hcju/isabelleps huggingface dataset:
abstract_primitive_in_doubt_deny_Deny2: fixes \<beta> :: "'c common_primitive \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'c common_primitive match_expr"   and p :: "('c, 'd) tagged_packet_scheme"   and disc :: "'c common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "\<not> matches (\<beta>, in_doubt_deny) m Accept p" shows "\<not> matches (\<beta>, in_doubt_deny) (abstract_primitive disc m) Accept p"
pcr_mapping_def: fixes T :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and Ta :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_mapping T Ta \<equiv> rel_fun T (rel_option Ta) OO cr_mapping"
in_hhomI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "B.arr \<mu>"   and "src\<^sub>B \<mu> = a"   and "trg\<^sub>B \<mu> = b" shows "B.in_hhom \<mu> a b"
\<phi>_img_\<psi>_img: fixes A' :: "'s set" assumes "A' \<subseteq> S.Univ" shows "\<psi> ` INV.\<psi> ` A' = A'"
preserves_dom: fixes f :: "'a" assumes "B.arr f" shows "B.dom (B.\<ll> f) = B.L (B.dom f)"
cond_prob_def: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "cond_prob M P Q = \<P>(\<omega> in M. P \<omega> \<and> Q \<omega>) / \<P>(\<omega> in M. Q \<omega>)"
distinct_adj_singleton: fixes x :: "'b" shows "distinct_adj [x]"
sym_transitive: shows "transitive A (sympartp (\<sqsubseteq>))"
lock_ok_las_may_lock: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and las :: "'a lock_actions"   and l :: "'a" assumes "lock_ok_las ls t las"   and "Lock \<in> set (las $ l)" shows "may_lock (ls $ l) t"
expands_to_uminus: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis" shows "((\<lambda>x. - f x) expands_to - F) basis"
jordan_nf_exists: fixes A :: "'b mat"   and n :: "nat"   and as :: "'b list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>as. [:- a, 1::'b:])" shows "\<exists>n_as. jordan_nf A n_as"
tee_local: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (local \<C>)"   and "local \<C> ! i = t" shows "\<C> \<turnstile> [Tee_local i] : [t] _> [t]"
mkArr_in_hom: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "A \<subseteq> Univ"   and "|A| <o \<AA>"   and "B \<subseteq> Univ"   and "|B| <o \<AA>"   and "F \<in> A \<rightarrow> B" shows "\<guillemotleft>mkArr A B F : mkIde A \<rightarrow> mkIde B\<guillemotright>"
cobounded1: fixes a :: "'b"   and b :: "'b" shows "a \<le> a \<squnion> b"
W_consts_4: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n false\<^sub>n = w \<Turnstile>\<^sub>n G\<^sub>n \<phi>"
msort_key_stable: fixes k :: "'k"   and xs :: "'a list" shows "filter (\<lambda>x. key x = k) (local.msort_key xs) = filter (\<lambda>x. key x = k) xs"
ownership_distinct_instructions_read_value_store_buffer_independent: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and sb' :: "'a memref list"   and p' :: "'a"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "all_acquired sb' \<subseteq> all_acquired sb" shows "ownership_distinct (ts\<^sub>s\<^sub>b[i := (p', is', \<theta>', sb', \<D>', \<O>, \<R>')])"
assms: shows "wfp_on P\<^sup>+\<^sup>+ A"
direct_prod_mem_eq: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>j\<in>I. aGroup (A j)"   and "f \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A)"   and "g \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A)"   and "\<forall>j\<in>I. (\<pi>\<^bsub>I,A,j\<^esub>) f = (\<pi>\<^bsub>I,A,j\<^esub>) g" shows "f = g"
targetsI: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "weak_composition.weak_unit V H b"   and "H b \<mu> \<noteq> partial_magma.null V" shows "b \<in> weak_composition.targets V H \<mu>"
norm_eq_sqrt_cinner: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and x :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "norm x = sqrt (cmod (cinner x x))"
linord_of_list_antisym: fixes xs :: "'b list" assumes "distinct xs" shows "antisym (linord_of_list xs)"
HSem_subst_expr: fixes e1 :: "exp"   and x :: "Vars.var"   and e2 :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "\<lbrakk> e1 \<rbrakk>\<^bsub>\<lbrace>(x, e2) # \<Gamma>\<rbrace>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e2 \<rbrakk>\<^bsub>\<lbrace>(x, e2) # \<Gamma>\<rbrace>\<rho>\<^esub>"   and "\<lbrakk> e2 \<rbrakk>\<^bsub>\<lbrace>(x, e1) # \<Gamma>\<rbrace>\<rho>\<^esub> \<sqsubseteq> \<lbrakk> e1 \<rbrakk>\<^bsub>\<lbrace>(x, e1) # \<Gamma>\<rbrace>\<rho>\<^esub>" shows "\<lbrace>(x, e1) # \<Gamma>\<rbrace>\<rho> = \<lbrace>(x, e2) # \<Gamma>\<rbrace>\<rho>"
lookup_eq_Some_iff: fixes vo :: "'d option"   and kvs :: "('c \<times> ('c, 'd) trie) list"   and ks :: "'c list"   and v :: "'d" assumes "invar_trie (Trie vo kvs)" shows "(lookup_trie (Trie vo kvs) ks = Some v) = (ks = [] \<and> vo = Some v \<or> (\<exists>k t ks'. ks = k # ks' \<and> (k, t) \<in> set kvs \<and> lookup_trie t ks' = Some v))"
r_list_encode_prim: fixes n :: "nat" shows "prim_recfn (Suc n) (r_list_encode n)"
sort_lt_var_INNER: shows "|UNIV| <o |UNIV|"
boundedUn: fixes A :: "(nat \<times> 'b) set"   and B :: "(nat \<times> 'b) set" shows "bounded (A \<union> B) = (bounded A \<and> bounded B)"
mult_cancel_left: fixes f1 :: "'d \<Rightarrow> 'e"   and F :: "'d filter"   and g1 :: "'d \<Rightarrow> 'e"   and f2 :: "'d \<Rightarrow> 'e"   and g2 :: "'d \<Rightarrow> 'e" assumes "f1 \<in> \<Theta>[F](g1)"   and "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'e)" shows "((\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)) = (f2 \<in> O[F](g2))"
map_disc_iff: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d llist" shows "lnull (lmap f a) = lnull a"
TUadmit_def: fixes \<sigma> :: "('a, 'b, 'c) subst"   and \<theta> :: "('a, 'c) trm"   and U :: "('c + 'c) set" shows "TUadmit \<sigma> \<theta> U = ((\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some f' \<Rightarrow> FVT f') \<inter> U = {})"
sqrt_sum_squares_le_sum_abs: fixes x :: "real"   and y :: "real" shows "sqrt (x\<^sup>2 + y\<^sup>2) \<le> \<bar>x\<bar> + \<bar>y\<bar>"
that_2: fixes a :: "'a"   and dl :: "'a dlist" assumes "y = DList.insert a dl" shows "thesis"
IH: assumes "infinite S" shows "card {a \<in> S. a < pick S n} = n"
hyps: assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> rel x y \<or> rel y x"   and "z \<in> set xs" shows "rel (min_list_param rel xs) z"
infinite_set_mset_mset_set: fixes A :: "'c set" assumes "infinite A" shows "set_mset (mset_set A) = {}"
h10: shows "(\<exists>(a', b', c')\<in>set d. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>- c' / b'. \<forall>x\<in>{- c' / b'<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0)) \<longrightarrow> (\<exists>x. (\<forall>(a, b, c)\<in>set a. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set b. a * x\<^sup>2 + b * x + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. a * x\<^sup>2 + b * x + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. a * x\<^sup>2 + b * x + c \<noteq> 0))"
a_Auth: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : \<tau>" shows "\<Gamma> \<turnstile> Auth e, Auth eP, Auth eV : AuthT \<tau>"
sequential_interchange: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d\<^sub>0 :: "'a::refinement_lattice"   and d\<^sub>1 :: "'a::refinement_lattice" shows "(c\<^sub>0::'a::refinement_lattice) ; (c\<^sub>1::'a::refinement_lattice) \<iinter> (d\<^sub>0::'a::refinement_lattice) ; (d\<^sub>1::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>0 \<iinter> d\<^sub>0) ; (c\<^sub>1 \<iinter> d\<^sub>1)"
a0: shows "local.adverse 0 i j (Suc xmin) \<down>= 0"
sSplice_es: fixes a :: "'a"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and s' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" shows "es (sSplice a s s') = es s"
preserves_runit_2: fixes f :: "'a" assumes "C.ide f" shows "F \<r>\<^sub>C\<^sup>-\<^sup>1[f] = \<Phi> (f, src\<^sub>C f) \<cdot>\<^sub>D (F f \<star>\<^sub>D unit (src\<^sub>C f)) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[F f]"
src_in_sdnets: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "\<not> member DenyAll x"   and "p \<in> dom (Cp x)" shows "subnetsOfAdr (src p) \<inter> fst_set (sdnets x) \<noteq> {}"
a: shows "k \<otimes> a \<oplus> combine Ks Us = \<zero>"
length_rev: fixes xs :: "'k list" shows "length (rev xs) = length xs"
rep_Gromov_completion_def: shows "rep_Gromov_completion \<equiv> quot_type.rep Rep_Gromov_completion"
rep_eq: fixes x :: "'a"   and xa :: "'a multiset" shows "count (add_mset x xa) = (\<lambda>b. if b = x then Suc (count xa b) else count xa b)"
non_trivial_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.time plus zero minus uminus less_eq less" shows "\<exists>x. less x zero"
unit_div_1_unit: 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 a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "dvd.dvd times (divide one a) one"
u\<^sub>\<tau>: shows "ide (u \<star> s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0)"
allUses_def_ass: fixes v :: "'val"   and g :: "'g"   and n :: "'node" assumes "v \<in> allUses g n"   and "n \<in> set (\<alpha>n g)" shows "defAss g n v"
map\<^sub>0_simps_3: fixes a :: "'c" assumes "C.obj a" shows "trg\<^sub>D (G.map\<^sub>0 a) = G.map\<^sub>0 a"
cmisi: shows "m\<inverse> O s\<inverse> \<subseteq> b\<inverse>"
sum_sqs_eq: fixes x :: "'a"   and y :: "'a" assumes "x * x + y * y = x * (y * (2::'a))" shows "y = x"
blinfun_apply_id_blinfun: shows "blinfun_apply 1\<^sub>L = (\<lambda>x. x)"
1: shows "YFtorNT C f \<in> mor\<^bsub>CatExp (Op C) SET\<^esub>"
regular_cover_4: fixes i :: "nat" assumes "i \<in> N" shows "codomain (\<psi> i) = ball (0::'b) 3"
trg_cod: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg\<^sub>B (B.cod \<mu>) = trg\<^sub>B \<mu>"
single_valued_insert: fixes rel :: "('a \<times> 'b) set"   and a :: "'a"   and b :: "'b" assumes "single_valued rel"   and "\<And>x y. \<lbrakk>(x, y) \<in> rel; x = a\<rbrakk> \<Longrightarrow> y = b" shows "single_valued (insert (a, b) rel)"
clinit: shows "P \<turnstile> C' sees clinit, Static :  []\<rightarrow>Void = (mxs', mxl', ins', xt') in C'"
atomize_eq: fixes x :: "'a"   and y :: "'a" shows "(x \<equiv> y) \<equiv> x = y"
h2: shows "h \<turnstile> set_disconnected_nodes owner_document (child # disc_nodes) \<rightarrow>\<^sub>h h2"
e_type_callcl_native: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and cl :: "cl"   and t1s' :: "t list"   and t2s' :: "t list"   and i :: "nat"   and tf :: "tf"   and ts :: "t list"   and es :: "b_e list" assumes "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : t1s' _> t2s'"   and "cl = Func_native i tf ts es" shows "\<exists>t1s t2s ts_c. t1s' = ts_c @ t1s \<and> t2s' = ts_c @ t2s \<and> tf = t1s _> t2s \<and> i < length (s_inst \<S>) \<and> (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ t1s @ ts, label := [t2s] @ label (s_inst \<S> ! i), return := Some t2s\<rparr> \<turnstile> es : [] _> t2s"
assms: shows "{x \<in> space (M1 \<Otimes>\<^sub>M M2). P (fst x) (snd x)} \<in> sets (M1 \<Otimes>\<^sub>M M2)"
step: shows "(is, \<theta>, sb, m, \<D>, \<O>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<S>')"
simps_1: fixes uu :: "'b ext" shows "(ext.Bot \<le> uu) = True"
from_mregex_to_mregex: assumes "safe_regex m g r" shows "(case to_mregex r of (x, xa) \<Rightarrow> from_mregex x xa) = r"
le_supI: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and x :: "'e"   and b :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq a x"   and "less_eq b x" shows "less_eq (sup a b) x"
equivp_gen_cong: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr" shows "equivp (cong.gen_cong rel eval R)"
Subst: fixes xvec :: "name list"   and Tvec :: "'a list"   and \<Psi> :: "'b" assumes "length xvec = length Tvec" shows "(\<Psi>, P[xvec::=Tvec], \<tau>.Q[xvec::=Tvec]) \<in> Rel"
intros_4: fixes a :: "'d"   and b :: "'d" assumes "a < b" shows "Le a \<prec> Lt b"
node_ptr_kinds_eq_h: shows "|h \<turnstile> node_ptr_kinds_M|\<^sub>r = |h2 \<turnstile> node_ptr_kinds_M|\<^sub>r"
less_llist_def: fixes xs :: "'b llist"   and ys :: "'b llist" shows "(xs < ys) = lstrict_prefix xs ys"
ipresIGVar_def: fixes h :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'g, 'h, 'i, 'j, 'c, 'k, 'l) model_scheme"   and MOD' :: "('m, 'n, 'g, 'o, 'p, 'j, 'd, 'q, 'r) model_scheme" shows "ipresIGVar h MOD MOD' \<equiv> \<forall>xs x. h (igVar MOD xs x) = igVar MOD' xs x"
2: shows "ground (subst_range (rm_vars (UNIV - X) \<I>))"
STA_bridge: fixes A :: "'b set"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and k :: "'b"   and i :: "'b"   and xs :: "(nat \<times> nat) set"   and j :: "'b" assumes "infinite UNIV"   and "A \<turnstile> branch"   and "descendants k i branch xs"   and "Nom k at j in' set branch"   and "finite A"   and "j \<in> A" shows "A \<turnstile> mapi_branch (bridge k j xs) branch"
lead_coeff_pCons_1: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "lead_coeff (pCons a p) = lead_coeff p"
prems_2: shows "r \<rightarrow>\<^sub>\<beta> s"
value_unlift: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a node \<times> 'a node \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>"   and "flow (flow_attainability.\<Delta>' \<Delta>) f" shows "value_flow \<Delta> (\<lambda>(x, y). f (Inner x, Inner y)) = value_flow (flow_attainability.\<Delta>' \<Delta>) f"
scale_left_distrib: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c"   and x :: "'d" assumes "module scale" shows "scale (a + b) x = scale a x + scale b x"
span_diff: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "module scale"   and "x \<in> module.span scale S"   and "y \<in> module.span scale S" shows "x - y \<in> module.span scale S"
assm: shows "\<parallel>len v ts c\<parallel> = 0 \<and> v=v1\<parallel>v2"
hyps: fixes u :: "'a" assumes "awalk u as v"   and "awalk u bs w"   and "hd as = hd bs"   and "as \<noteq> []"   and "bs \<noteq> []"   and "progressing as"   and "progressing bs"   and "set (inner_verts as) \<inter> V = {}"   and "set (inner_verts bs) \<inter> V = {}" shows "as = bs"
choose_arc_regular: fixes x :: "'a" shows "regular (choose_arc x)"
minusinf_bex: fixes p :: "fm"   and a :: "int"   and bs :: "real list" assumes "iszlfm p (real_of_int a # bs)" shows "(\<exists>x. Ifm (real_of_int x # bs) (minusinf p)) = (\<exists>x\<in>{1..\<delta> p}. Ifm (real_of_int x # bs) (minusinf p))"
induct: fixes P :: "(char list \<times> 'a) list \<Rightarrow> bool"   and a0 :: "(char list \<times> 'a) list" assumes "P []"   and "\<And>x a ps. P ps \<Longrightarrow> P ((x, a) # ps)" shows "P a0"
dual_lessI: fixes y :: "'a dual"   and x :: "'a dual" assumes "undual y < undual x" shows "x < y"
cases: fixes x :: "'o \<times> ('a \<times> 'c) list \<times> 'o" obtains ko :: "'o" and xs :: "('a \<times> 'c) list" and ox :: "'o" where "x = (ko, xs, ox)"
prime_int_nat_transfer: fixes k :: "int" shows "prime k = (0 \<le> k \<and> prime (nat k))"
snth_siterate: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and n :: "nat" shows "siterate f x !! n = (f ^^ n) x"
ideal_view_2_def: fixes x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'e"   and S :: "((bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf) \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> (bool \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ((bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf) \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> 'grp \<Rightarrow> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f \<Rightarrow> 'd spmf)"   and \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf)" shows "mal_def.ideal_view_2 x y z S \<A> = (let (S1, S2) = S in mal_def.ideal_game_2 x y z (S1 \<A>, S2 \<A>))"
mtx_vec_mult_add_rdistl: fixes A :: "'a sq_mtx"   and x :: "(real, 'a) vec"   and y :: "(real, 'a) vec" shows "A *\<^sub>V (x + y) = A *\<^sub>V x + A *\<^sub>V y"
WCR_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>a b c. \<lbrakk>a \<in> A; (a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> (b, c) \<in> r\<^sup>\<down>" shows "WCR_on r A"
leg1_composite: shows "fgh.\<mu>\<nu>.leg1 = fgh.\<mu>.leg1 \<cdot> fgh.\<mu>\<nu>.prj\<^sub>1"
derives1_def: fixes u :: "symbol list"   and v :: "symbol list" shows "derives1 u v = (\<exists>x y N \<alpha>. u = x @ [N] @ y \<and> v = x @ \<alpha> @ y \<and> is_sentence x \<and> is_sentence y \<and> (N, \<alpha>) \<in> \<RR>)"
metric_CauchyD: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'd"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "uniform_space.Cauchy uniformity X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
finsum_vec_empty: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b vec" shows "finsum_vec TYPE('b) n f {} = 0\<^sub>v n"
find_consistent_signs_at_roots_R_def: fixes p :: "real poly"   and qs :: "real poly list" shows "find_consistent_signs_at_roots_R p qs = (let (M, S, \<Sigma>) = calculate_data_R p qs in \<Sigma>)"
top_unique: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and top :: "'k"   and a :: "'k" assumes "class.order_top less_eq less top" shows "less_eq top a = (a = top)"
wens_single_finite_Suc: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat" assumes "single_valued act" shows "wens_single_finite act B (Suc k) = wens_single_finite act B k \<union> wp act (wens_single_finite act B k)"
float_defs_14: fixes a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "feq a b = (fcompare a b = Eq)"
cond2a: shows "\<forall>a. k < mod_type_class.to_nat a \<longrightarrow> A $ a $ mod_type_class.from_nat k = (0::'a)"
COD_in_cod: fixes f :: "'c local.term set" assumes "arr f" shows "COD f \<in> cod f"
w_cmpser_is_d_gchain: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "w_cmpser G n f" shows "d_gchain G n f"
i_shrink_eq_i_last_message_hold_shrink_last: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'a message_af" assumes "0 < k" shows "f \<div> k = f \<longmapsto> k \<div>\<^sub>l k"
circline_intersection_at_most_2_points: fixes H1 :: "circline"   and H2 :: "circline" assumes "H1 \<noteq> H2" shows "finite (circline_intersection H1 H2) \<and> card (circline_intersection H1 H2) \<le> 2"
lSup_LCons: fixes A :: "'c llist set"   and x :: "'c" assumes "A \<noteq> {}" shows "lSup (LCons x ` A) = LCons x (lSup A)"
abs_eq: fixes xb :: "'c"   and xa :: "'d"   and x :: "'c \<Rightarrow> 'd option" shows "Mapping.update xb xa (mapping.Mapping x) = mapping.Mapping (x(xb \<mapsto> xa))"
restrict_count_space: fixes B :: "'a set"   and A :: "'a set" shows "restrict_space (count_space B) A = count_space (A \<inter> B)"
assms: shows "Gromov_completion_rel u (\<lambda>n. x)"
d_weakly_future_consistent_def: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and R :: "'b \<Rightarrow> ('a list \<times> 'a list) set" shows "d_weakly_future_consistent P I D R \<equiv> \<forall>u\<in>range D \<inter> (- I) `` range D. \<forall>xs ys. (xs, ys) \<in> R u \<longrightarrow> (xs \<in> traces P) = (ys \<in> traces P) \<and> next_dom_events P D u xs = next_dom_events P D u ys"
is_sup_def: fixes x :: "'a"   and y :: "'a"   and s :: "'a" shows "is_sup x y s = (x \<sqsubseteq> s \<and> y \<sqsubseteq> s \<and> (\<forall>z. x \<sqsubseteq> z \<and> y \<sqsubseteq> z \<longrightarrow> s \<sqsubseteq> z))"
locally_open_map_image: fixes P :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and Q :: "'d set \<Rightarrow> bool" assumes "locally P S"   and "continuous_on S f"   and "\<And>T. openin (top_of_set S) T \<Longrightarrow> openin (top_of_set (f ` S)) (f ` T)"   and "\<And>T. \<lbrakk>T \<subseteq> S; P T\<rbrakk> \<Longrightarrow> Q (f ` T)" shows "locally Q (f ` S)"
square_free_multD_3: fixes f :: "'b poly"   and g :: "'b poly" assumes "square_free (f * g)" shows "square_free g"