hcju/isabelleps huggingface dataset:
comparator_of_def: fixes x :: "'a"   and y :: "'a" shows "c.comparator_of x y = (if c.lt x y then Lt else if x = y then Eq else Gt)"
foldl_cong: fixes a :: "'b"   and b :: "'b"   and l :: "'c list"   and k :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'b" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f a x = g a x" shows "foldl f a l = foldl g b k"
Lyndon_rec_all: fixes a :: "'a"   and w :: "'a list"   and n :: "nat" assumes "Lyndon_rec (a # w) \<^bold>|w\<^bold>|"   and "n < \<^bold>|a # w\<^bold>|"   and "0 < n" shows "Lyndon_rec (a # w) n"
assms_5: shows "freq t a = w\<^sub>a + w\<^sub>b"
polypoly_poly: fixes p :: "'b poly"   and n0 :: "nat"   and x :: "'b"   and bs :: "'b list" assumes "isnpolyh p n0" shows "Ipoly (x # bs) p = poly (polypoly (x # bs) p) x"
span_Basis: shows "vector_space_over_itself.span {1::'d} = UNIV"
fproductE1: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A \<times>\<^sub>\<bullet> B" obtains a :: "V" and b :: "V" where "x = [a, b]\<^sub>\<circ>" and "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> B"
strict_border_butlast: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<noteq> []"   and "strict_border xs ys" shows "strict_border (butlast xs) (butlast ys)"
monotone_snd: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "monotone (rel_prod orda ordb) ordb snd"
reads_consistent_snoc_Prog\<^sub>s\<^sub>b: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and p\<^sub>1 :: "'b"   and p\<^sub>2 :: "'b"   and mis :: "instr list" assumes "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m (sb @ [Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 mis])"
simps_7: fixes X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and F' :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F' - set X"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT ($f x1a x2a). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
map_of_filter_apply: fixes xs :: "('c \<times> 'd) list"   and P :: "'c \<times> 'd \<Rightarrow> bool"   and k :: "'c" assumes "distinct (list.map fst xs)" shows "map_of (List.filter P xs) k = (case map_of xs k of None \<Rightarrow> None | Some v \<Rightarrow> if P (k, v) then Some v else None)"
A_3: fixes S :: "'b set"   and x :: "'b" assumes "open S"   and "x \<in> S" shows "\<exists>i. A x i \<subseteq> S"
almost_never_le_\<alpha>: fixes p :: "nat \<Rightarrow> real"   and k :: "nat" assumes "\<forall>\<^sup>\<infinity>n. 0 < p n \<and> p n < 1"   and "0 < k"   and "\<forall>\<^sup>\<infinity>n. real (6 * k) * ln (real n) / real n \<le> p n" shows "(\<lambda>n. probGn p n (\<lambda>es. ereal (1 / 2 * real n / real k) \<le> ereal_of_enat (\<alpha> (edge_space.edge_ugraph n es)))) \<longlonglongrightarrow> 0"
abelian_groupE_4: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
wxK: shows "\<omega>_limit_set x \<subseteq> K"
comp_ide_self: fixes a :: "'a \<times> 'a" assumes "VoV.ide a" shows "VoV.comp a a = a"
when_neq_zero: fixes zero :: "'a"   and a :: "'a"   and P :: "bool" shows "(zero.when zero a P \<noteq> zero) = (P \<and> a \<noteq> zero)"
set_map_of_compr: fixes xs :: "('b \<times> 'c) list" assumes "distinct (map fst xs)" shows "set xs = {(k, v). map_of xs k = Some v}"
ln_Gamma_bounds_2: fixes x :: "real" assumes "1 \<le> x" shows "ln_Gamma x \<le> ln (2 * pi / x) / 2 + x * ln x - x + 1 / (12 * x)"
SetMem_def: shows "SetMem \<equiv> lmember\<cdot>(bpred (=))"
roifex_set_var_subtree_2: fixes b :: "'a ifex"   and v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and ass :: "'a \<Rightarrow> bool" assumes "ro_ifex b"   and "b = IF v t e" shows "val_ifex b (ass(v := False)) = val_ifex e ass"
fls_regpart_fps_trivial: fixes f :: "'b fps" shows "fls_regpart (fps_to_fls f) = f"
gdpr_EF: shows "gdpr_Kripke \<turnstile> EF sgdpr"
init_vars_def: fixes m :: "'a \<Rightarrow> 'b option" shows "init_vars m \<equiv> map_option (\<lambda>T. Null) \<circ> m"
IH_1: fixes M :: "('fun, 'var) Term.term set" assumes "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>c \<theta>" shows "\<lbrakk>M \<union> (ik\<^sub>s\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>); S'\<rbrakk>\<^sub>c \<theta>"
rtrancl_while_def: fixes p :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b list"   and x :: "'b" shows "While_Combinator.rtrancl_while p f x = while_option (While_Combinator.rtrancl_while_test p) (While_Combinator.rtrancl_while_step f) ([x], {x})"
simps: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "Optimal_BST2 w" shows "Optimal_BST2.opt_bst2 w i j = (if j < i then \<langle>\<rangle> else if i = j then \<langle>\<langle>\<rangle>, i, \<langle>\<rangle>\<rangle> else let left = root (Optimal_BST2.opt_bst2 w i (j - 1)); right = root (Optimal_BST2.opt_bst2 w (i + 1) j) in argmin (wpl.wpl w i j) (map (\<lambda>k. \<langle>Optimal_BST2.opt_bst2 w i (k - 1), k, Optimal_BST2.opt_bst2 w (k + 1) j\<rangle>) [left..right]))"
sequence_result_rvalD: fixes rs :: "('a, 'b) result list"   and vs :: "'a list" assumes "sequence_result rs = Rval vs" shows "rs = map Rval vs"
pmf_def: shows "pmf \<equiv> map_fun measure_pmf id (\<lambda>M x. Sigma_Algebra.measure M {x})"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "a +^ n + a = a + a +^ n"
full_checker_spec': shows "(uncurry2 full_checker, uncurry2 (\<lambda>spec A _. PAC_checker_specification spec A)) \<in> (Id \<times>\<^sub>r polys_rel) \<times>\<^sub>r Id \<rightarrow>\<^sub>f \<langle>{((st, G), st', G'). (st, st') \<in> status_rel \<and> (st \<noteq> FAILED \<longrightarrow> (G, G') \<in> polys_rel_full)}\<rangle>nres_rel"
lucas_lehmer_mult_cong_1: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "[fst (lucas_lehmer_mult m x y) = fst x * fst y + 3 * snd x * snd y] (mod m)"
gen_sets: fixes G :: "'b list set"   and S :: "'b list set" assumes "\<BB> G \<subseteq> S"   and "S \<subseteq> \<langle>G\<rangle>" shows "\<langle>S\<rangle> = \<langle>G\<rangle>"
case_option_collapse: fixes x :: "'c" shows "case_option x (\<lambda>_. x) = (\<lambda>_. x)"
is_subrun_def: fixes sub :: "'b run"   and r :: "'b run" shows "sub \<lless> r \<equiv> \<exists>f. dilating f sub r"
loop_orientation_diagonal: fixes y :: "'a"   and x :: "'a" assumes "loop_orientation x y" shows "y \<sqinter> y\<^sup>T = x \<sqinter> (1::'a)"
that: fixes t' :: "'a \<Rightarrow>\<^sub>0 nat" assumes "t' \<in> keys p"   and "t = monomial (poly_deg p - deg_pm t') x + t'" shows "thesis"
strict_prefixE': fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys" obtains z :: "'b" and zs :: "'b list" where "ys = xs @ z # zs"
index_one_mat_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "1\<^sub>m n $$ (i, j) = (if i = j then 1::'d else (0::'d))"
measurable_pair_swap: fixes f :: "'c \<times> 'd \<Rightarrow> 'e"   and M1 :: "'c measure"   and M2 :: "'d measure"   and M :: "'e measure" assumes "f \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M M" shows "(\<lambda>(x, y). f (y, x)) \<in> M2 \<Otimes>\<^sub>M M1 \<rightarrow>\<^sub>M M"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list"   and a2 :: "char list" assumes "subclsSp P a1 a2" obtains C :: "char list" and Bs :: "base list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" and D :: "char list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(Bs, rest)\<rfloor>" and "Shares D \<in> set Bs"
un_closed: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "1\<^sub>r\<^bsub>R\<^esub> \<in> carrier R"
rHom_mem: fixes f :: "'c \<Rightarrow> 'd"   and A :: "('c, 'e) Ring_scheme"   and R :: "('d, 'f) Ring_scheme"   and a :: "'c" assumes "f \<in> rHom A R"   and "a \<in> carrier A" shows "f a \<in> carrier R"
left_tree_separates: fixes s :: "'a"   and t :: "'a"   and u :: "'a"   and u' :: "'a" assumes "s \<rightarrow> t"   and "u \<in> left_tree s t"   and "u' \<in> left_tree t s" shows "s \<in> set (u \<leadsto> u')"
m_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "y \<in> carrier (poly_ring R)" shows "x \<oplus>\<^bsub>poly_ring R\<^esub> y \<in> carrier (poly_ring R)"
rewrite_negated_primitives_normalized_no_modification: fixes disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and m :: "'b common_primitive match_expr"   and a :: "'b common_primitive match_expr" assumes "wf_disc_sel (disc, sel) C"   and "\<not> has_disc_negated disc False m"   and "normalized_nnf_match m"   and "a \<in> set (normalize_match (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m))" shows "a = m"
real_cond_exp_charact: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "\<And>A. A \<in> sets F \<Longrightarrow> set_lebesgue_integral M A f = set_lebesgue_integral M A g"   and "integrable M f"   and "integrable M g"   and "g \<in> borel_measurable F" shows "AEeq M (real_cond_exp M F f) g"
bf_rec: fixes M :: "nat set"   and q :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and j :: "nat"   and i :: "nat" shows "bf M q (j, i) = (let R = case prev j i of None \<Rightarrow> M | Some u \<Rightarrow> snd (bf M q u) in grab R (q j i))"
less_prod_simp: fixes x1 :: "'a"   and y1 :: "'b"   and x2 :: "'a"   and y2 :: "'b" shows "((x1, y1) < (x2, y2)) = (x1 < x2 \<or> x1 \<le> x2 \<and> y1 < y2)"
assms_2: shows "B \<in> carrier_mat nr1 nc2"
norm_norm: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<parallel>\<^bold>\<parallel>t\<^bold>\<parallel>\<^bold>\<parallel> = \<^bold>\<parallel>t\<^bold>\<parallel>"
add_nonneg_nonneg: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "{#} \<subseteq># a"   and "{#} \<subseteq># b" shows "{#} \<subseteq># a + b"
trm_strong_cases: fixes M :: "'a trm"   and P :: "'a set \<Rightarrow> 'a trm \<Rightarrow> bool"   and S :: "'a set" assumes "M = Unit \<Longrightarrow> P S M"   and "\<And>x. M = Var x \<Longrightarrow> P S M"   and "\<And>A B. M = App A B \<Longrightarrow> P S M"   and "\<And>x T A. \<lbrakk>M = Fn x T A; x \<notin> S\<rbrakk> \<Longrightarrow> P S M"   and "\<And>A B. M = SimplyTyped.Pair A B \<Longrightarrow> P S M"   and "\<And>A. M = Fst A \<Longrightarrow> P S M"   and "\<And>A. M = Snd A \<Longrightarrow> P S M"   and "finite S" shows "P S M"
orderI: fixes a :: "'b::semilattice_sup"   and b :: "'b::semilattice_sup" assumes "(a::'b::semilattice_sup) = a \<squnion> (b::'b::semilattice_sup)" shows "(b::'b::semilattice_sup) \<le> (a::'b::semilattice_sup)"
pderiv_smult: fixes a :: "'a"   and p :: "'a poly" shows "pderiv (Polynomial.smult a p) = Polynomial.smult a (pderiv p)"
insert_before_list_in_set: fixes x :: "'b"   and v :: "'b"   and ref :: "'b option"   and xs :: "'b list" shows "(x \<in> set (insert_before_list v ref xs)) = (x = v \<or> x \<in> set xs)"
nn_integral_count_space_indicator: fixes X :: "'k set"   and f :: "'k \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = set_nn_integral (count_space UNIV) X f"
filterlim_pow_at_bot_odd: fixes n :: "nat"   and f :: "real \<Rightarrow> real"   and F :: "real filter" assumes "0 < n"   and "filterlim f at_bot F"   and "odd n" shows "LIM x F. f x ^ n :> at_bot"
AE: shows "AE x in lebesgue. indicat_real {a..c} x *\<^sub>R f x + indicat_real {c..b} x *\<^sub>R f x = indicat_real {a..b} x *\<^sub>R f x"
asList_def: fixes A :: "'b set" shows "asList A \<equiv> SOME as. set as = A"
inj_onD: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
mkArr_restrict_eq: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "S.arr (S.mkArr A B F)" shows "S.mkArr A B (restrict F A) = S.mkArr A B F"
prv_imp_cnj: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "bprv (imp \<phi> \<psi>)"   and "bprv (imp \<phi> \<chi>)" shows "bprv (imp \<phi> (cnj \<psi> \<chi>))"
Abs_cars_inject: fixes x :: "nat"   and y :: "nat" assumes "x \<in> {n. True}"   and "y \<in> {n. True}" shows "(Abs_cars x = Abs_cars y) = (x = y)"
a_rcos_zero: fixes I :: "'a set"   and i :: "'a" assumes "ideal I R"   and "i \<in> I" shows "I +> i = I"
filter_by_key_merge_all: fixes xss :: "'a list list"   and k :: "'k" assumes "\<forall>xs\<in>set xss. sorted (map key xs)" shows "filter (\<lambda>y. key y = k) (local.merge_all xss) = filter (\<lambda>y. key y = k) (concat xss)"
intros_15: fixes tf :: "tf"   and tn :: "t list"   and tm :: "t list"   and \<C> :: "t_context"   and es :: "b_e list" assumes "tf = tn _> tm"   and "\<C>\<lparr>label := [tm] @ label \<C>\<rparr> \<turnstile> es : tn _> tm" shows "\<C> \<turnstile> [Block tf es] : tn _> tm"
set_takeWhileD: fixes x :: "'g"   and P :: "'g \<Rightarrow> bool"   and xs :: "'g list" assumes "x \<in> set (takeWhile P xs)" shows "x \<in> set xs \<and> P x"
type_definition_enat: shows "type_definition Rep_enat Abs_enat UNIV"
exec_gpv_parametric: fixes S :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and CALL :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun CALL (rel_spmf (rel_prod (=) S)))) (rel_fun (rel_gpv A CALL) (rel_fun S (rel_spmf (rel_prod A S)))) exec_gpv exec_gpv"
IVT_strict: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and b :: "'b" assumes "y < f a \<and> f b < y \<or> f a < y \<and> y < f b"   and "a < b"   and "continuous_on {a..b} f" shows "\<exists>x>a. x < b \<and> f x = y"
IH_2: fixes t2 :: "('a, 'b) rbt" assumes "rbt_sorted t2" shows "rbt_lookup (RBT_Impl.fold rbt_insert r t2) = rbt_lookup t2 ++ map_of (rev (RBT_Impl.entries r))"
rep_def: fixes f :: "'c local.term set" shows "rep f \<equiv> \<^bold>\<parallel>SOME t. t \<in> f\<^bold>\<parallel>"
sign_changes_cong: fixes xs :: "'a list"   and ys :: "'a list" assumes "map sgn xs = map sgn ys" shows "sign_changes xs = sign_changes ys"
to_tree_order_ptrs_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and to :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "ptr' \<in> set to" shows "ptr' |\<in>| object_ptr_kinds h"
D: shows "Crypt (sesK (c * f)) (Number 0) \<in> synth (analz (A \<union> spies evsFR5))"
x2_sum: shows "(\<lambda>i. x i * x i) abs_summable_on A"
elims_3: fixes x :: "atom set \<times> 'a"   and xa :: "atom set \<times> 'a" assumes "\<not> x \<approx>abs_set xa" obtains bs :: "atom set" and xb :: "'a" and cs :: "atom set" and y :: "'a" where "x = (bs, xb)" and "xa = (cs, y)" and "\<nexists>p. (bs, xb) \<approx>set (=) supp p (cs, y)"
birkhoff_sum_abs: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and x :: "'a" shows "norm (birkhoff_sum f n x) \<le> birkhoff_sum (\<lambda>x. norm (f x)) n x"
big_step_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> int"   and x2a :: "char list \<Rightarrow> int"   and P :: "com \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "(x1a, x1b) \<Rightarrow> x2a"   and "\<And>s. P SKIP s s"   and "\<And>x a s. P (x ::= a) s (s(x := aval a s))"   and "\<And>c\<^sub>1 s\<^sub>1 s\<^sub>2 c\<^sub>2 s\<^sub>3. \<lbrakk>(c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2; P c\<^sub>1 s\<^sub>1 s\<^sub>2; (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3; P c\<^sub>2 s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) s\<^sub>1 s\<^sub>3"   and "\<And>b s c\<^sub>1 t c\<^sub>2. \<lbrakk>bval b s; (c\<^sub>1, s) \<Rightarrow> t; P c\<^sub>1 s t\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s t"   and "\<And>b s c\<^sub>2 t c\<^sub>1. \<lbrakk>\<not> bval b s; (c\<^sub>2, s) \<Rightarrow> t; P c\<^sub>2 s t\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s t"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P (WHILE b DO c) s s"   and "\<And>b s\<^sub>1 c s\<^sub>2 s\<^sub>3. \<lbrakk>bval b s\<^sub>1; (c, s\<^sub>1) \<Rightarrow> s\<^sub>2; P c s\<^sub>1 s\<^sub>2; (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3; P (WHILE b DO c) s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s\<^sub>1 s\<^sub>3" shows "P x1a x1b x2a"
inter_sorted_cons: fixes x :: "'a"   and xs :: "'a list" assumes "Sorted_Less.sorted (rev (x # xs))" shows "inter_sorted_rev (x # xs) xs = xs"
Elem_affine_of_ivl_ge: fixes e :: "nat \<Rightarrow> real"   and l :: "'b"   and u :: "'b" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "l \<le> u" shows "aform_val e (aform_of_ivl l u) \<le> u"
inject: fixes x21 :: "'b"   and x22 :: "'b"   and x23 :: "'c"   and y21 :: "'b"   and y22 :: "'b"   and y23 :: "'c" shows "(Maps.MkArr x21 x22 x23 = Maps.MkArr y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
eq: fixes y :: "'a" assumes "y \<in> space M" shows "f ` space M \<inter> {i. \<exists>x\<in>space M. i = f x \<and> g y = g x} = {v (g y)}"
A_2: shows "valid_abstraction A X (\<lambda>x. real (k x))"
case_cong_weak: fixes sum :: "'d + 'e"   and sum' :: "'d + 'e"   and f1 :: "'d \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'f" assumes "sum = sum'" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case sum' of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x)"
id_eq_prop_prop_3: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1"   and H :: "\<Pi>\<^sub>1" shows "[F \<^bold>= G \<^bold>& G \<^bold>= H \<^bold>\<rightarrow> F \<^bold>= H in v]"
ends_in_segment_1: fixes a :: "'b"   and b :: "'b" shows "a \<in> {a--b}"
minus_of_int: fixes x :: "'a"   and n :: "int" shows "cos (x - of_int n * ((2::'a) * of_real pi)) = cos x"
delete_conv': fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "map_of (AList.delete k al) = (map_of al)(k := None)"
right_total_rel_pred: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "right_unique A" shows "right_total (rel_pred A)"
permutation_self_in_orbit: fixes f :: "'c \<Rightarrow> 'c"   and s :: "'c" assumes "permutation f" shows "s \<in> orbit f s"
trancl_Int_mem: fixes S :: "'d"   and T :: "'d"   and R :: "('d \<times> 'd) set"   and A :: "'d set" assumes "(S, T) \<in> (Restr R A)\<^sup>+" shows "(S, T) \<in> Restr (R\<^sup>+) A"
trans_O_subset: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "r O r \<subseteq> r"
more_arith_simps_6: fixes a :: "'i" shows "a * (1::'i) = a"
1: shows "\<forall>d\<in>set Di. (t, s) \<cdot>\<^sub>p \<I> = d \<cdot>\<^sub>p \<I>"
tup_to_sum_app: fixes p :: "('d discr \<rightarrow> 'f) \<times> ('e discr \<rightarrow> 'f)" shows "tup_to_sum\<cdot>p = (\<Lambda> s. case p of (f, g) \<Rightarrow> case undiscr s of Inl x \<Rightarrow> f\<cdot>(Discr x) | Inr x \<Rightarrow> g\<cdot>(Discr x))"
below_meet2: fixes z :: "'a"   and x :: "'a" assumes "z \<sqsubseteq> x" shows "x \<sqinter> z = z"
funion_fminus_cancel2: fixes B :: "'a fset"   and A :: "'a fset" shows "B |-| A |\<union>| A = B |\<union>| A"
finite_identity_rules: fixes L :: "'b Standard_Constant set" assumes "finite L" shows "finite (identity_rules L)"
bi_unique_mat_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique R" shows "bi_unique (mat_rel R)"
lebesgue_measurable_imp_measurable_on: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "f \<in> borel_measurable lebesgue"   and "S \<in> sets lebesgue" shows "f measurable_on S"
IH: shows "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
cobounded2: fixes a :: "'b"   and b :: "'b" shows "order.greater_eq b (inf a b)"
subseq_Cons2_iff: fixes x :: "'d"   and xs :: "'d list"   and y :: "'d"   and ys :: "'d list" shows "subseq (x # xs) (y # ys) = (if x = y then subseq xs ys else subseq (x # xs) ys)"
eventually_sequentially: fixes P :: "nat \<Rightarrow> bool" shows "eventually P sequentially = (\<exists>N. \<forall>n\<ge>N. P n)"
congs_sym: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs \<cong> ys" shows "ys \<cong> xs"
wt_method_def: fixes P :: "'a prog"   and C :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "bytecode"   and xt :: "ex_table"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s \<equiv> 0 < length is \<and> length \<tau>s = length is \<and> check_types P mxs ((case b of Static \<Rightarrow> 0 | NonStatic \<Rightarrow> 1) + length Ts + mxl\<^sub>0) (map OK \<tau>s) \<and> wt_start P C b Ts mxl\<^sub>0 \<tau>s \<and> (\<forall>pc<length is. P,T\<^sub>r,mxs,length is,xt \<turnstile> is ! pc,pc :: \<tau>s)"
ev_iff_sdrop: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "ev P \<omega> = (\<exists>m. P (sdrop m \<omega>))"
queue_invar_simps_1: shows "SkewBinomialHeapStruc.queue_invar []"
ipcidr_union_set_uncurry: fixes ips :: "('b word \<times> nat) set" shows "ipcidr_union_set ips = \<Union> (uncurry ipset_from_cidr ` ips)"
arc_imp_simple_path: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "simple_path g"
upper_split_interval1: fixes X :: "'a interval"   and m :: "'a" shows "upper (fst (split_interval X m)) = max (lower X) m"
calc_x_axis_intersection_positive_abs': fixes A :: "real"   and B :: "real" assumes "A\<^sup>2 < B\<^sup>2"   and "A \<noteq> 0" shows "\<bar>sgn B * sqrt (B\<^sup>2 - A\<^sup>2) / A\<bar> < \<bar>- B / A\<bar>"
da_elim_cases_8: fixes Env :: "env"   and B :: "lname set"   and c1 :: "stmt"   and c2 :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In1r (c1;; c2)\<guillemotright> A" obtains C1 :: "assigned" and C2 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>c1\<rangle>\<guillemotright> C1" and "Env\<turnstile> nrm C1 \<guillemotright>\<langle>c2\<rangle>\<guillemotright> C2" and "nrm A = nrm C2" and "brk A = brk C1 \<Rightarrow>\<inter>  brk C2"
simps_2: fixes h :: "nat \<Rightarrow> nat option"   and x :: "nat"   and a :: "nat"   and as :: "nat list"   and y :: "nat" shows "Path h x (a # as) y = (x \<noteq> 0 \<and> a = x \<and> (\<exists>b. h x = Some b \<and> Path h b as y))"
closed_segment_imp_Re_Im_2: fixes x :: "complex"   and lb :: "complex"   and ub :: "complex" assumes "x \<in> closed_segment lb ub"   and "Im lb \<le> Im ub" shows "Im lb \<le> Im x \<and> Im x \<le> Im ub"
iMODb_add_neg: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and k :: "nat" shows "[ r, mod m, c ] \<oplus>- k = (if k \<le> r then [ r - k, mod m, c ] else if k \<le> r + m * c then [ (m + r mod m - k mod m) mod m, mod m, (r + m * c - k) div m ] else {})"
xX_simp3: shows "backflows (stateful \<union> X) - (backflows (X - {x}) - E) = backflows (stateful \<union> {x})"
Deq2_enabled: shows "\<turnstile> Enabled \<langle>Deq mid q2 out\<rangle>_vars = ($q2 \<noteq> #[])"
simps_1: fixes t :: "'a rose_tree" shows "tree_at t [] = t"
acquired_no_pending_write: fixes xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" shows "acquired False xs A = acquired False xs B"
primitive_remainder_sequence_axioms: shows "primitive_remainder_sequence F n \<delta> f k \<beta>"