hcju/isabelleps huggingface dataset:
induct: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> comp_res) \<Rightarrow> 'a list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'b \<Rightarrow> comp_res"   and a1 :: "'a list"   and a2 :: "'b list" assumes "\<And>cmp. P cmp [] []"   and "\<And>cmp v va. P cmp [] (v # va)"   and "\<And>cmp v va. P cmp (v # va) []"   and "\<And>cmp a l b m. (cmp a b = EQUAL \<Longrightarrow> P cmp l m) \<Longrightarrow> P cmp (a # l) (b # m)" shows "P a0 a1 a2"
r: shows "\<forall>x\<in>\<langle>gens\<rangle>. h x = h' x"
lcs_comm: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "class.lcs_powerprod plus minus zero lcs" shows "lcs s t = lcs t s"
atLeastAtMost_iff: fixes i :: "'a"   and l :: "'a"   and u :: "'a" shows "(i \<in> ord.atLeastAtMost (\<sqsubseteq>) l u) = (l \<sqsubseteq> i \<and> i \<sqsubseteq> u)"
lookup_pair_eq_valueI: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and v :: "'b" assumes "tc.oalist_inv_raw xs"   and "(k, v) \<in> set xs" shows "lookup_pair_tc xs k = v"
Inr_Inl_iff: fixes b :: "V"   and a :: "V" shows "(ZFC_Cardinals.Inr b = ZFC_Cardinals.Inl a) = False"
nummul_nz: fixes i :: "int"   and a :: "MIR.num" assumes "i \<noteq> 0"   and "nozerocoeff a" shows "nozerocoeff (nummul a i)"
yl: shows "y \<in> lambda_system \<Omega> M f"
xfer_reachable: fixes cs :: "bool list \<times> nat \<times> (nat \<Rightarrow> thread)" assumes "C.reachable cs" obtains as :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" where "sim_rel as cs" and "A.reachable as"
sel_3: fixes x21 :: "'b"   and x22 :: "'a"   and x23 :: "'c" shows "getGMUser (State x21 x22 x23) = x22"
icod: fixes xs :: "int list" shows "incr (Challenge1A.take xs) \<or> decr (Challenge1A.take xs)"
rel_init: shows "rel_fundefs (Finca_get F1) (Fubx_get F2)"
distinct_8: fixes x31 :: "'d"   and x32 :: "'b"   and x2 :: "('a, 'b, 'c, 'd) Tree" shows "tPred x31 x32 \<noteq> tNot x2"
simps: fixes a :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "(a \<in> fsubst) = (a = Var \<or> (\<exists>\<theta> v t. a = \<theta>(v := t) \<and> \<theta> \<in> fsubst \<and> v \<notin> subst_domain \<theta> \<and> t \<noteq> Var v))"
UN_constant_eq: fixes a :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c set"   and c :: "'c set" assumes "a \<in> A"   and "\<forall>y\<in>A. f y = c" shows "\<Union> (f ` A) = c"
cod_tensor: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a"   and d :: "'a" assumes "S.in_hom f a b"   and "S.in_hom g c d" shows "S.cod (M\<^sub>\<omega>.tensor f g) = M\<^sub>\<omega>.tensor b d"
FunctorM_axioms_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "FunctorM_axioms F \<equiv> \<forall>f X Y. f maps\<^bsub>CatDom F\<^esub> X to Y \<longrightarrow> F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y"
antisymp_less_eq: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "r \<le> s"   and "antisymp s" shows "antisymp r"
inf_def: fixes L :: "('c, 'd) gorder_scheme"   and A :: "'c set" shows "\<Sqinter>\<^bsub>L\<^esub>A = (SOME x. is_glb L x A)"
toplevel_summands_flatten_ACI_norm_image: fixes r :: "'c rexp" shows "toplevel_summands (PLUS (sorted_list_of_set (ACI_norm ` toplevel_summands r))) = ACI_norm ` toplevel_summands r"
inf_sup_distrib1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.distrib_lattice inf less_eq less sup" shows "inf x (sup y z) = sup (inf x y) (inf x z)"
elims_2: fixes x :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodAbs x" obtains xs :: "'varSort" and xa :: "'var" and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm" where "x = qAbs xs xa X" and "qGood X"
Cons_lenlex_iff: fixes m :: "'g"   and ms :: "'g list"   and n :: "'g"   and ns :: "'g list"   and r :: "('g \<times> 'g) set" shows "((m # ms, n # ns) \<in> lenlex r) = (length ms < length ns \<or> length ms = length ns \<and> (m, n) \<in> r \<or> m = n \<and> (ms, ns) \<in> lenlex r)"
delete_shadow_root_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and shadow_root_ptr' :: "(_) shadow_root_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "shadow_root_ptr \<noteq> shadow_root_ptr'" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr' getter) h h'"
quat_add_cnj_2: fixes q :: "quat" shows "Quaternions.cnj q + q = quat_of_real (2 * quat.Re q)"
of_complex_divide_of_complex: fixes z2 :: "complex"   and z1 :: "complex" assumes "z2 \<noteq> 0" shows "of_complex z1 :\<^sub>h of_complex z2 = of_complex (z1 / z2)"
bisimilar_equivp: shows "equivp (\<sim>\<cdot>\<^sub>S)"
eq: shows "number_partition_of A B F = image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}))"
x_def: shows "x = \<langle>a, a\<rangle>"
is_App_def: fixes "term" :: "Core.term" shows "is_App term = (\<exists>x51 x52. term = x51 $ x52)"
nn_integral_fst_count_space: fixes f :: "'h \<times> 'i \<Rightarrow> ennreal" shows "(\<Sum>\<^sup>+ x. \<Sum>\<^sup>+ y. f (x, y)) = integral\<^sup>N (count_space UNIV) f"
simps_3: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and more :: "'a" shows "signalsResp \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, \<dots> = more\<rparr> = signalsResp"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflexive A less_eq"   and "transitive A less_eq" shows "quasi_ordered_set A less_eq"
simps_3: fixes c :: "'b"   and c' :: "'b" shows "deriv c (Atom c') = (if c = c' then One else Zero)"
that: fixes u :: "'a \<Rightarrow> 'b" assumes "u \<in> [M]\<^bsub>v,n\<^esub>"   and "u c1 - u c2 = r" shows "thesis"
fresh_star_bij_2: fixes pi :: "(name \<times> name) list"   and b :: "name list"   and x :: "'d" shows "(pi \<bullet> b) \<sharp>* (pi \<bullet> x) = b \<sharp>* x"
eval_red2_Nml_Prim\<^sub>0: fixes t :: "'c local.term"   and a :: "'c" assumes "Ide t"   and "Nml t"   and "Src t = \<^bold>\<langle>a\<^bold>\<rangle>\<^sub>0" shows "\<lbrace>t \<^bold>\<Down> \<^bold>\<langle>a\<^bold>\<rangle>\<^sub>0\<rbrace> = \<r>[\<lbrace>t\<rbrace>]"
Complete_set_non_associates_def: fixes S :: "'b set" shows "Complete_set_non_associates S = (\<exists>f. ass_function f \<and> range f = S \<and> pairwise (\<lambda>a b. normalize a \<noteq> normalize b) S)"
arr_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "arr (f \<otimes> g)"
pt_rev_pi: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "rev pi \<bullet> pi \<bullet> x = x"
inv_iv_def: fixes S :: "'a set"   and X :: "'a set" shows "inv_iv S X = (independent_vertices E S \<and> X \<subseteq> V \<and> (\<forall>v1\<in>V - X. \<forall>v2\<in>S. {v1, v2} \<notin> E) \<and> S \<subseteq> X)"
nres_order_simps_4: fixes M :: "'a nres" shows "(FAIL \<le> M) = (M = FAIL)"
converse_rtranclp_into_rtranclp: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>*\<^sup>* a c"
n2_def: shows "n2 = (replicate (length (fst A)) False)[p := True]"
ciseq: shows "cis (arg y * real n) = cis (arg x)"
list: fixes l :: "real list" assumes "length l = x1" shows "drop (z + x1) l @ drop (z + x1 - length l) L = drop z L"
not_cong_4312: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Cong A B C D" shows "\<not> Cong D C A B"
dg_GRPH_ObjI: fixes \<alpha> :: "V"   and \<AA> :: "V" assumes "digraph \<alpha> \<AA>" shows "\<AA> \<in>\<^sub>\<circ> dg_GRPH \<alpha>\<lparr>Obj\<rparr>"
eq1: shows "(s + t) \<oplus> lt f = s \<oplus> (t \<oplus> lt f)"
F'_prop: shows "((\<lambda>x. F (a - (a \<bullet> base_vec) *\<^sub>R base_vec + x *\<^sub>R base_vec)) has_vector_derivative F') (at (a \<bullet> base_vec))"
minus_rat_cancel: fixes a :: "int"   and b :: "int" shows "Rat.Fract (- a) (- b) = Rat.Fract a b"
diff_frac_eq_1: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and b :: "real"   and a :: "real" assumes "PL_Indis cval indis"   and "b \<noteq> 0" shows "1 - a / b = (b - a) / b"
success_guardI: fixes P :: "heap \<Rightarrow> bool"   and h :: "heap"   and f :: "heap \<Rightarrow> 'a \<times> heap" assumes "P h" shows "success (guard P f) h"
Span_subgroup_props_1: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "R.Span K Us \<subseteq> carrier R"
semInt_prSwap: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "wlsSEM SEM" shows "prSwap (semInt SEM) SEM"
delete_keys: fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "map fst (delete k al) = removeAll k (map fst al)"
bind_spmf_const: fixes p :: "'e spmf"   and q :: "'d spmf" shows "p \<bind> (\<lambda>x. q) = scale_spmf (weight_spmf p) q"
ap_def: fixes f :: "('e, 'g \<Rightarrow> 'f) state"   and x :: "('e, 'g) state" shows "State_Monad.ap f x = State (\<lambda>s. case run_state f s of (g, s') \<Rightarrow> case run_state x s' of (y, s'') \<Rightarrow> (g y, s''))"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a multiset" assumes "\<And>x. Ra x x" shows "rel_mset Ra x x"
add_le_cancel_right: fixes a :: "'k multiset"   and c :: "'k multiset"   and b :: "'k multiset" shows "(a + c \<subseteq># b + c) = (a \<subseteq># b)"
filter_oseq_def: fixes p :: "'d \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'd option" shows "filter_oseq p s \<equiv> \<lambda>i. case s i of None \<Rightarrow> None | Some x \<Rightarrow> if p x then Some x else None"
finite_set_of_finite_funs_pred: fixes P :: "'d \<Rightarrow> bool" assumes "finite {x. True}"   and "finite {y. P y}" shows "finite {f. \<forall>x. P (f x)}"
llcp_eq_0_iff: fixes xs :: "'b llist"   and ys :: "'b llist" shows "(llcp xs ys = 0) = (lnull xs \<or> lnull ys \<or> lhd xs \<noteq> lhd ys)"
arr_dom: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.arr (Right_a.dom f)"
partList4_disjoint_list: fixes s :: "'b set"   and ts :: "'b set list" assumes "s \<subseteq> \<Union> (set ts)"   and "disjoint_list ts"   and "{} \<notin> set ts" shows "disjoint_list (partList4 s ts)"
card_of_Un_Plus_ordLeq: fixes A :: "'a set"   and B :: "'a set" shows "|A \<union> B| \<le>o |A <+> B|"
prob_compl: fixes A :: "'d set"   and M :: "'d pmf" assumes "A \<in> measure_pmf.events M" shows "measure_pmf.prob M (space (measure_pmf M) - A) = 1 - measure_pmf.prob M A"
combining_sys_satisfies_properties_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list" assumes "p \<noteq> 0"   and "0 < length qs1"   and "0 < length qs2"   and "satisfies_properties_R p qs1 (get_subsets_R (calculate_data_R p qs1)) (get_signs_R (calculate_data_R p qs1)) (get_matrix_R (calculate_data_R p qs1))"   and "satisfies_properties_R p qs2 (get_subsets_R (calculate_data_R p qs2)) (get_signs_R (calculate_data_R p qs2)) (get_matrix_R (calculate_data_R p qs2))" shows "satisfies_properties_R p (qs1 @ qs2) (get_subsets_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_signs_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2)))) (get_matrix_R (snd (combine_systems_R p (qs1, calculate_data_R p qs1) (qs2, calculate_data_R p qs2))))"
append_in_starI: fixes u :: "'b list"   and A :: "'b list set"   and v :: "'b list" assumes "u \<in> star A"   and "v \<in> star A" shows "u @ v \<in> star A"
space: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "Dynkin_system \<Omega> M" shows "\<Omega> \<in> M"
permutes_inv_o_2: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" assumes "p permutes S" shows "inv p \<circ> p = id"
left: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "[:row A i \<bullet> col B j:] * p = row (map_mat (\<lambda>a. [:a:] * p) A) i \<bullet> col (map_mat (\<lambda>a. [:a:]) B) j"
iso_cancel_right: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "VV.iso g"   and "VV.comp f g = VV.comp f' g"   and "VV.seq f g"   and "VV.iso g" shows "f = f'"
obj_sumE: fixes s :: "'g + 'h"   and P :: "bool" assumes "\<forall>x. s = Inl x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
assms: shows "\<forall>(a, b)\<in>A. \<forall>(c, d)\<in>A. b = c \<longrightarrow> (a, d) \<in> A"
simps_4: fixes A :: "'b"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" shows "(G \<^bold>\<and> H)[F/A] = G[F/A] \<^bold>\<and> H[F/A]"
hyps_3: obtains q :: "('t \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'b" where "p = r' + (\<Sum>g\<in>G. q g \<odot> g)" and "\<And>g. q g \<in> punit.dgrad_p_set d m" and "\<And>g. lt (q g \<odot> g) \<preceq>\<^sub>t lt p"
hyps: fixes a :: "'a list" assumes "x \<notin> set a" shows "list_diff a b' = list_diff a (removeAll x b')"
dvd_prodI: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "a \<in> A" shows "dvd.dvd times (f a) (comm_monoid_mult.prod times one f A)"
term_subst_eq_conv: fixes t :: "('d, 'f) Term.term"   and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term"   and \<tau> :: "'f \<Rightarrow> ('d, 'e) Term.term" shows "(t \<cdot> \<sigma> = t \<cdot> \<tau>) = (\<forall>x\<in>vars_term t. \<sigma> x = \<tau> x)"
assms: fixes i :: "nat" assumes "i < m * n" shows "a i = b i"
equivalence_one_closed: shows "(coreflexive (1::'a) \<and> transitive (1::'a)) \<and> symmetric (1::'a)"
frechet_derivative_norm: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "f differentiable at x"   and "f x \<noteq> (0::'c)" shows "frechet_derivative (\<lambda>x. norm (f x)) (at x) = (\<lambda>v. frechet_derivative f (at x) v \<bullet> sgn (f x))"
Jacobi_values: fixes p :: "int"   and q :: "int" shows "Jacobi p q \<in> {1, - 1, 0}"
get_parent_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_get_parent_wf: shows "l_get_parent_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_parent"
norm_triangle_ineq4: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) \<le> norm a + norm b"
face: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> \<F>"   and "T \<in> \<F>" shows "S \<inter> T face_of S"
eval_primfun'_at_top: fixes f :: "primfun" shows "filterlim (eval_primfun' f) at_top at_top"
special_subgroup_int_S: fixes T :: "'w set" assumes "T \<in> Pow S" shows "\<langle>T\<rangle> \<inter> S = T"
tllist_all2_mono: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and xs :: "('g, 'i) tllist"   and ys :: "('h, 'j) tllist"   and P' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Q' :: "'i \<Rightarrow> 'j \<Rightarrow> bool" assumes "tllist_all2 P Q xs ys"   and "\<And>x y. P x y \<Longrightarrow> P' x y"   and "\<And>x y. Q x y \<Longrightarrow> Q' x y" shows "tllist_all2 P' Q' xs ys"
gFreshClsRev_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshClsRev MOD \<equiv> gFreshGVarRev MOD \<and> gFreshGAbsRev MOD \<and> gFreshGOpRev MOD"
assms_2: shows "qs \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x))])"
assms_2: shows "(\<sigma>, s) \<in> oreachable p (otherwith S IPS (oarrivemsg I)) U"
seq_comp_secure_aux_1_case_1: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and xs :: "'d option list"   and y :: "'d option"   and ys :: "'d option list"   and X :: "'d option set"   and Q :: "'d option process" assumes "secure_termination I D"   and "sequential P"   and "secure P I D"   and "xs @ y # ys \<notin> sentences P"   and "(xs @ y # ys, X) \<in> failures P"   and "None \<noteq> y"   and "None \<notin> set xs"   and "None \<notin> set ys" shows "(xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys X) \<in> seq_comp_failures P Q"
prm_set_membership: fixes x :: "'b"   and S :: "'b set"   and \<pi> :: "'b prm" assumes "x \<in> S" shows "\<pi> $ x \<in> \<pi> {$} S"
star_rtc3: fixes x :: "('a \<times> 'a) set" shows "(Id \<union> x O x = x) = (x = x\<^sup>*)"
simps_1: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c" shows "equal_base.list_member equal [] y = False"
simps_6: fixes y :: "nat" shows "pratt1_max \<infinity>\<^sub>1 [y]\<^sub>1 = \<infinity>\<^sub>1"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> xs\<^sub>2 = vcard xs\<^sub>2"
subtype_x_Bag: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> Bag \<sigma>" obtains \<tau>' :: "'a OCL_Types.type" where "\<tau> = Bag \<tau>'" and "\<tau>' \<sqsubset> \<sigma>"
pair_sigma_finite_axioms: shows "pair_sigma_finite lborel lborel"
Restrp_semiattractive: shows "semiattractive UNIV ((\<lambda>x y. y \<sqsubseteq> x) \<restriction> A)"
MakeCat_def: fixes C :: "('a, 'b, 'c) Category_scheme" shows "MakeCat C \<equiv> \<lparr>Obj = obj\<^bsub>C\<^esub>, Mor = mor\<^bsub>C\<^esub>, Dom = restrict (Dom C) (mor\<^bsub>C\<^esub>), Cod = restrict (Cod C) (mor\<^bsub>C\<^esub>), Id = restrict (Category.Id C) (obj\<^bsub>C\<^esub>), Comp = \<lambda>f g. (\<lambda>(x, y)\<in>{(f, g) |f g. f \<approx>>\<^bsub>C\<^esub> g}. x ;;\<^bsub>C\<^esub> y) (f, g), \<dots> = Category.more C\<rparr>"
test3: fixes p :: "'a"   and q :: "'a" shows "\<iota> p \<cdot> \<iota> q \<cdot> ! (\<iota> p) = (0::'b)"
clop_wtrans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.cl_op less_eq cl_op" shows "less_eq (cl_op (cl_op x)) (cl_op x)"
1: shows "UNlift01 cl dl P F II = local.shift dl (pickT cl P II) (F (pickT cl P II) (back cl (pickT cl P II) II))"
Quotient3_ltln\<^sub>L: shows "Quotient3 (\<sim>\<^sub>L) abs_ltln\<^sub>L rep_ltln\<^sub>L"
map2_nth: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "n < length xs"   and "n < length ys" shows "map2 f xs ys ! n = f (xs ! n) (ys ! n)"
leg_simps_1: shows "C.arr \<nu>\<pi>.leg0"
simps_6: fixes t :: "'c local.term" shows "Cod \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Trg t \<^bold>\<star> Cod t"
subst_of_map_in_matchers_map: fixes d :: "'d \<Rightarrow> ('e, 'f) Term.term"   and m :: "'d \<Rightarrow> ('e, 'f) Term.term option" shows "subst_of_map d m \<in> matchers_map m"
distinct_59: fixes x2 :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Minus x2 \<noteq> x151 ^\<^sub>e x152"
lookup_def: shows "RBT_Mapping2.lookup \<equiv> map_fun mapping_rbt.impl_of id (rbt_comp_lookup ccomp)"
algebraic_extension: fixes x :: "'a" assumes "x \<in> carrier L" shows "(algebraic over K) x"
ctr: fixes x :: "quat"   and y :: "quat" shows "x - y = Quat (quat.Re x - quat.Re y) (Im1 x - Im1 y) (Im2 x - Im2 y) (Im3 x - Im3 y)"
l_span_sum_closedTr: fixes A :: "'b set"   and H :: "'a set"   and n :: "nat" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "\<forall>s f. s \<in> {j. j \<le> n} \<rightarrow> A \<and> f \<in> {j. j \<le> n} \<rightarrow> linear_span R M A H \<longrightarrow> \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s f j) n \<in> linear_span R M A H"
cet_cast_real: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e : BOOL" shows "\<Gamma> \<turnstile>\<^sub>c \<langle>e\<rangle>\<^sub>c : REAL"
vars_of_monom_in_alt_def: fixes xs :: "'a list"   and \<V> :: "'a set" shows "vars_of_monom_in xs \<V> = (set xs \<subseteq> \<V>)"
uminus_of_nat: fixes n :: "nat" shows "f (- of_nat n) = f (0::'a)"
gauss_jordan_single_3: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan_single A = C" shows "row_echelon_form C"
equivalence_def: fixes S :: "('a, 'b) eq_object_scheme" shows "equivalence S \<equiv> (\<forall>x. x \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y. x .=\<^bsub>S\<^esub> y \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> y .=\<^bsub>S\<^esub> x) \<and> (\<forall>x y z. x .=\<^bsub>S\<^esub> y \<longrightarrow> y .=\<^bsub>S\<^esub> z \<longrightarrow> x \<in> carrier S \<longrightarrow> y \<in> carrier S \<longrightarrow> z \<in> carrier S \<longrightarrow> x .=\<^bsub>S\<^esub> z)"
keys_subset_Keys: fixes f :: "'a \<Rightarrow>\<^sub>0 'b"   and F :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "f \<in> F" shows "keys f \<subseteq> Keys F"
sorted_map: fixes f :: "'b \<Rightarrow> 'a word"   and xs :: "'b list" shows "signed.sorted (map f xs) = sorted_wrt (\<lambda>x y. f x \<le>s f y) xs"
inject: fixes x1 :: "'b"   and x2 :: "'c"   and y1 :: "'b"   and y2 :: "'c" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
WTrt_binop_i_i_i_i_o_def: shows "WTrt_binop_i_i_i_i_o \<equiv> \<lambda>x xa xb xc. pred.Pred (WTrt_binop x xa xb xc)"
pw_ERETURN_2: fixes x :: "'d"   and e :: "'c" shows "\<not> inres (ERETURN x) (Inl e)"
index_swap_if_distinct: fixes xs :: "'e list"   and i :: "nat"   and j :: "nat"   and x :: "'e" assumes "distinct xs"   and "i < length xs"   and "j < length xs" shows "index (xs[i := xs ! j, j := xs ! i]) x = (if x = xs ! i then j else if x = xs ! j then i else index xs x)"
\<gamma>_def: shows "\<gamma> \<equiv> \<lambda>_ _. True"
in_set_spmf_exec_gpv_into_results_gpv: fixes x :: "'d"   and s' :: "'e"   and callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and gpv :: "('d, 'f, 'g) gpv"   and s :: "'e" assumes "(x, s') \<in> set_spmf (exec_gpv callee gpv s)"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "True" shows "x \<in> results_gpv \<I>_full gpv"
exhaust: fixes y :: "exetheory" obtains x1 :: "exesignature" and x2 :: "Core.term list" where "y = ExeTheory x1 x2"
\<Sigma>_same: fixes f :: "'d \<Rightarrow> 'c"   and j :: "int" shows "\<Sigma> f j j = (0::'c)"
rep_eq: fixes x :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xa :: "'d set_dlist" shows "DList_Set.fold x xa = List.fold x (list_of_dlist xa)"
IH: fixes i' :: "nat"   and i :: "nat"   and j' :: "nat"   and j :: "nat" assumes "cycle_free_up_to m k n"   and "i' \<le> i"   and "j' \<le> j"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "D m i' j' k = fw m n k i j i' j'"