hcju/isabelleps huggingface dataset:
asymp_equiv_plus_const_right: fixes c :: "real" shows "(\<lambda>n. real n + c) \<sim>[sequentially] real"
T_Mprefix: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b process" shows "T (Mprefix A P) = {s. s = [] \<or> (\<exists>a. a \<in> A \<and> s \<noteq> [] \<and> hd s = ev a \<and> tl s \<in> T (P a))}"
greater_bounded_def: fixes N :: "nat" shows "greater_bounded N \<equiv> inv_image less_than ((-) N)"
finx: shows "finite {i \<in> I. x i \<noteq> \<one>\<^bsub>A i\<^esub>}"
cod_eqI': fixes a :: "'d"   and f :: "'d" assumes "a \<in> D.codomains f" shows "a = D.cod f"
codomains_def: fixes f :: "'a1" shows "C1.codomains f \<equiv> {b. C1.ide b \<and> b \<cdot>\<^sub>1 f \<noteq> C1.null}"
qf3_cube_primelist_impl_cube_form: fixes ps :: "nat multiset"   and a :: "int"   and b :: "int" assumes "Multiset.Ball ps prime"   and "odd (int (\<Prod>i\<in>#ps. i))"   and "coprime a b"   and "a\<^sup>2 + 3 * b\<^sup>2 = int (\<Prod>i\<in>#ps. i) ^ 3" shows "is_cube_form a b"
insert_is_Un: fixes a :: "'c"   and A :: "'c set" shows "insert a A = {a} \<union> A"
not_in_add: fixes var :: "nat"   and p :: "real mpoly"   and q :: "real mpoly" assumes "var \<notin> vars p"   and "var \<notin> vars q" shows "var \<notin> vars (p + q)"
functions_of_subset_of: fixes A :: "'b set"   and F :: "('b \<Rightarrow> 'c) set"   and B :: "'c set" assumes "finite A"   and "F \<in> {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} // domain_permutation A B" shows "functions_of A (subset_of A F) = F"
c: fixes i :: "nat" shows "coeff (poly_of_list (map g [0..<k])) i = (if i < k then g i else 0)"
p_upper_bound: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "0 < n" shows "p n x \<le> 1 / (12 * x\<^sup>2)"
extend_cons_closed_aux: fixes ls :: "lit set" assumes "consistent ls"   and "G `` ls \<subseteq> ls" shows "consistent (extend ls) \<and> G `` extend ls \<subseteq> extend ls"
complete_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.metric_space dist uniformity open" shows "uniform_space.complete open uniformity S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> uniform_space.Cauchy uniformity f \<longrightarrow> (\<exists>l\<in>S. filterlim f (topological_space.nhds open l) sequentially))"
that_1: fixes x :: "'a" shows "A x \<longrightarrow> A' x"
hyps_3: fixes thesis :: "bool" assumes "\<And>c. \<lbrakk>c \<in> ideal B; (\<Prod>i\<in>I. (f i + g i) ^ m i) = (\<Prod>i\<in>I. f i ^ m i) + c\<rbrakk> \<Longrightarrow> thesis"   and "\<And>i. i \<in> I \<Longrightarrow> g i \<in> ideal B" shows "thesis"
altst_def: shows "altst = alternating_list (Suc (S_length w)) s t"
clause_semantics_def: fixes \<A> :: "'b \<Rightarrow> bool"   and C :: "'b literal set" shows "clause_semantics \<A> C \<equiv> \<exists>L\<in>C. lit_semantics \<A> L"
assms_4: shows "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
s_Prj1_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and v :: "Syntax.term" assumes "\<lless>\<pi>, Prj1 e\<ggreater> m\<rightarrow> \<lless>\<pi>', v\<ggreater>" obtains e' :: "Syntax.term" where "v = Prj1 e'" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   | v\<^sub>2 :: "Syntax.term" where "e = Syntax.Pair v v\<^sub>2" and "\<pi>' = \<pi>" and "value v" and "value v\<^sub>2"
F_anal_valid: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and two_chain :: "(real \<times> real \<Rightarrow> real \<times> real) set"   and s :: "(real \<times> real) set" assumes "green_typeI_chain i j F two_chain s" shows "\<forall>twoC\<in>two_chain. analytically_valid (cubeImage twoC) (\<lambda>x. F x \<bullet> i) j"
get_put: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c" assumes "wb_lens x" shows "put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
bind_commute_pmf: fixes A :: "'d pmf"   and B :: "'e pmf"   and C :: "'d \<Rightarrow> 'e \<Rightarrow> 'c pmf" shows "A \<bind> (\<lambda>x. B \<bind> C x) = B \<bind> (\<lambda>y. A \<bind> (\<lambda>x. C x y))"
bisim1InstanceOf: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and T :: "ty" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e instanceof T,h \<turnstile> (e' instanceof T, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
bounded_linear_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "bounded S"   and "bounded_linear f" shows "bounded (f ` S)"
bij_betw_vertices: shows "bij_betw f (\<Union> X) (\<Union> Y)"
fdts'_2: shows "FDTs = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs'"
b_def: shows "b \<equiv> coeff (pCons a (x mod y)) (degree y) / lead_coeff y"
d0: shows "v = unit_vec n i + c \<cdot>\<^sub>v unit_vec n j"
ctr_transfer_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "list_all2 R [] []"
ind_eq_fin: fixes u :: "'item list"   and v :: "'item list" assumes "Ind (set u) (set v)" shows "u @ v =\<^sub>F v @ u"
that: fixes h' :: "heap"   and r :: "'a" assumes "run f (Some h) (Some h') r" shows "thesis"
discovered_eq_finished_un_stack: shows "dom (discovered s) = dom (finished s) \<union> set (stack s)"
assms_4: shows "minimal_sequence_to_failure_extending V M1 M2 vs xs"
mayDiverge_Par_L: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and s :: "'b"   and c2 :: "('a, 'c) com" assumes "PL.mayDiverge tval aval c1 s" shows "PL.mayDiverge tval aval (c1 | c2) s"
implicit_dependent_with: shows "dependent.with (0::'a) (0::'b) (+) (*s) = local.dependent"
isFault_simps_2: fixes s :: "'e" shows "isFault (Abrupt s) = False"
Poly_eq_0: fixes as :: "'b list" shows "(Poly as = 0) = (\<exists>n. as = replicate n (0::'b))"
doctor_optimal_match_def: fixes ds :: "'b set"   and Y :: "'a set" shows "doctor_optimal_match ds Y = (stable_on ds Y \<and> (\<forall>X. \<forall>x\<in>X. stable_on ds X \<longrightarrow> (\<exists>y\<in>Y. (x, y) \<in> Pd (Xd x))))"
y2_prop: shows "q < y2 \<and> (case z of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y2}. d * x\<^sup>2 + e * x + f \<le> 0)"
simps_4: fixes va :: "'c"   and vb :: "('c, 'd) Term.term list" shows "simple\<^sub>s\<^sub>t\<^sub>p (send\<langle>Fun va vb\<rangle>\<^sub>s\<^sub>t) = False"
span_eq: fixes S :: "'d set"   and T :: "'d set" shows "(span S = span T) = (S \<subseteq> span T \<and> T \<subseteq> span S)"
F1_1: shows "set fs' \<subseteq> carrier_vec n"
wtA: fixes at :: "('a, 'b) atm" assumes "M.wtA at" shows "local.wtA at"
simps_2: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "color"   and l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "ord.rbt_sorted less (Branch c l k v r) = (ord.rbt_less less k l \<and> ord.rbt_greater less k r \<and> ord.rbt_sorted less l \<and> ord.rbt_sorted less r)"
prems_1: shows "continuous_on (convex hull {a, b, c}) f"
well_com_while: shows "well_com (While_P vars2 M0 M1 D)"
TAbs_type': fixes \<Gamma> :: "binding list"   and S :: "POPLmarkRecord.type"   and s :: "trm"   and U :: "POPLmarkRecord.type"   and U' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> (\<lambda><:S. s) : (\<forall><:U. U')" obtains S' :: "POPLmarkRecord.type" where "\<Gamma> \<turnstile> U <: S" and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'" and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'"
scale_cancel_left: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d"   and y :: "'d" assumes "vector_space scale" shows "(scale a x = scale a y) = (x = y \<or> a = (0::'c))"
acquired_reads_mono_in: fixes x :: "nat"   and pending_write :: "bool"   and xs :: "'b memref list"   and A :: "nat set"   and B :: "nat set" assumes "x \<in> acquired_reads pending_write xs A"   and "A \<subseteq> B" shows "x \<in> acquired_reads pending_write xs B"
e: shows "\<forall>w\<in>ball a e. w \<in> s \<and> (w \<noteq> a \<longrightarrow> w \<notin> insert p pts)"
smods_ext_prefix: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0"   and "q \<noteq> 0" shows "smods_ext p q = smods p q @ tl (smods_ext (last (smods p q)) (pderiv (last (smods p q))))"
prefs_from_table_eqI: fixes xs :: "('c \<times> 'd set list) list"   and ys :: "('c \<times> 'd set list) list" assumes "distinct (map fst xs)"   and "distinct (map fst ys)"   and "set xs = set ys" shows "prefs_from_table xs = prefs_from_table ys"
ran_concat: fixes m1 :: "'b \<Rightarrow> 'a option"   and m2 :: "'b \<Rightarrow> 'a option" shows "ran (m1 ++ m2) \<subseteq> ran m1 \<union> ran m2"
c_prod_idr: fixes x :: "'a" shows "x \<parallel> 1\<^sub>\<pi> = x"
prob_space_distr: fixes f :: "'a cfg stream \<Rightarrow> 'b"   and s :: "'a cfg"   and M' :: "'b measure" assumes "MC.T.random_variable s M' f" shows "prob_space (distr (MC.T s) M' f)"
a_get_child_nodes_tups_def: shows "a_get_child_nodes_tups = [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
iso_class_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "category C"   and "category.isomorphic C f g" shows "category.iso_class C f = category.iso_class C g"
fmrestrict_set_dom: fixes m :: "('d, 'e) fmap" shows "fmrestrict_set (fmdom' m) m = m"
pl_dvdm_imp_p_dvdm: fixes l :: "nat"   and a :: "int poly"   and b :: "int poly" assumes "l \<noteq> 0"   and "poly_mod.dvdm (p ^ l) a b" shows "p.dvdm a b"
set_insert: fixes xs :: "'b list"   and t :: "'b trie" shows "Trie_Fun.set (Trie_Fun.insert xs t) = Trie_Fun.set t \<union> {xs}"
fresh_abs_funI1: fixes x :: "'a"   and b :: "'b"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)"   and "b \<sharp> x"   and "a \<noteq> b" shows "b \<sharp> [a].x"
hyps_16: assumes "(\<lambda>f. f - monomial 1 x) ` S \<subseteq> .[X]"   and "monomial 1 x + t \<in> .[X]" shows "standard_decomp (deg_pm (monomial 1 x + t)) (snd (ps1, qs1))"
igSubstInp_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'b, 'i, 'j) Iteration.model_scheme"   and ys :: "'e"   and Y :: "'b"   and y :: "'h"   and inp :: "'a \<Rightarrow> 'b option" shows "igSubstInp MOD ys Y y inp \<equiv> lift (igSubst MOD ys Y y) inp"
elims: fixes x :: "'a list"   and xa :: "'a list"   and y :: "'a list" assumes "poly_add x xa = y" obtains p1 :: "'a list" and p2 :: "'a list" where "x = p1" and "xa = p2" and "y = (if length p2 \<le> length p1 then local.normalize (map2 (\<oplus>) p1 (replicate (length p1 - length p2) \<zero> @ p2)) else poly_add p2 p1)"
WELL: shows "Well_order r \<and> Well_order r' \<and> Well_order p"
under_ofilter: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "wo_rel r" shows "ofilter r (under r a)"
flatten_monomial: fixes t :: "'a"   and c :: "'a \<Rightarrow>\<^sub>0 'b" shows "flatten (monomial c t) = punit.monom_mult (1::'b) t c"
rcs_in_set_rcs: fixes H :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G" shows "H \<bullet> a \<in> set_rcs G H"
nfoldli_no_ctd: fixes ctd :: "'c \<Rightarrow> bool"   and s :: "'c"   and l :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres" assumes "\<not> ctd s" shows "nfoldli l ctd f s = RETURN s"
density_cong: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and f' :: "'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M"   and "AE x in M. f x = f' x" shows "density M f = density M f'"
PP_PPf: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla" shows "wrepr.PP \<langle>\<phi>\<rangle> = exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>)"
integrableI_bounded_set_indicator: fixes A :: "'d set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and B :: "real" assumes "A \<in> sets M"   and "f \<in> borel_measurable M"   and "emeasure M A < \<infinity>"   and "AE x\<in>A in M. norm (f x) \<le> B" shows "integrable M (\<lambda>x. indicat_real A x *\<^sub>R f x)"
nvC_Wax: fixes c :: "(('c, 'tp) efsym, 'd) lit list" assumes "c \<in> Wax" shows "nvC c = {}"
is_p_fun_def: fixes x :: "'a" shows "is_p_fun x \<equiv> x\<^sup>\<smile> ; x \<le> 1'"
mult1I: fixes M :: "'k multiset"   and a :: "'k"   and M0 :: "'k multiset"   and N :: "'k multiset"   and K :: "'k multiset"   and r :: "('k \<times> 'k) set" assumes "M = add_mset a M0"   and "N = M0 + K"   and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r" shows "(N, M) \<in> mult1 r"
even_bit_succ_iff: fixes a :: "'a"   and n :: "nat" assumes "even a" shows "bit ((1::'a) + a) n = (bit a n \<or> n = 0)"
Inf_between_def: fixes Inf :: "'b inference set"   and N :: "'b set"   and M :: "'b set" shows "inference_system.Inf_between Inf N M = inference_system.Inf_from Inf (N \<union> M) - inference_system.Inf_from Inf (N - M)"
unaffected_domains_def: fixes I :: "('a \<times> 'a) set"   and D :: "'b \<Rightarrow> 'a"   and U :: "'a set"   and xs :: "'b list" shows "unaffected_domains I D U xs \<equiv> {u \<in> range D. \<forall>v\<in>sinks_aux I D U xs. (v, u) \<notin> I}"
uniform_int_density_def: shows "uniform_int_density \<equiv> \<lambda>x xa. ennreal ((case x of (a, b) \<Rightarrow> \<lambda>x. if x \<in> {a..b} then inverse (real (nat (b - a + 1))) else 0) xa)"
inflsts_cases: fixes s :: "'b llist"   and A :: "'b set" assumes "s \<in> A\<^sup>\<omega>" obtains a :: "'b" and l :: "'b llist" where "l \<in> A\<^sup>\<omega>" and "a \<in> A" and "s = a ## l"
less_asym': fixes a :: "'a"   and b :: "'a"   and P :: "bool" assumes "lt a b"   and "lt b a" shows "P"
face_of_subset: fixes F :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "F face_of S"   and "F \<subseteq> T"   and "T \<subseteq> S" shows "F face_of T"
F2'_antimono: shows "antimono (\<lambda>XD. - Rf (CD_on (ds - {d'})) XD)"
invariance_of_domain_sphere_affine_set_gen: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and U :: "'a set" assumes "continuous_on S f"   and "inj_on f S"   and "f ` S \<subseteq> T"   and "bounded U"   and "convex U"   and "affine T"   and "aff_dim T < aff_dim U"   and "openin (top_of_set (rel_frontier U)) S" shows "openin (top_of_set T) (f ` S)"
nat_pow_comm: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (units_of R)" shows "x [^]\<^bsub>units_of R\<^esub> n \<otimes>\<^bsub>units_of R\<^esub> x [^]\<^bsub>units_of R\<^esub> m = x [^]\<^bsub>units_of R\<^esub> m \<otimes>\<^bsub>units_of R\<^esub> x [^]\<^bsub>units_of R\<^esub> n"
poly_add_length_le: fixes p1 :: "'a list"   and p2 :: "'a list" shows "length (poly_add p1 p2) \<le> max (length p1) (length p2)"
zero_sheaf_spec_in_sheaf_spec: fixes R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and U :: "'d set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring.zero_sheaf_spec R addition multiplication zero unit U \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"
\<omega>consistent_iffStd1: assumes "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> prv (imp (neg (neg \<phi>)) \<phi>)" shows "\<omega>consistent = \<omega>consistentStd1"
ee: fixes v :: "'a"   and e :: "'a" assumes "is_vector v"   and "e \<le> v ; - v\<^sup>T" shows "e ; e = 0"
lset_lmember: fixes x :: "'c"   and xs :: "'c llist" shows "(x \<in> lset xs) = lmember x xs"
reduced_GB_pmdl_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "pmdl (reduced_GB F) = pmdl F"
unit_def: fixes p :: "fm" shows "unit p = (let p' = zlfm p; l = \<zeta> p'; q = And (Dvd l (CN 0 1 (C 0))) (a_\<beta> p' l); d = \<delta> q; B = remdups (map simpnum (\<beta> q)); a = remdups (map simpnum (\<alpha> q)) in if length B \<le> length a then (q, B, d) else (mirror q, a, d))"
rep_eq: fixes x :: "'b"   and xa :: "'b poly" shows "coeff (Polynomial.smult x xa) = (\<lambda>n. x * coeff xa n)"
succs_mono: fixes p :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and i :: "instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty"   and \<tau>' :: "ty list \<times> ty err list"   and \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" assumes "wf_prog p P"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'" shows "set (succs i \<tau> pc) \<subseteq> set (succs i \<tau>' pc)"
v_5: fixes x :: "'a" shows "(SUP i. v i x) = g x"
shadow_root_same_host: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and host' :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"   and "h \<turnstile> get_shadow_root host' \<rightarrow>\<^sub>r Some shadow_root_ptr" shows "host = host'"
elims_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and x :: "('a, 'c) efsym" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl"   and "ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl x" obtains f :: "'a" where "x = Oldf f" and "wtFsym f"   | \<sigma> :: "'c" where "x = Wit \<sigma>" and "\<not> ProblemIkTpart.isRes wtFsym resOf \<sigma> \<or> protCl \<sigma>"
pair_loopfree_digraph_axioms_def: fixes G :: "'b pair_pre_digraph" shows "pair_loopfree_digraph_axioms G \<equiv> \<forall>e. e \<in> parcs G \<longrightarrow> fst e \<noteq> snd e"
contour_integral_continuous_on_linepath_2D: fixes U :: "complex set"   and F :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and a :: "complex"   and b :: "complex" assumes "open U"   and "\<And>w. w \<in> U \<Longrightarrow> F w contour_integrable_on linepath a b"   and "continuous_on (U \<times> U) (\<lambda>(x, y). F x y)"   and "closed_segment a b \<subseteq> U" shows "continuous_on U (\<lambda>w. contour_integral (linepath a b) (F w))"
has_domain_iff_arr: fixes f :: "('o, 'a) arr" shows "(domains f \<noteq> {}) = arr f"
integrable_const_ivl: fixes a :: "'a"   and b :: "'a"   and c :: "'b" shows "integrable (lebesgue_on {a..b}) (\<lambda>x. c)"
add: fixes A :: "(('b, 'd) vec, 'c) vec"   and b1 :: "('b, 'd) vec"   and b2 :: "('b, 'd) vec" shows "A *v (b1 + b2) = A *v b1 + A *v b2"
COMP_case: fixes g :: "nat list \<Rightarrow> nat"   and kg :: "nat"   and fs :: "(nat list \<Rightarrow> nat) list" assumes "\<forall>l. g l < ack kg (sum_list l)"   and "\<forall>f\<in>set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (sum_list l))" shows "\<exists>k. \<forall>l. COMP g fs l < ack k (sum_list l)"
corthogonal_inv: fixes vs :: "'b vec list"   and n :: "nat" assumes "corthogonal vs"   and "set vs \<subseteq> carrier_vec n" shows "inverts_mat (mat_of_rows n (map vec_inv vs)) (mat_of_cols n vs)"
float_round_up: fixes x :: "float"   and prec :: "nat" shows "real_of_float x \<le> real_of_float (float_round_up prec x)"
WT_converter_\<I>_full: fixes conv :: "('a, 'b, 'c, 'd) converter" shows "\<I>_full, \<I>_full \<turnstile>\<^sub>C conv \<surd>"
prog_unclocked_1: fixes env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and s' :: "'b state"   and r :: "(v sem_env, v) result" assumes "evaluate_prog False env s prog (s', r)" shows "r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s'"
mset_rev: fixes xs :: "'a list" shows "mset (rev xs) = mset xs"
PCF_consts_rel_def: fixes R :: "('b \<Rightarrow> ValD) set" shows "PCF_consts_rel R \<equiv> \<bottom> \<in> R \<and> (\<lambda>i. ValTT) \<in> R \<and> (\<lambda>i. ValFF) \<in> R \<and> (\<forall>n. (\<lambda>i. ValN\<cdot>n) \<in> R)"
uniqueness_of_extended_map_to_freeword_hom: fixes k :: "'a freeword \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "k \<circ> Abs_freeletter = f"   and "UGroupHom k" shows "k = (\<lambda>x. \<Sum>(s, b)\<leftarrow>freeword x. if b then f s else - f s)"
Mismatch: fixes P :: "pi"   and V :: "Early_Semantics.residual"   and a :: "name"   and b :: "name" assumes "P \<longmapsto>\<^sub>e V"   and "a \<noteq> b" shows "[a\<noteq>b]P \<longmapsto>\<^sub>e V"
reachable_trans: fixes q :: "'c"   and A :: "('d, 'c) nba"   and p :: "'c"   and s :: "'c" assumes "q \<in> reachable A p"   and "s \<in> reachable A q" shows "s \<in> reachable A p"
replacefacesAt_Nil: fixes f :: "face"   and fs :: "face list"   and F :: "face list list" shows "replacefacesAt [] f fs F = F"
le_minus_power_downI: fixes x :: "real"   and n :: "nat"   and a :: "real"   and prec :: "nat" assumes "0 \<le> x"   and "x ^ n \<le> - a" shows "a \<le> - power_down prec x n"
RSadj_cancel: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and x :: "'a"   and y :: "'a"   and \<nu> :: "(real, 'a) vec" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.RSadj x y (ids.RSadj x y \<nu>) = \<nu>"
equivalence_map_def: fixes f :: "'b" shows "D.equivalence_map f \<equiv> \<exists>g \<eta> \<epsilon>. D.internal_equivalence f g \<eta> \<epsilon>"