hcju/isabelleps huggingface dataset:
vcons_eqD'_1: fixes xs\<^sub>1 :: "V"   and xs\<^sub>2 :: "V"   and x\<^sub>1 :: "V"   and x\<^sub>2 :: "V" assumes "vfsequence xs\<^sub>1"   and "vfsequence xs\<^sub>2"   and "xs\<^sub>1 #\<^sub>\<circ> x\<^sub>1 = xs\<^sub>2 #\<^sub>\<circ> x\<^sub>2" shows "xs\<^sub>1 = xs\<^sub>2"
Rep_seqauto: fixes x :: "('d, 'e, 'f) seqauto" shows "Rep_seqauto x \<in> seqauto"
lemmaIOD: fixes U :: "'a set set"   and c :: "'a set"   and d :: "'a set" assumes "\<exists>T. T \<in> U \<and> c \<subseteq> T"   and "\<exists>T. T \<in> U \<and> d \<subseteq> T"   and "\<Union> U = c \<union> d"   and "\<And>T. T \<in> U \<Longrightarrow> T \<noteq> {}"   and "disjoint U"   and "\<nexists>T. U \<subseteq> {T}" shows "c \<in> U"
language_def: shows "Accessible.language \<equiv> {xs. Accessible.nextl (init Accessible_dfa) xs \<in> final Accessible_dfa}"
sum_power2: fixes k :: "nat" shows "sum ((^) 2) {0..<k} = 2 ^ k - 1"
inter_line_comm_line: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" assumes "A \<noteq> B" shows "Pascal_Property.inter (line A B) (line C D) = Pascal_Property.inter (line B A) (line C D)"
eq_f_restr_conv: fixes s :: "'c set"   and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and A :: "'c \<Rightarrow> 'd option" shows "(s \<subseteq> dom (f A) \<and> A = f A |` (- s)) = (A \<subseteq>\<^sub>m f A \<and> s = dom (f A) - dom A)"
sscan_snth: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b"   and i :: "nat" shows "sscan f xs a !! i = fold f (stake (Suc i) xs) a"
raw_converter_invariant_run_plossless_converter: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>" shows "raw_converter_invariant \<I> \<I>' run_converter (\<lambda>conv. plossless_converter \<I> \<I>' conv \<and> \<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>)"
d: fixes t :: "'b"   and f :: "'b" shows "(if d = 0 then t else f) = f"
is_Arg_def: fixes z :: "complex"   and r :: "real" shows "is_Arg z r \<equiv> z = complex_of_real (cmod z) * exp (\<i> * complex_of_real r)"
overlap_extensionality: fixes x :: "i"   and y :: "i" assumes "EM" shows "(x = y) = (\<forall>z. O z x = O z y)"
sign_def: fixes literal :: "'a literal" shows "sign literal = (\<exists>x11 x12. literal = Pos x11 x12)"
IHL: shows "real (time (mergesort_tm f l)) \<le> mergesort_recurrence (length l)"
opnet_deliver1: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and i :: "nat"   and d :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), i:deliver(d), \<sigma>', s') \<in> S" shows "((\<sigma>, SubnetS s t), i:deliver(d), \<sigma>', SubnetS s' t) \<in> opnet_sos S T"
glitOfC_def: fixes c :: "('fsym, 'psym) lit list" shows "glitOfC c \<equiv> \<Union> (set (map glitOfL c))"
vimage2p_mono: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'h"   and R :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'e"   and y :: "'g"   and S :: "'f \<Rightarrow> 'h \<Rightarrow> bool" assumes "vimage2p f g R x y"   and "R \<le> S" shows "vimage2p f g S x y"
assms_5: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
vsv_vinsert: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "a \<notin>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "vsv (vinsert \<langle>a, b\<rangle> r)"
rotate1_is_Nil_conv: fixes xs :: "'c list" shows "(rotate1 xs = []) = (xs = [])"
bexI: fixes P :: "'h \<Rightarrow> bool"   and x :: "'h"   and A :: "'h set" assumes "P x"   and "x \<in> A" shows "\<exists>x\<in>A. P x"
segment_subset: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and z :: "'b"   and w :: "'b" assumes "y \<in> segment f x z"   and "w \<in> segment f x y" shows "w \<in> segment f x z"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := P\<rparr>)" shows "Group.group (G\<lparr>carrier := P, carrier := H\<rparr>)"
assms_2: shows "Ftor F : Op C \<longrightarrow> SET"
load_sub2_def: fixes address :: "word32"   and asi :: "word8"   and rd :: "word5"   and curr_win :: "'b word"   and word0 :: "word32" shows "load_sub2 address asi rd curr_win word0 \<equiv> do write_reg word0 curr_win (rd AND 30); (result1, new_state1) \<leftarrow> gets (memory_read asi (address + 4)); if result1 = None then do raise_trap data_access_exception; return () od else do word1 \<leftarrow> gets (\<lambda>s. case result1 of Some v \<Rightarrow> v); modify (\<lambda>s. new_state1); write_reg word1 curr_win (rd OR 1); return () od od"
meas: shows "set_borel_measurable borel {A<..<B} f"
mult_if_delta: fixes P :: "bool"   and q :: "'d" shows "(if P then 1::'d else (0::'d)) * q = (if P then q else (0::'d))"
nth_w2p_same: fixes n :: "nat" shows "(2 ^ n) !! n = (n < LENGTH('b))"
pax3: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> \<circle>\<box>F"
Init_def: fixes CS :: "'b list set set" shows "Init CS \<equiv> {(X, Init_rhs CS X) |X. X \<in> CS}"
hyps: fixes k :: "nat" shows "(1 + fps_X) ^ n $ k = of_nat (n choose k)"
sfirst_finiteI_recurrent: fixes x :: "'s"   and y :: "'s" assumes "recurrent x"   and "(x, y) \<in> acc" shows "AE \<omega> in T x. sfirst (HLD {y}) \<omega> < \<infinity>"
A: assumes "\<exists>u\<in>U. (u, D x) \<in> I" shows "ipurge_fail_aux_t_aux \<lparr>Pol = I, Map = D, Doms = U, List = ipurge_tr I D (D x) xs, ListOp = Some ys, Set = ipurge_ref I D (D x) xs X\<rparr> \<in> ipurge_fail_aux_t_set \<lparr>Pol = I, Map = D, Doms = U, List = ipurge_tr I D (D x) xs, ListOp = Some ys, Set = ipurge_ref I D (D x) xs X\<rparr>"
rev_in_lists: fixes u :: "'b list"   and A :: "'b set" assumes "rev u \<in> lists A" shows "u \<in> lists A"
mult_mono_nonpos_nonneg: fixes a :: "'a"   and c :: "'a"   and d :: "'a"   and b :: "'a" assumes "a \<le> c"   and "a \<le> (0::'a)"   and "(0::'a) \<le> d"   and "d \<le> b" shows "a * b \<le> c * d"
siso_imp_Sbis: fixes c :: "('test, 'atom, 'choice) cmd" assumes "siso c" shows "c \<approx>s c"
one_neq_neg_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "one \<noteq> uminus (numeral.numeral one plus n)"
basic_trans_rules_19: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y < z" shows "x < z"
plist: shows "a \<in> set list \<longrightarrow> prod_list list = prod_list (remove1 a list) * a"
intros: fixes C :: "'b literal multiset"   and n :: "nat"   and A :: "'b"   and D :: "'b literal multiset" shows "unord_resolve (C + replicate_mset (Suc n) (Pos A)) (add_mset (Neg A) D) (C + D)"
leof_RES_rule_1: fixes \<Psi> :: "'c \<Rightarrow> bool"   and \<Phi> :: "'c \<Rightarrow> bool" assumes "\<And>x. \<Psi> x \<Longrightarrow> \<Phi> x" shows "SPEC \<Psi> \<le>\<^sub>n SPEC \<Phi>"
exec_gpv_def: fixes callee :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'f) spmf" shows "exec_gpv callee \<equiv> curry (spmf.fixp_fun (\<lambda>exec_gpv (c, s). the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y))))"
trans: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "subseq b a"   and "subseq c b" shows "subseq c a"
borel_measurable_stake: fixes f :: "bool stream \<Rightarrow> 'c"   and n :: "nat"   and w :: "bool stream"   and y :: "bool stream" assumes "f \<in> borel_measurable (nat_filtration n)"   and "stake n w = stake n y" shows "f w = f y"
subspace_UNIV: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Modules.module scale" shows "module.subspace scale UNIV"
rotater1_rev': fixes xs :: "'b list" shows "rotater1 (rev xs) = rev (rotate1 xs)"
apd2: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) \<le> ad (x \<cdot> ad (ad y))"
cosh_real_ge_1: fixes x :: "real" shows "1 \<le> cosh x"
dX_def: fixes X :: "'a set"   and d :: "'b" shows "Singleton_for_d.dX X d = {x \<in> X. Xd x = d}"
S: fixes i :: "nat" shows "S i \<in> sets (completion M)"
assms_1: shows "x \<notin> \<int>\<^sub>\<le>\<^sub>0"
connected: fixes G :: "('c, 'd) graph" assumes "connected_graph G" shows "\<forall>v\<in>nodes G. \<forall>v'\<in>nodes G. nodes_connected G v v'"
eval_elim_cases_10: fixes G :: "prog"   and s :: "st"   and vn :: "lname"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In2 (LVar vn)\<succ>\<rightarrow> (v, s')" obtains "v = In2 (lvar vn s)" and "s' = Norm s"
image_of_unitor_2: fixes g :: "'a" assumes "ide g" shows "S.UP \<r>[g] = (S.UP g \<star>\<^sub>S S.inv (UP.unit (src g))) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (g, src g))"
semiring_normalization_rules_22: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "plus a (plus c d) = plus c (plus a d)"
case_1: fixes f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
qAFreshAll_simps_2: fixes xs :: "'a"   and x :: "'b"   and delta :: "'e"   and inp :: "'c \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm option"   and binp :: "'d \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qAbs option" shows "qAFresh xs x (qOp delta inp binp) = (liftAll (qAFresh xs x) inp \<and> liftAll (qAFreshAbs xs x) binp)"
ennreal_neq_top: fixes r :: "real" shows "ennreal r \<noteq> top"
assms: fixes x :: "\<nu>" shows "[\<^bold>\<A>(\<phi> x \<^bold>\<equiv> \<psi> x) in v]"
\<Delta>'_sel_4: fixes \<Delta> :: "('a, 'b) network_scheme" assumes "flow_attainability \<Delta>" shows "sink (flow_attainability.\<Delta>' \<Delta>) = Inner (sink \<Delta>)"
tendsto_Im1: fixes g :: "'a \<Rightarrow> quat"   and a :: "quat"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im1 (g x)) \<longlongrightarrow> Im1 a) F"
widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty"   and C :: "char list" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C))"
intro: fixes n :: "int"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a" assumes "mod_type n Rep Abs" shows "mod_ring n Rep Abs"
cs_done_implies_both_snapshotted_1: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "trace init t final"   and "snd (cs (S t i) cid) = Done"   and "i < length t"   and "channel cid = Some (p, q)" shows "ps (S t i) p \<noteq> None"
rep_eq: shows "eval\<Pi>\<^sub>1 E! = (\<lambda>u s w. case u of \<omega>\<upsilon> x \<Rightarrow> ConcreteInWorld x w | \<sigma>\<upsilon> \<sigma> \<Rightarrow> False)"
parenthesis_contained: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (finished s)"   and "w \<in> dom (discovered s)"   and "\<delta> s v < \<delta> s w"   and "\<delta> s w < \<phi> s v" shows "w \<in> dom (finished s) \<and> \<phi> s w < \<phi> s v"
decomp_closure_analyzed: fixes M :: "('fun, 'var) Term.term set"   and M' :: "('fun, 'var) Term.term set" assumes "decomp_closure M M'" shows "analyzed M'"
set2_S_def: shows "set2_S = set2_G \<circ> Rep_S"
reflp_def: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
hom_add: fixes c :: "'b"   and x :: "'b"   and y :: "'b" shows "c * (x + y) = c * x + c * y"
neutral_sds_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "neutral_sds agents alts sds \<equiv> social_decision_scheme agents alts sds \<and> neutral_sds_axioms agents alts sds"
Ide_implies_Arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Ide C t" shows "monoidal_language.Arr C t"
finsertE: fixes a :: "'l"   and b :: "'l"   and A :: "'l fset" assumes "a |\<in>| finsert b A" obtains "a = b"   | "a |\<in>| A"
map_option_case: fixes f :: "'c \<Rightarrow> 'b"   and y :: "'c option" shows "map_option f y = (case y of None \<Rightarrow> None | Some x \<Rightarrow> Some (f x))"
greaterThan_non_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.no_top less_eq less" shows "ord.greaterThan less x \<noteq> {}"
IH_2: fixes as :: "accent list"   and bs :: "accent list"   and cs :: "accent list" assumes "set as \<subseteq> {Grave}"   and "set bs \<subseteq> {Macron}"   and "set cs \<subseteq> {Acute}"   and "d # ds = as @ bs @ cs" shows "thesis"
inf_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
Cd_range: fixes d :: "'d"   and X :: "'x set" shows "Cd d X \<subseteq> X \<inter> Field (Pd d)"
ring_simprules_12: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
while_bot_unfold: fixes b :: "'b upred"   and P :: "'b hrel" shows "while\<^sub>\<bottom> b do P od = P ;; while\<^sub>\<bottom> b do P od \<triangleleft> b \<triangleright>\<^sub>r II"
c: shows "rawpsubst (eqv \<phi>1 \<phi>2) (zip (map Var us) (map snd txs)) = eqv (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (rawpsubst \<phi>2 (zip (map Var us) (map snd txs)))"
not_greatest_aligned: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" assumes "x < y"   and "is_aligned x n"   and "is_aligned y n" shows "x + 2 ^ n \<noteq> 0"
no_root: shows "\<forall>p\<in>set (smods p q). \<forall>x. a < x \<and> x \<le> a' \<or> b' \<le> x \<and> x < b \<longrightarrow> poly p x \<noteq> 0"
prems: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "finite (B x)"
minus_Box_def: fixes x :: "'a Box"   and y :: "'a Box" shows "x - y \<equiv> bliftM2 (\<Lambda> a b. a - b)\<cdot>x\<cdot>y"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow>\<^sub>\<tau> (a6,a7)" obtains ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" where "a1 = Sa__" and "a2 = f__" and "a3 = ms__" and "a4 = s__" and "a5 = []" and "a6 = ms__" and "a7 = s__" and "length ms__ = length s__"   | S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and a :: "'edge" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = a # as" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')" and "S,f \<turnstile> (ms',s') =as\<Rightarrow>\<^sub>\<tau> (ms'',s'')"
inner_prod_with_itself_eq0: fixes u :: "complex Matrix.vec" assumes "u = 0\<^sub>v (dim_vec u)" shows "\<langle>u|u\<rangle> = 0"
subdegree_power_eqI: fixes f :: "'a fps"   and n :: "nat" assumes "fps_nth f (subdegree f) ^ n \<noteq> (0::'a)" shows "subdegree (f ^ n) = n * subdegree f"
union_fold_insert: fixes A :: "'b set"   and B :: "'b set" assumes "finite A" shows "A \<union> B = Finite_Set.fold insert B A"
elims_2: fixes x :: "'b ltln set"   and xa :: "'b ltln" assumes "x \<Turnstile>\<^sub>P xa" obtains \<A> :: "'b ltln set" where "x = \<A>__" and "xa = true\<^sub>n"   | \<A> :: "'b ltln set" and \<phi> :: "'b ltln" and \<psi> :: "'b ltln" where "x = \<A>'__" and "xa = \<phi>__ and\<^sub>n \<psi>__" and "\<A>'__ \<Turnstile>\<^sub>P \<phi>__ \<and> \<A>'__ \<Turnstile>\<^sub>P \<psi>__"   | \<A> :: "'b ltln set" and \<phi> :: "'b ltln" and \<psi> :: "'b ltln" where "x = \<A>''__" and "xa = \<phi> or\<^sub>n \<psi>" and "\<A>''__ \<Turnstile>\<^sub>P \<phi> \<or> \<A>''__ \<Turnstile>\<^sub>P \<psi>"   | \<A> :: "'b ltln set" and v :: "'b" where "x = \<A>'''__" and "xa = prop\<^sub>n(v__)" and "prop\<^sub>n(v__) \<in> \<A>'''__"   | \<A> :: "'b ltln set" and v :: "'b" where "x = \<A>''''__" and "xa = nprop\<^sub>n(va__)" and "nprop\<^sub>n(va__) \<in> \<A>''''__"   | \<A> :: "'b ltln set" and v :: "'b ltln" where "x = \<A>'''''__" and "xa = X\<^sub>n vb__" and "X\<^sub>n vb__ \<in> \<A>'''''__"   | \<A> :: "'b ltln set" and v :: "'b ltln" and va :: "'b ltln" where "x = \<A>''''''__" and "xa = vc__ U\<^sub>n vaa__" and "vc__ U\<^sub>n vaa__ \<in> \<A>''''''__"   | \<A> :: "'b ltln set" and v :: "'b ltln" and va :: "'b ltln" where "x = \<A>'''''''__" and "xa = vd__ R\<^sub>n vab__" and "vd__ R\<^sub>n vab__ \<in> \<A>'''''''__"   | \<A> :: "'b ltln set" and v :: "'b ltln" and va :: "'b ltln" where "x = \<A>''''''''__" and "xa = ve__ W\<^sub>n vac__" and "ve__ W\<^sub>n vac__ \<in> \<A>''''''''__"   | \<A> :: "'b ltln set" and v :: "'b ltln" and va :: "'b ltln" where "x = \<A>" and "xa = v M\<^sub>n va" and "v M\<^sub>n va \<in> \<A>"
sameDom_and_liftAll2_iff: fixes inp :: "'d \<Rightarrow> 'e option"   and inp' :: "'d \<Rightarrow> 'f option"   and phi :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "(sameDom inp inp' \<and> liftAll2 phi inp inp') = (\<forall>i. inp i = None \<and> inp' i = None \<or> (\<exists>v v'. inp i = Some v \<and> inp' i = Some v' \<and> phi v v'))"
seq_char: fixes g :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr" shows "S.seq g f = (S.arr f \<and> S.arr g \<and> S.Dom g = S.Cod f)"
fold_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and t :: "('d, 'e) rbt" shows "RBT_Impl.fold f t = List.fold (\<lambda>(x, y). f x y) (RBT_Impl.entries t)"
lm092: fixes L :: "'c list"   and F :: "'c \<Rightarrow> 'd list"   and G :: "'c \<Rightarrow> 'd set" assumes "\<forall>x\<in>set L. set (F x) = G x" shows "set (concat (map F L)) = \<Union> (G ` set L)"
basic_identity\<^sub>2_def: shows "(\<^bold>=\<^sub>2) \<equiv> \<lambda>F G. \<^bold>\<forall>x. (\<^bold>\<lambda>y. \<lparr>F,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>=\<^sub>1 (\<^bold>\<lambda>y. \<lparr>G,x\<^sup>P,y\<^sup>P\<rparr>) \<^bold>& (\<^bold>\<lambda>y. \<lparr>F,y\<^sup>P,x\<^sup>P\<rparr>) \<^bold>=\<^sub>1 (\<^bold>\<lambda>y. \<lparr>G,y\<^sup>P,x\<^sup>P\<rparr>)"
fun_app_lifted_def: fixes f\<^sub>T :: "['f \<Rightarrow> ['e]]"   and x\<^sub>T :: "['f]" shows "f\<^sub>T . x\<^sub>T \<equiv> f\<^sub>T \<bind> (\<bind>) x\<^sub>T"
noAbrupt_start: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and s' :: "'e" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "\<forall>t'. t \<noteq> Abrupt t'" shows "s \<noteq> Abrupt s'"
red_simps_1: fixes t :: "'a local.term" assumes "Ide t" shows "Src (t\<^bold>\<down>) = Src t"
in_rel: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (\<exists>z. z \<in> UNIV \<and> id z = a \<and> id z = b)"
ln_less_cancel_iff: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "(ln x < ln y) = (x < y)"
sel: fixes x :: "'a" shows "run_option (OptionT x) = x"
fds_deriv_def: fixes f :: "'a fds" shows "fds_deriv f = fds (\<lambda>n. - ln (real n) *\<^sub>R fds_nth f n)"
match_Node_def: shows "match_Node \<equiv> \<Lambda> x k. tree_case\<cdot>Fixrec.fail\<cdot>k\<cdot>x"
suminf_def: fixes f :: "nat \<Rightarrow> 'b" shows "suminf f = (THE s. f sums s)"
arr_dom: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr f" shows "A_BxA.arr (A_BxA.dom f)"
scf_list_map: fixes scf :: "nat \<Rightarrow> nat"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "scf_list scf (map f xs) = map f (scf_list scf xs)"
hyps_2: fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and \<xi> :: "nat \<Rightarrow> 'a set" assumes "ipath gba.E \<sigma>"   and "gba.is_acc \<sigma>"   and "\<forall>i. local.gba.L (\<sigma> i) (\<xi> i)"   and "\<eta> \<in> old (\<sigma> 0)" shows "\<xi> \<Turnstile>\<^sub>r \<eta>"
fact_split: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and k :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero"   and "k \<le> n" shows "semiring_char_0.fact one plus zero n = times (semiring_1.of_nat one plus zero (prod Suc {n - k..<n})) (semiring_char_0.fact one plus zero (n - k))"
a_r_coset_def': fixes G :: "('c, 'd) ring_scheme"   and H :: "'c set"   and a :: "'c" shows "H +>\<^bsub>G\<^esub> a \<equiv> \<Union>h\<in>H. {h \<oplus>\<^bsub>G\<^esub> a}"
expand: fixes idx_interval :: "'a idx_interval"   and idx_interval' :: "'a idx_interval" assumes "idx_interval.int idx_interval = idx_interval.int idx_interval' \<and> idx idx_interval = idx idx_interval'" shows "idx_interval = idx_interval'"
lift_spmf_bind: fixes r :: "'g spmf"   and f :: "'g \<Rightarrow> ('d, 'e, 'f) gpv" shows "lift_spmf r \<bind> f = GPV (r \<bind> the_gpv \<circ> f)"
Upper_antimono: fixes A :: "'c set"   and B :: "'c set"   and L :: "('c, 'd) gorder_scheme" assumes "A \<subseteq> B" shows "Upper L B \<subseteq> Upper L A"
1: fixes i :: "nat" assumes "i < length (arOf f)" shows "TE.tpOfV (xl ! i) = arOf f ! i"
Disjunctuous_def: fixes H :: "'d upred \<Rightarrow> 'd upred" shows "Disjunctuous H = (\<forall>P Q. H (P \<sqinter> Q) = H P \<sqinter> H Q)"
pseudoinverse: fixes \<omega> :: "real"   and x :: "real" assumes "0 < \<omega>"   and "\<omega> < 1" shows "(\<omega> \<le> C x) = (I \<omega> \<le> x)"
cfE_ss_invE: shows "f.cf.E \<subseteq> E \<union> E\<inverse>"
inverse_inverse: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "group f z inverse" shows "inverse (inverse a) = a"
impl_of: shows "impl_of hm \<in> {hm. ArrayHashMap_Impl.ahm_invar hm}"
Fract_of_int_eq: fixes k :: "int" shows "Fract k 1 = of_int k"
cases: fixes a1 :: "'d \<Rightarrow> 'e tree option"   and a2 :: "'d set"   and a3 :: "('d, 'e) ta_rule \<Rightarrow> nat option"   and \<delta> :: "('d, 'e) ta_rule set" assumes "(a1, a2, a3) \<in> brw_initial \<delta>" obtains rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" and Q :: "'d \<Rightarrow> 'e tree option" where "a1 = Q" and "a2 = br_iq \<delta>" and "a3 = rcm" and "\<And>r. r \<in> \<delta> \<Longrightarrow> rcm r = Some (card (set (rhsq r)))" and "Q \<in> brw_iq \<delta>"
def_scheme'': fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" shows "f X Y = (\<lambda>\<tau>. if d\<^sub>x X \<tau> = true \<tau> \<and> d\<^sub>y Y \<tau> = true \<tau> then g (X \<tau>) (Y \<tau>) else invalid \<tau>)"
bin_def: fixes I :: "item set"   and k :: "nat" shows "bin I k = {x \<in> I. item_end x = k}"
possible_trans: fixes n :: "nat"   and p :: "'person"   and w1 :: "'person \<Rightarrow> 'color"   and w2 :: "'person \<Rightarrow> 'color"   and w3 :: "'person \<Rightarrow> 'color" assumes "local.possible n p w1 w2"   and "local.possible n p w2 w3" shows "local.possible n p w1 w3"
authKeys_are_not_AKcryptSK: fixes K :: "nat"   and evs :: "event list" assumes "K \<in> authKeys evs \<union> range shrK"   and "evs \<in> kerbIV" shows "\<forall>SK. \<not> AKcryptSK SK K evs \<and> K \<in> symKeys"