hcju/isabelleps huggingface dataset:
antisym_conv2: fixes x :: "'t"   and y :: "'t" assumes "x \<preceq>\<^sub>t y" shows "(\<not> x \<prec>\<^sub>t y) = (x = y)"
cd_not_pd: fixes l :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<pi> l \<noteq> \<pi> k" shows "\<not> \<pi> l pd\<rightarrow> \<pi> k"
pmf_binomial: fixes k :: "nat" shows "pmf local.binomial_pmf k = real (n choose k) * p ^ k * (1 - p) ^ (n - k)"
sec_prik_eq: fixes n :: "nat"   and m :: "nat"   and m' :: "nat" assumes "{Tok_PriK n, Sec_PriK m, Rev_PriK} = {Tok_PriK n, Sec_PriK m', Rev_PriK}" shows "m' = m"
real_cond_exp_int_2: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integral\<^sup>L M (real_cond_exp M Invariants f) = integral\<^sup>L M f"
parts_msg_crypt: fixes K :: "key"   and X :: "msg" shows "parts_msg (Crypt K X) = insert (Crypt K X) (parts_msg X)"
check_prefix_bd_def: fixes xs :: "'a list"   and ys :: "'a list" shows "check_prefix_bd xs ys \<equiv> if length xs \<le> length ys then WHILE\<^sub>T\<^bsup>case_sum (\<lambda>x. True) (\<lambda>s. s \<le> length xs \<and> take s xs = take s ys)\<^esup> (case_sum (\<lambda>x. False) (\<lambda>s. s < length xs)) (\<lambda>s. ASSERT (\<not> isl s) \<bind> (\<lambda>_. let s = projr s in ASSERT (s < length xs \<and> s < length ys) \<bind> (\<lambda>_. if xs ! s = ys ! s then RETURN (Inr (s + 1)) else RETURN (Inl ())))) (Inr 0) \<bind> (\<lambda>x. case x of Inl e \<Rightarrow> RETURN (Inl e) | Inr x \<Rightarrow> RETURN (Inr ())) else RETURN (Inl ())"
nullable_iff: fixes r :: "'a rexp" shows "nullable r = ([] \<in> lang r)"
embed_measure_eq_distr: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" assumes "inj f" shows "embed_measure M f = distr M (embed_measure M f) f"
ord_option_simps_1: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option" shows "ord_option ord None x = True"
simps: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and C :: "'b literal set" shows "validate_ground_clause I C = (\<exists>L. L \<in> C \<and> validate_ground_lit I L)"
FG_proj_in_FG: fixes x :: "('f, 'g) aezfun" shows "FG_proj x \<in> FG"
assms_1: shows "op = \<phi>\<^sub>O \<Psi> op'"
has_derivative_def: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and F :: "'d filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (Lim F (\<lambda>x. x)) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> (0::'e)) F)"
linorder_not_le: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<le> y) = (y < x)"
ih_13: shows "InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
f_inc_mono: fixes f :: "nat \<Rightarrow> nat" assumes "\<forall>x. f x < f (x + 1)" shows "\<forall>x y. x < y \<longrightarrow> f x < f y"
AE_restr_to_subalg: fixes M :: "'b measure"   and F :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "subalgebra M F"   and "almost_everywhere (restr_to_subalg M F) P" shows "almost_everywhere M P"
subcocycle_avg_real_ereal: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "- \<infinity> < subcocycle_avg_ereal u" shows "subcocycle_avg_ereal u = ereal (subcocycle_avg u)"
Anj: shows "A $ n $ j \<noteq> (0::'a) \<and> i \<le> n"
UnI1: fixes c :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "c \<in> A" shows "c \<in> A \<union> B"
valid_3: shows "valid_history program_step ts\<^sub>s\<^sub>b'"
assms: shows "x \<in>\<^sub>\<circ> r `\<^sub>\<bullet> A"
seq'_simps_2: fixes x :: "'a"   and i :: "nat" shows "seq' x (Suc i) = f (seq' x i)"
sup_V_def: fixes x :: "V"   and y :: "V" shows "x \<squnion> y \<equiv> set (elts x \<union> elts y)"
allUses_def: fixes g :: "'a"   and n :: "'node" shows "i.allUses g n \<equiv> uses2 g n \<union> i.phiUses g n"
funcset_image: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "f \<in> A \<rightarrow> B" shows "f ` A \<subseteq> B"
zero_less_Fract_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(0 < Rat.Fract a b) = (0 < a)"
indests: fixes dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and rsn :: "nat" assumes "\<forall>ip\<in>dom dests. ip \<in> kD rt \<and> sqn rt ip \<le> the (dests ip)"   and "dests ip = Some rsn" shows "ip \<in> kD rt \<and> sqn rt ip \<le> rsn"
of_nat_le_iff: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
zero_uint32_def: shows "0 \<equiv> Abs_uint32 0"
ord_pos: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "finite_comm_group G"   and "x \<in> carrier G" shows "0 < group.ord G x"
simps_2: fixes carrier :: "'a set"   and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and zero :: "'a"   and add :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and monom :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and coeff :: "'a \<Rightarrow> nat \<Rightarrow> 'b"   and more :: "'c" shows "coeff \<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, smult = smult, monom = monom, coeff = coeff, \<dots> = more\<rparr> = coeff"
sup_bot_right: fixes x :: "'a" shows "x + (zero_class.zero::'a) = x"
strict_order_iff: fixes a :: "'a"   and b :: "'a" shows "(a < b) = (a = inf_class.inf a b \<and> a \<noteq> b)"
connected_component_in: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "connected_component S x y" shows "x \<in> S \<and> y \<in> S"
subset_antisym: fixes A :: "'b set"   and B :: "'b set" assumes "(A::'b set) \<subseteq> (B::'b set)"   and "(B::'b set) \<subseteq> (A::'b set)" shows "(A::'b set) = (B::'b set)"
filterlim_at_top_int_iff_filterlim_real: fixes f :: "'a \<Rightarrow> int"   and F :: "'a filter" shows "filterlim f at_top F = (LIM x F. real_of_int (f x) :> at_top)"
ir_Skip_sym: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com" assumes "\<And>t t'. Q t' t \<Longrightarrow> \<exists>s'. P s' t \<and> (c', s') \<Rightarrow> t'" shows "ir_hoare P c' SKIP Q"
distinct_2: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and x1 :: "'e" shows "Fun x21 x22 \<noteq> Var x1"
tl_Nil: fixes xs :: "'a list" shows "(tl xs = []) = (xs = [] \<or> (\<exists>x. xs = [x]))"
set_awalk_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" shows "set (pawalk_verts u p) = {u} \<union> set (map fst p) \<union> set (map snd p)"
matches_add_match_MatchNot_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match (MatchNot m) rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[], s\<rangle> \<Rightarrow> t"
pdens: shows "has_density M (stock_measure (PRODUCT INTEG INTEG)) \<delta>"
cases: fixes x :: "acom" obtains "x = SKIP"   | xa :: "char list" and a :: "aexp" where "x = xa ::= a"   | C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = C\<^sub>1__;; C\<^sub>2__"   | b :: "bexp" and C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = IF b__ THEN C\<^sub>1 ELSE C\<^sub>2"   | uu_ :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" and uv_ :: "char list" and b :: "bexp" and C :: "acom" where "x = {uu_/uv_} WHILE b DO C"
fpr_mid_path': fixes P :: "cld list"   and ctx :: "ctx"   and cl :: "cl"   and path' :: "(ctx \<times> cld) list"   and path :: "(ctx \<times> cld) list"   and ctxcld :: "ctx \<times> cld"   and path'' :: "(ctx \<times> cld) list" assumes "find_path_rec_f P ctx cl path' = Some path"   and "ctxcld \<in> set path" shows "ctxcld \<in> set path' \<or> (\<forall>path_fqn. find_path_rec_f P (fst ctxcld) (cl_fqn (fqn_def (class_name_f (snd ctxcld)))) path'' = Some path_fqn \<longrightarrow> (\<forall>path'''. path_fqn = path'' @ path''' \<longrightarrow> (\<exists>path_rest. path = path_rest @ path''')))"
update_simps_6: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "ip \<in> kD rt"   and "\<pi>\<^sub>2 r \<le> sqn rt ip"   and "\<pi>\<^sub>3 r = kno"   and "sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val" shows "update rt ip r = rt(ip \<mapsto> the (rt ip))"
squash_def: fixes x :: "'b" shows "squash x = (if (0::'b) < x then 1 else if x < (0::'b) then - 1 else 0)"
tendsto_componentwise1: fixes b :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>L 'd"   and a :: "'b \<Rightarrow>\<^sub>L 'd"   and F :: "'c filter" assumes "\<And>j. j \<in> Basis \<Longrightarrow> ((\<lambda>n. blinfun_apply (b n) j) \<longlongrightarrow> blinfun_apply a j) F" shows "(b \<longlongrightarrow> a) F"
monom_mult_dist_right: fixes c :: "'d"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" shows "monom_mult c t (p + q) = monom_mult c t p + monom_mult c t q"
card_of_cong: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r =o r'" shows "|Field r| =o |Field r'|"
nuc_fres_closed: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f (f x \<leftarrow> f y) = f x \<leftarrow> f y"
ideD_2: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "\<F>C.dom a = a"
intros_1: fixes \<C> :: "t_context"   and b_es :: "b_e list"   and tf :: "tf"   and \<S> :: "s_context" assumes "\<C> \<turnstile> b_es : tf" shows "\<S>\<bullet>\<C> \<turnstile> $* b_es : tf"
self_append_conv2: fixes ys :: "'b list"   and xs :: "'b list" shows "(ys = xs @ ys) = (xs = [])"
power_tendsto_0_iff: fixes n :: "nat"   and f :: "'b \<Rightarrow> real"   and F :: "'b filter" assumes "0 < n" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> 0) F"
splitAt_split: fixes ram :: "'a"   and vs :: "'a list"   and a :: "'a list"   and b :: "'a list" assumes "ram \<in> set vs"   and "(a, b) = splitAt ram vs" shows "vs = a @ ram # b"
succ_in_omega: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>" shows "ZFC_in_HOL.succ n \<in>\<^sub>\<circ> \<omega>"
AssertAllState_axioms: shows "AssertAllState (AssertAllState''.assert_all_state init_state assert_bound_code check_code)"
less_restrictive_correct: fixes R1 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and R2 :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" assumes "less_restrictive R1 R2"   and "Sound R1" shows "Sound R2"
lexordp_append_leftI: fixes us :: "'a list"   and vs :: "'a list"   and xs :: "'a list" assumes "lexordp us vs" shows "lexordp (xs @ us) (xs @ vs)"
ZObisL_length: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list" assumes "ZObisL cl dl" shows "length cl = length dl"
clock_numbering_1: shows "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"
moebius_mu_1: shows "moebius_mu 1 = (1::'b)"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "CCC.arr f = (arr (fst f) \<and> CC.arr (snd f))"
sp_mono: fixes z :: "real"   and x :: "real"   and y :: "real" assumes "z \<in> {0..1}"   and "x \<le> y" shows "sp z x \<le> sp z y"
inA_is_sorted: fixes xs :: "nat list" assumes "xs \<in> A" shows "Sorted_Less.sorted (rev xs)"
it': shows "map_iterator_genord iti (map_of lc) (\<lambda>x y. R (case x of (k, v) \<Rightarrow> (k, \<alpha> v)) (case y of (k, v) \<Rightarrow> (k, \<alpha> v)))"
Inr_Inl_False: fixes x :: "'h"   and y :: "'g" shows "(Inr x = Inl y) = False"
span_superset: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "Modules.module scale" shows "S \<subseteq> module.span scale S"
distinct_disc_6: fixes event :: "('a, 'b, 'c) event" assumes "isSnapshot event" shows "\<not> isTrans event"
mult_imp_msetext_dersh_rel: fixes ys :: "'b list"   and A :: "'b set"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_mset (mset ys) \<subseteq> A"   and "set_mset (mset xs) \<subseteq> A"   and "(mset xs, mset ys) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> gt y x}"   and "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x" shows "msetext_dersh gt ys xs"
partn_lst_def: fixes \<beta> :: "'b set"   and \<alpha> :: "nat list"   and \<gamma> :: "nat" shows "partn_lst \<beta> \<alpha> \<gamma> \<equiv> \<forall>f\<in>[\<beta>]\<^bsup>\<gamma>\<^esup> \<rightarrow> {..<length \<alpha>}. \<exists>i<length \<alpha>. \<exists>H\<in>[\<beta>]\<^bsup>(\<alpha> ! i)\<^esup>. f ` [H]\<^bsup>\<gamma>\<^esup> \<subseteq> {i}"
construction_of_all_possible_states_lemma: fixes v :: "'a"   and X :: "'a set" assumes "v \<notin> X" shows "{s. fmdom' s = insert v X} = fmupd v True ` {s. fmdom' s = X} \<union> fmupd v False ` {s. fmdom' s = X}"
intensional_simps_22: fixes P :: "'b \<Rightarrow> bool" shows "\<not> P \<and> P \<equiv> #False"
assms_2: shows "(oid, op1) \<in> set op_list"
grdOf_def: fixes c :: "'a"   and l :: "'b"   and x :: "Preliminaries.var" shows "grdOf c l x = grdLit x"
tendsto_arsinh: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. arsinh (f x)) \<longlongrightarrow> arsinh a) F"
no_leading_def: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "no_leading P xs = (xs \<noteq> [] \<longrightarrow> \<not> P (hd xs))"
Mp_smult_1: fixes m :: "int"   and a :: "int"   and f :: "int poly" shows "poly_mod.Mp m (Polynomial.smult (poly_mod.M m a) f) = poly_mod.Mp m (Polynomial.smult a f)"
select_convs_1: fixes pverts :: "'b set"   and parcs :: "('b \<times> 'b) set"   and more :: "'c" shows "pverts \<lparr>pverts = pverts, parcs = parcs, \<dots> = more\<rparr> = pverts"
that: fixes ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> defNode g v-ns\<rightarrow>m"   and "var g v \<in> oldUses g m"   and "\<And>x. x \<in> set (tl ns) \<Longrightarrow> var g v \<notin> oldDefs g x" shows "thesis"
not_nonpos_Int: fixes t :: "complex" assumes "0 < Re t" shows "t \<notin> \<int>\<^sub>\<le>\<^sub>0"
get_exists_matching_src_ips_subset: fixes m :: "'a common_primitive match_expr"   and a :: "action"   and iface :: "iface" assumes "normalized_nnf_match m" shows "{ip. \<exists>p. matches (common_matcher, in_doubt_allow) m a (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>)} \<subseteq> get_exists_matching_src_ips iface m"
assms_3: shows "\<forall>X\<in>A // R. finite X"
U_consts_1: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n true\<^sub>n"
pdevs_val_partition: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and xs :: "'b list"   and p :: "'b \<Rightarrow> bool" assumes "e \<in> UNIV \<rightarrow> I" obtains f :: "nat \<Rightarrow> real" and g :: "nat \<Rightarrow> real" where "pdevs_val e (pdevs_of_list xs) = pdevs_val f (pdevs_of_list (filter p xs)) + pdevs_val g (pdevs_of_list (filter (Not \<circ> p) xs))" and "f \<in> UNIV \<rightarrow> I" and "g \<in> UNIV \<rightarrow> I"
comp_permute: fixes f :: "'a"   and g :: "'a"   and k :: "'a"   and l :: "'a"   and h :: "'a" assumes "f \<cdot> g = k \<cdot> l"   and "C.seq f g"   and "C.seq g h" shows "f \<cdot> g \<cdot> h = k \<cdot> l \<cdot> h"
stat_ref_comp_imp_red_stat_ref_comp: assumes "statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F" shows "reducedly_statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
F4: shows "\<forall>x\<^sub>2 x\<^sub>3. \<bar>x\<^sub>3\<bar> * \<bar>x\<^sub>2\<bar> = \<bar>x\<^sub>3 * x\<^sub>2\<bar>"
inv_f_f: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "inj f" shows "inv f (f x) = x"
nat_diff_split_asm: fixes P :: "nat \<Rightarrow> bool"   and a :: "nat"   and b :: "nat" shows "P (a - b) = (\<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d)))"
degree_m_mult_eq: fixes p :: "int poly"   and q :: "int poly" assumes "monic p"   and "lead_coeff q mod m \<noteq> 0"   and "1 < m" shows "degree_m (p * q) = degree p + degree q"
hyps_2: shows "\<exists>m>0. \<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
deformation_retract: fixes S :: "'a set"   and T :: "'a set" shows "(\<exists>r. homotopic_with_canon (\<lambda>x. True) S S id r \<and> retraction S T r) = (T retract_of S \<and> (\<exists>f. homotopic_with_canon (\<lambda>x. True) S S id f \<and> f ` S \<subseteq> T))"
hcomplex_of_hypreal_approx_iff: fixes x :: "real star"   and z :: "real star" shows "(hcomplex_of_hypreal x \<approx> hcomplex_of_hypreal z) = (x \<approx> z)"
prj0_simps_1: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.arr \<p>\<^sub>0[f, g]"
abs_le_iff: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) b = (less_eq a b \<and> less_eq (uminus a) b)"
taylor_expansion_def: fixes R :: "('e, 'f) ring_scheme"   and a :: "'e"   and p :: "nat \<Rightarrow> 'e" shows "taylor_expansion R a p = Cring_Poly.compose R p (X_poly_plus R a)"
diagonalize_is_strict_monoidal_functor: shows "strict_monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<F>C.D"
sum_ideals_la1: fixes I1 :: "'a set"   and I2 :: "'a set" assumes "ideal R I1"   and "ideal R I2" shows "I1 \<subseteq> I1 \<minusplus> I2"
0: shows "map (I.int (kE \<xi>)) Tl = map2 fromFull (arOf f) (map (F.int \<xi>) Tl)"
lex_triple_def: shows "lex_triple \<equiv> less_than <*lex*> lex_pair"
not_different_AKcryptSK: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and authK' :: "nat" assumes "AKcryptSK authK servK evs"   and "authK' \<noteq> authK"   and "evs \<in> kerbIV_gets" shows "\<not> AKcryptSK authK' servK evs \<and> servK \<in> symKeys"
normalize_rules_dnf_correct: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state" assumes "wf_ruleset \<gamma> p rs" shows "approximating_bigstep_fun \<gamma> p (normalize_rules_dnf rs) s = approximating_bigstep_fun \<gamma> p rs s"
Sigma_empty_iff: fixes I :: "'a set"   and X :: "'a \<Rightarrow> 'b set" shows "(Sigma I X = {}) = (\<forall>i\<in>I. X i = {})"
circ_sup_9: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = (x\<^sup>\<star> * y)\<^sup>\<star> * x\<^sup>\<star>"
refine: fixes ai :: "'b list"   and a :: "'c set"   and sctni :: "real list sctn"   and sctn :: "'c sctn" assumes "(ai, a) \<in> appr_rel"   and "(sctni, sctn) \<in> \<langle>lv_rel\<rangle>sctn_rel" shows "(RETURN (intersects_impl ai sctni), op_intersects $ a $ sctn) \<in> \<langle>bool_rel\<rangle>nres_rel"
prefix_match_to_CIDR_def2: shows "prefix_match_to_CIDR = (\<lambda>pfx. (pfxm_prefix pfx, pfxm_length pfx))"
xn_def: shows "xn = x + real_of_int n * (2 * pi)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>uu_. P [uu_]"   and "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)" shows "P a0"
simps: fixes a1 :: "'a list"   and a2 :: "'a list" shows "lexordp a1 a2 = ((\<exists>y ys. a1 = [] \<and> a2 = y # ys) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> x < y) \<or> (\<exists>x y xs ys. a1 = x # xs \<and> a2 = y # ys \<and> \<not> x < y \<and> \<not> y < x \<and> lexordp xs ys))"
presSubstAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presSubstAll h hA MOD \<equiv> presSubst h MOD \<and> presSubstAbs h hA MOD"
num_aform_def: fixes f :: "'a" shows "num_aform f = (f, zero_pdevs)"
simps_30: fixes v :: "'b rexp"   and va :: "'b rexp"   and vk :: "'b rexp"   and vl :: "'b rexp" shows "le_rexp (Times v va) (Plus vk vl) = False"
get_writer_def: fixes get :: "('g \<Rightarrow> 'f) \<Rightarrow> 'f"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) writerT" shows "get_writer get f = WriterT (get (\<lambda>s. run_writer (f s)))"
ProofStepInduct_NonTrivial_axioms: shows "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size"
1594: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> (y \<squnion> z)) \<sqinter> (x \<sqinter> z) = bot"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<sqsubseteq> b) = (a = a \<sqinter> b)"
same_prefix_prefix: fixes xs :: "[:'a:]"   and ys :: "[:'a:]"   and zs :: "[:'a:]" assumes "xs \<noteq> \<bottom>" shows "prefix\<cdot>(xs :@ ys)\<cdot>(xs :@ zs) = prefix\<cdot>ys\<cdot>zs"
pathint_def: fixes x :: "complex"   and y :: "complex" shows "pathint x y \<equiv> contour_integral (linepath x y) f"
semantics_fresh: fixes i :: "'b"   and p :: "('c, 'b) fm"   and M :: "('d, 'c) model"   and g :: "'b \<Rightarrow> 'd"   and w :: "'d"   and v :: "'d" assumes "i \<notin> nominals p" shows "(M, g, w \<Turnstile> p) = (M, g(i := v), w \<Turnstile> p)"
"1_3": shows "(ptr, child) \<in> (parent_child_rel h \<union> local.a_host_shadow_root_rel h)\<^sup>*"
avl_insert: fixes t :: "'b tree"   and x :: "'b" assumes "avl t" shows "avl (AVL.insert x t)"
Lower_is_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "is_closed (Lower L A)"
SAT_def: fixes \<Phi> :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list set" shows "GE.SAT \<Phi> \<equiv> \<forall>\<xi>. GE.wtE \<xi> \<longrightarrow> GE.satPB \<xi> \<Phi>"
idem: fixes a :: "'c DBMEntry" shows "linordered_monoid.max a a = a"
powr_gt_zero: fixes x :: "real"   and a :: "real" shows "(0 < x powr a) = (x \<noteq> 0)"
that: fixes as' :: "'edge list"   and pex :: "'node" assumes "n -as'\<rightarrow>\<^sub>\<iota>* pex"   and "method_exit pex"   and "set (sourcenodes as') \<subseteq> set (sourcenodes as)" shows "thesis"
Formal_in_SDG_Def: fixes n :: "'node SDG_node"   and m :: "'node"   and x :: "nat"   and p :: "'pname"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var" assumes "valid_SDG_node n"   and "n = Formal_in (m, x)"   and "get_proc m = p"   and "(p, ins, outs) \<in> set procs"   and "V = ins ! x" shows "V \<in> Def\<^bsub>SDG\<^esub> n"
real_of_u_inj: fixes x :: "mini_alg_unique"   and y :: "mini_alg_unique" shows "(real_of_u x = real_of_u y) = (x = y)"
Obj: shows "X \<in> obj\<^bsub>CatDom (YFtor C @@ X)\<^esub>"
P: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'a measure"   and M :: "nat \<Rightarrow> 'a measure"   and i :: "nat" assumes "Ionescu_Tulcea P M" shows "P i \<in> Pi\<^sub>M {0..<i} M \<rightarrow>\<^sub>M subprob_algebra (M i)"
hcomp_def: fixes G :: "hf"   and F :: "hf" shows "hcomp G F = \<lbrace>XZ \<^bold>\<in> hdomain F * hrange G. hsnd XZ = happ G (happ F (hfst XZ))\<rbrace>"