hcju/isabelleps huggingface dataset:
qFreshEnv_def: fixes zs :: "'a"   and z :: "'b"   and rho :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm option" shows "qFreshEnv zs z rho \<equiv> rho zs z = None \<and> (\<forall>xs. liftAll (qFresh zs z) (rho xs))"
lnull_lfilter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "(lfilter P xs = LNil) = (\<forall>x\<in>lset xs. \<not> P x)"
greaterThanAtMost_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and k :: "'c" assumes "class.order less_eq less"   and "less_eq l k" shows "ord.greaterThanAtMost less_eq less k l = {}"
w: fixes w :: "'a" assumes "w \<noteq> (0::'a)" shows "(\<Sum>i\<le>n. c (Suc i) * w ^ i) = (0::'a)"
connected_by_path: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>* w" obtains xs :: "'a list" where "v \<leadsto>xs\<leadsto> w"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c p. P (CN c 0 p)"   and "\<And>v. P (C v)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)"   and "\<And>v vc vb. P (CN v (Suc vc) vb)" shows "P a0"
takeWhile_void: fixes x :: "'a"   and evs :: "'a list" assumes "x \<notin> set evs" shows "takeWhile (\<lambda>z. z \<noteq> x) evs = evs"
cobounded1: fixes a :: "'d"   and b :: "'d" shows "a \<le> sup a b"
simps_2: fixes x21 :: "bool"   and x22 :: "trie \<times> trie" shows "Lf \<noteq> Nd x21 x22"
birkhoff_sum_mono: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real"   and n :: "nat"   and x :: "'a" assumes "\<And>x. f x \<le> g x" shows "birkhoff_sum f n x \<le> birkhoff_sum g n x"
one_stalk_def: fixes V :: "'a set" shows "stk1.one_stalk V \<equiv> stk1.class_of V (one_str1 V)"
eq_split_subst: fixes x :: "'d \<Longrightarrow> 'e"   and P :: "('f, 'e) uexpr"   and Q :: "('f, 'e) uexpr" assumes "vwb_lens x" shows "(P = Q) = (\<forall>v. P\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk> = Q\<lbrakk>\<guillemotleft>v\<guillemotright>/x\<rbrakk>)"
map_insort_key_fst: fixes ps :: "('c \<times> 'd) list"   and k :: "'c"   and f :: "'c \<Rightarrow> 'e"   and v :: "'d" assumes "distinct (map fst ps)"   and "k \<notin> set (map fst ps)"   and "inj f"   and "mono f" shows "map (\<lambda>(k, v). (f k, v)) (insort_key fst (k, v) ps) = insort_key fst (f k, v) (map (\<lambda>(k, v). (f k, v)) ps)"
igSubstAllIPresIGWlsAllSTR_imp_igSubstAllIPresIGWlsAll: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstAllIPresIGWlsAllSTR Delta MOD" shows "FixSyn.igSubstAllIPresIGWlsAll Delta MOD"
simps_10: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and v :: "'a gexp"   and va :: "'a gexp" shows "local.less_gexp_aux (Eq e1 e2) (Nor v va) = True"
fps_nth_fps_const: fixes c :: "'a"   and n :: "nat" shows "fps_nth (fps_const c) n = (if n = 0 then c else (0::'a))"
filter_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "filter P (linorder.sort_key less_eq f xs) = linorder.sort_key less_eq f (filter P xs)"
certificate_merEK_valid_lemma: fixes i :: "nat"   and M :: "agent"   and merEK :: "nat"   and evs :: "event list" assumes "Crypt (priSK (CA i)) \<lbrace>Agent M, Key merEK, onlyEnc\<rbrace> \<in> parts (knows Spy evs)"   and "CA i \<notin> bad"   and "CA i \<noteq> RCA"   and "evs \<in> set_mr" shows "\<exists>X Y Z. Says (CA i) M \<lbrace>X, Y, cert M merEK onlyEnc (priSK (CA i)), Z\<rbrace> \<in> set evs"
weakBisimParPresAux: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>R :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi"   and A\<^sub>R :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q"   and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q" shows "\<Psi> \<rhd> P \<parallel> R \<approx> Q \<parallel> R"
closed\<^sub>T1: shows "T\<^sub>1 closed_in \<Gamma>"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "expr list"   and a3 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a4 :: "expr list"   and a5 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,a1 \<turnstile> \<langle>a2,a3\<rangle> [\<Rightarrow>] \<langle>a4,a5\<rangle>" obtains E :: "char list \<Rightarrow> ty option" and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "a1 = E__" and "a2 = []" and "a3 = s" and "a4 = []" and "a5 = s"   | E :: "char list \<Rightarrow> ty option" and e :: "expr" and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and v :: "val" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es :: "expr list" and es' :: "expr list" and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" where "a1 = Ea__" and "a2 = e__ # es__" and "a3 = s\<^sub>0__" and "a4 = Val v # es'" and "a5 = s\<^sub>2" and "P,Ea__ \<turnstile> \<langle>e__,s\<^sub>0__\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1__\<rangle>" and "P,Ea__ \<turnstile> \<langle>es__,s\<^sub>1__\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   | E :: "char list \<Rightarrow> ty option" and e :: "expr" and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and e' :: "expr" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" and es :: "expr list" where "a1 = E" and "a2 = e # es" and "a3 = s\<^sub>0" and "a4 = throw e' # es" and "a5 = s\<^sub>1" and "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
D5_1: fixes \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and r :: "\<upsilon> \<Rightarrow> j \<Rightarrow> i \<Rightarrow> bool"   and o\<^sub>1 :: "\<nu>"   and x :: "\<kappa>"   and w :: "i" assumes "IsProperInX \<phi>" shows "Some r = d\<^sub>1 (\<^bold>\<lambda>x. \<phi> (x\<^sup>P)) \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<longrightarrow> (o\<^sub>1 \<in> ex1 r w) = [\<phi> x in w]"
TC': fixes x :: "V" shows "TC x = x \<union>\<^sub>\<circ> TC (\<Union>\<^sub>\<circ> x)"
quasi_inverses_are_adjoint_pair: fixes f :: "'a"   and g :: "'a" assumes "quasi_inverses f g" shows "adjoint_pair f g"
sep_conjD: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" shows "\<exists>x y. sep_disj x y \<and> h = plus x y \<and> P x \<and> Q y"
assms_2: shows "c \<turnstile> ev \<mapsto> c''"
afWrite_def: fixes afile :: "nat \<times> (nat \<Rightarrow> byte)"   and byteIndex :: "nat"   and "value" :: "byte" shows "afWrite afile byteIndex value \<equiv> if byteIndex div blockSize < numBlocks then Some (raWrite afile byteIndex value fillByte) else None"
sint_uint: fixes w :: "'a word" shows "sint w = signed_take_bit (LENGTH('a) - Suc 0) (uint w)"
insert_correct: fixes e :: "'a"   and a :: "'b"   and q :: "('a, 'b) SkewBinomialHeap" shows "SkewBinomialHeap_loc.to_mset (SkewBinomialHeap_loc.insert e a q) = SkewBinomialHeap_loc.to_mset q + {#(e, a)#}"
right_mult_one_mat': fixes A :: "'b mat"   and n :: "nat" assumes "dim_col A = n" shows "A * 1\<^sub>m n = A"
fff: fixes n :: "nat" shows "f (f n) < f (Suc n)"
wcode_halt_case: fixes m :: "nat"   and ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main stp = (0, Bk # ires, Bk # Oc # Bk \<up> ln @ Bk # Bk # Oc \<up> Suc rs @ Bk \<up> rn)"
unsigned_of_int: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and k :: "int" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (word_of_int k) = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('b) k))"
nb\<^sub>3: fixes op\<^sub>1 :: "('variable, 'domain) sas_plus_operator"   and op\<^sub>1' :: "('variable \<times> 'domain) strips_operator"   and op\<^sub>2 :: "('variable, 'domain) sas_plus_operator"   and op\<^sub>2' :: "('variable \<times> 'domain) strips_operator" assumes "op\<^sub>1 \<in> set (map (strips_op_to_sasp \<Psi>) ops')"   and "op\<^sub>1' \<in> set ops'"   and "op\<^sub>1' = \<phi>\<^sub>O \<Psi> op\<^sub>1"   and "is_valid_operator_sas_plus \<Psi> op\<^sub>1"   and "op\<^sub>2 \<in> set (map (strips_op_to_sasp \<Psi>) ops')"   and "op\<^sub>2' \<in> set ops'"   and "op\<^sub>2' = \<phi>\<^sub>O \<Psi> op\<^sub>2"   and "is_valid_operator_sas_plus \<Psi> op\<^sub>2" shows "\<forall>(v, a)\<in>set (add_effects_of op\<^sub>1'). \<forall>(v', a')\<in>set (add_effects_of op\<^sub>2'). v \<noteq> v' \<or> a = a'"
is_slim_def: fixes V :: "'a set" shows "H.is_slim V \<equiv> (\<forall>v\<in>pverts H. v \<in> V \<or> in_degree (with_proj H) v \<le> 2 \<and> (\<exists>x p y. H.gen_iapath V x p y \<and> v \<in> set (pawalk_verts x p))) \<and> (\<forall>e\<in>parcs H. fst e \<noteq> snd e \<and> (\<exists>x p y. H.gen_iapath V x p y \<and> e \<in> set p)) \<and> (\<forall>u v p q. H.gen_iapath V u p v \<and> H.gen_iapath V u q v \<longrightarrow> p = q) \<and> V \<subseteq> pverts H"
p: shows "p represents x \<and> irreducible p \<and> 0 < lead_coeff p"
almost: assumes "Vagree \<nu> \<nu>' (FVT (Times t1 t2))" shows "frechet I (Times t1 t2) (fst \<nu>) (snd \<nu>) = frechet I (Times t1 t2) (fst \<nu>') (snd \<nu>')"
simps: fixes I :: "'b set"   and v :: "nat \<Rightarrow> 'c"   and as :: "('b \<times> 'c atom) set" shows "(I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>a\<^sub>s (snd ` (as \<inter> I \<times> UNIV))"
l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M append_child insert_before"
ipurge_l_def: fixes execs :: "'dom_t \<Rightarrow> 'action_t list list"   and u :: "'dom_t" shows "ipurge_l execs u \<equiv> \<lambda>d. if intermediary d u then [] else if d = u then remove_gateway_communications u (execs u) else execs d"
assms_4: shows "trm_rep (t \<lhd> \<sigma>) S \<noteq> t \<lhd> \<sigma>"
sr: shows "sgn (poly (real_of_int_poly p) (real_of_1 (p, l, r) + d)) = sgn (poly (real_of_int_poly p) (real_of_rat r))"
compl_one: shows "NOT (- (1::'a)) = (0::'a)"
UN_absorb: fixes k :: "'e"   and I :: "'e set"   and A :: "'e \<Rightarrow> 'f set" assumes "k \<in> I" shows "A k \<union> \<Union> (A ` I) = \<Union> (A ` I)"
mult_onel: fixes x :: "'a" shows "(1::'a) \<odot> x = x"
assms_2: fixes xs :: "nat list" assumes "length xs = arity f" shows "eval f xs = eval g xs"
eventually_nhds_metric: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "class.metric_space dist uniformity open" shows "eventually P (topological_space.nhds open a) = (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
d_cp0: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "d\<^sub>x X \<tau> = d\<^sub>x (\<lambda>_. X \<tau>) \<tau>"
rel_distinct_2: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and y21 :: "'e"   and y22 :: "'e list" shows "\<not> list_all2 R (y21 # y22) []"
hv_eq_zero: fixes v :: "(real, 7) vec" shows "(Hv v = 0) = (v = 0)"
S4: shows "c_fst (list_to_nat ls - 1) = length ls - 1"
simps_3: shows "negUni FalseFUni = TrueFUni"
smult_minus_right: fixes a :: "'a"   and p :: "'a poly" shows "Polynomial.smult a (- p) = - Polynomial.smult a p"
same_limit_imp_Gromov_product_tendsto_infinity: fixes z :: "'a Gromov_completion"   and u :: "nat \<Rightarrow> 'a"   and v :: "nat \<Rightarrow> 'a"   and C :: "real"   and a :: "'a" assumes "z \<in> Gromov_boundary"   and "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> z"   and "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> z" shows "\<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. C \<le> Gromov_product_at a (u n) (v m)"
left_le_key: fixes c :: "color"   and lt :: "('b, 'c) RBT_Impl.rbt"   and k :: "'b"   and v :: "'c"   and rt :: "('b, 'c) RBT_Impl.rbt"   and x :: "'b" assumes "rbt_sorted (Branch c lt k v rt)"   and "x \<in> set (RBT_Impl.keys lt)" shows "x \<le> k"
elims_2: fixes x :: "nat list \<Rightarrow> bool"   and xa :: "nat \<times> nat list" assumes "x abc_holds_for xa" obtains P :: "nat list \<Rightarrow> bool" and s :: "nat" and lm :: "nat list" where "x = P" and "xa = (s, lm)" and "P lm"
inv_def: shows "inv \<equiv> \<lambda>x. if x \<le> 0 then \<infinity> else inverse x"
uadmit_dterm_ntadjoint': fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<theta> :: "('sf + 'b, 'sz) trm" assumes "\<And>i. dsafe (\<sigma> i)"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT \<theta>}. FVT (\<sigma> i))"   and "dsafe \<theta>" shows "dterm_sem (adjointFO I \<sigma> \<nu>) \<theta> = dterm_sem (adjointFO I \<sigma> \<omega>) \<theta>"
antisym: fixes y :: "'a list"   and x :: "'a list" assumes "y \<le>lex x"   and "x \<le>lex y" shows "x = y"
card_Plus_conv_if: fixes A :: "'a set"   and B :: "'b set" shows "card (A <+> B) = (if finite A \<and> finite B then card A + card B else 0)"
If: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and b :: "state \<Rightarrow> bool"   and c1 :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c2 :: "com" assumes "C \<turnstile> {\<lambda>z s. P z s \<and> b s} c1 {Q}"   and "C \<turnstile> {\<lambda>z s. P z s \<and> \<not> b s} c2 {Q}" shows "C \<turnstile> {P} IF b THEN c1 ELSE c2 {Q}"
add_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.comm_semiring_1_cancel_crossproduct plus minus zero times one" shows "(b = plus b a) = (a = zero)"
plus_up_def: fixes prec :: "nat"   and x :: "real"   and y :: "real" shows "plus_up prec x y = truncate_up prec (x + y)"
is_open_from_ind_is_open: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and U :: "'c set" assumes "ind_topology X is_open S"   and "is_open S"   and "ind_topology.ind_is_open X is_open S U" shows "is_open U"
p: shows "p = smult c (\<Prod>a\<in>A. [:- a, 1::'a:] ^ m a)"
elims_2: fixes x :: "'a list" assumes "sorted x" obtains "xb__ = []"   | x :: "'a" and ys :: "'a list" where "xb__ = x # ys" and "Ball (set ys) ((\<le>) x) \<and> sorted ys"
combining_to_smash_R: fixes p :: "real poly"   and qs1 :: "real poly list"   and m1 :: "rat mat"   and sub1 :: "(nat list \<times> nat list) list"   and sgn1 :: "rat list list"   and qs2 :: "real poly list"   and m2 :: "rat mat"   and sub2 :: "(nat list \<times> nat list) list"   and sgn2 :: "rat list list" shows "combine_systems_R p (qs1, m1, sub1, sgn1) (qs2, m2, sub2, sgn2) = smash_systems_R p qs1 qs2 sub1 sub2 sgn1 sgn2 m1 m2"
red2_in_Hom: fixes a :: "'a local.term"   and b :: "'a local.term" assumes "Ide a"   and "Diag a"   and "Ide b"   and "Diag b" shows "a \<^bold>\<Down> b \<in> Hom (a \<^bold>\<otimes> b) \<^bold>\<lfloor>a \<^bold>\<otimes> b\<^bold>\<rfloor>"
rep_eq: fixes x :: "real"   and xa :: "'a \<Rightarrow>\<^sub>L 'b" shows "blinfun_apply (x *\<^sub>R xa) = (\<lambda>xb. x *\<^sub>R blinfun_apply xa xb)"
inj_on_impl_disjoint_family_on_singleton: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "disjoint_family_on (\<lambda>x. {f x}) A"
dom_eqI: fixes a :: "'a"   and f :: "'a" assumes "Right_a.ide a"   and "Right_a.seq f a" shows "Right_a.dom f = a"
lunit'_in_hom_2: fixes f :: "'a" assumes "ide f" shows "\<guillemotleft>\<l>\<^sup>-\<^sup>1[f] : f \<Rightarrow> trg f \<star> f\<guillemotright>"
is_aligned_shiftl_self: fixes p :: "'b word"   and n :: "nat" shows "is_aligned (p << n) n"
gen_igbg_impl_rel_eext_def: fixes Rm :: "('k \<times> 'm) set"   and Racc :: "('j \<times> ('l \<Rightarrow> nat set)) set" shows "\<langle>Rm, Racc\<rangle>gen_igbg_impl_rel_eext = {(\<lparr>igbgi_num_acc = num_acci, igbgi_acc = acci, \<dots> = mi\<rparr>, \<lparr>igbg_num_acc = num_acc, igbg_acc = acc, \<dots> = m\<rparr>) |num_acci acci mi num_acc acc m. (num_acci, num_acc) \<in> nat_rel \<and> (acci, acc) \<in> Racc \<and> (mi, m) \<in> Rm}"
BVG_test: fixes \<phi> :: "fml" shows "BVG (? \<phi>) = {}"
ae: shows "AE x in M. (x \<in> A) = (x \<in> B)"
eqD_14: fixes y :: "cmsg"   and A :: "agent" assumes "eq y (cAgent A)" shows "y = cAgent A"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a"   and trm :: "trm" assumes "\<And>name coname z. P z (Ax name coname)"   and "\<And>coname trm1 name trm2 z. \<lbrakk>coname \<sharp> z; name \<sharp> z; coname \<sharp> trm2; name \<sharp> trm1; \<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (Cut <coname>.trm1 name.trm2)"   and "\<And>name trm coname z. \<lbrakk>name \<sharp> z; name \<sharp> coname; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (NotR name.trm coname)"   and "\<And>coname trm name z. \<lbrakk>coname \<sharp> z; coname \<sharp> name; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (NotL <coname>.trm name)"   and "\<And>coname1 trm1 coname2 trm2 coname3 z. \<lbrakk>coname1 \<sharp> z; coname2 \<sharp> z; coname2 \<noteq> coname1; coname1 \<sharp> trm2; coname1 \<sharp> coname3; coname2 \<sharp> trm1; coname2 \<sharp> coname3; \<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (AndR <coname1>.trm1 <coname2>.trm2 coname3)"   and "\<And>name1 trm name2 z. \<lbrakk>name1 \<sharp> z; name1 \<sharp> name2; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (AndL1 name1.trm name2)"   and "\<And>name1 trm name2 z. \<lbrakk>name1 \<sharp> z; name1 \<sharp> name2; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (AndL2 name1.trm name2)"   and "\<And>coname1 trm coname2 z. \<lbrakk>coname1 \<sharp> z; coname1 \<sharp> coname2; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (OrR1 <coname1>.trm coname2)"   and "\<And>coname1 trm coname2 z. \<lbrakk>coname1 \<sharp> z; coname1 \<sharp> coname2; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (OrR2 <coname1>.trm coname2)"   and "\<And>name1 trm1 name2 trm2 name3 z. \<lbrakk>name1 \<sharp> z; name2 \<sharp> z; name2 \<noteq> name1; name1 \<sharp> trm2; name1 \<sharp> name3; name2 \<sharp> trm1; name2 \<sharp> name3; \<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (OrL name1.trm1 name2.trm2 name3)"   and "\<And>name coname1 trm coname2 z. \<lbrakk>name \<sharp> z; coname1 \<sharp> z; name \<sharp> coname2; coname1 \<sharp> coname2; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (ImpR name.<coname1>.trm coname2)"   and "\<And>coname trm1 name1 trm2 name2 z. \<lbrakk>coname \<sharp> z; name1 \<sharp> z; coname \<sharp> trm2; coname \<sharp> name2; name1 \<sharp> trm1; name1 \<sharp> name2; \<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (ImpL <coname>.trm1 name1.trm2 name2)" shows "P z trm"
inter_fset: fixes xs :: "'b fset"   and ys :: "'b fset" shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
B_def: fixes m :: "'b option"   and \<alpha> :: "'b \<Rightarrow> bool" shows "m \<rightharpoonup> \<alpha> \<equiv> case m of None \<Rightarrow> True | Some a \<Rightarrow> \<alpha> a"
restrict_SN_trancl_simp: fixes A :: "('b \<times> 'b) set" shows "(restrict_SN A A)\<^sup>+ = restrict_SN (A\<^sup>+) A"
extremum_uniqueI: fixes a :: "'b list" assumes "a \<le>p \<epsilon>" shows "a = \<epsilon>"
distinct_load_tmps_append: fixes xs :: "instr list"   and ys :: "instr list" shows "distinct_load_tmps (xs @ ys) = (distinct_load_tmps xs \<and> distinct_load_tmps ys \<and> load_tmps xs \<inter> load_tmps ys = {})"
iT_div_5: fixes k :: "nat"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < k"   and "m mod k = 0" shows "[ r, mod m, c ] \<oslash> k = [ r div k, mod m div k, c ]"
fresh_fmdrop_fset: fixes x :: "atom"   and A :: "'a fset"   and B :: "('a, 'b) fmap" assumes "x \<sharp> A"   and "x \<sharp> B" shows "x \<sharp> fmdrop_fset A B"
Lim_trivial_limit: fixes net :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and l :: "'d" assumes "net = bot" shows "(f \<longlongrightarrow> l) net"
nres_order_simps_20: fixes x :: "'c"   and y :: "'c" shows "(RETURN x \<le> RETURN y) = (x = y)"
\<Z>_\<omega>\<omega>: shows "\<Z> (\<omega> + \<omega>)"
preserves_iso: fixes f :: "'a" assumes "B.iso f" shows "B.iso (src\<^sub>B f)"
devBC_act: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat" assumes "\<not> Option.is_none (devBC t n nid n')" shows "\<parallel>the (devBC t n nid n')\<parallel>\<^bsub>t n'\<^esub>"
vimage_empty: fixes f :: "'c \<Rightarrow> 'd" shows "f -` {} = {}"
grep3_def: shows "grep3 \<equiv> prod.snd (prod.snd (prod.snd (fix_syn (\<lambda>(root3, op3, rep3, grep3). (\<Lambda> pat. rep3\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root3\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op3\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. grep3\<cdot>pat\<cdot>(case cfst\<cdot>usvs of [::] \<Rightarrow> Null | u :# us \<Rightarrow> sfoldl\<cdot>(op3\<cdot>pat)\<cdot>(root3\<cdot>pat)\<cdot>us)\<cdot>usvs, \<Lambda> pat l usvs. Node\<cdot>usvs\<cdot>(next\<cdot>(csnd\<cdot>usvs)\<cdot>l)\<cdot>(case csnd\<cdot>usvs of [::] \<Rightarrow> Null | v :# vs \<Rightarrow> rep3\<cdot>pat\<cdot>(cfst\<cdot>usvs :@ [:v:], vs)))))))"
front_tickFree_def: fixes s :: "'a event list" shows "front_tickFree s = (s = [] \<or> tickFree (tl (rev s)))"
lim_mono: fixes N :: "nat"   and X :: "nat \<Rightarrow> 'a"   and Y :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "\<And>n. N \<le> n \<Longrightarrow> X n \<le> Y n"   and "X \<longlonglongrightarrow> x"   and "Y \<longlonglongrightarrow> y" shows "x \<le> y"
subexp_induct: fixes R :: "'c exp \<Rightarrow> bool"   and Rs :: "'c exp list \<Rightarrow> bool"   and e :: "'c exp" assumes "\<And>e. subexp e = {} \<Longrightarrow> R e"   and "\<And>e. (\<And>e'. subexp_of e' e \<Longrightarrow> R e') \<Longrightarrow> R e"   and "\<And>es. (\<And>e'. e' \<in> subexps es \<Longrightarrow> R e') \<Longrightarrow> Rs es" shows "(\<forall>e'. subexp_of e' e \<longrightarrow> R e') \<and> R e"