hcju/isabelleps huggingface dataset:
assms: shows "\<Gamma> \<turnstile> (\<lambda><:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet>\<^sub>\<tau> T\<^sub>2 : T"
op_nres_ASSERT_bnd_def: fixes \<Phi> :: "bool"   and m :: "'a nres" shows "op_nres_ASSERT_bnd \<Phi> m \<equiv> ASSERT \<Phi> \<bind> (\<lambda>_. m)"
has_field_derivative_imp_has_derivative: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and F :: "'c filter" assumes "(f has_field_derivative D) F" shows "(f has_derivative (*) D) F"
rep_eq: fixes x :: "plane_vec"   and xa :: "plane_vec" shows "plane_vec_eq x xa = plane_vec_eq_r4 (Rep_plane_vec x) (Rep_plane_vec xa)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext_trans ext"   and "ext_trans_before_irrefl_axioms ext" shows "ext_trans_before_irrefl ext"
cases: fixes x :: "('c \<times> 'd) tree" obtains "x = \<langle>\<rangle>"   | l :: "('c \<times> 'd) tree" and a :: "'c" and uu_ :: "'d" and r :: "('c \<times> 'd) tree" where "x = \<langle>l, (a, uu_), r\<rangle>"
inf_mono_right: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and c :: "'b::type"   and d :: "'b::type"   and a :: "'b::type" assumes "class.refinement_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (c::'b::type) (d::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (c::'b::type)) (inf a (d::'b::type))"
Rep_inject: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and x :: "'c"   and y :: "'c" assumes "type_definition Rep Abs A" shows "(Rep x = Rep y) = (x = y)"
rev_path_rev_path: fixes p :: "('b \<times> 'b) list" shows "rev_path (rev_path p) = p"
MapsToObj_1: fixes f :: "'b"   and X :: "'a"   and Y :: "'a" assumes "f maps X to Y" shows "X \<in> obj"
cong_exists: fixes P :: "conc_interface"   and Q :: "conc_interface"   and x :: "char list" assumes "P \<simeq> Q" shows "Exists_int_conc x P \<simeq> Exists_int_conc x Q"
a3: fixes hbc :: "nat \<Rightarrow> nat"   and dbc :: "nat \<Rightarrow> nat"   and hmining :: "nat \<Rightarrow> bool"   and dmining :: "nat \<Rightarrow> bool"   and sbc :: "nat"   and cnf :: "nat" assumes "bounded_growth hbc dbc hmining dmining sbc cnf" shows "dbc 0 < sbc"
remdups_nil_noteq_cons_2: fixes h :: "'b"   and t :: "'b list" shows "remdups [] \<noteq> remdups (h # t)"
induct: fixes P :: "('a, 'b) rbt \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> ('a, 'b) rbt \<Rightarrow> bool"   and a0 :: "('a, 'b) rbt"   and a1 :: "'a"   and a2 :: "'b"   and a3 :: "('a, 'b) rbt" assumes "\<And>a k x b s y c. P (Branch R a k x b) s y c"   and "\<And>k x a s y b. P Empty k x (Branch B a s y b)"   and "\<And>va vb vc vd k x a s y b. P (Branch B va vb vc vd) k x (Branch B a s y b)"   and "\<And>k x a s y b t z c. P Empty k x (Branch R (Branch B a s y b) t z c)"   and "\<And>va vb vc vd k x a s y b t z c. P (Branch B va vb vc vd) k x (Branch R (Branch B a s y b) t z c)"   and "\<And>k x. P Empty k x Empty"   and "\<And>k x vb vc vd. P Empty k x (Branch R Empty vb vc vd)"   and "\<And>k x ve vf vg vh vb vc vd. P Empty k x (Branch R (Branch R ve vf vg vh) vb vc vd)"   and "\<And>va vb vc vd k x. P (Branch B va vb vc vd) k x Empty"   and "\<And>va vb vc vd k x vf vg vh. P (Branch B va vb vc vd) k x (Branch R Empty vf vg vh)"   and "\<And>va vb vc vd k x vi vj vk vl vf vg vh. P (Branch B va vb vc vd) k x (Branch R (Branch R vi vj vk vl) vf vg vh)" shows "P a0 a1 a2 a3"
simps_1: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "shd (siterate f x) = x"
par_comp_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and \<A> :: "('d strand_label \<times> ('a, 'b) strand_step) list"   and Secrets :: "('a, 'b) Term.term set" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2" shows "labeled_typed_model.par_comp arity public Ana \<Gamma> \<A> Secrets \<equiv> (\<forall>l1 l2. l1 \<noteq> l2 \<longrightarrow> labeled_typed_model.GSMP_disjoint arity public Ana \<Gamma> (trms\<^sub>l\<^sub>s\<^sub>t (proj l1 \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l2 \<A>)) Secrets) \<and> (\<forall>s\<in>Secrets. \<forall>s'\<in>subterms s. intruder_model.intruder_synth arity public {} s' \<or> s' \<in> Secrets) \<and> ground Secrets"
\<S>_reps_S: fixes s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "s \<in> \<S>" shows "reps s \<in> S"
that: fixes y' :: "'a" assumes "foldr_option f xs e = Some y'"   and "f x y' = Some y" shows "thesis"
lm061: fixes bids :: "'b \<times> 'c set \<Rightarrow> 'a"   and a :: "('b \<times> 'c set) set" shows "sum snd (summedBid bids ` a) = sum (snd \<circ> summedBid bids) a"
asm2: shows "(n, U) \<in> set ((m, T) # \<Gamma>)"
inj_onI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. \<lbrakk>s2p A x; s2p A y; f x = f y\<rbrakk> \<Longrightarrow> x = y" shows "inj_on f A"
poly_add_sign_def: fixes m :: "nat"   and n :: "nat" shows "poly_add_sign m n = signof (\<lambda>i. if i < n then m + i else if i < m + n then i - n else i)"
dom_cod: fixes f :: "'b" assumes "C'.arr f" shows "C'.dom (C'.cod f) = C'.cod f"
Cons: shows "map_of (filter (\<lambda>a. case a of (x', y) \<Rightarrow> P x') ys) x = map_of ys x"
mtf2_q_after: fixes A :: "'c list"   and q :: "'c"   and n :: "nat" assumes "distinct A"   and "q \<in> set A" shows "index (mtf2 n q A) q = index A q - n"
parabolic_imp_not_elliptic_loxodromic_1: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "\<not> elliptic_isometry f"
pdivmod_via_pseudo_divmod: fixes f :: "'b poly"   and g :: "'b poly" shows "pdivmod f g = (if g = 0 then (0, f) else let ilc = inverse (lead_coeff g); h = Polynomial.smult ilc g; (q, r) = pseudo_divmod f h in (Polynomial.smult ilc q, r))"
gcd_fib_diff: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "gcd (fib m) (fib (n - m)) = gcd (fib m) (fib n)"
index_by_eq_SomeD: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'a list"   and n :: "'b"   and x :: "'a" assumes "index_by f l n = Some x" shows "x \<in> set l \<and> f x = n"
lift_SD_iff_1: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and agents' :: "'a set"   and alts' :: "'b set"   and p :: "'b pmf"   and q :: "'b pmf"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "agents \<subseteq> agents'"   and "alts \<subseteq> alts'"   and "alts \<noteq> {}"   and "agents \<noteq> {}"   and "finite alts'"   and "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts"   and "i \<in> agents" shows "SD (lift_pref_profile agents alts agents' alts' R i) p q = SD (R i) p q"
tendsto_eq_rhs: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and F :: "'b filter"   and y :: "'a" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open x) F"   and "x = y" shows "filterlim f (topological_space.nhds open y) F"
Exit_no_postdominator: fixes n :: "'node" assumes "(_Exit_) postdominates n" shows "False"
subset_ffilter: fixes P :: "'b \<Rightarrow> bool"   and A :: "'b fset"   and Q :: "'b \<Rightarrow> bool" shows "(ffilter P A |\<subseteq>| ffilter Q A) = (\<forall>x. x |\<in>| A \<longrightarrow> P x \<longrightarrow> Q x)"
finite_has_maximal: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq m b \<longrightarrow> m = b"
strict_sorted_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b list" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less l = (linorder.sorted less_eq l \<and> distinct l)"
assms_2: fixes \<pi> :: "perm" assumes "supp (- \<pi>) \<sharp>* fv (Let as body)"   and "supp \<pi> \<sharp>* Let as body" shows "F (\<pi> \<bullet> as) (\<pi> \<bullet> body) (Let as body) = F as body (Let as body)"
generate_subgroup_id: fixes H :: "int set" assumes "subgroup H (Z 1)" shows "generate (Z 1) H = H"
eq1: fixes c :: "real"   and x :: "'a" assumes "a + c *\<^sub>R x \<in> S"   and "0 \<le> c"   and "a + x \<in> S" shows "c = 1"
FinalAllows_subseteq_in_doubt_deny: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs" shows "_Collect p ((\<beta>, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow) \<subseteq> _Collect p (\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow)"
line_eq_iff: fixes a :: "'b"   and b :: "'b"   and x :: "real"   and y :: "real" shows "({a -- b}\<^bsub>x\<^esub> = {a -- b}\<^bsub>y\<^esub>) = (x = y \<or> a = b)"
span_li_not_depend_2: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N V" shows "LinearCombinations.module.lin_dep K (vs N) S = lin_dep S"
composite_pseudofunctor_axioms: shows "composite_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C G \<Phi>\<^sub>G F \<Phi>\<^sub>F"
comp_ide_self: fixes a :: "'a" assumes "B.ide a" shows "a \<cdot> a = a"
learn_consE_2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option"   and f :: "nat \<Rightarrow> nat option" assumes "learn_cons \<psi> U s"   and "f \<in> U" shows "\<exists>i n\<^sub>0. \<psi> i = f \<and> (\<forall>n\<ge>n\<^sub>0. s (f \<triangleright> n) \<down>= i)"
hyps: fixes t' :: "('p, 's, 'm) event list" assumes "trace c t c'"   and "t = t' @ t''" shows "\<exists>c''. trace c t' c'' \<and> trace c'' t'' c'"
size_def: fixes m :: "('c, 'd) mapping" shows "Mapping.size m = (if finite (keys m) then card (keys m) else 0)"
sints32: shows "sints 32 = {i. - (2 ^ 31) \<le> i \<and> i < 2 ^ 31}"
WHILEIT_refine_new_invar: fixes I' :: "'c \<Rightarrow> bool"   and x' :: "'c"   and x :: "'d"   and R :: "('d \<times> 'c) set"   and I :: "'d \<Rightarrow> bool"   and b :: "'d \<Rightarrow> bool"   and b' :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'c nres" assumes "I' x' \<Longrightarrow> (x, x') \<in> R"   and "\<lbrakk>I' x'; (x, x') \<in> R\<rbrakk> \<Longrightarrow> I x"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; I x; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R (f' x')"   and "\<And>x x'. \<lbrakk>nofail (f x); (x, x') \<in> R; b x; b' x'; I x; I' x'\<rbrakk> \<Longrightarrow> f x \<le> SPEC I" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f x \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I'\<^esup> b' f' x')"
gba_rename_ecnv_def: fixes ecnv :: "('d, 'b, 'e) gba_rec_scheme \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'a"   and G :: "('d, 'b, 'e) gba_rec_scheme" shows "gba_rename_ecnv ecnv f G \<equiv> \<lparr>gba_L = \<lambda>q l. if q \<in> f ` g_V G then gba_L G (the_inv_into (g_V G) f q) l else False, \<dots> = ecnv G\<rparr>"
Y: fixes y :: "'b" assumes "y \<in> topspace Y" shows "g y \<in> topspace X \<and> f (g y) = y"
map_fst_map_const_snd: fixes b :: "'c"   and xs :: "'b list" shows "map fst (map (\<lambda>s. (s, b)) xs) = xs"
g1_path_continuous: shows "continuous_on (cbox a b) (\<lambda>x. (x, g1 x))"
inf_idem: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.semilattice_inf inf less_eq less" shows "inf x x = x"
sym_gfp: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "mono Retr"   and "\<And>theta. sym theta \<Longrightarrow> sym (Retr theta)" shows "sym (gfp Retr)"
foundation15: fixes \<tau> :: "'a state \<times> 'a state"   and A :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> A \<triangleq> invalid) = (\<tau> \<Turnstile> not (\<upsilon> A))"
f'_def: shows "f' = (\<lambda>x. x powr (- c - 1) * (1 - c * ln x))"
eval: shows "\<langle>c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>"
i_set_mult_subset_i_set0_mult: fixes k :: "nat" shows "i_set_mult k \<subseteq> i_set0_mult k"
AtomicP_Eq: fixes x :: "tm"   and y :: "tm" shows "{TermP x, TermP y} \<turnstile> AtomicP (Q_Eq x y)"
ltakeWhile_all: fixes xs :: "'c llist"   and P :: "'c \<Rightarrow> bool" assumes "\<forall>x\<in>lset xs. P x" shows "ltakeWhile P xs = xs"
ptr_kinds_M_pure: fixes h :: "(_) heap" shows "pure document_ptr_kinds_M h"
DERIV_caratheodory_within: fixes f :: "'a \<Rightarrow> 'a"   and l :: "'a"   and x :: "'a"   and S :: "'a set" shows "(f has_field_derivative l) (at x within S) = (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> continuous (at x within S) g \<and> g x = l)"
IH1_3: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M1" shows "subterms f \<subseteq> Mapping.keys M1"
boundedE: fixes b :: "'a \<times> 'b"   and c :: "'a \<times> 'b"   and a :: "'a \<times> 'b" assumes "less_eq_prod (sup_prod b c) a" obtains "less_eq_prod b a" and "less_eq_prod c a"
swap_events: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list" assumes "i < j"   and "j < length t"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "postrecording_event t i"   and "prerecording_event t j"   and "trace init t final" shows "trace init (swap_events i j t) final \<and> (\<forall>k\<ge>j + 1. S (swap_events i j t) k = S t k) \<and> (\<forall>k\<le>i. S (swap_events i j t) k = S t k) \<and> prerecording_event (swap_events i j t) i \<and> postrecording_event (swap_events i j t) (i + 1) \<and> (\<forall>k. i + 1 < k \<and> k < j + 1 \<longrightarrow> \<not> regular_event (swap_events i j t ! k))"
eucl_rel_poly_0_iff: fixes y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "eucl_rel_poly 0 y (q, r) = (q = 0 \<and> r = 0)"
beukers_nn_integral1_altdef: fixes r :: "nat"   and s :: "nat" shows "beukers_nn_integral1 r s = \<integral>\<^sup>+x\<in>{0<..<1} \<times> {0<..<1} \<times> {0<..<1}. (case x of (w, xa, y) \<Rightarrow> ennreal (1 / (1 - (1 - xa * y) * w) * xa ^ r * y ^ s))\<partial>lborel"
LocalSystem_def: fixes init :: "(nat \<Rightarrow> 'a) set"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and q :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and x :: "nat \<Rightarrow> 'b" shows "LocalSystem init p r q x = (\<not> fail_sys init p r x \<and> (\<forall>u y. u \<in> init \<and> run r u x y \<longrightarrow> q y))"
suminf_eq_SUP_real: fixes X :: "nat \<Rightarrow> real" assumes "summable X"   and "\<And>i. 0 \<le> X i" shows "suminf X = (\<Squnion>i. sum X {..<i})"
set_intros_2: fixes y :: "'b"   and x22 :: "'b llist"   and x21 :: "'b" assumes "y \<in> lset x22" shows "y \<in> lset (LCons x21 x22)"
arity_var_ne_infinity_if_\<delta>_gt_0: fixes x :: "'v" assumes "0 < \<delta>" shows "arity_var x \<noteq> \<infinity>"
Aj_of_def: fixes A :: "'a" shows "Aj_of A = image_mset atm_of {#L \<in># CA_of A. L = Pos A#}"
fg: fixes x :: "'n" assumes "x \<in> S" shows "(\<lambda>k. f k x) \<longlonglongrightarrow> g x"
scaleR_add_right: fixes a :: "real"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>R (x + y) = a *\<^sub>R x + a *\<^sub>R y"
outputBoundTrans: fixes a :: "name"   and b :: "name"   and P :: "pi"   and c :: "subject"   and x :: "name"   and P' :: "pi" assumes "a{b}.P \<longmapsto> c\<guillemotleft>x\<guillemotright> \<prec> P'" shows "False"
subprob_measurableD_2: fixes N :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and S :: "'d measure"   and x :: "'c" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra S"   and "x \<in> space M" shows "sets (N x) = sets S"
indoc_appendI1: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "traces.indoc ind a u" shows "traces.indoc ind a (u @ v)"
edom_AnalBinds: fixes exp :: "exp \<Rightarrow> 'b \<rightarrow> 'a"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" shows "edom (ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae) \<subseteq> domA \<Gamma> \<inter> edom ae"
r_tuple_nth_prim: shows "prim_recfn 3 r_tuple_nth"
bound1b: shows "real_of_int (sint NEG_INF) < real_of_int (sint w1)"
assms: shows "z + (1::'a) \<notin> \<int>\<^sub>\<le>\<^sub>0"
count_image_mset': fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d multiset"   and x :: "'c" shows "count (image_mset f A) x = sum (count A) {x'. x' \<in># A \<and> f x' = x}"
lfilter_llist_of: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "lfilter P (llist_of xs) = llist_of (filter P xs)"
ie3: shows "\<gamma>1 \<delta>S \<le> \<delta>S"
hyps: assumes "poly_inv p" shows "poly_inv (poly_mult p q)"
hnorm_scaleR: fixes a :: "real"   and x :: "'a star" shows "hnorm (a *\<^sub>R x) = \<bar>hypreal_of_real a\<bar> * hnorm x"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post"   and "ProofStepInduct_y_neq_new_last_axioms v1 P_new y" shows "ProofStepInduct_y_neq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
decr_grading_p_monomial: fixes d :: "'a \<Rightarrow> nat"   and n :: "nat"   and v :: "'t"   and c :: "'c" shows "decr_grading_p d n (monomial c v) = monomial c (decr_grading_term d n v)"
df_2: fixes b :: "bool" shows "degree (Polynomial.smult (\<beta> i) (F i)) < degree (F (i - 1)) \<or> b"
continuous_map_fst_of: fixes Z :: "'b topology"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'b \<Rightarrow> 'c \<times> 'd" assumes "continuous_map Z (prod_topology X Y) f" shows "continuous_map Z X (fst \<circ> f)"
sorted_list_of_set_nonempty: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "sorted_list_of_set A = Min A # sorted_list_of_set (A - {Min A})"
generate_pow_card: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.group G"   and "a \<in> carrier G" shows "group.ord G a = card (generate G {a})"
bla: shows "per_union R x y `` {y} = per_union R x y `` {x}"
antisym: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a"   and "a \<le> b" shows "a = b"
map_add_def: fixes m1 :: "'e \<Rightarrow> 'f option"   and m2 :: "'e \<Rightarrow> 'f option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>y\<rfloor>)"
zero_le_iff_zero_nprt: 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 inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less_eq zero a = (lattice_ab_group_add.nprt zero inf a = zero)"
inf_sup_ord_3: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
times_def: fixes x :: "'a"   and y :: "'a" shows "x * y = x \<sqinter> y"
part12_pred_def: fixes V :: "'a set"   and V1_V2 :: "'a set \<times> 'a set" shows "part12_pred V V1_V2 \<equiv> V = fst V1_V2 \<union> snd V1_V2 \<and> fst V1_V2 \<inter> snd V1_V2 = {} \<and> infinite (fst V1_V2) \<and> infinite (snd V1_V2)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) local.map (T \<circ> \<Delta>.map) \<delta>"
tdghm_eqI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and \<AA>' :: "V"   and \<BB>' :: "V"   and \<FF>' :: "V"   and \<GG>' :: "V"   and \<NN>' :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<NN>' : \<FF>' \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>' : \<AA>' \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>'"   and "\<NN>\<lparr>NTMap\<rparr> = \<NN>'\<lparr>NTMap\<rparr>"   and "\<FF> = \<FF>'"   and "\<GG> = \<GG>'"   and "\<AA> = \<AA>'"   and "\<BB> = \<BB>'" shows "\<NN> = \<NN>'"
ideD_1: fixes a :: "'a \<times> 'a" assumes "CC.ide a" shows "CC.arr a"
maxl_funTr: fixes x :: "'a" assumes "x \<in> carrier D"   and "\<exists>y. y \<in> upper_bounds D {x} \<and> y \<noteq> x" shows "(SOME z. z \<in> upper_bounds D {x} \<and> z \<noteq> x) \<in> carrier D"
Borsukian_continuous_logarithm_circle_real: fixes S :: "'a set" shows "Borsukian S = (\<forall>f. continuous_on S f \<and> f ` S \<subseteq> sphere 0 1 \<longrightarrow> (\<exists>g. continuous_on S (complex_of_real \<circ> g) \<and> (\<forall>x\<in>S. f x = exp (\<i> * complex_of_real (g x)))))"
FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_def: fixes n :: "nat" shows "FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m n = P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>F n,Up\<^sub>F\<rbrakk>"
le_supE: fixes a :: "'b"   and b :: "'b"   and x :: "'b" assumes "a \<sqinter> b \<le> x" obtains "a \<le> x" and "b \<le> x"
norm_le_componentwise_cart: fixes x :: "('c, 'b) Finite_Cartesian_Product.vec"   and y :: "('d, 'b) Finite_Cartesian_Product.vec" assumes "\<And>i. norm (x $h i) \<le> norm (y $h i)" shows "norm x \<le> norm y"
Iod_self: fixes D :: "('c, 'd) Order_scheme" assumes "Order D" shows "D = Iod D (carrier D)"
inext_less_mono: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I"   and "\<exists>x\<in>I. b < x" shows "inext a I < inext b I"
intros_3: fixes t :: "('b, 'c, 'd) Tree"   and \<alpha> :: "'d" shows "(t, tAct \<alpha> t) \<in> Tree_wf"
simps_1: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" shows "cond_action_oks' s t [] = True"
hyps: fixes s :: "nat"   and r :: "nat \<Rightarrow>f value option" assumes "accepts_trace e s r (t @ t')" shows "accepts_trace e s r t"
t: shows "\<guillemotleft>t : local.map a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] local.map a'\<guillemotright>"