hcju/isabelleps huggingface dataset:
S'_P'AQ': shows "S' = P' * Mod_Type_Connect.from_hma\<^sub>m A * Q'"
exhaust: fixes y :: "Location" obtains x11 :: "CAttId" and x12 :: "nat" where "y = objLoc x11 x12"   | x2 :: "AttId" where "y = staticLoc x2"   | x31 :: "Arraytype" and x32 :: "nat" where "y = arrLenLoc x31 x32"   | x41 :: "Arraytype" and x42 :: "nat" and x43 :: "nat" where "y = arrLoc x41 x42 x43"
cnj_add_mult_eq_Re: fixes z :: "complex"   and w :: "complex" shows "z * cnj w + cnj z * w = complex_of_real (2 * Re (z * cnj w))"
multlist_closed: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list" assumes "Group.monoid G"   and "set fs \<subseteq> carrier G" shows "foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<in> carrier G"
cases: fixes x :: "D" obtains xa :: "bool" where "x = C xa"   | v :: "int" where "x = A v__"   | v :: "char list" where "x = B v"
insert_index_image: fixes i :: "nat"   and n :: "nat" assumes "i < Suc n" shows "insert_index i ` {0..<n} = {0..<Suc n} - {i}"
assms: shows "ideal_generated {a, b} = ideal_generated {1::'a}"
simps_27: fixes v :: "'a rexp"   and va :: "'a rexp"   and vf :: "'a rexp" shows "local.le_rexp (Times v va) (Star vf) = False"
terminal_char2: fixes t :: "'t" shows "S'.terminal t = (S'.ide t \<and> S'.set t = {t})"
a: shows "\<forall>y\<in>H. h y \<le> p y"
conjI_PL: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b"   and A :: "'b set"   and q :: "'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "p \<in> Cn A"   and "q \<in> Cn A" shows "conj_PL p q \<in> Cn A"
WT_callee: fixes s :: "'s" assumes "I s" shows "\<I> \<turnstile>c callee s \<surd>"
singleton_group: fixes a :: "'b" shows "Group.group (singleton_group a)"
beta_upright_def: fixes va :: "'c zmultiset"   and vb :: "'c zmultiset" shows "beta_upright va vb = (\<forall>t. 0 < zcount va t \<longrightarrow> (\<exists>s<t. zcount va s < 0 \<or> zcount vb s < 0))"
bij_betw_quotient_domain_permutation_eq_empty: fixes A :: "'c set"   and B :: "'d set" assumes "card A \<noteq> card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_permutation A B = {}"
compl_less_compl_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less (uminus x) (uminus y) = less y x"
abs_eq: shows "1 = abs_fract (1::'b, 1::'b)"
isVal_Aeta_expand: fixes e :: "exp"   and a :: "Arity" assumes "isVal e" shows "isVal (Aeta_expand a e)"
contraposition_1: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> in v] = [\<^bold>\<not>\<psi> \<^bold>\<rightarrow> \<^bold>\<not>\<phi> in v]"
normalize_length_eq: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R"   and "lead_coeff p \<noteq> \<zero>\<^bsub>R\<^esub>" shows "length (ring.normalize R p) = length p"
IH: assumes "A - Max_wrt_among le A \<subseteq> carrier"   and "x \<in> A - Max_wrt_among le A"   and "y \<in> A - Max_wrt_among le A" shows "of_weak_ranking (weak_ranking_aux (A - Max_wrt_among le A)) x y = restrict_relation (A - Max_wrt_among le A) le x y"
tsubst_simulates_subst_typ': fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "Core.term" shows "subst_typ' insts T = tsubst T (\<lambda>idn S. the_default (Tv idn S) (lookup (\<lambda>x. x = (idn, S)) insts))"
simp_num_pair_ci: fixes bs :: "real list"   and t :: "MIR.num"   and n :: "int" shows "(case simp_num_pair (t, n) of (t, n) \<Rightarrow> Inum bs t / real_of_int n) = (case (t, n) of (t, n) \<Rightarrow> Inum bs t / real_of_int n)"
powser_uniform_limit: fixes r :: "real"   and a :: "nat \<Rightarrow> 'b"   and \<xi> :: "'b" assumes "ereal r < conv_radius a" shows "uniform_limit (cball \<xi> r) (\<lambda>n x. \<Sum>i<n. a i * (x - \<xi>) ^ i) (\<lambda>x. \<Sum>i. a i * (x - \<xi>) ^ i) sequentially"
red_shext_incr: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "P,E,h,sh \<turnstile> e : T" shows "sh \<unlhd>\<^sub>s sh'"
in_pdata_pairs_to_listI1: fixes f :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and g :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" assumes "(f, g) \<in> set ps" shows "monom_mult ((1::'d) / lc (fst f)) (lcs (lp (fst f)) (lp (fst g)) - lp (fst f)) (fst f) \<in> set (pdata_pairs_to_list ps)"
subspace_span: fixes S :: "'a set" shows "ideal.subspace (ideal S)"
leg_simps_3: shows "C.arr \<nu>.cod.leg1"
cospan_\<nu>\<pi>: shows "Maps.cospan SPN_fgh.\<nu>.leg0 SPN_fgh.\<pi>.leg1"
F_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.F p q = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0}}"
bv_msb_bv_not: fixes w :: "bit list" assumes "0 < length w" shows "bv_msb (bv_not w) = (\<not>\<^sub>b bv_msb w)"
nth_def: fixes a :: "'a array"   and i :: "nat" shows "Array.nth a i = Heap_Monad.guard (\<lambda>h. i < Array.length h a) (\<lambda>h. (Array.get h a ! i, h))"
freeFreshChainDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "distinct (bn \<alpha>)"   and "xvec \<sharp>* P"   and "xvec \<sharp>* \<alpha>" shows "xvec \<sharp>* P'"
simps_2: fixes Obj :: "'e set"   and Mor :: "'d set"   and Dom :: "'d \<Rightarrow> 'e"   and Cod :: "'d \<Rightarrow> 'e"   and Id :: "'e \<Rightarrow> 'd"   and Comp :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'f" shows "mor\<^bsub>\<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr>\<^esub> = Mor"
INT_simps_4: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "(\<Inter>x\<in>C. A - B x) = (if C = {} then UNIV else A - \<Union> (B ` C))"
shiftl_lt_0: fixes i :: "int"   and n :: "nat" shows "(i << n < 0) = (i < 0)"
induct: fixes x1 :: "nat"   and x2 :: "com"   and P :: "nat \<Rightarrow> com \<Rightarrow> bool" assumes "x1 \<turnstile>' x2"   and "\<And>l. P l SKIP"   and "\<And>x a l. \<lbrakk>sec a \<le> sec x; l \<le> sec x\<rbrakk> \<Longrightarrow> P l (x ::= a)"   and "\<And>l c\<^sub>1 c\<^sub>2. \<lbrakk>l \<turnstile>' c\<^sub>1; P l c\<^sub>1; l \<turnstile>' c\<^sub>2; P l c\<^sub>2\<rbrakk> \<Longrightarrow> P l (c\<^sub>1;; c\<^sub>2)"   and "\<And>b l c\<^sub>1 c\<^sub>2. \<lbrakk>sec b \<le> l; l \<turnstile>' c\<^sub>1; P l c\<^sub>1; l \<turnstile>' c\<^sub>2; P l c\<^sub>2\<rbrakk> \<Longrightarrow> P l (IF b THEN c\<^sub>1 ELSE c\<^sub>2)"   and "\<And>b c. \<lbrakk>sec b = 0; 0 \<turnstile>' c; P 0 c\<rbrakk> \<Longrightarrow> P 0 (WHILE b DO c)"   and "\<And>l c l'. \<lbrakk>l \<turnstile>' c; P l c; l' \<le> l\<rbrakk> \<Longrightarrow> P l' c" shows "P x1 x2"
apex_def: shows "\<eta>.cod.apex \<equiv> C.dom \<eta>.cod.leg0"
inf_absorb1: fixes x :: "'g multiset"   and y :: "'g multiset" assumes "x \<subseteq># y" shows "x \<inter># y = x"
g: fixes y :: "'b" shows "monotone ord (ord_spmf (=)) (g y)"
diff_gr0_ennreal: fixes b :: "ennreal"   and a :: "ennreal" assumes "b < a" shows "0 < a - b"
PPsI: fixes t :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set" assumes "keys t \<subseteq> X" shows "t \<in> .[X]"
hseq_char': fixes \<nu> :: "'b"   and \<mu> :: "'b" shows "D.hseq \<nu> \<mu> = (\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null)"
guessed_frame_dom_spec: fixes R :: "fid_t" shows "dom (sklvl1.guessed_frame R) = sklvl1.domain (role (guessed_runs R))"
side: shows "s\<^sub>1 r'' = Some (\<sigma>', \<tau>', VE v)"
inf_dres_addsimps_3: fixes x :: "'b dres" shows "inf x dFAILi = x"
f_2: shows "convert_atom var f (xs' @ x # xs) = Some f'"
mult_1s_1: fixes a :: "'a" shows "Numeral1 * a = a"
simps_1: fixes x :: "'a" shows "del_tm x \<langle>\<rangle> =1 return None"
assms_2: shows "exception_\<I> \<I>, exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>\<^sub>C cnv \<surd>"
dom_add_anno_rel: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and v :: "'Var"   and m :: "Mode" shows "\<forall>x\<in>dom \<Gamma> \<oplus>\<^sub>\<S> (v -=\<^sub>m m). \<Gamma> \<oplus>\<^sub>\<S> (v -=\<^sub>m m) x = \<Gamma> x"
Simul_def: fixes S :: "'e \<Rightarrow> 'd"   and b :: "('c, 'e) alpha_rp_scheme" shows "Simul S b = alpha_rp.extend (alpha_rp.make (ok b) (wait b) (tr b) (ref b)) (S (alpha_rp.more b))"
returns_heap_def: fixes h :: "'a"   and p :: "('a, 'b, 'c) prog"   and h' :: "'a" shows "h \<turnstile> p \<rightarrow>\<^sub>h h' = (case h \<turnstile> p of Inl x \<Rightarrow> False | Inr (x, h'') \<Rightarrow> h' = h'')"
sympartpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "sympartp r x y" obtains "r x y" and "r y x"
path_split_1: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>* n'" shows "n -as\<rightarrow>* sourcenode a"
assoc'_simps_4: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "local.dom \<a>\<^sup>-\<^sup>1[f, g, h] = local.dom f \<star> local.dom g \<star> local.dom h"
sets: shows "sets (embed_measure M f \<Otimes>\<^sub>M embed_measure N g) = sets (embed_measure (M \<Otimes>\<^sub>M N) (map_prod f g))"
ball_divide_subset_numeral: fixes x :: "'a"   and e :: "real"   and w :: "num" shows "ball x (e / numeral w) \<subseteq> ball x e"
card_of_ordLeq2: fixes A :: "'a set"   and B :: "'b set" assumes "A \<noteq> {}" shows "(\<exists>g. g ` B = A) = (|A| \<le>o |B|)"
isLimOrd_aboveS: fixes r :: "('a \<times> 'a) set"   and i :: "'a" assumes "wo_rel r"   and "isLimOrd r"   and "i \<in> Field r" shows "aboveS r i \<noteq> {}"
axis: fixes x :: "complex" assumes "is_real x"   and "0 \<le> Re x"   and "Re x < 1" shows "P 0\<^sub>h (of_complex x)"
lossless_init_AND: fixes X :: "'g \<times> 'a"   and W :: "'h \<times> 'b" shows "lossless_spmf (init_AND X W)"
rel_T_pos_distr_cond_def: fixes Co1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Co1' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Co2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2' :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and Contra1 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1' :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra2 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and Contra2' :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and uu :: "('n \<times> 'o \<times> 'p \<times> 'q) itself" shows "rel_T_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu = (\<forall>L1 L1'. rel_T L1 Co1 Co2 Contra1 Contra2 OO rel_T L1' Co1' Co2' Contra1' Contra2' \<le> rel_T (L1 OO L1') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2'))"
emb_step_under_args_num_args: fixes p :: "dir list"   and d :: "dir"   and t :: "('a, 'b) tm" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p" shows "num_args (emb_step_at p d t) = num_args t"
Domain_nodes_connected: shows "Domain {(x, y) |x y. nodes_connected G x y} = V"
poly_pfd_simple_def: fixes x :: "'a poly"   and cs :: "('a \<times> nat) list" shows "poly_pfd_simple x cs = (if cs = [] then (x, []) else let zs = map (\<lambda>i. let (c, n) = cs ! i in normalise_decomp_poly (decompose x (map (\<lambda>(c, n). [:1::'a, - c:] ^ Suc n) cs) ! i) [:1::'a, - c:] (n + 1)) [0..<length cs] in (sum_list (map fst zs), map (map (\<lambda>p. coeff p 0) \<circ> snd) zs))"
ab_rationality_imp: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "\<forall>n. 0 < a n"   and "\<forall>\<^sub>F n in sequentially. 1 < a n"   and "(\<lambda>n. real_of_int \<bar>b n\<bar> / real_of_int (a (n - 1) * a n)) \<longlonglongrightarrow> 0"   and "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) \<in> \<rat>" shows "\<exists>B>0. \<exists>c. (\<forall>\<^sub>F n in sequentially. B * b n = c n * a n - c (n + 1) \<and> real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2) \<and> (\<lambda>n. real_of_int (c (Suc n)) / real_of_int (a n)) \<longlonglongrightarrow> 0"
better_right: fixes N :: "trm"   and x :: "name"   and a :: "coname"   and M :: "trm" assumes "\<not> fin N x" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>c N{x:=<a>.M}"
brw_invar_initial: fixes \<delta> :: "('d, 'e) ta_rule set" shows "brw_initial \<delta> \<subseteq> brw_invar_add \<delta>"
is_tm_tdghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<Z> \<alpha>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^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>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
subspace_add: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "subspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
ZeroHeightMultiElimAux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and ll :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, ll, r) \<in> MStep"   and "0 = k" shows "r = s \<and> ll = l"
are_all_operators_applicable_def: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list" shows "are_all_operators_applicable s ops \<equiv> list_all (is_operator_applicable_in s) ops"
fun_add_eval_comm: fixes a :: "'c"   and x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "x \<in> carrier F"   and "y \<in> carrier F" shows "(x \<oplus>\<^bsub>F\<^esub> y) a = (y \<oplus>\<^bsub>F\<^esub> x) a"
IH: shows "\<forall>\<^sub>F x in at y within Y. x \<notin> F"
Max_ge: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "x \<sqsubseteq> linorder.Max (\<sqsubseteq>) A"
eq: fixes U :: "'a set"   and V :: "'a set"   and s :: "'b" assumes "U \<in> neighborhoods x"   and "V \<in> neighborhoods x"   and "V \<subseteq> U"   and "s \<in> \<FF> U" shows "(\<psi> V \<circ> \<rho> U V) s = \<psi> U s"
codomains_comp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'b" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.codomains C (C g f) = partial_magma.codomains C g"
E_Abs: fixes v\<^sub>2 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and t\<^sub>1\<^sub>2 :: "trm" assumes "v\<^sub>2 \<in> value" shows "(\<lambda>:T\<^sub>1\<^sub>1. t\<^sub>1\<^sub>2) \<bullet> v\<^sub>2 \<longmapsto> t\<^sub>1\<^sub>2[0 \<mapsto> v\<^sub>2]"
summarize_pdevs_def: fixes p :: "nat"   and I :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "nat"   and x :: "'a pdevs" shows "summarize_pdevs p I d x = (let t = tdev' p (filter_pdevs (- I) x) in msum_pdevs d (filter_pdevs I x) (pdevs_of_centered_ivl t))"
eq': fixes t :: "'a" assumes "t \<in> cspan G" shows "A t = B t"
redlove: fixes p :: "fm"   and bs :: "real list" assumes "qfree p" shows "(\<exists>x. Ifm (real_of_int x # bs) p) = Ifm bs (redlove p) \<and> qfree (redlove p)"
mk_linear_orders_range: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and r :: "'a list"   and C :: "'a set"   and B :: "'a set" assumes "f_range_on A f"   and "r \<in> mk_linear_orders f C B"   and "B \<subseteq> A" shows "set r \<subseteq> B"
QB_Bnd: shows "Q + B \<subseteq> Bounded_vec nBnd"
has_field_derivative_at_within_union: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "(f has_field_derivative D) (at x within A)"   and "(f has_field_derivative D) (at x within B)" shows "(f has_field_derivative D) (at x within A \<union> B)"
fglb_def: fixes A :: "'b fset set"   and t :: "'b fset" shows "fglb A t = (if A = {} then t else Abs_fset (\<Inter> (fset ` A)))"
pochhammer_rGamma: fixes z :: "'a"   and n :: "nat" shows "rGamma z = pochhammer z n * rGamma (z + of_nat n)"
ipathD: fixes E :: "('a \<times> 'a) set"   and r :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "ipath E r" shows "(r i, r (Suc i)) \<in> E"
sup_Assertion_def: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "p \<squnion> q = abs_wpt ({\<cdot> p } \<squnion> {\<cdot> q })"
periodic_simps_4: fixes nc :: "nat"   and x :: "real"   and n :: "int" shows "pbernpoly nc (x - real_of_int n) = pbernpoly nc x"
erf_remainder_integral_conv_Suc: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "erf_remainder_integral n x = exp (- x\<^sup>2) / (2 * x ^ (2 * n + 1)) - real (2 * n + 1) / 2 * erf_remainder_integral (Suc n) x"
comp_funpow: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c" shows "(\<circ>) f ^^ n = (\<circ>) (f ^^ n)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>n. P 0 n"   and "\<And>m. P m 1 \<Longrightarrow> P (Suc m) 0"   and "\<And>m n. \<lbrakk>P (Suc m) n; P m (ack (Suc m) n)\<rbrakk> \<Longrightarrow> P (Suc m) (Suc n)" shows "P a0 a1"
signof_permutation_insert: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "p permutes {0..<n}"   and "i < Suc n"   and "j < Suc n" shows "signof (permutation_insert i j p) = (- (1::'b)) ^ (i + j) * signof p"
comp_char: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" shows "C.VxV.comp g f = (if C.seq (fst g) (fst f) \<and> C.seq (snd g) (snd f) then (fst g \<cdot>\<^sub>C fst f, snd g \<cdot>\<^sub>C snd f) else (C.null, C.null))"
that: fixes n :: "nat" assumes "u \<le>p r \<^sup>@ n"   and "n \<noteq> 0" shows "thesis"
starfun_ln_inverse: fixes x :: "real star" assumes "0 < x" shows "(*f* real_ln) (inverse x) = - (*f* real_ln) x"
Hadamard's_inequality_int: fixes A :: "int mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "real_of_int \<bar>det A\<bar> \<le> sqrt (real_of_int (prod_list (map sq_norm (rows A))))"
shiftr1_irrelevant_lsb: fixes x :: "'b word" shows "x !! 0 \<or> x >> 1 = x + 1 >> 1"
abs_INV_transfer: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme"   and I :: "'a set" assumes "PO_refines R Ta Tc"   and "reach Ta \<subseteq> I" shows "reach Tc \<subseteq> R `` I"
that: fixes C :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite C"   and "disjnt C U"   and "card C \<le> card \<F>"   and "continuous_on (\<Union> \<F> - C) g"   and "g ` (\<Union> \<F> - C) \<subseteq> T"   and "\<And>x. x \<in> (\<Union> \<F> - C) \<inter> K \<Longrightarrow> g x = h x" shows "thesis"
lex_ext_unbounded_map: fixes ss :: "'b list"   and ts :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<And>i. \<lbrakk>i < length ss; i < length ts; fst (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> fst (r (map f ss ! i) (map f ts ! i))"   and "\<And>i. \<lbrakk>i < length ss; i < length ts; snd (r (ss ! i) (ts ! i))\<rbrakk> \<Longrightarrow> snd (r (map f ss ! i) (map f ts ! i))" shows "(fst (lex_ext_unbounded r ss ts) \<longrightarrow> fst (lex_ext_unbounded r (map f ss) (map f ts))) \<and> (snd (lex_ext_unbounded r ss ts) \<longrightarrow> snd (lex_ext_unbounded r (map f ss) (map f ts)))"
control_spec: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and s0 :: "'a"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and realistic_execution :: "'b list list \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and AS_set :: "'b list set"   and invariant :: "'a \<Rightarrow> bool"   and AS_precondition :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and aborting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and waiting :: "'a \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool" assumes "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting" shows "\<forall>s d aseqs. case control s d aseqs of (a, aseqs', s') \<Rightarrow> (aseqs = [] \<or> aseqs = [[]]) \<and> (a, aseqs') = (None, []) \<or> aseqs \<noteq> [] \<and> hd aseqs \<noteq> [] \<and> \<not> aborting s' d (the a) \<and> \<not> waiting s' d (the a) \<and> (a, aseqs') = (Some (hd (hd aseqs)), tl (hd aseqs) # tl aseqs) \<or> aseqs \<noteq> [] \<and> hd aseqs \<noteq> [] \<and> waiting s' d (the a) \<and> (a, aseqs', s') = (Some (hd (hd aseqs)), aseqs, s) \<or> (a, aseqs') = (None, tl aseqs)"
analz_Un_implConfidSet: fixes G :: "msg set"   and H :: "msg set"   and Ag :: "(agent \<times> agent) set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({sym.implConfid A B M |A B M. (A, B) \<in> Ag \<and> M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
integral_map_pmf: fixes g :: "'f \<Rightarrow> 'e"   and p :: "'f pmf"   and f :: "'e \<Rightarrow> 'd" shows "measure_pmf.expectation (map_pmf g p) f = measure_pmf.expectation p (\<lambda>x. f (g x))"
that_2: fixes c :: "'form"   and "is" :: "nat list" assumes "c \<in> set conclusions"   and "is \<in> it_paths (it' c)"   and "v = (c, 0 # is)" shows "thesis"
headR_def: fixes t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.headR t = (case FingerTreeStruc.viewR t of Some (a, x) \<Rightarrow> a)"
cases: fixes a1 :: "'loc"   and a2 :: "'loc"   and a3 :: "('loc \<times> 'sum \<times> 'loc) list" assumes "path a1 a2 a3" obtains l1 :: "'loc" and l2 :: "'loc" where "a1 = l1__" and "a2 = l2__" and "a3 = []" and "l1__ = l2__"   | l1 :: "'loc" and l2 :: "'loc" and xs :: "('loc \<times> 'sum \<times> 'loc) list" and lbl :: "'sum" and l3 :: "'loc" where "a1 = l1" and "a2 = l3" and "a3 = xs @ [(l2, lbl, l3)]" and "path l1 l2 xs" and "lbl \<in>\<^sub>A summary l2 l3"
commute: fixes a :: "'e"   and b :: "'e" shows "a + b = b + a"
assms_2: shows "sp_fundef (Fubx_get F) fd (body fd) = Ok [None]"
apath_def: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and v :: "'b" shows "pre_digraph.apath G u p v \<equiv> pre_digraph.awalk G u p v \<and> distinct (pre_digraph.awalk_verts G u p)"
max_used_load_tmps: fixes t :: "nat"   and e :: "expr" shows "t + used_tmps e \<notin> load_tmps (issue_expr t e)"