hcju/isabelleps huggingface dataset:
deleteAround_separated: fixes g :: "graph"   and a :: "nat"   and f :: "face"   and ps :: "(nat \<times> nat) list" assumes "minGraphProps g"   and "final g"   and "a \<in> \<V> g"   and "|vertices f| \<le> 4"   and "f \<in> set (facesAt g a)" shows "\<V> f \<inter> set (map fst (deleteAround g a ps)) \<subseteq> {a}"
that: assumes "continuous_on (cball z r) g"   and "g holomorphic_on ball z r" shows "thesis"
sigma_finite_measure_axioms: shows "sigma_finite_measure (interval_measure (\<lambda>x. x))"
succ_chain_vars_id': fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i \<le> j"   and "j < length l" shows "tvars (\<T> (l ! i)) = tvars (\<T> (l ! j))"
subformulas_in_subformulas_1: fixes G :: "'a formula"   and H :: "'a formula"   and F :: "'a formula" assumes "G \<^bold>\<and> H \<in> set (subformulae F)" shows "G \<in> set (subformulae F) \<and> H \<in> set (subformulae F)"
glb_special_subset_coset_ex: fixes w :: "'w"   and T :: "'w set"   and T' :: "'w set" assumes "w \<in> W"   and "T \<in> Pow S"   and "T' \<in> Pow S" shows "\<exists>B. dual_order.glbound_in_of \<P> \<langle>T\<rangle> (w +o \<langle>T'\<rangle>) B"
left_tree_union_V: fixes s :: "'c"   and t :: "'c" assumes "s \<rightarrow>\<^bsub>T\<^esub> t" shows "T.left_tree s t \<union> T.left_tree t s = V\<^bsub>T\<^esub>"
Bet_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C \<or> Bet C B A" shows "Bet A B C"
tfr_stp_all_split_1: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) strand_step list" assumes "list_all tfr\<^sub>s\<^sub>t\<^sub>p (x # S)" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p [x]"
integrable_def: shows "integrable \<equiv> \<lambda>M f. lfp (\<lambda>p. \<exists>x. has_bochner_integral M f x)"
square_minus_one_cong_one: fixes n :: "nat"   and x :: "nat" assumes "1 < n"   and "[x = n - 1] (mod n)" shows "[x\<^sup>2 = 1] (mod n)"
discI_5: fixes rexp :: "'b rexp"   and x51 :: "'b rexp"   and x52 :: "'b rexp" assumes "rexp = Times x51 x52" shows "is_Times rexp"
\<mu>_stable_def: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<mu>_stable \<phi> w = (\<G>\<F> \<phi> w = \<F> \<phi> w)"
infinite_partition_on: fixes A :: "'b set" assumes "infinite A" shows "infinite {P. partition_on A P}"
isstop_def: fixes F :: "'b formula" shows "isstop (F::'b formula) \<equiv> F = \<^bold>\<not> \<bottom> \<or> F = \<top>"
def_prg_Acts: fixes F :: "'b program"   and init :: "'b set"   and acts :: "('b \<times> 'b) set set"   and allowed :: "('b \<times> 'b) set set" assumes "F = mk_total_program (init, acts, allowed)" shows "Acts F = insert Id (totalize_act ` acts)"
ded_right: fixes a :: "'a G"   and b :: "'a G"   and c :: "'a G" shows "a * b r\<rightarrow> c = b r\<rightarrow> a r\<rightarrow> c"
da_elim_cases_53: fixes Env :: "env"   and B :: "lname set"   and accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and e :: "expr"   and fn :: "vname"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>{accC,statDeclC,stat}e..fn\<rangle>\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
induct: fixes P :: "('a, 'b) atomic \<Rightarrow> bool"   and atomic :: "('a, 'b) atomic" assumes "\<And>x. P (Fo x)"   and "\<And>x. P (Z x)"   and "\<And>x1a x2a. P (Less x1a x2a)"   and "\<And>x1a x2a. P (In x1a x2a)" shows "P atomic"
in_empty_table: fixes x :: "'b" shows "x \<notin> empty_table"
accepts_def: fixes A :: "'b \<times> ('c \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and w :: "'c list" shows "NA.accepts A w = (\<exists>q\<in>delta A w (start A). fin A q)"
fnth_indexed_fmember: fixes i :: "nat"   and s :: "'b fset" assumes "i < size s" shows "s |!| i |\<in>|\<^bsub>i\<^esub> s"
last_message_hold_init_snoc: fixes xs :: "'b message_af list"   and x :: "'b message_af"   and m :: "'b message_af" shows "last_message_hold_init (xs @ [x]) m = last_message_hold_init xs m @ [if x = NoMsg then last_message (m # xs) else x]"
auto: fixes x :: "'a"   and s :: "real"   and t :: "real" assumes "x \<in> X" shows "f s x = f t x"
inefficient_list_intersect_correct: fixes a :: "'b list"   and b :: "'b list" shows "set (inefficient_list_intersect a b) = set a \<inter> set b"
monoid_add_hom_0_axioms: shows "monoid_add_hom_0 hom"
factored_char_poly_norm_bound: fixes A :: "'b mat"   and n :: "nat"   and as :: "'b list"   and N :: "nat" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'b:])"   and "\<exists>n_as. jordan_nf A n_as"   and "\<And>a. a \<in> set as \<Longrightarrow> norm a \<le> 1"   and "\<And>a. \<lbrakk>a \<in> set as; norm a = 1\<rbrakk> \<Longrightarrow> length (filter ((=) a) as) \<le> N" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ (N - 1))"
dSk: shows "ket_zero_k (Suc k) \<in> carrier_vec (2 ^ Suc (Suc k))"
linepath_def: fixes a :: "'a"   and b :: "'a" shows "linepath a b = (\<lambda>x. (1 - x) *\<^sub>R a + x *\<^sub>R b)"
unit_simps_4: fixes a :: "'a" assumes "C.obj a" shows "D.dom (F.unit a) = F.map\<^sub>0 a"
ik\<^sub>e\<^sub>s\<^sub>t_append_subst_2: fixes \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "ik\<^sub>e\<^sub>s\<^sub>t (A @ B) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = (ik\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) \<union> (ik\<^sub>e\<^sub>s\<^sub>t B \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>)"
unit_simps_5: fixes a :: "'c" assumes "C.obj a" shows "D.cod (H.unit a) = H a"
restrict_pos: fixes x :: "'a option list" assumes "wf_tuple n C x"   and "P x" shows "list_all (\<lambda>A. P (restrict A x)) A_pos"
wt_ParT_WtFT_pickFT_0: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'a) cmd list"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "properL cl"   and "WtFT cl = 1" shows "PL.wt aval tval cval (ParT cl) s (PL.brnL cl (pickFT cl)) = 1"
thetaIfZO_ZOretr: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.thetaIfZO tval aval indis \<subseteq> PL_Indis.ZOretr tval aval indis (PL_Indis.thetaIfZO tval aval indis \<union> PL_Indis.ZObis tval aval indis)"
ih_8: shows "InvariantWatchListsUniq (getWatchList state')"
deriv_downwards: fixes n :: "nat"   and x :: "(nat \<times> form) list"   and s :: "(nat \<times> form) list" assumes "(Suc n, x) \<in> deriv s" shows "\<exists>y. (n, y) \<in> deriv s \<and> x \<in> set (subs y) \<and> \<not> is_axiom (s_of_ns y)"
simps: fixes s :: "'b"   and p :: "'c \<rightarrow> tr"   and h :: "'b \<rightarrow> ('c, 'b) Step" assumes "s \<noteq> \<bottom>" shows "filterS\<cdot>p\<cdot>(Stream\<cdot>h\<cdot>s) = Stream\<cdot>(filterStep\<cdot>p\<cdot>h)\<cdot>s"
callee: fixes x :: "'call"   and s :: "'s" assumes "x \<in> outs_\<I> \<I>"   and "I s" shows "plossless_gpv \<I>' (callee s x)"
fromArr_toArr: fixes f :: "(bool, bool list) concrete_category.arr" assumes "arr f"   and "\<not> ide f" shows "fromArr (toArr f) = f"
exist_conjugateI2: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "\<exists>g. conjugation_pair g f"
range_disjointed_sets: fixes A :: "nat \<Rightarrow> 'c set"   and M :: "'c measure" assumes "range A \<subseteq> sets M" shows "range (disjointed A) \<subseteq> sets M"
monom_mult_list_inj: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('c \<times> nat) list"   and m1 :: "('c \<times> nat) list"   and m2 :: "('c \<times> nat) list" assumes "class.linorder less_eq less"   and "linorder.monom_inv less_eq m"   and "linorder.monom_inv less_eq m1"   and "linorder.monom_inv less_eq m2"   and "linorder.monom_mult_list less m m1 = linorder.monom_mult_list less m m2" shows "m1 = m2"
map_tl: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "map f (tl xs) = tl (map f xs)"
singular_simplex_singleton: fixes X :: "'c topology"   and a :: "'c"   and p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'c" assumes "topspace X = {a}" shows "singular_simplex p X f = (f = (\<lambda>x\<in>standard_simplex p. a))"
le_ant_def: fixes z :: "ant"   and w :: "ant" shows "z \<le> w \<equiv> if snd (Rep_Ainteg w) = 1 then True else if snd (Rep_Ainteg w) = 0 then if snd (Rep_Ainteg z) = 1 then False else if snd (Rep_Ainteg z) = 0 then fst (Rep_Ainteg z) \<le> fst (Rep_Ainteg w) else True else if snd (Rep_Ainteg z) = - 1 then True else False"
mult_closed: fixes r :: "('f, 'g) aezfun"   and s :: "('f, 'g) aezfun" assumes "r \<in> FG"   and "s \<in> FG" shows "r * s \<in> FG"
set_eqI: fixes A :: "'f set"   and B :: "'f set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
assms_2: shows "\<FF>f = \<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr>"
H': shows "\<Gamma>\<turnstile>T<:U \<rightarrow> U'"
inj_vec_to_iarray: shows "inj vec_to_iarray"
F_partially_differentiable: shows "\<forall>a\<in>Dy_pair. has_partial_vector_derivative (\<lambda>x. F x \<bullet> i) j (partial_vector_derivative (\<lambda>a. F a \<bullet> i) j a) a"
alg_dvd_trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x alg_dvd y"   and "y alg_dvd z" shows "x alg_dvd z"
assms_2: fixes s :: "'a" shows "left (f s) = f (LEFT s)"
col_upd_col_If: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and x :: "('c, 'd) vec"   and j :: "'d" shows "col (upd_col M i x) j = (if i = j then x else col M j)"
derivation_ge_ladder: fixes index :: "nat" assumes "index < length L" shows "length \<delta> \<le> ladder_i L index \<and> length \<delta> \<le> ladder_j L index"
empty_eq_Plus_conv: fixes A :: "'a set"   and B :: "'b set" shows "({} = A <+> B) = (A = {} \<and> B = {})"
hcomplex_hcnj_hcnj: fixes z :: "complex star" shows "hcnj (hcnj z) = z"
tautology_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K .\<bottom>. \<phi> = {}" shows "\<gamma>\<^sub>T\<^sub>R K \<phi> = {K}"
IH: shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
A1': fixes a :: "'p"   and b :: "'p" shows "a b \<equiv> b a"
surj_def: fixes f :: "'b \<Rightarrow> 'a" shows "top_pres ((`) f) = (\<forall>y. \<exists>x. y = f x)"
const_space: fixes m :: "'a measure"   and n :: "'a measure" assumes "m \<in> M"   and "n \<in> M" shows "space m = space n"
simps_8: fixes t :: "MIR.num" shows "not (Eq t) = NEq t"
inf_drop_def: fixes i :: "nat"   and s :: "nat \<Rightarrow> 'd" shows "inf_drop i s \<equiv> \<lambda>j. s (i + j)"
yun_factorization_induct: fixes P :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> bool"   and bn :: "'c poly"   and p :: "'c poly"   and cn :: "'c poly" assumes "\<And>bn cn. bn = 1 \<Longrightarrow> P bn cn"   and "\<And>bn cn. \<lbrakk>bn \<noteq> 1; P (bn div gcd bn (cn - pderiv bn)) ((cn - pderiv bn) div gcd bn (cn - pderiv bn))\<rbrakk> \<Longrightarrow> P bn cn"   and "bn = p div gcd p (pderiv p)"   and "cn = pderiv p div gcd p (pderiv p)"   and "monic p" shows "P bn cn"
to_list_correct: fixes empty :: "'c"   and get :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and add :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and to_list :: "'c \<Rightarrow> ('d \<times> 'e) list"   and e :: "'c" assumes "env empty get add to_list" shows "map_of (to_list e) = get e"
card_1_singletonE: fixes A :: "'k set" assumes "card A = 1" obtains x :: "'k" where "A = {x}"
sum: shows "gs.M.sumlist (map (\<lambda>j. fs.gs.\<mu> i j \<cdot>\<^sub>v fs.gs.gso j) [0..<i]) \<in> Rn"
a2: shows "\<forall>n''\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. n'' < n' \<longrightarrow> (\<exists>i\<ge>n''. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> (\<exists>n'''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n''\<^esub>. n''' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n''\<^esub> \<and> eval c t t' n''' \<gamma>') \<or> \<not> (\<exists>i\<ge>n''. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> eval c t t' n'' \<gamma>'"
wfP_subset: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "wfP r"   and "p \<le> r" shows "wfP p"
Red1AAssNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and i :: "'addr val"   and e :: "'addr val"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<lfloor>Val i\<rceil> := Val e,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
bst_insert: fixes t :: "('a \<times> color) tree"   and x :: "'a" assumes "Tree2.bst t" shows "Tree2.bst (insert_rbt x t)"
solve_factored_ratfps: fixes cs :: "('a \<times> nat) list"   and p :: "'a poly" assumes "distinct (map fst cs)"   and "(0::'a) \<notin> fst ` set cs" shows "fps_of_poly p / fps_of_poly (\<Prod>(c, n)\<leftarrow>cs. [:1::'a, - c:] ^ Suc n) = Abs_fps (interp_ratfps_solution (solve_factored_ratfps p cs))"
enres_additional_laws_1: fixes e :: "'c"   and f :: "'e \<Rightarrow> ('c + 'd) nres" shows "ebind (THROW e) f = THROW e"
strict_iff_order: fixes b :: "'b list"   and a :: "'b list" shows "(b <p a) = (b \<le>p a \<and> a \<noteq> b)"
mem_upD: fixes f :: "nat \<Rightarrow> 'c"   and R :: "('c, 'd) ring_scheme"   and n :: "nat" assumes "f \<in> up R" shows "f n \<in> carrier R"
cases: fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "a \<in> support_flow f" obtains e :: "'a \<times> 'a" where "a = e" and "0 < f e"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "yoneda_lemma C S \<phi> F a"   and "set_valued_transformation (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) F \<tau>" shows "yoneda_lemma_fixed_\<tau> C S \<phi> F a \<tau>"
bullet_incremental: fixes t :: "CL" shows "(t, bullet t) \<in> red\<^sup>*"
fps_cutoff_def: fixes n :: "nat"   and f :: "'a fps" shows "fps_cutoff n f = Abs_fps (\<lambda>i. if i < n then fps_nth f i else (0::'a))"
isCont_def: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "isCont f a = f \<midarrow>a\<rightarrow> f a"
intensional_simps_16: fixes y :: "'b \<Rightarrow> bool" shows "y \<and> #True \<equiv> y"
vimage_func1: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd" shows "(-`) (f \<circ> g) = (-`) g \<circ> (-`) f"
renumber': fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and r :: "('a \<times> 'a) set"   and b :: "'a" assumes "\<forall>i. (a, f i) \<in> r\<^sup>* \<and> (f i, f (Suc i)) \<in> r"   and "(a, b) \<in> r\<^sup>*"   and "b = f 0" shows "\<exists>f. f 0 = a \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
wf_cdecl'_def: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" shows "wf_cdecl' wf_md P = (\<lambda>(C, D, fs, ms). (\<forall>f\<in>set fs. wf_fdecl P f) \<and> distinct_fst fs \<and> (\<forall>m\<in>set ms. wf_mdecl wf_md P C m) \<and> distinct_fst ms \<and> (C \<noteq> Object \<longrightarrow> is_class P D \<and> (\<forall>m\<in>set ms. wf_overriding P D m)) \<and> (C = Thread \<longrightarrow> (\<exists>m. (run, [], Void, m) \<in> set ms)))"
ub: fixes n :: "nat" shows "f n \<le>\<^sub>L A"
prop_rules_def: shows "prop_rules = cycle [allI, allE, impI, impE]"
prime_nth_prime: fixes n :: "nat" shows "prime (nth_prime n)"
addrow_mat: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and l :: "nat"   and a :: "'b"   and k :: "nat" assumes "A \<in> carrier_mat n nc"   and "l < n" shows "addrow a k l A = addrow_mat n a k l * A"
R: fixes S :: "'a literal set set"   and C :: "'a literal set"   and D :: "'a literal set"   and k :: "'a" assumes "S \<turnstile> C"   and "S \<turnstile> D"   and "k\<^sup>+ \<in> C"   and "k\<inverse> \<in> D" shows "S \<turnstile> C - {k\<^sup>+} \<union> (D - {k\<inverse>})"
that: fixes c4 :: "real" assumes "\<And>x i. \<lbrakk>x\<^sub>1 \<le> x; i < k\<rbrakk> \<Longrightarrow> g_approx i x \<le> c4 * g x"   and "0 < c4" shows "thesis"
weakCongTransitionWeakTransition: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
rel_nabla_coinduct: fixes P :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" assumes "P \<subseteq> relfdia X P \<union> rdom Q" shows "P \<subseteq> rel_nabla X \<union> relfdia (X\<^sup>*) Q"
assms_2: shows "A' \<in> set (tr\<^sub>p\<^sub>c A [])"
assms: shows "summable (\<lambda>n. f (n + k))"
inv_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> kD rt"   and "the (flag rt ip) = Aodv_Basic.inv" shows "nsqn rt ip = sqn rt ip - 1"
orderE: fixes b :: "'d multiset"   and a :: "'d multiset" assumes "b \<subseteq># a" obtains "a = a \<union># b"
insert_remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "mset_set (insert x A) = add_mset x (mset_set (A - {x}))"
Mapping_filter_keys: fixes m :: "('a, 'b) mapping"   and P :: "'b option \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>k\<in>Mapping.keys m. P (Mapping.lookup m k)" shows "\<forall>k\<in>Mapping.keys (Mapping.filter f m). P (Mapping.lookup (Mapping.filter f m) k)"
cardinal_eqpoll: fixes a :: "V" shows "vcard a \<approx>\<^sub>\<circ> a"
mono: shows "mono (\<lambda>p x1 x2. \<exists>\<omega> s. x1 = s \<and> x2 = \<omega> \<and> p (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
strassen_measure_odd: fixes A1 :: "'b mat"   and A2 :: "'b mat"   and A3 :: "'b mat"   and A4 :: "'b mat"   and A :: "'b mat"   and B1 :: "'c mat"   and B2 :: "'c mat"   and B3 :: "'c mat"   and B4 :: "'c mat"   and B :: "'c mat" assumes "(A1, A2, A3, A4) = split_block A (dim_row A div 2 * 2) (dim_col A div 2 * 2)"   and "(B1, B2, B3, B4) = split_block B (dim_col A div 2 * 2) (dim_col B div 2 * 2)"   and "\<not> strassen_even A B" shows "strassen_measure (A1, B1) < strassen_measure (A, B)"
chart_inverse_inv_chart: fixes y :: "'c"   and c :: "('d, 'c) chart" assumes "y \<in> codomain c" shows "apply_chart c (inv_chart c y) = y"
B'_def: shows "B' = (\<lambda>(t', s). if t' \<le> j then K.lim_stream (t', s) \<bind> (\<lambda>\<omega>. K.lim_stream (j, trace_at s \<omega> j) \<bind> (\<lambda>\<omega>'. return (stream_space S) (merge_at \<omega> j \<omega>'))) else K.lim_stream (t', s))"
eq_def: fixes c :: "int"   and t :: "Ferrack.num" shows "eq c t = (if c = 0 then Eq t else if 0 < c then Eq (CN 0 c t) else Eq (CN 0 (- c) (Neg t)))"
lub_singleton_def: fixes lub :: "'b set \<Rightarrow> 'b" shows "lub_singleton lub = (\<forall>a. lub {a} = a)"
word_add_rbl: fixes v :: "'a word"   and vbl :: "bool list"   and w :: "'a word"   and wbl :: "bool list" assumes "to_bl v = vbl"   and "to_bl w = wbl" shows "to_bl (v + w) = rev (rbl_add (rev vbl) (rev wbl))"
comp_left_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z = x \<sqinter> (y \<squnion> z)"
domsMT_notND_DD4: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set" assumes "x \<in> dom (Cp (DenyAllFromTo a b))"   and "x \<in> dom (Cp (DenyAllFromTo c d))" shows "\<not> netsDistinct b d"
smult_diff_right: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "smult a (p - q) = smult a p - smult a q"
simps: fixes a :: "'a list" shows "list_of_array (Array a) = a"
fun_ord_le: shows "fun_ord (\<le>) = (\<le>)"
sp_rewrite_prog: fixes f :: "nat \<Rightarrow> 'l \<Rightarrow> Unboxed.type option list \<Rightarrow> ('f, ('g, 'h, 'i, 'op, 'opinl, 'opubx, 'j, 'k) Ubx.instr \<times> Unboxed.type option list) result"   and n :: "nat"   and p1 :: "'l list"   and \<Sigma>1 :: "Unboxed.type option list"   and p2 :: "('g, 'h, 'i, 'op, 'opinl, 'opubx, 'j, 'k) Ubx.instr list"   and \<Sigma>2 :: "Unboxed.type option list"   and F :: "'i \<Rightarrow> 'm fundef option" assumes "rewrite_prog f n p1 \<Sigma>1 = Ok (p2, \<Sigma>2)"   and "\<And>x \<Sigma>1 n x' \<Sigma>2. f n x \<Sigma>1 = Ok (x', \<Sigma>2) \<Longrightarrow> Subx.sp_instr F x' \<Sigma>1 = Ok \<Sigma>2" shows "Subx.sp F p2 \<Sigma>1 = Ok \<Sigma>2"
intro: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" assumes "blinding_of_on UNIV h bo"   and "merge_on_axioms A h bo m" shows "merge_on A h bo m"
simps_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c" shows "shift (Suc n) f = poly_shift (shift n f)"
adapt: shows "P \<subseteq> {s. \<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t)}"
is_embedding_functor: shows "embedding_functor (\<cdot>) C S.map"
pos: shows "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> < (0::'a)"
mat_kernel_def: fixes A :: "'b mat" shows "mat_kernel A = {v \<in> carrier_vec (dim_col A). A *\<^sub>v v = 0\<^sub>v (dim_row A)}"
inject_3: fixes x3 :: "'e"   and y3 :: "'e" shows "(Fault x3 = Fault y3) = (x3 = y3)"
dgrad_p_setD: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and v :: "'c" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<in> gd_term.dgrad_p_set pair_of_term d m"   and "v \<in> keys p" shows "d (term_powerprod.pp_of_term pair_of_term v) \<le> m"
ereal_minus_6: fixes x :: "ereal"   and y :: "ereal" shows "x - - y = x + y"
mult_left_le: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times a c) a"
ap_spmf_comp: fixes u :: "('f \<Rightarrow> 'd) spmf"   and v :: "('e \<Rightarrow> 'f) spmf"   and w :: "'e spmf" shows "return_spmf (\<circ>) \<diamondop> u \<diamondop> v \<diamondop> w = u \<diamondop> (v \<diamondop> w)"
zip_eq_Nil_iff: fixes xs :: "'g list"   and ys :: "'h list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
mem_WWa_then: fixes W :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "W \<in> WWa D g a" shows "W \<subseteq> carrier D \<and> Worder (Iod D W) \<and> a \<in> W \<and> (\<forall>x\<in>W. a \<preceq> x) \<and> (\<forall>x\<in>W. if ExPre (Iod D W) x then g (Pre (Iod D W) x) = x else if a \<noteq> x then Algebra2.Sup D (segment (Iod D W) x) = x else a = a)"
isDERIV_DERIV_floatarith: fixes v :: "nat"   and f :: "floatarith"   and vs :: "real list" assumes "isDERIV v f vs" shows "isDERIV v (DERIV_floatarith v f) vs"
abs_conv_abscissa_norm: fixes f :: "'a fds" shows "abs_conv_abscissa (fds_norm f) = abs_conv_abscissa f"
res: shows "(if n = 0 then (1001, primes_1000) else (n + 30, [n, n + 2, n + 6, n + 8, n + 12, n + 18, n + 20, n + 26])) = (m, ps)"