hcju/isabelleps huggingface dataset:
intro: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "(cl, f) \<in> CLF_set" shows "CLF_axioms cl f"
Basic: fixes i :: "'b"   and l :: "'a FOL_Fitting.term list"   and G :: "('a, 'b) form list" shows "\<turnstile> Pred i l # Neg (Pred i l) # G"
monotone_ennreal_add1: fixes y :: "ennreal" shows "monotone (\<le>) (\<le>) (\<lambda>x. x + y)"
has_derivative_vec_nth: fixes f :: "real \<Rightarrow> ('b, 'c) vec"   and f' :: "real \<Rightarrow> ('b, 'c) vec"   and x :: "real"   and T :: "real set"   and i :: "'c" assumes "D f \<mapsto> (\<lambda>h. h *\<^sub>R f' x) at x within T" shows "D (\<lambda>t. f t $ i) \<mapsto> (\<lambda>h. h *\<^sub>R f' x $ i) at x within T"
dom_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "C.ide a"   and "C.ide b"   and "C.ide c" shows "C.dom \<a>\<^sub>C[a, b, c] = (a \<otimes>\<^sub>C b) \<otimes>\<^sub>C c"
22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
le_cSup_finite: fixes X :: "'b multiset set"   and x :: "'b multiset" assumes "finite X"   and "x \<in> X" shows "x \<subseteq># Sup X"
iso_restrict: fixes G :: "('c, 'd) monoid_scheme"   and F :: "('e, 'f) monoid_scheme"   and H :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group F"   and "subgroup H G"   and "\<phi> \<in> Group.iso G F" shows "restrict \<phi> H \<in> Group.iso (G\<lparr>carrier := H\<rparr>) (F\<lparr>carrier := \<phi> ` H\<rparr>)"
strict_sorted_sorted_wrt: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less = sorted_wrt less"
lderivs_Zero: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and project :: "'c \<Rightarrow> 'c"   and embed :: "'c \<Rightarrow> 'c list"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "Pi_Derivatives.embed \<Sigma> project embed" shows "embed.lderivs lookup embed xs Zero = Zero"
assms: shows "run_one_step d i (s, vs, ves, $Drop) = (s', vs', res)"
is_node_ptr_kind_obtains: fixes ptr :: "(_) object_ptr" assumes "is_node_ptr_kind ptr" obtains node_ptr :: "(_) node_ptr" where "cast ptr = Some node_ptr"
group_left_neutral: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and inverse :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "group f z inverse" shows "f z a = a"
Merge_defined: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and A :: "'a set" assumes "merkle_interface h bo m"   and "finite A"   and "A \<noteq> {}"   and "\<forall>a\<in>A. \<forall>b\<in>A. h a = h b" shows "merkle_interface.Merge m A \<noteq> None"
csa_list_copr_rel: fixes p :: "real poly"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q" shows "characterize_consistent_signs_at_roots p qs = characterize_consistent_signs_at_roots_copr p qs"
const_term_not_zero: fixes p :: "'a list" assumes "const_term p \<noteq> \<zero>" shows "p \<noteq> []"
rangeE: fixes b :: "'h"   and f :: "'i \<Rightarrow> 'h" assumes "b \<in> range f" obtains x :: "'i" where "b = f x"
map_cong0: fixes x :: "'c multiset"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>z. z \<in># x \<Longrightarrow> f z = g z" shows "image_mset f x = image_mset g x"
fr_rename_ext_def: fixes ecnv :: "('c, 'd) graph_rec_scheme \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a"   and G :: "('c, 'd) graph_rec_scheme" shows "fr_rename_ext ecnv f G \<equiv> \<lparr>g_V = f ` g_V G, g_E = rename_E f (g_E G), g_V0 = f ` g_V0 G, \<dots> = ecnv G\<rparr>"
simps: fixes a1 :: "com \<times> (char list \<Rightarrow> int)"   and a2 :: "com \<times> (char list \<Rightarrow> int)" shows "a1 \<rightarrow> a2 = ((\<exists>x a s. a1 = (x ::= a, s) \<and> a2 = (SKIP, s(x := aval a s))) \<or> (\<exists>c\<^sub>2 s. a1 = (SKIP;; c\<^sub>2, s) \<and> a2 = (c\<^sub>2, s)) \<or> (\<exists>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. a1 = (c\<^sub>1;; c\<^sub>2, s) \<and> a2 = (c\<^sub>1';; c\<^sub>2, s') \<and> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')) \<or> (\<exists>b s c\<^sub>1 c\<^sub>2. a1 = (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<and> a2 = (c\<^sub>1, s) \<and> bval b s) \<or> (\<exists>b s c\<^sub>1 c\<^sub>2. a1 = (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<and> a2 = (c\<^sub>2, s) \<and> \<not> bval b s) \<or> (\<exists>b c s. a1 = (WHILE b DO c, s) \<and> a2 = (IF b THEN c;; WHILE b DO c ELSE SKIP, s)))"
point_in_vector_or_complement: fixes p :: "'a"   and v :: "'a" assumes "point p"   and "is_vector v" shows "p \<le> v \<or> p \<le> - v"
intros_1: fixes f :: "'a \<Rightarrow> 'b" shows "hom_graph f {} \<bottom> \<bottom>"
act4: fixes x :: "'a"   and P :: "'b set" shows "\<alpha> x (\<Squnion>P) = \<Squnion>{\<alpha> x p |p. p \<in> P}"
derived_is_subgroup: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "H \<subseteq> carrier G" shows "subgroup (derived G H) G"
ZFfunApp_def: fixes f :: "ZF"   and x :: "ZF" shows "f |@| x \<equiv> Snd f \<acute> x"
subst_new_all: fixes a :: "'a"   and cs :: "'a list"   and p :: "('a, 'b) form"   and m :: "nat" assumes "a \<notin> set cs"   and "list_all (\<lambda>c. new c p) cs" shows "list_all (\<lambda>c. new c (p[App a []/m])) cs"
delay: fixes st :: "'s \<times> ('c \<Rightarrow> real)"   and l :: "'s"   and u :: "'c \<Rightarrow> real"   and t :: "real" assumes "st \<in> S"   and "st = (l, u)"   and "0 \<le> t"   and "u \<oplus> t \<turnstile> PTA.inv_of A l" shows "return_pmf (l, u \<oplus> t) \<in> K st"
Sigma_mset_Diff_distrib1: fixes I :: "'b multiset"   and J :: "'b multiset"   and C :: "'b \<Rightarrow> 'c multiset" shows "Sigma_mset (I - J) C = Sigma_mset I C - Sigma_mset J C"
ipresCons_imp_ipresSubstAll: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "ipresCons h hA MOD"   and "igSubstCls MOD"   and "igConsIPresIGWls MOD"   and "igFreshCls MOD" shows "ipresSubstAll h hA MOD"
char_to_digit_div_mod: fixes c :: "char" assumes "is_digit_char c" shows "char_to_digit c < 10"
bisim1CallThrow: fixes ps :: "'addr expr1 list"   and vs :: "'addr val list"   and P :: "'a prog"   and h :: "'heap"   and obj :: "'addr expr1"   and M :: "String.literal"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val" assumes "length ps = length vs" shows "P,obj\<bullet>M(ps),h \<turnstile> (Throw a, xs) \<leftrightarrow> (vs @ [v], xs, length (compE2 obj) + length (compEs2 ps), \<lfloor>a\<rfloor>)"
pseudo_mod_main_list: fixes l :: "'b"   and q :: "'b list"   and xs :: "'b list"   and ys :: "'b list"   and n :: "nat" shows "snd (pseudo_divmod_main_list l q xs ys n) = pseudo_mod_main_list l xs ys n"
Entry_empty: shows "lDef NewEntry = {} \<and> lUse NewEntry = {}"
assms_2: fixes y :: "'a"   and p :: "'a"   and q :: "'a"   and r :: "'a" shows "(y \<in> set (depressed_solver p q r)) = (y ^ 4 + p * y\<^sup>2 + q * y + r = (0::'a))"
sel_2: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat option"   and d :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and i :: "(String.literal \<times> value list) stream" shows "stl (make_full_observation e s d p i) = make_full_observation e (fst (ltl_step e s d (shd i))) (snd (snd (ltl_step e s d (shd i)))) (fst (snd (ltl_step e s d (shd i)))) (stl i)"
\<sigma>_\<rho>_cong: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and t :: "MIR.num"   and t' :: "MIR.num"   and c :: "int" assumes "iszlfm p (a # bs)"   and "Inum (a # bs) t = Inum (a # bs) t'" shows "Ifm (a # bs) (\<sigma>_\<rho> p (t, c)) = Ifm (a # bs) (\<sigma>_\<rho> p (t', c))"
length_strip_while_le: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (strip_while P xs) \<le> length xs"
induced_group_hom': fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and h :: "'d \<Rightarrow> 'f"   and I :: "'d set" assumes "group_hom G H h"   and "subgroup I G" shows "group_hom (G\<lparr>carrier := I\<rparr>) H h"
qSwapped_qSkel: fixes X :: "('a, 'b, 'c, 'd, 'e) qTerm"   and Y :: "('a, 'b, 'c, 'd, 'e) qTerm" assumes "(X, Y) \<in> qSwapped" shows "qSkel Y = qSkel X"
normalize_idem_imp_unit_factor_eq: fixes a :: "'a" assumes "normalize a = a" shows "unit_factor a = of_bool (a \<noteq> (0::'a))"
R28_4: shows "pmf (sds R28) d = 1 / 2"
profile_mono_scheme_defined_axioms: shows "profile_mono_scheme_defined f g"
o_def: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "f \<cdot> g = (\<lambda>x. f (g x))"
Iod_le: fixes D :: "('c, 'd) Order_scheme"   and T :: "'c set"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T" shows "a \<preceq>\<^bsub>Iod D T\<^esub> b = a \<preceq>\<^bsub>D\<^esub> b"
simps_2: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" shows "del_lo \<langle>l, a, r\<rangle> = merge l r"
case_eint_def: shows "case_eint \<equiv> rec_eint"
transfer_fun_app_lifted: fixes R0 :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and R1 :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "(crel_vs (R0 ===>\<^sub>T R1) ===> crel_vs R0 ===>\<^sub>T R1) App (.)"
pp_dist_sum: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.stone_relation_algebra inf less_eq less sup bot top uminus one times conv" shows "comm_monoid_add.sum sup bot (\<lambda>k. uminus (uminus (f k))) {k. True} = uminus (uminus (comm_monoid_add.sum sup bot f {k. True}))"
HNatInfinite_def: shows "HNatInfinite = {n. n \<notin> \<nat>}"
OAlist_ntm_def: fixes xs :: "('a \<times> 'b) list \<times> 'a nat_term_order" shows "OAlist_ntm xs = Abs_oalist_ntm (sort_oalist_ko_ntm xs)"
mult_if_delta: fixes P :: "bool"   and q :: "'b" shows "(if P then 1::'b else (0::'b)) * q = (if P then q else (0::'b))"
iMODb_iTILL_subset: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and n :: "nat" assumes "r' + m' * c' \<le> n" shows "[ r', mod m', c' ] \<subseteq> [\<dots>n]"
zero_gt_all: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P 0"   and "\<And>n. 0 < n \<Longrightarrow> P n" shows "P n"
Units_closed: fixes x :: "'a mat" assumes "x \<in> Units (ring_mat TYPE('a) n b)" shows "x \<in> carrier (ring_mat TYPE('a) n b)"
assms_2: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
perfect_sec_P2_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'e spmf"   and S2 :: "'b \<Rightarrow> 'f \<Rightarrow> 'e spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf"   and m1 :: "'a"   and m2 :: "'b" assumes "sim_det_def R1 S1 R2 S2 funct" shows "sim_det_def.perfect_sec_P2 R2 S2 funct m1 m2 \<equiv> R2 m1 m2 = funct m1 m2 \<bind> (\<lambda>(s1, s2). S2 m2 s2)"
eq_iff_iszero_diff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.ring_1 one times plus zero minus uminus" shows "(x = y) = ring_1.iszero zero (minus x y)"
1: shows "can f (f \<^bold>\<star> Src f) = \<r>[\<lbrace>f\<rbrace>]"
hom_char: fixes a :: "'a"   and b :: "'a" shows "L.hom a b = hom a b \<inter> Collect (left g)"
Un_Int_assoc_eq: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "(A \<inter> B \<union> C = A \<inter> (B \<union> C)) = (C \<subseteq> A)"
a_group: fixes G :: "('b, 'c) ring_scheme" assumes "abelian_group G" shows "Group.group (add_monoid G)"
HMSet_le: fixes M :: "hmultiset multiset"   and N :: "hmultiset multiset" shows "(HMSet M \<le> HMSet N) = (M \<le> N)"
circ_left_edge_neq_top: shows "(- 1, \<lambda>y. (- (d / 2), 0)) \<noteq> (- 1, \<lambda>x. ((x - 1 / 2) * d, d * sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2))))"
bound_le: fixes \<eta> :: "nat" shows "bound' \<eta> * max (bound_conv \<eta>) 1 \<le> bound \<eta>"
walk_iarray_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c iarray"   and x :: "'b" shows "walk_iarray f a x = walk_iarray' f a x (IArray.length a) 0"
unat_less_power: fixes sz :: "nat"   and k :: "'b word" assumes "sz < LENGTH('b)"   and "k < 2 ^ sz" shows "unat k < 2 ^ sz"
maximal_independent_subset: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "cindependent B" and "V \<subseteq> cspan B"
sel_rews_6: fixes a :: "'b"   and list :: "['b]" shows "tail\<cdot>(a : list) = list"
laplace_integrand_absolutely_integrable_on_Icc: fixes a :: "real"   and b :: "real"   and B :: "real"   and s :: "complex" assumes "AE x\<in>{a..b} in lebesgue. cmod (f x) \<le> B"   and "f integrable_on {a..b}" shows "complex_set_integrable lebesgue {a..b} (local.laplace_integrand s)"
fags_ipp_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e" shows "fags_ipp f z i a = (\<lambda>x\<in>fags_carr f z i a. (- (THE n. x = n\<Odot>a\<^bsub>f,i,z\<^esub>))\<Odot>a\<^bsub>f,i,z\<^esub>)"
set_2: fixes x2 :: "'f" shows "set_option (Some x2) = {x2}"
cancel_comm_semigroup_axioms: shows "cancel_comm_semigroup (\<oplus>\<^sub>a) (\<ominus>\<^sub>a)"
sym_preproc_aux_term_def: fixes d :: "'a \<Rightarrow> nat" shows "sym_preproc_aux_term d = sym_preproc_aux_term1 d \<inter> sym_preproc_aux_term2 d"
scc_of_in_sccs_verts: fixes u :: "'a" assumes "u \<in> verts G" shows "G.scc_of u \<in> G.sccs_verts"
assms_2: fixes s :: "(name \<times> name) list" shows "Q[
] \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P[
]"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm T2"
x_hx: shows "(x, h x) \<in> graph H'' h''"
not_iszero_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "\<not> ring_1.iszero zero one"
even_of_nat: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and a :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (semiring_1.of_nat one plus zero a) = even a"
a_star_ImpR: fixes M :: "trm"   and M' :: "trm"   and x :: "name"   and a :: "coname"   and b :: "coname" assumes "M \<longrightarrow>\<^sub>a* M'" shows "ImpR x.<a>.M b \<longrightarrow>\<^sub>a* ImpR x.<a>.M' b"
partition_without_covers: fixes elem :: "'b"   and P :: "'b set set" shows "\<Union> (partition_without elem P) = \<Union> P - {elem}"
integrable_max: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. max (f x) (g x))"
DERIV_unique: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b"   and E :: "'b" assumes "(f has_field_derivative D) (at x)"   and "(f has_field_derivative E) (at x)" shows "D = E"
maxindex_coefficients: fixes p :: "'b poly" shows "\<forall>c\<in>set (coefficients p). maxindex c \<le> maxindex p"
induct: fixes P :: "'s \<Rightarrow> 's list \<Rightarrow> 't list \<Rightarrow> bool"   and a0 :: "'s"   and a1 :: "'s list"   and a2 :: "'t list" assumes "\<And>s. P s [] []"   and "\<And>s t xs o os. P t xs os \<Longrightarrow> P s (t # xs) (o # os)"   and "\<And>a v va. P a (v # va) []"   and "\<And>a v va. P a [] (v # va)" shows "P a0 a1 a2"
coboundedI: fixes A :: "'b set"   and a :: "'b" assumes "finite A"   and "a \<in> A" shows "a \<le> Max A"
lower: fixes t :: "real" assumes "t \<in> T"   and "t0 < t" shows "v' t < f t (v t)"
vars_mult_Const: fixes n :: "int"   and a :: "int mpoly" shows "vars (Const n * a) = (if n = 0 then {} else vars a)"
of_bl_rev_eq: fixes bs :: "bool list" shows "of_bl (rev bs) = horner_sum of_bool 2 bs"
eq_neutr_iff: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<union># b = {#}) = (a = {#} \<and> b = {#})"
simps_5: fixes v :: "'a rexp"   and va :: "'a rexp" shows "pnPlus (Times v va) Zero = Times v va"
simps_1: fixes \<sigma> :: "nat \<Rightarrow> val"   and v :: "val" shows "cexpr_sem \<sigma> (CVal v) = v"
converse_rtranclp_into_rtranclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>*\<^sup>* a c"
mat_col_eqI: fixes B :: "'a mat"   and A :: "'a mat" assumes "\<And>i. i < dim_col B \<Longrightarrow> col A i = col B i"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
rep_eq: fixes x :: "'b upred" shows "`x` = All \<lbrakk>x\<rbrakk>\<^sub>e"
sgnx_able_poly_2: fixes p :: "real poly"   and a :: "real" shows "poly p sgnx_able at_left a"
prems_2: shows "limit = nat (root_nat_floor 3 n)"
complex_poly_decompose: fixes p :: "complex poly" shows "smult (lead_coeff p) (\<Prod>z\<in>{z. poly p z = 0}. [:- z, 1:] ^ order z p) = p"
split_def: shows "case_prod = (\<lambda>c p. c (fst p) (snd p))"
of_nat_0: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 0 = zero"
reflE: fixes x :: "'a" assumes "x \<in> A" shows "(x, x) \<in> r"
prems_2: shows "density_context V V' \<Gamma> \<delta>"
vconstructor_value_def: shows "vconstructor_value \<equiv> value_pred.pred (\<lambda>_ _. True) (\<lambda>name. name |\<in>| C) (\<lambda>_. True)"
rA'': shows "local.mute q (Suc (l + k)) l (r', A') = (r'', A'')"
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 more :: "'b" shows "(\<oplus>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub>) = add"
BinOpThrow1: fixes e :: "char list exp"   and bop :: "bop"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>throw e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
neg_numeral_less_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus (numeral.numeral one plus m)) (uminus (numeral.numeral one plus n)) = (n < m)"
indspaceD_range: fixes f :: "('f, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('f, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> BaseRep.indspace G FH V" shows "range f \<subseteq> V"
Mp_smult_m_0: fixes m :: "int"   and f :: "int poly" shows "poly_mod.Mp m (Polynomial.smult m f) = 0"
norm_default_preservation: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "(\<forall>i>n. \<forall>j. M i j = \<one>) \<and> (\<forall>j>n. \<forall>i. M i j = \<one>)" shows "(\<forall>i>n. \<forall>j. DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n i j = \<one>) \<and> (\<forall>j>n. \<forall>i. DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n i j = \<one>)"
Sup_image_mono: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lessa :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.ccpo luba orda lessa"   and "monotone orda (\<le>) f"   and "Complete_Partial_Order.chain orda A"   and "A \<noteq> {}" shows "\<Squnion> (f ` A) \<le> f (luba A)"
rel_F_neg_distr: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co3 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co3' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra3 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and uu :: "('ah \<times> 'ai \<times> 'aj \<times> 'ak \<times> 'al \<times> 'am \<times> 'an \<times> 'ao \<times> 'ap \<times> 'aq) itself"   and L1 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and L1' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and L2 :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and L2' :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and L3 :: "'an \<Rightarrow> 'ao \<Rightarrow> bool"   and L3' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool" assumes "rel_F_neg_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' uu" shows "rel_F (L1 OO L1') (L2 OO L2') (L3 OO L3') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3') \<le> rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 OO rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3'"
bounded_clinearI: fixes f :: "'a \<Rightarrow> 'b"   and K :: "real" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>C b) = r *\<^sub>C f b"   and "\<forall>x. norm (f x) \<le> norm x * K" shows "bounded_clinear f"
set_borel_measurable_def: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "set_borel_measurable M A f \<equiv> (\<lambda>x. indicat_real A x *\<^sub>R f x) \<in> borel_measurable M"
emeasure_finite: fixes A :: "'a set" shows "emeasure N A \<noteq> top"
assms_7: shows "negligible (\<lambda>\<eta>. UPF.advantage1 \<eta> (guessing_many_one.reduction (q' \<eta>) (\<lambda>_. reduction_upf \<eta> (\<A> \<eta>)) ()))"
whr_inv_auto_6: fixes p :: "trm"   and q :: "trm"   and t :: "trm" assumes "App p q \<leadsto> t" obtains x :: "name" and t\<^sub>1 :: "trm" where "p = Lam [x].t\<^sub>1" and "t = t\<^sub>1[x::=q]"   | t\<^sub>1' :: "trm" where "t = App t\<^sub>1' q" and "p \<leadsto> t\<^sub>1'"
zero_mod_ring_def: shows "0 \<equiv> Abs_mod_ring 0"
lpo_with_assms_axioms: shows "lpo_with_assms prc prl"