hcju/isabelleps huggingface dataset:
intro: fixes envVal :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and jview :: "'b \<Rightarrow> 'd Trace \<Rightarrow> 'f"   and jkbp :: "'b \<Rightarrow> ('b, 'c, 'g) GC list"   and envInit :: "'d list"   and envAction :: "'d \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'g) \<Rightarrow> 'd \<Rightarrow> 'd"   and simf :: "'d Trace \<Rightarrow> 'e"   and simRels :: "'b \<Rightarrow> ('e \<times> 'e) set"   and simVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "sim (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview)) (mkKripke (simf ` PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview) simRels simVal) simf" shows "SimIncrEnvironment_axioms jkbp envInit envAction envTrans envVal jview simf simRels simVal"
Keys_to_list_sorted_wrt: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ps :: "('c \<Rightarrow>\<^sub>0 'f) list" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "sorted_wrt ord_term_strict\<inverse>\<inverse> (ordered_term.Keys_to_list ord_term ord_term_strict ps)"
less_irrefl: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "class.preorder less_eq less" shows "\<not> less x x"
finite_quotient_iff: fixes A :: "'a set"   and R :: "('a \<times> 'a) set" assumes "equiv A R" shows "finite A = (finite (A // R) \<and> (\<forall>X\<in>A // R. finite X))"
assms: shows "xs \<in> (rev \<circ> sorted_list_of_set) ` Pow (set nodes)"
diff_numeral_special_4: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (uminus (numeral.numeral one plus m)) one = uminus (numeral.numeral one plus (m + num.One))"
leftderives_implies_LeftDerivation: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.leftderives \<NN> \<TT> \<RR> a b" shows "\<exists>D. CFG.LeftDerivation \<NN> \<TT> \<RR> a D b"
set_btree_inorder: fixes t :: "'b btree" shows "set (inorder t) = set_btree t"
wf'_2: shows "free_vars f \<subseteq> shift_var_set (set vs')"
cbox_right_un: fixes x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and x\<^sub>2 :: "real"   and y\<^sub>0 :: "real"   and y\<^sub>1 :: "real" assumes "x\<^sub>0 \<le> x\<^sub>1"   and "x\<^sub>1 \<le> x\<^sub>2" shows "cbox (x\<^sub>0, y\<^sub>0) (x\<^sub>1, y\<^sub>1) \<union> cbox (x\<^sub>1, y\<^sub>0) (x\<^sub>2, y\<^sub>1) = cbox (x\<^sub>0, y\<^sub>0) (x\<^sub>2, y\<^sub>1)"
c: fixes \<Delta>1 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and \<Delta>2 :: "'state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "(\<Delta>1, \<Delta>2) \<in> allConsec \<Delta>s" shows "unwind_cont \<Delta>1 {\<Delta>1, \<Delta>2, \<Delta>e}"
not_greater_Greatest: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "(GREATEST x. P x) < k" shows "\<not> P k"
arrI: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and a :: "'c"   and b :: "'c" assumes "partial_magma C"   and "partial_magma.in_hom C f a b" shows "partial_magma.arr C f"
wt_subst_trm'': fixes \<sigma> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>" shows "\<Gamma> t = \<Gamma> (t \<cdot> \<sigma>)"
Cons_2: fixes y :: "'a" assumes "y \<in> set (local.filter_min_aux (x0 # xs) ys)"   and "rel y x" shows "thesis"
compositionSym: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d" assumes "assertion SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'" shows "assertionAux.AssertionStatEq SImp (SCompose \<Psi>'' \<Psi>) (SCompose \<Psi>'' \<Psi>')"
order_hom: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a"   and f :: "'a poly" assumes "map_poly_inj_idom_divide_hom hom" shows "order (hom x) (map_poly hom f) = order x f"
set_pmf_not_empty: fixes M :: "'c pmf" shows "set_pmf M \<noteq> {}"
MsgInTrace'': shows "msg \<in> set (drop (length trace) trace'')"
AE_not_in: fixes N :: "'a set"   and M :: "'a measure" assumes "N \<in> null_sets M" shows "AE x in M. x \<notin> N"
\<eta>_eq: fixes x :: "'a" assumes "x \<in> M" shows "\<eta>' x = \<eta> x"
Y_X_closed: fixes y :: "('s \<times> ('c \<Rightarrow> real)) stream"   and x :: "('s \<times> ('c \<Rightarrow> real) set) stream" assumes "Q y"   and "smap abss y = x"   and "y \<in> Y" shows "x \<in> X \<and> P x"
subst_upd_comp: fixes \<rho> :: "'g \<Rightarrow> 'f"   and x :: "'h \<Longrightarrow> 'f"   and v :: "('h, 'g) uexpr"   and \<sigma> :: "'e \<Rightarrow> 'g" shows "\<rho>(x \<mapsto>\<^sub>s v) \<circ> \<sigma> = (\<rho> \<circ> \<sigma>)(x \<mapsto>\<^sub>s \<sigma> \<dagger> v)"
sorted_Cons_imp_sorted: fixes cmp :: "'b comparator"   and x :: "'b"   and xs :: "'b list" assumes "Sorting_Algorithms.sorted cmp (x # xs)" shows "Sorting_Algorithms.sorted cmp xs"
tensors_ht_l0: fixes j :: "nat"   and r0 :: "nat"   and M :: "nat" assumes "j < r0" shows "tensors_from_net (Conv (id_matrix r0 M) (Input M)) $ j = (if j < M then unit_vec M j else tensor0 [M])"
get_tag_name_def: shows "get_tag_name \<equiv> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name"
isolate_islimpt_iff: fixes x :: "'b"   and S :: "'b set" shows "x isolate S = (\<not> x islimpt S \<and> x \<in> S)"
ts'_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [v] : ts' _> ts @ [t]"
record_simps_4: fixes f :: "nat \<Rightarrow> nat"   and x :: "'b state" shows "defined_types (update_clock f x) = defined_types x"
intros: fixes a :: "'c"   and b :: "'d"   and r :: "('c \<times> 'd) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
chain: fixes x :: "'b" shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>f. f x) ` ccpo.iterates (fun_lub Sup) (fun_ord (\<le>)) F)"
uint64_mod_def: fixes x :: "uint64"   and y :: "uint64" shows "uint64_mod x y = snd (uint64_divmod x y)"
elims_3: fixes x :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xa :: "'e option"   and xb :: "'e option" assumes "\<not> option_le x xa xb" obtains P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" and xc :: "'e" where "x = P__" and "xa = Some xc__" and "xb = None"   | P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" and xc :: "'e" and y :: "'e" where "x = P" and "xa = Some xc" and "xb = Some y" and "\<not> P xc y"
mem_samp_primes_prime: fixes prime_set :: "nat set"   and x :: "nat" assumes "rsa_base prime_set"   and "x \<in> set_spmf (rsa_base.sample_primes prime_set)" shows "prime x"
downset_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ord.downset less_eq = ord.downset_set less_eq \<circ> (\<lambda>x. {x})"
h_minus_mono: fixes i :: "nat"   and e :: "'v \<times> 'v" shows "h_minus i e \<le> h_minus (Suc i) e"
rel_mono: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_sp\<^sub>\<mu> R1 R2 \<le> rel_sp\<^sub>\<mu> R1a R2a"
Sura_Bura: fixes S :: "'a set"   and C :: "'a set" assumes "locally compact S"   and "C \<in> components S"   and "compact C" shows "C = \<Inter> {K. C \<subseteq> K \<and> compact K \<and> openin (top_of_set S) K}"
dist_self: fixes x :: "fbint" shows "local.dist x x = 0"
x_in: shows "x \<in> {b} \<union> parents d (child b dir d) p"
distinct_44: fixes pi1' :: "pi"   and pi2' :: "pi"   and pi :: "pi" shows "pi1' \<parallel> pi2' \<noteq> \<tau>.pi"
l3_inv8_derived: fixes implem :: "chan \<Rightarrow> msg" assumes "dhlvl3 implem" shows "l3_inv2 \<inter> dhlvl3.l3_inv3 implem \<inter> l3_inv4 \<subseteq> dhlvl3.l3_inv8 implem"
Cons_4: fixes k' :: "'a" assumes "k' \<in> fst ` set xs" shows "lt k'__ k'"
power_eq_if: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and p :: "'d"   and m :: "nat" shows "power.power one times p m = (if m = 0 then one else times p (power.power one times p (m - 1)))"
exhaust: fixes y :: "subject" obtains x1 :: "name" where "y = InputS x1"   | x2 :: "name" where "y = BoundOutputS x2"
sum_characters: fixes x :: "'a" assumes "x \<in> carrier G" shows "(\<Sum>\<chi>\<in>characters G. \<chi> x) = (if x = \<one> then of_nat (order G) else 0)"
simps: fixes a :: "'a"   and uu :: "'b" shows "fs (Pd (a, uu)) = a"
f2: fixes x :: "'a" assumes "x \<in> sphere a r" shows "f2 x = x"
E: assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c R,B" shows "S"
lve_Entry_edge: fixes e :: "'f LDCFG_node \<times> 'g edge_kind \<times> 'f LDCFG_node"   and E :: "'f"   and valid_edge :: "'h \<Rightarrow> bool"   and src :: "'h \<Rightarrow> 'f"   and trg :: "'h \<Rightarrow> 'f"   and knd :: "'h \<Rightarrow> 'g edge_kind"   and X :: "'f" assumes "e = (NewEntry, (\<lambda>s. True)\<^sub>\<surd>, Node E)" shows "lift_valid_edge valid_edge src trg knd E X e"
simps_3: fixes e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "Astack (Alts e1 e2 # S) = 0"
AE_one_D_integrals_eq: shows "AE x in lborel. LBINT y. f (x, y) = integral UNIV (\<lambda>y. f (x, y))"
map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" shows "map \<alpha> S T \<equiv> \<alpha> \<in> S \<rightarrow>\<^sub>E T"
multi_member_split: fixes x :: "'f"   and M :: "'f multiset" assumes "x \<in># M" shows "\<exists>A. M = add_mset x A"
rotate_eq: fixes xs :: "'a list"   and ys :: "'a list" assumes "rotate1 xs = rotate1 ys" shows "xs = ys"
LTb: shows "LT1 ars r (s, [\<alpha>_step]) (s, [\<beta>_step]) \<delta>1 \<delta>2 \<delta>3"
cinner_zero_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x zero = 0"
simps_8: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'c" shows "(case EQ of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f2"
less_bot_def: fixes x :: "'a bot"   and y :: "'a bot" shows "(x < y) = (case y of Value y \<Rightarrow> case x of Value x \<Rightarrow> x < y | Bot \<Rightarrow> True | Bot \<Rightarrow> False)"
Un_Int_eq_2: fixes S :: "'a set"   and T :: "'a set" shows "(S \<union> T) \<inter> T = T"
that: fixes r :: "'a"   and s :: "'a" assumes "r \<in> R"   and "s \<in> R"   and "s \<notin> I"   and "X = local.frac r s" shows "thesis"
c: shows "a < c \<and> c < b \<and> ((\<lambda>x. (f b - f a) * g x - (g b - g a) * f x) has_real_derivative l) (at c) \<and> (f b - f a) * g b - (g b - g a) * f b - ((f b - f a) * g a - (g b - g a) * f a) = (b - a) * l"
fag_single_additionTr5: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<forall>m<Suc n. (THE x. x \<in> add_set f {a} (Suc n)) \<^sub>f+ (THE x. x \<in> add_set f {\<^sub>i- a} m) = (THE x. x \<in> add_set f {a} (n - m))"
seq_semI': fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp"   and \<Gamma> :: "('sf, 'sc, 'sz) formula list"   and \<Delta> :: "('sf, 'sc, 'sz) formula list" assumes "\<nu> \<in> fml_sem I (foldr (&&) \<Gamma> TT) \<Longrightarrow> \<nu> \<in> fml_sem I (foldr (||) \<Delta> FF)" shows "\<nu> \<in> seq_sem I (\<Gamma>, \<Delta>)"
ord_recp_def: fixes T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf"   and x :: "hf"   and y :: "hf" shows "ord_recp T G H x y = (if x = 0 then y = T else if Ord x then \<exists>s. ord_rec_Seq T G s x y else y = H x)"
Diff_triv: fixes A :: "'b set"   and B :: "'b set" assumes "A \<inter> B = {}" shows "A - B = A"
uint_minus_simple_alt: fixes y :: "'b word"   and x :: "'b word" shows "(y \<le> x) = (uint (x - y) = uint x - uint y)"
kk_squared: shows "\<k>\<^sup>2 = - 1"
bin_last_integer_nbe: fixes i :: "integer" shows "bin_last_integer i = (i mod 2 \<noteq> 0)"
QuoteP_sf: fixes t :: "tm"   and u :: "tm" shows "Sigma_fm (QuoteP t u)"
prv_cnj_imp: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp (cnj \<phi> \<chi>) \<psi>)" shows "prv (imp \<phi> (imp \<chi> \<psi>))"
trace_core'_def: fixes core :: "('k, 'l, 'm, 'o, 'n, 'p) core" shows "trace_core' core \<equiv> \<lambda>uu uua. rec_list (\<lambda>S. (\<lambda>e. weight_spmf' (S \<bind> (\<lambda>s. cpoke core s e)), \<lambda>ia. S \<bind> (\<lambda>s. map_spmf fst (cfunc_adv core s ia)), \<lambda>iu. S \<bind> (\<lambda>s. map_spmf fst (cfunc_usr core s iu)))) (\<lambda>obs tr tra S. case obs of Inl e \<Rightarrow> tra (mk_lossless (S \<bind> (\<lambda>s. cpoke core s e))) | Inr (Inl (ia, oa)) \<Rightarrow> tra (cond_spmf_fst (S \<bind> (\<lambda>s. cfunc_adv core s ia)) oa) | Inr (Inr (iu, ou)) \<Rightarrow> tra (cond_spmf_fst (S \<bind> (\<lambda>s. cfunc_usr core s iu)) ou)) uua uu"
fresh_3: fixes a :: "atom"   and fmla :: "fmla" shows "a \<sharp> Robinson_Arithmetic.neg fmla = a \<sharp> fmla"
natural_transformation_axioms: shows "natural_transformation A B G H \<psi>"
pr1_in_hom': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr1 a b : local.dom (pr0 a b) \<rightarrow> a\<guillemotright>"
surjective_respects_domain_and_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. f ` A = B) respects domain_and_range_permutation A B"
gre1I_nat: fixes n :: "nat" assumes "n = 0 \<Longrightarrow> False" shows "Suc 0 \<le> n"
simps_2: fixes l :: "bool list bdd"   and r :: "bool list bdd" shows "quantify_bdd 0 (Branch l r) = bdd_binop bv_or l r"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and Value :: "Value" shows "P (case Value of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = ((\<forall>x1. Value = boolV x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. Value = intgV x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. Value = shortV x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. Value = byteV x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x51 x52. Value = objV x51 x52 \<longrightarrow> P (f5 x51 x52)) \<and> (\<forall>x61 x62. Value = arrV x61 x62 \<longrightarrow> P (f6 x61 x62)) \<and> (Value = nullV \<longrightarrow> P f7))"
assms_4: shows "u \<in> [reset' M n cs v d]\<^bsub>v,n\<^esub>"
Diff_iff: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
not_eq_extremum: fixes a :: "'c" shows "(a \<noteq> \<top>) = (a < \<top>)"
insert_ops_subset_last: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops (xs @ [x])"   and "insert_ops ys"   and "set ys \<subseteq> set (xs @ [x])"   and "x \<in> set ys" shows "x = last ys"
empty: fixes x :: "'b set" assumes "finite x"   and "card x = 0" shows "x = {}"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "P []"   and "\<And>head. P [head]"   and "\<And>head x xs. P (x # xs) \<Longrightarrow> P (head # x # xs)" shows "P a0"
is_aligned_add_or: fixes p :: "'b word"   and n :: "nat"   and d :: "'b word" assumes "is_aligned p n"   and "d < 2 ^ n" shows "p + d = p OR d"
is_Zp_cauchy_imp_has_limit: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "padic_integers.is_Zp_cauchy p s"   and "a = res_lim s" shows "padic_integers.Zp_converges_to p s a"
chaum_ped_\<Sigma>_base_axioms: shows "chaum_ped_\<Sigma>_base \<G>"
res: shows "res = (int (y - k1 * p), int (y - k2 * q))"
nz: shows "\<parallel>fs ! i\<parallel>\<^sup>2 \<noteq> 0"
runit_char_2: fixes f :: "'a" assumes "ide f" shows "R \<r>[f] = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]"
i_def: shows "i = nat \<lceil>(f 0 - a) / \<delta>\<rceil>"
power_less_power_Suc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "less (power.power one times a n) (times a (power.power one times a n))"
llist_of_stream_stream_of_llist: fixes xs :: "'c llist" assumes "\<not> lfinite xs" shows "llist_of_stream (stream_of_llist xs) = xs"
disc_7: fixes x21 :: "variable"   and x22 :: "typ" shows "is_Fv (Fv x21 x22)"
float_le_zero: fixes a :: "int"   and b :: "int" shows "(float (a, b) \<le> 0) = (a \<le> 0)"
curr_val_satisfies_state_def: fixes s :: "('a, 'b) state" shows "\<Turnstile> s \<equiv> \<langle>\<V> s\<rangle> \<Turnstile>\<^sub>s s"
funpow_swap1: fixes f :: "'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "'a" shows "f ((f ^^ n) x) = (f ^^ n) (f x)"
f_1_lemma_2: fixes x :: "int" assumes "x \<in> local.S1" shows "local.f_1 (local.f_1 x) = x"
abs_power_minus: 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 a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times (uminus a) n) = abs (power.power one times a n)"
alg_dvd_triv_right: fixes x :: "'b"   and y :: "'b" assumes "algebraic_int x" shows "y alg_dvd x * y"
complete_market_def: shows "complete_market = (\<forall>matur. \<forall>der\<in>borel_measurable (G matur). \<exists>p. replicating_portfolio p der matur)"
to_RED: fixes s :: "trm"   and \<sigma> :: "ty"   and x :: "name"   and t :: "trm"   and \<tau> :: "ty" assumes "s \<in> RED (T \<sigma>)"   and "\<forall>p\<in>RED \<sigma>. t[x::=p] \<in> RED (T \<tau>)" shows "s to x in t \<in> RED (T \<tau>)"
C: shows "x ^ o * \<bottom> \<squnion> (1::'a) = x ^ o"
nxt_monotone: fixes s :: "nat \<Rightarrow> nat option"   and b :: "nat"   and b' :: "nat" assumes "total_cons s"   and "eval (total_cons.r_next s) [b] \<down>= b'" shows "e_length b < e_length b'"
l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.to_tree_order Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs Shadow_DOM_Components.get_component Shadow_DOM_Components.is_strongly_dom_component_safe Shadow_DOM_Components.is_weakly_dom_component_safe Shadow_DOM.get_root_node Shadow_DOM.get_root_node_locs Shadow_DOM.get_ancestors Shadow_DOM.get_ancestors_locs get_disconnected_nodes Shadow_DOM.get_element_by_id Shadow_DOM.get_elements_by_tag_name"
prefix_fininf_conc: fixes u :: "'a list"   and v :: "'a stream" shows "u \<le>\<^sub>F\<^sub>I u @- v"
adjacentsetD_adj: fixes y :: "'a set"   and x :: "'a set" assumes "y \<in> adjacentset x" shows "x \<sim> y"
simps_3: fixes uv :: "'a simple_rule"   and v :: "'a simple_rule"   and va :: "'a simple_rule list" shows "has_default_policy (uv # v # va) = has_default_policy (v # va)"
simps_2: fixes t :: "'a Trace"   and s :: "'a" shows "tLength (t \<leadsto> s) = 1 + tLength t"
recvmsg_simps_7: fixes P :: "'d \<Rightarrow> bool"   and m :: "'d" shows "recvmsg P (receive m) = P m"
Eq\<kappa>I: fixes x :: "\<kappa>"   and y :: "\<kappa>"   and v :: "i" assumes "\<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2" shows "[x \<^bold>=\<^sub>\<kappa> y in v]"
subst_typ'_comp_single_left: fixes single :: "(variable \<times> String.literal set) \<times> typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and t :: "Core.term" shows "subst_typ' [single] (subst_typ' insts t) = subst_typ' (map (apsnd (subst_typ [single])) insts @ [single]) t"
axioms_3: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_fail_alt return bind fail alt" shows "monad_fail_alt_axioms fail alt"
eq_refl: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>L \<phi>"
in_space_T: fixes x :: "(real \<times> 'a) stream" shows "x \<in> space (stream_space S)"
iso_dghm_Arr_HomDom_if_Arr_HomCod: fixes g :: "V" assumes "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" obtains f :: "V" where "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" and "g = \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>"
constant_rep_design_axioms_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and design_rep_number :: "int" shows "constant_rep_design_axioms point_set block_collection design_rep_number \<equiv> \<forall>x. x \<in> point_set \<longrightarrow> block_collection rep x = design_rep_number"
card_eq_0_iff: fixes A :: "'c set" shows "(card A = 0) = (A = {} \<or> infinite A)"