigma> (G'.attractor p W) W"
range_vars_alt_def: fixes s :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "range_vars s \<equiv> fv\<^sub>s\<^sub>e\<^sub>t (subst_range s)"
refl_ax_holds: fixes e :: "(name, hf) finfun" shows "eval_fm e refl_ax"
sign_simps_8: fixes ab :: "'a"   and bb :: "'a"   and cb :: "'a" shows "ab + (bb - cb) = ab + bb - cb"
h2: shows "h \<turnstile> adopt_node owner_document node \<rightarrow>\<^sub>h h2"
rsplit0_complete: fixes x :: "real"   and t :: "MIR.num"   and bs :: "real list" assumes "0 \<le> x"   and "x < 1" shows "\<exists>(p, n, s)\<in>set (rsplit0 t). Ifm (x # bs) p"
shuffle_to_top_def: fixes Y :: "'a set" shows "TruePref.shuffle_to_top Y = (\<lambda>d. Pd d - ManiPref'.dX Y d \<times> UNIV \<union> (Domain (Pd d) \<union> ManiPref'.dX Y d) \<times> ManiPref'.dX Y d)"
list_of_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "lfinite xs"   and "lfinite ys" shows "list_of (lappend xs ys) = list_of xs @ list_of ys"
vsubst_preserves_good: fixes Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x :: "'var" assumes "good Y" shows "good (Y #[x1 // x]_xs)"
image_cong_simp: fixes M :: "'d set"   and N :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "M = N"   and "\<And>x. x \<in> N =simp=> f x = g x" shows "f ` M = g ` N"
shared_leD: fixes m\<^sub>1 :: "nat \<Rightarrow> bool option"   and m\<^sub>2 :: "nat \<Rightarrow> bool option" assumes "m\<^sub>1 \<subseteq>\<^sub>s m\<^sub>2" shows "m\<^sub>1 \<subseteq>\<^sub>m m\<^sub>2 \<and> read_only m\<^sub>1 \<subseteq> read_only m\<^sub>2"
remember_init_state_def: shows "\<doteq> \<equiv> \<lambda>Y. (=)"
B_1: shows "a \<in># A \<or> a \<in># B"
relH_in_rangeI_1: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap" assumes "relH as h h'" shows "in_range (h, as)"
key_map_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<pp> :: "'c set"   and U :: "'c set set" assumes "key_map R addition multiplication zero unit \<pp>" shows "key_map.key_map R addition multiplication zero unit \<pp> U \<equiv> \<lambda>s\<in>comm_ring.sheaf_spec R addition multiplication zero unit U. s \<pp>"
less_ack1: fixes i :: "nat"   and j :: "nat" shows "i < ack i j"
eval_fds_const: fixes c :: "'a" shows "eval_fds (fds_const c) = (\<lambda>_. c)"
inf_continuousD: fixes F :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> 'c" assumes "inf_continuous F"   and "decseq M" shows "F (Inf (range M)) = (INF i. F (M i))"
prems_2: shows "weak_sharing_consistent \<O> (x # sb)"
convergent_prod_LIMSEQ: fixes f :: "nat \<Rightarrow> 'a" assumes "convergent_prod f" shows "(\<lambda>n. prod f {..n}) \<longlonglongrightarrow> prodinf f"
simps: fixes x :: "'a"   and y :: "'a list" shows "cons (x, y) = x # y"
af_to_af_opt_1: fixes \<phi> :: "'b ltl"   and w :: "'b set list" shows "Unf (af \<phi> w) = af\<^sub>\<UU> (Unf \<phi>) w"
disjnt_empty1: fixes A :: "'c set" shows "disjnt {} A"
assms_1: shows "otherwith Q I P \<sigma> \<sigma>' a"
generate: shows "\<exists>i\<in>carrier R. I = Idl {i}"
sort_key_simps_1: fixes f :: "'c \<Rightarrow> 'a" shows "sort_key f [] = []"
Domainp_rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Domainp (rel_pmf R) = pred_pmf (Domainp R)"
WSE_3: shows "length ms\<^sub>1 = length s\<^sub>1"
cut_less_empty: fixes t :: "'a" shows "{} \<down>< t = {}"
hyps_3: fixes gpv :: "('a, 'call, 'ret) gpv" assumes "gpv \<in> sub_gpvs \<I> (GPV p)" shows "lossless_gpv \<I> gpv"
lex_append_leftI: fixes ys :: "'a list"   and zs :: "'a list"   and r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "(ys, zs) \<in> lex r" shows "(xs @ ys, xs @ zs) \<in> lex r"
simps_9: fixes \<phi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_letter (F \<phi>) \<nu> = F \<phi> or af_letter \<phi> \<nu>"
fin_set: shows "finite {x. \<exists>q\<in>set qs. rpoly q x = (0::'a)}"
jkbpT_def: fixes T :: "'s Trace set" shows "jkbpT T \<equiv> \<Union>n. jkbpT\<^bsub>n\<^esub> T"
partition_onD2: fixes A :: "'b set"   and P :: "'b set set" assumes "partition_on A P" shows "disjoint P"
add_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (add_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
eq_def: fixes c :: "int"   and t :: "MIR.num" shows "eq c t = (if c = 0 then Eq t else if 0 < c then Eq (CN 0 c t) else Eq (CN 0 (- c) (Neg t)))"
cla_def: fixes Z :: "('c \<Rightarrow> real) set" shows "Closure\<^sub>\<alpha> Z = \<Union> {R \<in> \<R>. R \<inter> Z \<noteq> {}}"
tendsto_ln_over_powr: fixes a :: "real" assumes "0 < a" shows "((\<lambda>x. ln x / x powr a) \<longlongrightarrow> 0) at_top"
add_diff_inverse: fixes a :: "'e"   and b :: "'e" assumes "a \<le> b" shows "a + (b - a) = b"
cases: fixes a :: "'s Trace"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "a \<in> jpTraces jp" obtains s :: "'s" where "a = tInit s__" and "s__ \<in> set envInit"   | t :: "'s Trace" and eact :: "'eAct" and aact :: "'a \<Rightarrow> 'aAct" and s :: "'s" where "a = t \<leadsto> s" and "t \<in> jpTraces jp" and "eact \<in> set (envAction (tLast t))" and "\<And>a. aact a \<in> set (actJP jp t a)" and "s = envTrans eact aact (tLast t)"
AE_K: fixes x :: "real \<times> 'a" shows "AE y in K x. fst x < fst y \<and> snd y \<in> set_pmf (J (snd x))"
not_iszero_neg_1: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "\<not> ring_1.iszero zero (uminus one)"
line_extension_hom: fixes K :: "'a set"   and a :: "'a"   and E :: "'a set" assumes "K \<subseteq> carrier R"   and "a \<in> carrier R"   and "E \<subseteq> carrier R" shows "S.line_extension (h ` K) (h a) (h ` E) = h ` R.line_extension K a E"
new_untame: fixes g :: "graph"   and g' :: "graph" assumes "P g"   and "g' \<in> set (f g)"   and "g' \<notin> set (f' g)" shows "U g'"
dom_subid_aux2'': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and domain_op :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.domain_pre_dioid_one plus times one less_eq less domain_op" shows "less_eq (times x (domain_op y)) x"
cases: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" obtains t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" where "x = [t__]"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and v :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and va :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # v # va"   | "x = []"
exec_gpv_restrict_gpv_invariant: fixes gpv :: "('d, 'a, 'b) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "exec_gpv callee (restrict_gpv \<I> gpv) s = exec_gpv callee gpv s"
polymap_apply: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and v :: "'b"   and p :: "'a poly" assumes "VectorSpaceEnd smult V T"   and "v \<in> V" shows "VectorSpaceEnd.polymap smult V T p v = scalar_mult.lincomb smult (coeffs p) (map (\<lambda>S. S v) (map (VectorSpaceEnd.endpow V T) [0..<Suc (degree p)]))"
word_ao_absorbs_6: fixes x :: "'b word"   and y :: "'b word" shows "x || x && y = x"
add_0_left: fixes a :: "'c" shows "top \<sqinter> a = a"
simps_2: fixes e :: "event"   and s :: "event list" shows "hotel (e # s) = (hotel s \<and> (case e of Check_in g r (k, k') \<Rightarrow> k = currk s r \<and> k' \<notin> issued s | Enter g r (k, k') \<Rightarrow> (k, k') \<in> cards s g \<and> roomk s r \<in> {k, k'} | Exit g r \<Rightarrow> g \<in> isin s r))"
run_reachable: fixes r :: "nat \<Rightarrow> 'Q" assumes "is_run r" shows "range r \<subseteq> reachable"
span_Un: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and T :: "'b set" assumes "module scale" shows "module.span scale (S \<union> T) = {x + y |x y. x \<in> module.span scale S \<and> y \<in> module.span scale T}"
ord_spmf_None: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c spmf" shows "ord_spmf ord (return_pmf None) x"
pmf_cond: fixes p :: "'b pmf"   and s :: "'b set"   and x :: "'b" assumes "set_pmf p \<inter> s \<noteq> {}" shows "pmf (cond_pmf p s) x = (if x \<in> s then pmf p x / measure_pmf.prob p s else 0)"
prefix_conc_length: fixes w :: "'c list"   and w' :: "nat \<Rightarrow> 'c" shows "prefix (length w) (w \<frown> w') = w"
R'_def: fixes x :: "'a"   and y :: "'b" shows "R' x y = ((x, y) \<in> AB)"
T1_eqI: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier"   and "\<And>u v. \<lbrakk>u \<in> nhds x; v \<in> nhds y; y \<notin> u\<rbrakk> \<Longrightarrow> x \<in> v" shows "x = y"