hcju/isabelleps huggingface dataset:
distinct_1: fixes x11 :: "name"   and x12 :: "value list"   and x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap" shows "Vconstr x11 x12 \<noteq> Vabs x21 x22"
in_derived_set_of: fixes x :: "'c"   and X :: "'c topology"   and S :: "'c set" shows "(x \<in> X derived_set_of S) = (x \<in> topspace X \<and> (\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T)))"
is_ubD: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and u :: "'a"   and x :: "'a" assumes "class.po below"   and "po.is_ub below S u"   and "x \<in> S" shows "below x u"
prime_divisors_induct: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "P zero"   and "\<And>x. dvd.dvd times x one \<Longrightarrow> P x"   and "\<And>p x. \<lbrakk>normalization_semidom.prime zero times one normalize p; P x\<rbrakk> \<Longrightarrow> P (times p x)" shows "P x"
finite_set_of_finite_maps: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "finite {m. dom m = A \<and> ran m \<subseteq> B}"
RedCallNone: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and P :: "(char list list \<times> char list exp) prog"   and M :: "char list"   and vs :: "val list"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "h a = \<lfloor>(C, fs)\<rfloor>"   and "\<nexists>b Ts T m D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>addr a\<bullet>M(map Val vs),(h, l, sh),b\<rangle> \<rightarrow> \<langle>THROW NoSuchMethodError,(h, l, sh),b\<rangle>"
simps_5: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = \<ll> \<lbrace>t\<rbrace>"
CRR_market_axioms: shows "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G"
bis_raw_coind: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set"   and theta :: "('b \<times> 'b) set" assumes "Bisim.mono Retr"   and "sym theta"   and "theta \<subseteq> Retr theta" shows "theta \<subseteq> bis Retr"
cartesian_product2: fixes D :: "(('a \<times> 'b) \<times> 'c) set"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite D"   and "{(a, b). \<exists>c. g a b c \<noteq> (0::'d)} \<times> {c. \<exists>a b. g a b c \<noteq> (0::'d)} \<subseteq> D" shows "(\<Sum>(a, b). Sum_any (g a b)) = (\<Sum>(a, b, c). g a b c)"
oCons: fixes m2 :: "'a list list" shows "map (\<lambda>uv. vec_plus (fst uv) (snd uv)) (zip m1 m2) = map (\<lambda>uv. vec_plus (fst uv) (snd uv)) (zip m2 m1)"
assert_loop_def: fixes as :: "('i \<times> 'a atom) list"   and s :: "('i, 'a) state" shows "assert_loop as s \<equiv> foldl (\<lambda>s' a. if \<U> s' then s' else assert a s') s as"
UN_constant: fixes c :: "'h set"   and A :: "'i set" shows "(\<Union>y\<in>A. c) = (if A = {} then {} else c)"
opeU: shows "openin (top_of_set T) (f ` (S \<inter> U))"
vc_foreach_ignore: fixes S :: "'v set"   and V :: "'v set"   and v :: "'v"   and it :: "'v set"   and V' :: "'v set"   and v' :: "'v" assumes "fe_inv S V v it (V', False)"   and "v' \<in> V'" shows "fe_inv S V v (it - {v'}) (V', False)"
exec_move_CondI3: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1"   and e1 :: "'addr expr1" assumes "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (if (e) e1 else e2) h (stk, loc, Suc (Suc (length (compE2 e) + length (compE2 e1) + pc)), xcp) ta h' (stk', loc', Suc (Suc (length (compE2 e) + length (compE2 e1) + pc')), xcp')"
unfold_optimize_ruleset_CHAIN: fixes \<Gamma> :: "(char list \<times> 'a rule list) list"   and chain_name :: "char list"   and default_action :: "action"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and optimize :: "'a match_expr \<Rightarrow> 'a match_expr"   and p :: "'b"   and rs :: "'a rule list"   and s :: "state"   and t :: "state" assumes "sanity_wf_ruleset \<Gamma>"   and "chain_name \<in> set (map fst \<Gamma>)"   and "default_action = action.Accept \<or> default_action = action.Drop"   and "\<And>m. Semantics.matches \<gamma> (optimize m) p = Semantics.matches \<gamma> m p"   and "unfold_optimize_ruleset_CHAIN optimize chain_name default_action (map_of \<Gamma>) = Some rs" shows "map_of \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t = map_of \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule MatchAny (Call chain_name), Rule MatchAny default_action], s\<rangle> \<Rightarrow> t"
obtain_set_qFresh_card_of: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>W. infinite W \<and> W \<inter> V = {} \<and> (\<forall>x\<in>W. \<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
insertI1: fixes a :: "'l"   and B :: "'l set" shows "a \<in> insert a B"
fdia_one: fixes x :: "'a" shows " |(1::'a)\<rangle> x = ad (ad x)"
chinese_remainder_unique_poly: fixes A :: "'b set"   and m :: "'b \<Rightarrow> 'c poly"   and u :: "'b \<Rightarrow> 'c poly" assumes "\<forall>i\<in>A. m i \<noteq> 0"   and "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> algebraic_semidom_class.coprime (m i) (m j)"   and "0 < degree (prod m A)" shows "\<exists>!x. degree x < (\<Sum>i\<in>A. degree (m i)) \<and> (\<forall>i\<in>A. [x = u i] (mod m i))"
subst_elimD': fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a" assumes "subst_elim \<sigma> v" shows "\<sigma> v \<noteq> Var v"
f_consequent_def: fixes r :: "'rule" shows "f_consequent r = fset_from_list (consequent r)"
lift_option_simps_2: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and b' :: "'c option" shows "lift_option f None b' = None"
mex_def: fixes P :: "'a \<Rightarrow> bool" shows "mex P = Ex P"
has_subprob_densityD_1: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "f \<in> borel_measurable N"
is_scc_unique: fixes E :: "('a \<times> 'a) set"   and scc :: "'a set"   and scc' :: "'a set"   and v :: "'a" assumes "is_scc E scc"   and "is_scc E scc'"   and "v \<in> scc"   and "v \<in> scc'" shows "scc = scc'"
cv_def: shows "cv \<equiv> (`) (inv_into U (ordermap U r))"
isolate_var_one: fixes x :: "nat" shows "isolate_variable_sparse (Var x) x 1 = 1"
ipt_iprange_compress: fixes l :: "'a ipt_iprange negation_type list" shows "\<Union> (ipt_iprange_to_set ` set (ipt_iprange_compress l)) = \<Inter> (ipt_iprange_to_set ` set (getPos l)) - \<Union> (ipt_iprange_to_set ` set (getNeg l))"
mult_exH_k_left: fixes k :: "nat" assumes "Suc k < n" shows "hadamard_on_i (Suc k) * exH_k k = exH_k (Suc k)"
irreducibleD: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "irreducible p"   and "p = a * b" shows "a dvd (1::'a) \<or> b dvd (1::'a)"
quality_increases_msg_fresh: fixes \<sigma> :: "nat \<Rightarrow> state"   and \<sigma>' :: "nat \<Rightarrow> state"   and m :: "msg" assumes "\<forall>j. quality_increases (\<sigma> j) (\<sigma>' j)"   and "msg_fresh \<sigma> m" shows "msg_fresh \<sigma>' m"
rev_conv_fold: fixes xs :: "'f list" shows "rev xs = fold (#) xs []"
Healthy_Idempotent: fixes H :: "'a upred \<Rightarrow> 'a upred"   and P :: "'a upred" assumes "Idempotent H" shows "H P is H"
nxt: fixes q :: "hf"   and x :: "'a" assumes "q \<in> nfa.states \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>" shows "nfa.nxt \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr> q x \<subseteq> nfa.states \<lparr>nfa.states = insert q0 (dfa.states MS), init = {q0}, final = {q0}, nxt = \<lambda>q x. if q \<in> dfa.states MS then {dfa.nxt MS q x} else {}, eps = insert (q0, dfa.init MS) ((\<lambda>q. (q, q0)) ` dfa.final MS)\<rparr>"
chambersub_imp_sub: fixes X :: "'b set set"   and Y :: "'b set set" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y" shows "order.greater_eq X Y \<and> SimplicialComplex Y"
restrict_indets_pp_plus: fixes s :: "'b option \<Rightarrow>\<^sub>0 nat"   and t :: "'b option \<Rightarrow>\<^sub>0 nat" shows "restrict_indets_pp (s + t) = restrict_indets_pp s + restrict_indets_pp t"
s_t_stop: shows "ego2.s_stop = ego2.p_max"
high_proc\<^sub>1_correct: fixes p :: "prog" assumes "vars p = vars\<^sub>0" shows "high_proc\<^sub>1 p = high_proc p"
invertible_left_cancel: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "monoid M composition unit"   and "monoid.invertible M composition unit x"   and "x \<in> M"   and "y \<in> M"   and "z \<in> M" shows "(composition x y = composition x z) = (y = z)"
cases: fixes a :: "nat \<times> nat" assumes "zerozero a" obtains "a = (0, 0)"
vss: fixes \<phi> :: "'fmla" shows "vs \<phi> = getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)"
closed_Proj_rel: shows "closed {(x, y). Proj x = Proj y}"
aforms_of_ivls: fixes ls :: "real list"   and us :: "real list"   and xs :: "real list" assumes "length ls = length us"   and "length xs = length ls"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i \<in> {ls ! i..us ! i}" shows "xs \<in> Joints (aforms_of_ivls ls us)"
list_all2_trans: fixes P1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and P3 :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and as :: "'a list"   and bs :: "'b list"   and cs :: "'c list" assumes "\<And>a b c. \<lbrakk>P1 a b; P2 b c\<rbrakk> \<Longrightarrow> P3 a c"   and "list_all2 P1 as bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 as cs"
c_def: shows "c \<equiv> \<lambda>k x. (\<i> * complex_of_real t) ^ k / fact k * complex_of_real (x ^ k)"
extreme_point_of_translation_eq: fixes a :: "'b"   and x :: "'b"   and S :: "'b set" shows "(a + x extreme_point_of (+) a ` S) = (x extreme_point_of S)"
cimage_cong: fixes M :: "'f cset"   and N :: "'f cset"   and f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g" assumes "M = N"   and "\<And>x. cin x N \<Longrightarrow> f x = g x" shows "cimage f M = cimage g N"
nfs: fixes x :: "'a formula"   and n :: "nat" shows "\<not> fin_sat (x \<triangleright> extender n S)"
psubset_imp_subset: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subset> B" shows "A \<subseteq> B"
discI: fixes option :: "'d option"   and x2 :: "'d" assumes "option = Some x2" shows "option \<noteq> None"
fps_right_inverse: fixes f :: "'a fps"   and y :: "'a" assumes "f $ 0 * y = (1::'a)" shows "f * fps_right_inverse f y = 1"
yoneda_functor_fixed_object_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a" shows "yoneda_functor_fixed_object C S \<phi> a \<equiv> yoneda_functor C S \<phi> \<and> yoneda_functor_fixed_object_axioms C a"
span_is_monotone: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and S :: "'d set"   and T :: "'d set" assumes "module R M"   and "S \<subseteq> T" shows "LinearCombinations.module.span R M S \<subseteq> LinearCombinations.module.span R M T"
delay_bisimulation_final_base_def: fixes trsys1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and trsys2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and bisim :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and \<tau>move1 :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move2 :: "'i \<Rightarrow> 'j \<Rightarrow> 'i \<Rightarrow> bool"   and final1 :: "'g \<Rightarrow> bool"   and final2 :: "'i \<Rightarrow> bool" shows "delay_bisimulation_final_base trsys1 trsys2 bisim \<tau>move1 \<tau>move2 final1 final2 \<equiv> (\<forall>s1 s2. bisim s1 s2 \<longrightarrow> final1 s1 \<longrightarrow> (\<exists>s2'. (\<tau>trsys.silent_move trsys2 \<tau>move2)\<^sup>*\<^sup>* s2 s2' \<and> bisim s1 s2' \<and> final2 s2')) \<and> (\<forall>s1 s2. bisim s1 s2 \<longrightarrow> final2 s2 \<longrightarrow> (\<exists>s1'. (\<tau>trsys.silent_move trsys1 \<tau>move1)\<^sup>*\<^sup>* s1 s1' \<and> bisim s1' s2 \<and> final1 s1'))"
WHILE_annotI_def: fixes I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "WHILE_annotI I \<equiv> While"
Nset_0: shows "{j. j \<le> 0} = {0}"
UN_E: fixes b :: "'a"   and B :: "'d \<Rightarrow> 'a set"   and A :: "'d set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'d" where "x \<in> A" and "b \<in> B x"
r_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes> (x \<oplus> y) = z \<otimes> x \<oplus> z \<otimes> y"
degree_power_eq: fixes p :: "'a poly"   and n :: "nat" assumes "p \<noteq> 0" shows "Polynomial.degree (p ^ n) = n * Polynomial.degree p"
llist_all2_lzipI: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist"   and P' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs' :: "'e llist"   and ys' :: "'f llist" assumes "llist_all2 P xs ys"   and "llist_all2 P' xs' ys'" shows "llist_all2 (rel_prod P P') (lzip xs xs') (lzip ys ys')"
select_convs_1: fixes init :: "'e set"   and trans :: "('e \<times> 'e) set"   and obs :: "'e \<Rightarrow> 'f"   and more :: "'g" shows "obs \<lparr>init = init, trans = trans, obs = obs, \<dots> = more\<rparr> = obs"
util_def_conf: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "(u y \<le> u x) = x \<succeq>[relation] y"
suminf_eq_SUP: fixes f :: "nat \<Rightarrow> 'a" shows "suminf f = (SUP n. sum f {..<n})"
to_fun_one: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp" shows "\<one>\<^bsub>UP Zp\<^esub> \<bullet> a = \<one>"
bangPres: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set" assumes "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>[Rel] S"   and "eqvt Rel" shows "!P \<leadsto>[bangRel Rel] !Q"
secure_while_aux: fixes n :: "nat"   and k :: "nat"   and m :: "nat"   and s :: "Var \<Rightarrow> Val"   and b :: "BExpr"   and c :: "IMP"   and ss :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val"   and tt :: "Var \<Rightarrow> Val" assumes "n \<le> k"   and "m \<le> k"   and " s , While b c \<rightarrow>\<^sub>n  ss "   and " t , While b c \<rightarrow>\<^sub>m  tt "   and "secure c"   and "\<And>s ss.  s \<approx> ss  \<Longrightarrow> evalB b s = evalB b ss"   and " s \<approx> t " shows " ss \<approx> tt "
nonunit_contained_maxid: fixes a :: "'a" assumes "\<not> zeroring R"   and "a \<in> carrier R"   and "\<not> Unit R a" shows "\<exists>mx. maximal_ideal R mx \<and> a \<in> mx"
ln_float_intervalI: fixes x :: "real"   and X :: "float interval"   and p :: "nat" assumes "x \<in>\<^sub>r X" shows "ln x \<in> set_of' (ln_float_interval p X)"
simps_4: fixes v :: "String.literal"   and va :: "typ"   and uu :: "nat"   and uv :: "Core.term list" shows "subst_bvs1' (Ct v va) uu uv = Ct v va"
1: shows "{brnL cl (pickFT cl)..<brnL cl (pickFT cl) + brn (cl ! pickFT cl)} = (+) (brnL cl (pickFT cl)) ` {..<brn (cl ! pickFT cl)}"
reach_k_x: shows "s\<langle>l := z\<rangle>\<turnstile> k reachable_from x"
Abs_key_order_inverse: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "y \<in> {compare. comparator compare}" shows "key_compare (Abs_key_order y) = y"
norm_compile_instr: fixes instr :: "('d, 'e, 'f, 'op) Std.instr" shows "norm_instr (compile_instr instr) = instr"
depth_def: shows "depth \<equiv> pi_rec 0 (\<lambda>a b P. (+) 1) (\<lambda>P. (+) 1) (\<lambda>a x P. (+) 1) (\<lambda>a b P Pa. Pa) (\<lambda>a b P Pa. Pa) (\<lambda>P Q. max) (\<lambda>P Q. (+)) (\<lambda>x P Pa. Pa) (\<lambda>P Pa. Pa)"
assms_3: shows "M \<subseteq> sigma_sets \<Omega> E"
Basis: fixes F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<in> A ensures B" shows "(A, B) \<in> leads F"
to_tree_order_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (Shadow_DOM.to_tree_order ptr) h"
app: fixes t :: "'b"   and u :: "'b" shows "no_abs (app t u) = (no_abs t \<and> no_abs u)"
assms_2: shows "Lower_Semicontinuous.convex_on C f"
OUTfromVCorrect1_data24: shows "OUTfromVCorrect1 data24"
locate2_def: fixes f :: "'a \<Rightarrow> nat"   and al :: "'a list"   and i :: "nat" shows "locate2 f al i \<equiv> snd (locate f al i)"
issued_app: fixes s :: "event list"   and s' :: "event list" shows "issued (s @ s') = issued s \<union> issued s'"
remainder_theorem: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and g :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "g = UP_root_div f a"   and "r = UP_root_rem f a" shows "f = r \<oplus>\<^bsub>P\<^esub> X_poly_minus R a \<otimes>\<^bsub>P\<^esub> g"
last_in_set: fixes as :: "'c list" assumes "as \<noteq> []" shows "last as \<in> set as"
simps_2: fixes uv :: "char list"   and xs :: "pat list" shows "is_cupcake_pat (Pcon (Some (Short uv)) xs) = list_all is_cupcake_pat xs"
relation_of_Ndet: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "action.relation_of (P \<sqinter> Q) = (action.relation_of P \<or> action.relation_of Q)"
less_eq_def: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and s :: "'e"   and s' :: "'e" shows "pre_RDR.less_eq \<delta> s s' \<equiv> \<exists>rs. s' = pre_RDR.exec \<delta> s rs"
compl_unique: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> y = \<bottom>"   and "x \<squnion> y = \<top>" shows "- x = y"
le_iff_subset: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "le_rep p1 p2 = (\<gamma>_rep p1 \<subseteq> \<gamma>_rep p2)"
brec_inner_step_brw_desc: fixes \<Sigma> :: "('a, 'b tree) HashMap.hashmap \<times> ('a list \<times> 'a list) \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap \<times> 'a option"   and Qi :: "('a, unit) HashMap.hashmap"   and r :: "('a, 'b) ta_rule" assumes "\<Sigma> \<in> brec_invar_inner (hs.\<alpha> Qi)" shows "(brec_\<alpha> \<Sigma>, brec_\<alpha> (brec_inner_step Qi r \<Sigma>)) \<in> brw_inner_step r"
f': fixes s :: "real" assumes "s \<in> {0..t}" shows "(f has_derivative f' s) (at s within {0..t})"
l11_10: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and C' :: "'p"   and D' :: "'p"   and F' :: "'p" assumes "A B C CongA D E F"   and "B Out A' A"   and "B Out C' C"   and "E Out D' D"   and "E Out F' F" shows "A' B C' CongA D' E F'"
antipar_1: shows "trg (src \<mu>) = src (src \<mu>)"
Ps_refl_on_low_mds_eq: fixes i :: "nat"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and mem\<^sub>C :: "'Var\<^sub>C \<Rightarrow> 'Val"   and mem\<^sub>C' :: "'Var\<^sub>C \<Rightarrow> 'Val"   and c\<^sub>C :: "'Com\<^sub>C" assumes "i < length cms"   and "conc.low_mds_eq mds\<^sub>C mem\<^sub>C mem\<^sub>C'" shows "(\<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C'\<rangle>\<^sub>C) \<in> P_rel (cms ! i)"
bind_ASSERT_eq_if: fixes \<Phi> :: "bool"   and m :: "'a nres" shows "ASSERT \<Phi> \<bind> (\<lambda>_. m) = (if \<Phi> then m else FAIL)"
coinductive_eq_I: fixes a :: "'c llist"   and b :: "'c llist" assumes "llength a = llength b \<and> (\<forall>n. enat n < llength a \<longrightarrow> a $ n = b $ n)" shows "a = b"
set_intros_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "x21 mem (x21 # x22)"
assms_1: shows "\<forall>x\<in>M. x = y"
fls_lr_inverse_one_one_1: shows "fls_left_inverse 1 (1::'a) = 1"
favorites_no_agent: fixes i :: "'agent" assumes "i \<notin> agents" shows "favorites R i = {}"
no_wrt: shows "no_write_to_read_only_memory \<S> sb"
msubstle_def: fixes c :: "poly"   and t :: "tm"   and d :: "poly"   and s :: "tm"   and a :: "poly"   and r :: "tm" shows "msubstle c t d s a r = evaldjf (\<lambda>(x, y). Parametric_Ferrante_Rackoff.conj x y) [let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Le (tm.Add (tm.Mul (~\<^sub>p a) (tm.Add (tm.Mul d t) (tm.Mul c s))) (tm.Mul ((2)\<^sub>p *\<^sub>p cd) r))), let cd = c *\<^sub>p d in (lt (CP cd), Le (tm.Sub (tm.Mul a (tm.Add (tm.Mul d t) (tm.Mul c s))) (tm.Mul ((2)\<^sub>p *\<^sub>p cd) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Le (tm.Add (tm.Mul (~\<^sub>p a) t) (tm.Mul ((2)\<^sub>p *\<^sub>p c) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP c)) (Eq (CP d)), Le (tm.Sub (tm.Mul a t) (tm.Mul ((2)\<^sub>p *\<^sub>p c) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Le (tm.Add (tm.Mul (~\<^sub>p a) s) (tm.Mul ((2)\<^sub>p *\<^sub>p d) r))), (Parametric_Ferrante_Rackoff.conj (lt (CP d)) (Eq (CP c)), Le (tm.Sub (tm.Mul a s) (tm.Mul ((2)\<^sub>p *\<^sub>p d) r))), (Parametric_Ferrante_Rackoff.conj (Eq (CP c)) (Eq (CP d)), Le r)]"
select_convs_1: fixes edge :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and more :: "'l" shows "edge \<lparr>edge = edge, \<dots> = more\<rparr> = edge"
length_apath_less: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "apath u p v" shows "length p < card (verts G)"
prod_mset_prime_factorization: fixes x :: "'b poly" assumes "x \<noteq> 0" shows "\<Prod>\<^sub># (field_poly.prime_factorization x) = smult (inverse (lead_coeff x)) x"
bin_sign_cat: fixes n :: "nat"   and y :: "int"   and x :: "int" shows "bin_sign (concat_bit n y x) = bin_sign x"
map_rw1: shows "map (sq_norm \<circ> row (sylvester_mat f g)) [0..<degree g] = replicate k \<parallel>f\<parallel>\<^sup>2"
lex_diff1: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex (a - b) c = Counterclockwise_2D_Arbitrary.lex a (c + b)"
subspace_span: fixes S :: "'b set" shows "csubspace (cspan S)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) Dom_f.map Cod_f.map Fun_f.map"
l_get_ancestors_wf_axioms: shows "l_get_ancestors_wf heap_is_wellformed parent_child_rel DocumentClass.known_ptr DocumentClass.known_ptrs DocumentClass.type_wf get_ancestors get_ancestors_locs get_child_nodes get_parent"
eq_newpkt_def: fixes m :: "msg" shows "eq_newpkt m \<equiv> case m of Newpkt d dip \<Rightarrow> True | _ \<Rightarrow> False"
that: assumes "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs"   and "x \<in>\<^sub>\<circ> A"   and "n = vcard (xs #\<^sub>\<circ> x)" shows "thesis"
suffix_same_cases: fixes xs\<^sub>1 :: "'b list"   and ys :: "'b list"   and xs\<^sub>2 :: "'b list" assumes "xs\<^sub>1 \<le>s ys"   and "xs\<^sub>2 \<le>s ys" shows "xs\<^sub>1 \<le>s xs\<^sub>2 \<or> xs\<^sub>2 \<le>s xs\<^sub>1"
distinct_11: fixes x41 :: "nat"   and x42 :: "e list"   and x43 :: "e list" shows "Trap \<noteq> Label x41 x42 x43"
Int_empty_right: fixes A :: "'k set" shows "A \<inter> {} = {}"
not_le: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(\<not> x \<preceq> y) = (y \<prec> x)"
sdiv_def: fixes a :: "int"   and b :: "int" shows "a sdiv b = sgn a * sgn b * (\<bar>a\<bar> div \<bar>b\<bar>)"