hcju/isabelleps huggingface dataset:
complete_lattice_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a" shows "class.complete_lattice Inf Sup inf less_eq less sup bot top \<equiv> class.lattice inf less_eq less sup \<and> class.complete_lattice_axioms Inf Sup less_eq bot top"
quasinorm_onI: fixes F :: "'c set"   and N :: "'c \<Rightarrow> ennreal"   and C :: "real" assumes "\<And>x y. \<lbrakk>x \<in> F; y \<in> F\<rbrakk> \<Longrightarrow> x + y \<in> F"   and "\<And>x y. \<lbrakk>x \<in> F; y \<in> F\<rbrakk> \<Longrightarrow> N (x + y) \<le> ennreal C * N x + ennreal C * N y"   and "\<And>c x. \<lbrakk>c \<noteq> 0; x \<in> F\<rbrakk> \<Longrightarrow> c *\<^sub>R x \<in> F"   and "\<And>c x. \<lbrakk>c \<noteq> 0; x \<in> F\<rbrakk> \<Longrightarrow> N (c *\<^sub>R x) \<le> ennreal \<bar>c\<bar> * N x"   and "(0::'c) \<in> F"   and "N (0::'c) = 0"   and "1 \<le> C" shows "quasinorm_on F C N"
TS_d'': fixes x :: "nat"   and y :: "nat"   and x0 :: "nat"   and y0 :: "nat"   and s :: "(nat list \<times> nat list) pmf"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "TS_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom x])" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h0)) s qs = 0"
VPi_memberD: fixes f :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V"   and p :: "V" assumes "f \<in> elts (VPi A B)"   and "p \<in> elts f" obtains x :: "V" where "x \<in> elts A" and "p = \<langle>x, app f x\<rangle>"
fst_snd_Gauss_Jordan_column_k_det_P_eq_fst_snd_Gauss_Jordan_column_k_PA: fixes n :: "'b"   and i :: "nat"   and A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and P :: "(('b, 'd) vec, 'd) vec" shows "fst (snd (Gauss_Jordan_column_k_det_P (n, i, A) k)) = fst (snd (Gauss_Jordan_column_k_PA (P, i, A) k))"
assms_3: shows "S\<lparr>ArrCod\<rparr> = B"
RedCall: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and vs :: "val list" assumes "hp s a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"   and "length vs = length pns"   and "length Ts = length pns" shows "P \<turnstile> \<langle>addr a\<bullet>M(map Val vs),s\<rangle> \<rightarrow> \<langle>blocks (this # pns, Class D # Ts, Addr a # vs, body),s\<rangle>"
cfE_ss_invE: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "NPreflow c s t f" shows "Graph.E (residualGraph c f) \<subseteq> Graph.E c \<union> (Graph.E c)\<inverse>"
qf1pt: shows "is_sum2sq_nat ((4 * k + 1) * t)"
simps: fixes a1 :: "('c \<times> 'd) option"   and a2 :: "('c \<times> 'd) option"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool" shows "((a1, a2) \<in> whileLoop_results C B) = ((\<exists>r s. a1 = Some (r, s) \<and> a2 = Some (r, s) \<and> \<not> C r s) \<or> (\<exists>r s. a1 = Some (r, s) \<and> a2 = None \<and> C r s \<and> snd (B r s)) \<or> (\<exists>r s r' s' z. a1 = Some (r, s) \<and> a2 = z \<and> C r s \<and> (r', s') \<in> fst (B r s) \<and> (Some (r', s'), z) \<in> whileLoop_results C B))"
naturality: fixes f :: "J.Arr" assumes "J.arr f" shows "\<chi> (J.cod f) \<cdot> \<chi>.A.map f = local.map f \<cdot> \<chi> (J.dom f)"
bits_div_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide zero a = zero"
One_subst: fixes t1 :: "lam"   and t2 :: "lam"   and s1 :: "lam"   and s2 :: "lam"   and x :: "name" assumes " t1 \<longrightarrow>\<^sub>1 t2"   and " s1 \<longrightarrow>\<^sub>1 s2" shows " t1[x::=s1] \<longrightarrow>\<^sub>1 t2[x::=s2]"
content_lt_nz: fixes a :: "'a"   and b :: "'a" shows "(0 < content (cbox a b)) = (content (cbox a b) \<noteq> 0)"
igSwapAbsIPresIGWlsAbs_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme" shows "igSwapAbsIPresIGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A. isInBar (us, s) \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A)"
is_inv: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f" assumes "iso_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms" shows "\<exists>\<psi>. morphism_presheaves_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<FF> \<rho> b add_str mult_str zero_str one_str \<psi> \<and> (\<forall>U. is_open U \<longrightarrow> (\<forall>x\<in>\<FF>' U. (fam_morphisms U \<circ> \<psi> U) x = x) \<and> (\<forall>x\<in>\<FF> U. (\<psi> U \<circ> fam_morphisms U) x = x))"
actions_ok2_ex_actions_ok1: fixes s2 :: "('d, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'd \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'e wait_set_status option) \<times> 't set"   and t :: "'t"   and ta2 :: "('d, 't, 'x2, 'm2, 'e, 'f) thread_action"   and ta1 :: "('d, 't, 'x1, 'm1, 'e, 'f) thread_action" assumes "r2.actions_ok s2 t ta2"   and "ta1 \<sim>m ta2" obtains s1 :: "('d, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'd \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'e wait_set_status option) \<times> 't set" where "r1.actions_ok s1 t ta1"
ask_env_def: fixes f :: "'c \<Rightarrow> ('c, 'm) envT" shows "local.ask_env f = EnvT (\<lambda>r. run_env (f r) r)"
assms_1: fixes c :: "int" assumes "e = PExpr1 (PCnst c)" shows "P"
hcomp_simps_2: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "hseq \<nu> \<mu>" shows "trg (\<nu> \<star> \<mu>) = trg \<nu>"
termFSwSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b, 'c) model_scheme" shows "termFSwSbImorph h hA MOD \<equiv> termFSwImorph h hA MOD \<and> ipresSubstAll h hA MOD"
small_lt: fixes u :: "real"   and n :: "int"   and m :: "int" assumes "0 \<le> u"   and "u < 1" shows "(real_of_int n < real_of_int m - u) = (n < m)"
projs_2: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "\<pi>\<^sub>3 (dsn, dsk, flag, hops, nhip, pre) = dsk"
filtered_alt: fixes X :: "'a set" shows "filtered X = (X \<noteq> {} \<and> (\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. z \<le> x \<and> z \<le> y))"
abelian_monoidE_4: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "abelian_monoid R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
X_def: shows "X = Pascal_Property.inter (line A C') (line R B)"
has_derivative_scaleR_right: fixes g :: "'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and r :: "real" assumes "(g has_derivative g') F" shows "((\<lambda>x. r *\<^sub>R g x) has_derivative (\<lambda>x. r *\<^sub>R g' x)) F"
assms_2: fixes v :: "'a" shows "P\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk> = Q\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk>"
the_\<nu>_props_1: shows "\<guillemotleft>the_\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> chine\<guillemotright>"
weight_spmf_nonneg: fixes p :: "'c spmf" shows "0 \<le> weight_spmf p"
vector_derivative_unique_at: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and f'' :: "'a" assumes "(f has_vector_derivative f') (at x)"   and "(f has_vector_derivative f'') (at x)" shows "f' = f''"
sup_state_refl: fixes P :: "'a prog"   and s :: "ty list \<times> ty err list" shows "P \<turnstile> s \<le>\<^sub>i s"
givenBy_o_eq_extend_set: fixes v :: "'a \<Rightarrow> 'e" shows "givenBy (v \<circ> f) = extend_set h ` givenBy v"
ipd_def: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" shows "IFC_def.ipd prob x = (THE y. IFC_def.is_ipd prob y x)"
assms: shows "equal_mod_timpls TI (Fun f T) (Fun g S)"
comp_simp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.seq (fst g) (fst f)"   and "B.seq (snd g) (snd f)" shows "B.VxV.comp g f = (fst g \<cdot>\<^sub>B fst f, snd g \<cdot>\<^sub>B snd f)"
Comp_Hom_Id: fixes A :: "'o"   and f :: "'a"   and B :: "'o" assumes "A \<in> Obj"   and "f \<in> Hom A B" shows "Comp B A A f (Id A) = f"
distinct_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "distinct (linorder.sort_key less_eq f xs) = distinct xs"
bind_pmf_cong: fixes p :: "'a pmf"   and q :: "'a pmf"   and f :: "'a \<Rightarrow> 'b pmf"   and g :: "'a \<Rightarrow> 'b pmf" assumes "p = q"   and "\<And>x. x \<in> set_pmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
finite_imp_fold_graph: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "finite A" shows "\<exists>x. fold_graph f z A x"
carrier_direct_lim_def: shows "stk1.carrier_direct_lim \<equiv> e1.Partition"
green_typeI_cube_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" shows "green_typeI_cube i j twoC F \<equiv> R2 i j \<and> green_typeI_cube_axioms i j twoC F"
empty_store_buffers_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "empty_store_buffers ts \<equiv> \<forall>i p is xs sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, xs, sb, \<D>, \<O>, \<R>) \<longrightarrow> sb = []"
lconjby_add: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus (plus (plus x y) z) (plus x y) = minus (plus x (minus (plus y z) y)) x"
set_category_given_img_axioms: shows "set_category_given_img (\<cdot>\<^sub>S) S.some_img (cardSuc (cmax |UNIV| natLeq))"
induct: fixes P :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a1 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>t. P \<langle>\<rangle> t"   and "\<And>v va vb. P \<langle>v, va, vb\<rangle> \<langle>\<rangle>"   and "\<And>t1 a uu_ t2 t3 c uv_ t4. P t2 t3 \<Longrightarrow> P (R uu_ t1 a t2) (R uv_ t3 c t4)"   and "\<And>t1 a uw_ t2 t3 c ux_ t4. P t2 t3 \<Longrightarrow> P (B uw_ t1 a t2) (B ux_ t3 c t4)"   and "\<And>v vc vf vb t2 a uy_ t3. P (B vf v vc vb) t2 \<Longrightarrow> P (B vf v vc vb) (R uy_ t2 a t3)"   and "\<And>t1 a uz_ t2 v vc vf vb. P t2 (B vf v vc vb) \<Longrightarrow> P (R uz_ t1 a t2) (B vf v vc vb)" shows "P a0 a1"
arrow_to_functor_axioms: shows "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y (Fo y) (\<eta>o y)"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(allow x1 = allow y1) = (x1 = y1)"
list_cases4: fixes x :: "'a list" obtains "x = []"   | y :: "'a" where "x = [ya__]"   | y :: "'a" and z :: "'a" where "x = [yb__, za__]"   | w :: "'a" and y :: "'a" and z :: "'a" and zs :: "'a list" where "x = w # y # z # zs"
comp_ide_self: fixes a :: "'a" assumes "Adj.C.ide a" shows "a \<cdot>\<^sub>C a = a"
qstar_def: fixes x :: "'a" shows "qstar x = \<Squnion> range ((^) x)"
IH_1: fixes f :: "nat \<Rightarrow> real" assumes "\<And>x. x < count_weights False (remove_weights m1) \<Longrightarrow> f x = extract_weights False m1 x" shows "m1 = insert_weights False (remove_weights m1) f"
hyps: fixes env :: "('a, 'b, 'c) env" assumes "lookup env xs = None" shows "lookup env (xs @ ys) = None"
stutter_free_suffix: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and k :: "nat" assumes "stutter_free \<sigma>" shows "stutter_free (suffix k \<sigma>)"
elims_3: fixes x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "\<not> Derivation x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = ab__" and "xa = []" and "xb = ba__" and "ab__ \<noteq> ba__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<nexists>x. Derives1 a (fst d) (snd d) x \<and> Derivation x D b"
f_def: shows "f \<equiv> \<lambda>i j. eval rf [i, j]"
PfP_implies_ModPon_PfP: fixes H :: "fm set"   and x :: "tm"   and y :: "tm" assumes "H \<turnstile> PfP (Q_Imp x y)"   and "H \<turnstile> PfP x" shows "H \<turnstile> PfP y"
row_def: fixes A :: "'a mat"   and i :: "nat" shows "row A i = Matrix.vec (dim_col A) (\<lambda>j. A $$ (i, j))"
pivot_fun_init: fixes A :: "'c mat" shows "pivot_fun A (\<lambda>_. 0) 0"
is_regular_spairD2: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "rep_list q \<noteq> 0"
INF_eq: 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 :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (g j) (f i)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (f i) (g j)" shows "Inf (f ` A) = Inf (g ` B)"
measurable_on_scaleR: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f measurable_on S"   and "g measurable_on S" shows "(\<lambda>x. f x *\<^sub>R g x) measurable_on S"
in_segmentE: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and z :: "'b" assumes "y \<in> segment f x z"   and "z \<in> orbit f x" obtains "(f ^^ funpow_dist1 f x y) x = y" and "funpow_dist1 f x y < funpow_dist1 f x z"
vec_of_list_coeffs_nth: fixes i :: "nat"   and h :: "'d poly" assumes "i \<in> {..degree h}"   and "h \<noteq> 0" shows "vec_of_list (coeffs h) $ i = coeff h i"
length_induced_play: fixes A :: "'c llist set"   and N :: "enat"   and s :: "'d list \<Rightarrow> 'd"   and p :: "'d list" assumes "GSgame A N" shows "llength (GSgame.induced_play N s p) = 2 * N"
coef_hd_cong: fixes \<zeta> :: "('s, 'v) hd"   and A :: "'v pvar \<Rightarrow> zhmultiset"   and B :: "'v pvar \<Rightarrow> zhmultiset"   and i :: "nat" assumes "\<forall>x\<in>vars_hd \<zeta>. \<forall>i. A (PCoef x i) = B (PCoef x i)" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta> i)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta> i))"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "carrier G \<subseteq> Units G" shows "Group.group_axioms G"
someinit_def: fixes i :: "nat" shows "op0.someinit i \<equiv> SOME x. x \<in> (fst \<circ> Fun.id) ` init (ptoy i)"
ts_def: shows "ts = (pos, re, cl, dyn, ps, sd)"
filter_by_key_sequences: fixes key :: "'a \<Rightarrow> 'b"   and k :: "'b"   and xs :: "'a list" shows "filter (\<lambda>y. key y = k) (concat (sequences key xs)) = filter (\<lambda>y. key y = k) xs"
closure_def: fixes as :: "'a list" shows "closure as = rtrancl_while (\<lambda>(p, q). final p = final q) (\<lambda>(p, q). map (\<lambda>a. (delta a p, delta a q)) as)"
abs_summable_on_finite_diff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "f abs_summable_on A"   and "A \<subseteq> B"   and "finite (B - A)" shows "f abs_summable_on B"
PSQ_D: shows "(P', S, Q') = diagonal_to_Smith_PQ_JNF D bezout"
psums_Cons: fixes x :: "'a"   and xs :: "'a list" shows "psums (x # xs) = x # map ((+) x) (psums xs)"
dom_MkArr: fixes A :: "bool"   and B :: "bool"   and F :: "'arr list" assumes "arr (MkArr A B F)" shows "local.dom (MkArr A B F) = MkIde A"
get_proc_return: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "get_proc (sourcenode a) = p"
summarize_ASSERT_conv: fixes \<Phi> :: "bool"   and \<Psi> :: "bool"   and m :: "'a nres" shows "ASSERT \<Phi> \<bind> (\<lambda>_. ASSERT \<Psi> \<bind> (\<lambda>_. m)) = ASSERT (\<Phi> \<and> \<Psi>) \<bind> (\<lambda>_. m)"
enc_def: fixes enc_alg :: "'c \<Rightarrow> 'a \<Rightarrow> 'b spmf" shows "cipher.enc enc_alg \<equiv> CNV (stateless_callee (\<lambda>m. Pause (Inl ()) Done \<bind> (\<lambda>k. lift_spmf (enc_alg (projl k) m) \<bind> (\<lambda>c. Pause (Inr c) Done \<bind> (\<lambda>_. Done ()))))) ()"
f_get: fixes x :: "'k"   and g :: "('k \<Rightarrow> 'a option) \<Rightarrow> 'a \<Rightarrow> 'm" shows "get (\<lambda>s. bind (f x) (g s)) = bind (f x) (\<lambda>a. get (\<lambda>s. g s a))"
InfStruct: shows "IInfStruct I_intT I_intF I_intP"
norm_fst_le: fixes x :: "'d"   and y :: "'e" shows "norm x \<le> norm (x, y)"
lists_mono: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "lists A \<subseteq> lists B"
simps_2: fixes x :: "'b"   and a :: "'b"   and xs :: "'b list" shows "del_list x (a # xs) = (if x = a then xs else a # del_list x xs)"
path_component_path_image_pathstart: fixes p :: "real \<Rightarrow> 'b"   and x :: "'b" assumes "path p"   and "x \<in> path_image p" shows "path_component (path_image p) (pathstart p) x"
algebraic_closure_axioms_def: fixes L :: "('c, 'd) ring_scheme"   and K :: "'c set" shows "algebraic_closure_axioms L K \<equiv> (\<forall>x. x \<in> carrier L \<longrightarrow> ((\<lambda>K x. \<not> ring.transcendental L K x) over K) x) \<and> (\<forall>P. P \<in> carrier (K [X]\<^bsub>L\<^esub>) \<longrightarrow> ring.splitted L P)"
verit_minus_simplify_3: fixes b :: "'c" shows "(0::'c) - b = - b"
emp_in: shows "{#} \<in> wrp.Q_of_wstate (lnth wSts k)"
H2_def: shows "H2 = h ` {a..dist x z}"
fps_XD'_power: fixes a :: "'b fps"   and b :: "'b fps"   and n :: "nat" shows "fps_XD' a (b ^ n) = of_nat n * b ^ (n - 1) * fps_XD' a b"
leftmost_exists: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and k :: "nat" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.is_sentence \<NN> \<TT> a"   and "k < length a"   and "CFG.is_nonterminal \<NN> (a ! k)" shows "\<exists>i. CFG.leftmost \<NN> \<TT> i a \<and> i \<le> k"
chkchop_def: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "chkchop gt t s = (is_Hd s \<or> gt t (chop s))"
rec_minus_def: shows "rec_minus = Pr 1 (recf.id 1 0) (Cn 3 UF.rec_pred [recf.id 3 2])"
inner_dirsum_Cons: fixes A :: "'a set"   and As :: "'a set list" assumes "add_independentS (A # As)" shows "(\<Oplus>X\<leftarrow>A # As. X) = A \<oplus> (\<Oplus>X\<leftarrow>As. X)"
extremum_uniqueI: fixes a :: "'b list" assumes "prefix a []" shows "a = []"
assms: shows "trace_rest_eq rest1 rest2 IA IU p q"
greaterThanAtMost_empty_iff2: fixes k :: "'b multiset"   and l :: "'b multiset" shows "({} = subset_mset.greaterThanAtMost k l) = (\<not> k \<subset># l)"
infinite_complement_partition_finite: assumes "finite UNIV" shows "infinite_complement_partition = UNIV"
rel_Grp: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "rel_blindable\<^sub>h (Grp A f) = Grp {x. set_blindable\<^sub>h x \<subseteq> A} (map_blindable\<^sub>h f)"
le: shows "outs_\<I> \<I> \<subseteq> outs_\<I> \<I>'"
comp_eq_dest: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'e \<Rightarrow> 'c"   and d :: "'b \<Rightarrow> 'e"   and v :: "'b" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
ordering_mem: fixes x :: "ordinal" shows "OrdinalVeblen.ordering A x \<in> A"
fmimage_filter: fixes P :: "'d \<Rightarrow> bool"   and m :: "('d, 'c) fmap"   and A :: "'d fset" shows "fmimage (fmfilter P m) A = fmimage m (ffilter P A)"
congruent_triangleD_18: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "angle b1 c1 a1 = angle a2 c2 b2"
arith_prog_altdef: fixes a :: "int"   and b :: "nat" shows "arith_prog a b = range (\<lambda>n. a + int b * n)"
subst_bvs1'_subst_bv2: fixes i :: "nat"   and j :: "nat"   and t :: "Core.term"   and vs :: "Core.term list"   and u :: "Core.term" assumes "i < j + 1" shows "subst_bv2 (subst_bvs1' t (Suc j) (map (\<lambda>v. lift v i) vs)) i (subst_bvs1' u j vs) = subst_bvs1' (subst_bv2 t i u) j vs"
C3l: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> y" shows "x\<^sup>\<star> \<cdot> y \<le> y"
that: fixes F :: "'a formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "F, \<Gamma>' \<Rightarrow> \<Delta>'"   and "fst (pn_atoms F) \<subseteq> (patoms_mset \<Gamma> \<union> natoms_mset \<Delta>) \<inter> (natoms_mset \<Gamma>' \<union> patoms_mset \<Delta>')"   and "snd (pn_atoms F) \<subseteq> (natoms_mset \<Gamma> \<union> patoms_mset \<Delta>) \<inter> (patoms_mset \<Gamma>' \<union> natoms_mset \<Delta>')" shows "thesis"
zeros_rtrancl: fixes M :: "nat bdd list \<times> bool list"   and x :: "nat"   and v :: "nat"   and y :: "nat" shows "(\<exists>n. dfa_reach M x (zeros n v) y) = ((x, y) \<in> (succsr (rquot_succs M v))\<^sup>*)"
case_1: fixes f1 :: "'d option \<Rightarrow> 'e \<Rightarrow> 'c"   and f2 :: "'d \<Rightarrow> 'c"   and x11 :: "'d option"   and x12 :: "'e" shows "(case Insert x11 x12 of Insert x xa \<Rightarrow> f1 x xa | Delete x \<Rightarrow> f2 x) = f1 x11 x12"