hcju/isabelleps huggingface dataset:
ctor_induct2: fixes phi1 :: "'d IF1 \<Rightarrow> 'e IF1 \<Rightarrow> bool"   and phi2 :: "'d IF2 \<Rightarrow> 'e IF2 \<Rightarrow> bool"   and a1 :: "'d IF1"   and b1 :: "'e IF1"   and a2 :: "'d IF2"   and b2 :: "'e IF2" assumes "\<And>x y. \<lbrakk>\<And>a b. \<lbrakk>a \<in> F1set2 x; b \<in> F1set2 y\<rbrakk> \<Longrightarrow> phi1 a b; \<And>a b. \<lbrakk>a \<in> F1set3 x; b \<in> F1set3 y\<rbrakk> \<Longrightarrow> phi2 a b\<rbrakk> \<Longrightarrow> phi1 (ctor1 x) (ctor1 y)"   and "\<And>x y. \<lbrakk>\<And>a b. \<lbrakk>a \<in> F2set2 x; b \<in> F2set2 y\<rbrakk> \<Longrightarrow> phi1 a b; \<And>a b. \<lbrakk>a \<in> F2set3 x; b \<in> F2set3 y\<rbrakk> \<Longrightarrow> phi2 a b\<rbrakk> \<Longrightarrow> phi2 (ctor2 x) (ctor2 y)" shows "phi1 a1 b1 \<and> phi2 a2 b2"
paths_with_new_def: shows "paths_with_new \<equiv> insert P_new paths"
quotE: fixes R :: "'b set" assumes "R \<in> quot" obtains a :: "'b" where "R = {x. a \<sim> x}"
downset_set_ext: shows "id \<le> ord.downset_set (\<lambda>x y. y \<le> x)"
EWP: fixes x :: "'a" shows "ewp x = (\<exists>y. x = (1::'a) + y \<and> \<not> ewp y)"
contractL: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, F, \<Gamma> \<Rightarrow> \<Delta>" shows "F, \<Gamma> \<Rightarrow> \<Delta>"
p_inf_pp: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> - - y) = - (x \<sqinter> y)"
inf_S_next_stimulus: fixes s :: "'b"   and t :: "'b"   and a :: "'a" assumes "s \<le>\<^sub>\<S> t" shows "\<lambda> (s, a) \<le>\<^sub>\<S> \<lambda> (t, a)"
csifi: shows "s\<inverse> O f\<inverse> \<subseteq> d\<inverse>"
infdist_triangle_side: fixes G :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "geodesic_segment_between G x y" shows "infdist z G \<le> Gromov_product_at z x y + 2 * deltaG TYPE('a)"
wffp_iff_map_Idle: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl" shows "\<exists>n st. 0 < n \<and> sl = map Idle (replicate n st) \<or> (\<exists>st1 u1 c1 sl1. sl = map Idle (replicate n st) @ [State st1 u1 c1] @ sl1)"
subring_add_pow: fixes S :: "'a set"   and a :: "'a"   and n :: "nat" assumes "subring S R"   and "a \<in> S" shows "[n] \<cdot>\<^bsub>R\<lparr>carrier := S\<rparr>\<^esub> a = [n] \<cdot> a"
IFF: fixes b :: "BExpr"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c1 :: "OBJ"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and c2 :: "OBJ"   and \<Psi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "BExpr_low b"   and "G \<rhd> c1 : Sec \<Phi>"   and "G \<rhd> c2 : Sec \<Psi>" shows "G \<rhd> Iff b c1 c2 : Sec (\<lambda>(s, t, \<beta>). (evalB b (fst t) \<longrightarrow> \<Phi> (s, t, \<beta>)) \<and> (\<not> evalB b (fst t) \<longrightarrow> \<Psi> (s, t, \<beta>)))"
order_trans: fixes less_eq :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and less :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and x :: "'p"   and y :: "'p"   and z :: "'p" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
jmpl_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (jmpl_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
natLeq_on_ofilter_less_eq: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "ofilter (natLeq_on m) {0..<n}"
SMVT_lemma2: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real" assumes "f integrable_on {a..b}"   and "\<And>x y. x \<le> y \<Longrightarrow> g x \<le> g y" shows "(\<Union>y. {\<lambda>x. if y \<le> g x then f x else 0}) equiintegrable_on {a..b}"
self_codomain_iff_ide: fixes a :: "('o, 'a) arr" shows "(a \<in> codomains a) = ide a"
abs_mult_self_eq: fixes a :: "'a" shows "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
distinct_mapI: fixes f :: "'c \<Rightarrow> 'b"   and l :: "'c list" assumes "distinct (map f l)" shows "distinct l"
convert_extTA_simps_6: fixes f :: "'i \<Rightarrow> 'e"   and ta :: "('d, 'c, 'i, 'f, 'g, 'h) thread_action" shows "\<lbrace>convert_extTA f ta\<rbrace>\<^bsub>i\<^esub> = \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
bres_fres_eq: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 x :: "'b"   and y :: "'b" assumes "class.ab_quantale times Inf Sup inf less_eq less sup bot top" shows "proto_near_quantale.bres Sup less_eq times x y = proto_near_quantale.fres Sup less_eq times y x"
vfsequence_vremove_vcons_vfsequence: fixes xs' :: "V"   and x :: "V" assumes "T = xs' #\<^sub>\<circ> x" shows "vfsequence xs'"
containsCall_indirection: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and prog :: "cmd"   and qs :: "char list list"   and q :: "char list"   and c :: "cmd"   and ps :: "char list list"   and p :: "char list"   and ins :: "char list list"   and outs :: "char list list" assumes "containsCall procs prog qs q"   and "containsCall procs c ps p"   and "(q, ins, outs, c) \<in> set procs" shows "containsCall procs prog (qs @ q # ps) p"
Dom_Cod: fixes t :: "'c local.term" assumes "Arr t" shows "Dom (Cod t) = Cod t"
lt_lbound_def: fixes c :: "'a"   and b :: "'a option" shows "c <\<^sub>l\<^sub>b b = \<lhd>\<^sub>l\<^sub>b (<) c b"
bijective_map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "bijective_map \<alpha> S T \<equiv> Set_Theory.map \<alpha> S T \<and> bijective \<alpha> S T"
Cons_2: fixes m :: "(name, 'a) fmap" shows "map_option ((++\<^sub>f) m) (matchs xs ys) = map_option (foldl (++\<^sub>f) m) (those (Term_Utils.map2 match xs ys))"
Cauchy: fixes f :: "'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'b" assumes "bounded_linear f"   and "Cauchy X" shows "Cauchy (\<lambda>n. f (X n))"
simple_rotations_fix: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "\<exists>v. r v = v"
simps: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "class.monoid_mult one times" shows "monoid_mult.binary_power one times x n = (if n = 0 then one else let (d, r) = divmod_nat n 2; rec = monoid_mult.binary_power one times (times x x) d in if r = 0 then rec else times rec x)"
assms: shows "Z \<in> ipurge_fail_aux_t_set Y"
assms_1: fixes \<eta> :: "nat" shows "constructive_security_obsf (real_resource \<eta>) (ideal_resource \<eta>) (sim \<eta>) (\<I>_real \<eta>) (\<I>_ideal \<eta>) (\<I>_common \<eta>) (\<A> \<eta>) (adv \<eta>)"
mask_twice: fixes x :: "'a word"   and n :: "nat"   and m :: "nat" shows "(x && mask n) && mask m = x && mask (min m n)"
fill_index_none: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and x :: "'c"   and mi :: "'d"   and ma :: "'d"   and ns :: "nat list"   and ub :: "nat"   and xs :: "'c list" assumes "index_less index key"   and "key x \<in> {mi..ma}"   and "ns \<noteq> []"   and "offs_pred ns ub (x # xs) index key mi ma"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "fill xs (ns[index key x (length ns) mi ma := Suc (ns ! index key x (length ns) mi ma)]) index key ub mi ma ! (ns ! index key x (length ns) mi ma) = None"
neg_numeral_le_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n)"
fds_abs_converges_scaleR: fixes f :: "'b fds"   and s :: "'b"   and c :: "real" assumes "fds_abs_converges f s" shows "fds_abs_converges (c *\<^sub>R f) s"
refined_intro: fixes tag :: "'a list \<Rightarrow> 'b"   and A :: "'a list set" assumes "\<And>x y z. \<lbrakk>x =tag= y; x @ z \<in> A\<rbrakk> \<Longrightarrow> y @ z \<in> A" shows "=tag= \<subseteq> \<approx>A"
n: shows "(\<Prod>p\<in>prime_factors n. p ^ multiplicity p n) = n"
load_word_mem_select_8: fixes s1 :: "'b sparc_state"   and addra :: "word32"   and s2 :: "'c sparc_state" assumes "fst (case load_word_mem s1 addra 8 of None \<Rightarrow> (None, s1) | Some w \<Rightarrow> (Some w, add_instr_cache s1 addra w 15)) = fst (case load_word_mem s2 addra 8 of None \<Rightarrow> (None, s2) | Some w \<Rightarrow> (Some w, add_instr_cache s2 addra w 15))" shows "load_word_mem s1 addra 8 = load_word_mem s2 addra 8"
isVal_Lam: fixes x :: "Vars.var"   and e :: "exp" shows "isVal (Lam [x]. e)"
right_idem: fixes a :: "'b"   and b :: "'b" shows "sup_class.sup (sup_class.sup a b) b = sup_class.sup a b"
basic_trans_rules_31: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A \<subseteq> B"   and "c \<in> A" shows "c \<in> B"
nonpos_Reals_zero_I: shows "(0::'a) \<in> \<real>\<^sub>\<le>\<^sub>0"
PPs_closed_adds: fixes s :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]"   and "t adds s" shows "t \<in> .[X]"
mult_comp_val_process_measurable: fixes Mkt2 :: "('a, 'c) discrete_market"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat"   and qty :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "val_process Mkt2 pf n \<in> borel_measurable (F n)"   and "portfolio pf"   and "qty (Suc n) \<in> borel_measurable (F n)" shows "val_process Mkt2 (qty_mult_comp pf qty) n \<in> borel_measurable (F n)"
null_sets_iff_AE: fixes A :: "'a set" shows "(A \<in> null_sets M) = ((AE x in M. x \<notin> A) \<and> A \<subseteq> space M)"
antisym: fixes b :: "'d list"   and a :: "'d list" assumes "prefix b a"   and "prefix a b" shows "a = b"
equivp_retr: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr"   and "equivp R" shows "equivp (retr R)"
union_mset_add_mset_right: fixes A :: "'a multiset"   and a :: "'a"   and B :: "'a multiset" shows "A + add_mset a B = add_mset a (A + B)"
comp_cod_arr: fixes f :: "'c"   and b :: "'c" assumes "C.arr f"   and "C.cod f = b" shows "C b f = f"
neqE: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
prems_1: shows "p \<notin> snd (pn_atoms (F \<^bold>\<or> G))"
max_list_set: fixes xs :: "nat list" shows "max_list xs = (if set xs = {} then 0 else THE x. x \<in> set xs \<and> (\<forall>y\<in>set xs. y \<le> x))"
rep_eq: fixes x :: "integer" shows "Rep_uint64 (Uint64 x) = word_of_int (int_of_integer x)"
wf_trms_subst_rm_vars: fixes \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and M :: "('fun, 'b) Term.term set"   and X :: "'b set" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (M \<cdot>\<^sub>s\<^sub>e\<^sub>t rm_vars X \<delta>)"
Smith_prod_zero_imp_last_zero: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat"   and i :: "nat" assumes "A \<in> carrier_mat m n"   and "Smith_normal_form_mat A"   and "(\<Prod>j = 0..<Suc i. A $$ (j, j)) = (0::'b)"   and "i < min m n" shows "A $$ (i, i) = (0::'b)"
suppSubst: fixes subst :: "'c \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'c"   and M :: "'c"   and xvec :: "name list"   and Tvec :: "'d list" assumes "substType subst" shows "supp (subst M xvec Tvec) \<subseteq> supp M \<union> supp xvec \<union> supp Tvec"
dim_eq: fixes X :: "((('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'a" assumes "X \<subseteq> cotangent_space p" shows "cotangent_space.dim p X = dim X"
tuple_ext: fixes f :: "'a"   and g :: "'a" assumes "\<not> span f g" shows "\<langle>f, g\<rangle> = null"
IH: fixes i :: "nat"   and q :: "'Q"   and n :: "nat" assumes "r i = (q, n)"   and "ofs < num_acc" shows "\<exists>j\<ge>i. \<exists>q'. r j = (q', (n + ofs) mod num_acc) \<and> (n + ofs) mod num_acc \<in> acc q'"
induct: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>tyvrs. P (Tvar tyvrs)"   and "P Top"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)"   and "\<And>tyvrs ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (\<forall>tyvrs<:ty2. ty1)" shows "P ty"
random_list_def: shows "random_list \<equiv> pmf_of_set (permutations_of_set A)"
InitBlockReds: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and v :: "val"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "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 T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V \<mapsto> v))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>" shows "P \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow>* \<langle>{V:T; V:=Val (the (l' V));; e'},(h', l'(V := l V))\<rangle>"
restrict_complement_singleton_eq: fixes f :: "'d \<Rightarrow> 'e option"   and x :: "'d" shows "f |` (- {x}) = f(x := None)"
spec_ops_def: fixes ops :: "('b \<times> 'c) list"   and deps :: "'c \<Rightarrow> 'b set" shows "spec_ops ops deps \<equiv> sorted (map fst ops) \<and> distinct (map fst ops) \<and> (\<forall>oid oper ref. (oid, oper) \<in> set ops \<and> ref \<in> deps oper \<longrightarrow> ref < oid)"
eval_Tensor: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u" shows "\<lbrace>t \<^bold>\<otimes> u\<rbrace> = D.tensor \<lbrace>t\<rbrace> \<lbrace>u\<rbrace>"
holds_def: shows "Code_Generator.holds \<equiv> (\<lambda>x. PROP x \<equiv> \<lambda>x. PROP x)"
calculus_axioms: shows "calculus Bot_F Inf_F (\<Turnstile>\<inter>\<G>) empty_ord.Red_I Red_F_\<G>"
linear: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (x::'b) (y::'b) \<or> less_eq y x"
inject_5: fixes x61 :: "'b formula"   and x62 :: "'b formula"   and y61 :: "'b formula"   and y62 :: "'b formula" shows "((x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula) = (y61::'b formula) \<^bold>\<rightarrow> (y62::'b formula)) = (x61 = y61 \<and> x62 = y62)"
individually_rational_on_empty: fixes ds :: "D2 set" shows "StableNoDecomp.individually_rational_on ds {}"
mapping_of_pmf_pmf_of_list: fixes xs :: "('b \<times> real) list" assumes "\<And>x. x \<in> snd ` set xs \<Longrightarrow> 0 < x"   and "sum_list (map snd xs) = 1" shows "mapping_of_pmf (pmf_of_list xs) = Mapping.tabulate (remdups (map fst xs)) (\<lambda>x. sum_list (map snd (filter (\<lambda>z. fst z = x) xs)))"
prod_in_hom: fixes f :: "'a"   and a :: "'a"   and c :: "'a"   and g :: "'a"   and b :: "'a"   and d :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> c\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>f \<otimes> g : a \<otimes> b \<rightarrow> c \<otimes> d\<guillemotright>"
Cons_change_css_step: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate"   and xss :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list" assumes "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css' @ css, t)" shows "\<Gamma>\<turnstile> (cs, xss, s) \<rightarrow> (cs', css' @ xss, t)"
VectorSpaceHom_scalar_mul: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c"   and a :: "'a" assumes "VectorSpaceHom smult V smult' T" shows "VectorSpaceHom smult V smult' (\<lambda>v. smult' a (T v))"
that: fixes j :: "nat"   and zs :: "'a list" assumes "xs = take j ys @ zs"   and "j < length ys"   and "\<forall>k>j. \<forall>zs'. xs \<noteq> take k ys @ zs'" shows "thesis"
vec_def: fixes x :: "'b" shows "Finite_Cartesian_Product.vec x = (\<chi>i. x)"
faceListAt_len_def: fixes g :: "graph" shows "faceListAt_len g \<equiv> |faceListAt g| = countVertices g"
rtrancl_mono_mp: fixes U :: "('a \<times> 'a) set"   and V :: "('a \<times> 'a) set"   and x :: "'a \<times> 'a" assumes "U \<subseteq> V"   and "x \<in> U\<^sup>*" shows "x \<in> V\<^sup>*"
elims_1: fixes x :: "('c, 'd, 'e, 'f) edge_kind"   and xa :: "(('c \<Rightarrow> 'd option) \<times> 'e) list"   and y :: "bool" assumes "local.CFG.pred x xa = y" obtains f :: "('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = \<Up>f__" and "xa = cf__ # cfs__" and "y"   | Q :: "('c \<Rightarrow> 'd option) \<Rightarrow> bool" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = (Q__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "y = Q__ (fst cfa__)"   | Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool" and r :: "'e" and p :: "'f" and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "y = Qa__ (fst cfb__, r)"   | 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" and cf :: "('c \<Rightarrow> 'd option) \<times> 'e" and cfs :: "(('c \<Rightarrow> 'd option) \<times> 'e) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "y = (Q cf \<and> cfs \<noteq> [])"   | et :: "('c, 'd, 'e, 'f) edge_kind" where "x = et" and "xa = []" and "\<not> y"
awalk_empty_ends: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and v :: "'b" assumes "wf_digraph G"   and "pre_digraph.awalk G u [] v" shows "u = v"
tendsto_bot: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open a) bot"
mrbst_diff_correct: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "random_bst A \<bind> (\<lambda>t1. random_bst B \<bind> mrbst_diff t1) = random_bst (A - B)"
lin_span_sub_carrier: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "linear_span R M A H \<subseteq> carrier M"
elims_2: fixes x :: "'a ns_constraint" assumes "is_leq_ns x" obtains p :: "linear_poly" and c :: "'a" where "x = LEQ_ns p c"
fset_of_list_elem: fixes x :: "'c"   and xs :: "'c list" shows "(x |\<in>| fset_of_list xs) = (x \<in> set xs)"
as_top_code: fixes s :: "'c array \<times> nat" shows "as_top s = (let (a, n) = s in array_get a (n - 1))"
simple_list_index_equality: fixes a :: "'c list"   and n :: "nat"   and b :: "'c list" assumes "length a = n"   and "length b = n"   and "\<forall>i<n. a ! i = b ! i" shows "a = b"
F2set2_image_Lev2: fixes xe :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xb :: "'e"   and xd :: "bd_type_F"   and x :: "'e" assumes "xe \<in> Lev2 s1 s2 n xb"   and "xe @ [Inl xd] \<in> Lev2 s1 s2 (Suc n) xb"   and "rv2 s1 s2 xe xb = Inr x" shows "xd \<in> tobd_F22 s2 x ` F2set2 (s2 x)"
lm048: fixes a :: "('a \<times> 'b set) set" assumes "runiq a" shows "pseudoAllocation a = {(x, Y) |Y x. Y \<in> finestpart (a ,, x) \<and> x \<in> Domain a}"
order_trans_rules_21: fixes x :: "'d"   and y :: "'d"   and z :: "'d" assumes "x \<le> y"   and "y < z" shows "x < z"
a: shows "\<forall>i. (\<tau> ^^ i) {} \<subseteq> (\<tau> ^^ (i + 1)) {}"
\<tau>exec_move_def: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap" shows "\<tau>exec_move ci P t e h = (\<lambda>(stk, loc, pc, xcp) s'. exec_move ci P t e h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s' \<and> \<tau>move2 P h stk e pc xcp)"
indRelR_def: shows "indRelR \<equiv> _Collect (x, y) (indRelRp x y)"
assms_4: shows "h \<turnstile> get_tag_name host \<rightarrow>\<^sub>r tag"
iterate_def: fixes m :: "nat"   and n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "iterate m n f x = map (\<lambda>n. (f ^^ n) x) [m..<n]"
partial_mon_sg_laction_axioms: shows "partial_mon_sg_laction \<alpha> Dla"
l7_17_bis: fixes A :: "'p"   and P :: "'p"   and P' :: "'p"   and B :: "'p" assumes "A Midpoint P P'"   and "B Midpoint P' P" shows "A = B"
intro_spec_refine: fixes X :: "'a set"   and f :: "'a \<Rightarrow> 'b nres"   and R :: "('b \<times> 'c) set"   and M :: "'c nres" assumes "\<And>x. x \<in> X \<Longrightarrow> f x \<le> \<Down> R M" shows "RES X \<bind> f \<le> \<Down> R M"
bst_mrbst_diff: fixes t' :: "'a tree"   and t1 :: "'a tree"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_diff t1 t2)"   and "bst t1"   and "bst t2" shows "bst t'"
mtf2_effect: fixes q :: "'c"   and xs :: "'c list"   and i :: "nat"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs" shows "(index xs q < i \<and> i < length xs \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index xs q < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < length xs) \<and> (index xs q = i \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs q - n \<and> index (mtf2 n q xs) (xs ! i) = index xs q - n) \<and> (index xs q - n \<le> i \<and> i < index xs q \<longrightarrow> index (mtf2 n q xs) (xs ! i) = Suc (index xs (xs ! i)) \<and> index xs q - n < index (mtf2 n q xs) (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) \<le> index xs q) \<and> (i < index xs q - n \<longrightarrow> index (mtf2 n q xs) (xs ! i) = index xs (xs ! i) \<and> index (mtf2 n q xs) (xs ! i) < index xs q - n)"