hcju/isabelleps huggingface dataset:
quicksort_ref_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" shows "quicksort_ref R h = (\<lambda>(lo, hi, xs0). REC\<^sub>T (\<lambda>f (lo, hi, xs). ASSERT (lo \<le> hi \<and> hi < length xs0 \<and> mset xs = mset xs0) \<bind> (\<lambda>_. partition_between_ref R h lo hi xs \<bind> (\<lambda>(xs, p). ASSERT (mset xs = mset xs0 \<and> lo \<le> p \<and> p < length xs0) \<bind> (\<lambda>_. (if p - 1 \<le> lo then RETURN xs else f (lo, p - 1, xs)) \<bind> (\<lambda>xs. ASSERT (mset xs = mset xs0) \<bind> (\<lambda>_. if hi \<le> p + 1 then RETURN xs else f (p + 1, hi, xs))))))) (lo, hi, xs0))"
mod_self: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a a = zero"
26682: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> y) \<sqinter> x = x"
the_pmf_def: fixes p :: "'b pmf" shows "the_pmf p = (THE x. p = return_pmf x)"
FL_valid_eqvt: fixes P :: "'state"   and x :: "('c, 'pred, 'act, 'effect) formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
in_homI: fixes f :: "('a, 'a) CC.arr"   and a :: "('a, 'a) CC.arr"   and b :: "('a, 'a) CC.arr" assumes "CC.arr f"   and "CC.dom f = a"   and "CC.cod f = b" shows "CC.in_hom f a b"
I_valid: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "wf_ast_problem problem" shows "ast_problem.I problem \<in> ast_problem.valid_states problem"
drop_bit_eq_div: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n a = divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
max_subgraph_subg_eq: fixes P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and H1 :: "('a, 'b) pre_digraph"   and H2 :: "('a, 'b) pre_digraph" assumes "max_subgraph P H1"   and "max_subgraph P H2"   and "subgraph H1 H2" shows "H1 = H2"
cardinal: shows "Card_order \<AA> \<and> infinite (Field \<AA>)"
dterms_not_choice: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "q \<in> dterms \<Gamma> p" shows "not_choice q"
augment_edge_cf_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'a" shows "augment_edge_cf cf \<equiv> \<lambda>(u, v) \<Delta>. cf((u, v) := cf (u, v) - \<Delta>, (v, u) := cf (v, u) + \<Delta>)"
is_ortho_set_def: fixes S :: "'b set" shows "is_ortho_set S = ((\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> is_orthogonal x y) \<and> (0::'b) \<notin> S)"
getFreshN: fixes V :: "'var set"   and n :: "nat" assumes "finite V" shows "set (getFreshN V n) \<subseteq> var \<and> set (getFreshN V n) \<inter> V = {} \<and> length (getFreshN V n) = n \<and> distinct (getFreshN V n)"
card_2_iff: fixes S :: "'d set" shows "(card S = 2) = (\<exists>x y. S = {x, y} \<and> x \<noteq> y)"
field_differentiable_at_Arctan: fixes z :: "complex" assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1" shows "Arctan field_differentiable at z"
pdom_res_swap: fixes A :: "'a set"   and f :: "('a, 'b) pfun"   and B :: "'b set" shows "A \<lhd>\<^sub>p f \<rhd>\<^sub>p B = A \<lhd>\<^sub>p (f \<rhd>\<^sub>p B)"
antisym: fixes b :: "'a Word.word"   and a :: "'a Word.word" assumes "b \<le>s a"   and "a \<le>s b" shows "a = b"
nrename_AndL1': fixes R :: "trm"   and x :: "name"   and y :: "name"   and v :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = AndL1 v.N u"   and "v \<sharp> (R, u, x, y)"   and "x \<noteq> y" shows "(\<exists>N'. R = AndL1 v.N' u \<and> N'[x\<turnstile>n>y] = N) \<or> (\<exists>N'. R = AndL1 v.N' x \<and> y = u \<and> N'[x\<turnstile>n>y] = N)"
ds2_commute: fixes M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and M :: "('e, 'a, 'f) Module_scheme" assumes "R module M1"   and "R module M2"   and "R module M"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2" shows "M2 \<Oplus>\<^bsub>R,M\<^esub> M1"
insert_ran_def: fixes p :: "'b \<Rightarrow> nat"   and j :: "nat" shows "insert_ran p j \<equiv> \<lambda>i. insert_index j (p i)"
coboundedI2: fixes c :: "'c"   and b :: "'c"   and a :: "'c" assumes "c \<le> b" shows "c \<le> max a b"
in_language_Zero: fixes xs :: "'b list" shows "\<not> in_language Zero xs"
checkDeadProcs_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme" assumes "gState_inv prog g" shows "(g, checkDeadProcs g) \<in> gState_progress_rel prog"
lead_coeff_1: shows "lead_coeff 1 = (1::'a)"
countable_basis: shows "countable (SOME B. countable B \<and> topological_basis B)"
strip_list_comb: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a"   and ys :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (term.list_comb app f ys) = (fst (term.strip_comb unapp f), snd (term.strip_comb unapp f) @ ys)"
delete_def: shows "delete \<equiv> map_fun id (map_fun impl_of Alist) AList.delete"
integrable_spike_set_eq: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible (sym_diff S T)" shows "(f integrable_on S) = (f integrable_on T)"
fmupd_lookup: fixes a :: "'d"   and b :: "'c"   and m :: "('d, 'c) fmap"   and a' :: "'d" shows "fmlookup (fmupd a b m) a' = (if a = a' then Some b else fmlookup m a')"
assms_2: shows "alphaAbs_ex_equal_or_qAFresh xs x X xs' x' X'"
CFG_call_edge_length: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "length fs = length ins"
strand_vars_split_2: fixes S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>t (S @ S') = wfrestrictedvars\<^sub>s\<^sub>t S \<union> wfrestrictedvars\<^sub>s\<^sub>t S'"
inj_imp_surj_inv: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "top_pres ((`) (inv f))"
pair_in_swap_image: fixes y :: "'c"   and x :: "'d"   and A :: "('d \<times> 'c) set" shows "((y, x) \<in> prod.swap ` A) = ((x, y) \<in> A)"
conforms_def: fixes xs :: "state"   and E :: "prog \<times> (lname \<Rightarrow> ty option)" shows "xs\<Colon>\<preceq>E = (let (G, L) = E; s = snd xs; l = locals s in (\<forall>r. ! obj:globs s r: G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r) \<and> G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L \<and> (\<forall>a. fst xs = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)) \<and> (fst xs = Some (Jump Ret) \<longrightarrow> l Result \<noteq> None))"
ideD_3: fixes a :: "('a, 'a set) Maps.arr" assumes "Maps.ide a" shows "Maps.cod a = a"
cancel_imp_equal: fixes u :: "'a"   and v :: "'a" assumes "u - v = \<zero>"   and "u \<in> R"   and "v \<in> R" shows "u = v"
no_rotate_prim: fixes w :: "'b list" assumes "w \<noteq> \<epsilon>"   and "\<And>n. \<lbrakk>0 < n; n < \<^bold>|w\<^bold>|\<rbrakk> \<Longrightarrow> rotate n w \<noteq> w" shows "primitive w"
simps: fixes \<I> :: "('e, 'f) \<I>"   and callee :: "'e \<Rightarrow> ('f \<times> 'g) spmf" shows "\<I> \<turnstile>c callee \<surd> = (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (callee x) \<longrightarrow> xa \<in> responses_\<I> \<I> x)"
simps_1: fixes t\<^sub>1 :: "sterm"   and t\<^sub>2 :: "sterm" shows "pre_strong_term_class.wellformed (t\<^sub>1 $\<^sub>s t\<^sub>2) = (pre_strong_term_class.wellformed t\<^sub>1 \<and> pre_strong_term_class.wellformed t\<^sub>2)"
mantissa10_def: fixes float10 :: "float10" shows "mantissa10 float10 = (case float10 of x1 \<e> x2 \<Rightarrow> x1)"
preserves_ide: fixes a :: "'a" assumes "B.ide a" shows "B.ide (trg\<^sub>B a)"
sum_inverse_squares_remainder_bigo: fixes N :: "nat" assumes "0 < N" shows "R N \<in> O(\<lambda>n. 1 / real n ^ (2 * N + 2))"
fw_shortest_path: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and i' :: "nat"   and i :: "nat"   and j' :: "nat"   and j :: "nat"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "linordered_ab_monoid_add.cycle_free plus less_eq neutral m n"   and "i' \<le> i"   and "j' \<le> j"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "linordered_ab_monoid_add.D plus less_eq m i' j' k = linordered_ab_monoid_add.fw plus less_eq m n k i j i' j'"
minus_coset_filter: fixes A :: "'g set"   and xs :: "'g list" shows "A - List.coset xs = set (filter (\<lambda>x. x \<in> A) xs)"
set_of_map_finite: fixes m :: "'c \<Rightarrow> 'd option" assumes "finite (dom m)" shows "finite (set_of_map m)"
R_def: shows "R = all zz (imp (LLq (Var zz) (Var yy)) (all xx' (imp (NN (Var xx) (Var xx')) (neg (PPf (Var zz) (Var xx'))))))"
dghm_ObjMap_app_in_HomCod_Obj: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<HH>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
cong_exp_mod_1: fixes a :: "'a"   and n :: "'a"   and b :: "nat"   and c :: "'a" shows "[(a mod n) ^ b = c] (mod n) = [a ^ b = c] (mod n)"
higher_differentiable_on_imp_continuous_on: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "higher_differentiable_on S f n" shows "continuous_on S f"
intro: fixes Rep :: "'c \<Rightarrow> 'd"   and A :: "'d set"   and Abs :: "'d \<Rightarrow> 'c" assumes "\<And>x. Rep x \<in> A"   and "\<And>x. Abs (Rep x) = x"   and "\<And>y. y \<in> A \<Longrightarrow> Rep (Abs y) = y" shows "type_definition Rep Abs A"
upper_triangular_upt_k_def: fixes A :: "(('c, 'd) vec, 'd) vec"   and k :: "nat" shows "upper_triangular_upt_k A k = (\<forall>i j. j < i \<and> mod_type_class.to_nat j < k \<longrightarrow> A $ i $ j = (0::'c))"
method_typing_aux: fixes CT :: "nat \<Rightarrow> classDef option"   and m :: "nat"   and D :: "nat"   and Cs :: "nat list"   and C :: "nat" shows "(\<forall>Ds D0. mtype(CT,m,D) = Ds \<rightarrow> D0 \<longrightarrow> Cs = Ds \<and> C = D0) = (\<not> method_typing_aux CT m D Cs C)"
tauChainRefl: fixes P :: "ccs" shows "P \<Longrightarrow>\<^sub>\<tau> P"
in_homE: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
inf_vec_def: fixes x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "inf x y = (\<chi>i. inf (x $ i) (y $ i))"
rec_fact_def: shows "rec_fact = CN rec_fact_aux [recf.Id 1 0, recf.Id 1 0]"
newvar: fixes vs :: "nat list" shows "newvar vs \<notin> set vs"
monoid_axioms: shows "Group_Theory.monoid M' (\<cdot>') \<one>'"
exhaust: fixes y :: "bval" obtains x1 :: "nat" where "y = BNat x1"   | x21 :: "nat" and x22 :: "exp" and x23 :: "(nat \<times> bval) list" where "y = BClos x21 x22 x23"
add_Real: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "Real.cauchy Y" shows "Real.Real X + Real.Real Y = Real.Real (\<lambda>n. X n + Y n)"
abelian_subgroupI3: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme" assumes "additive_subgroup H G"   and "abelian_group G" shows "abelian_subgroup H G"
divide_left_mono: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq b a"   and "less_eq zero c"   and "less zero (times a b)" shows "less_eq (divide c a) (divide c b)"
table_of_Some_in_set: fixes l :: "('d \<times> 'c) list"   and k :: "'d"   and x :: "'c" assumes "table_of l k = Some x" shows "(k, x) \<in> set l"
exit_validFrom: fixes vl :: "'value list"   and s :: "'state"   and tr :: "('state, 'act, 'out) trans list" assumes "vl \<noteq> []"   and "exit s (hd vl)"   and "validFrom s tr"   and "V tr = vl"   and "never T tr" shows "False"
2: shows "cod\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
nth_map_out_of_bound: fixes xs :: "'a list"   and i :: "nat"   and f :: "'a \<Rightarrow> 'b" assumes "length xs \<le> i" shows "map f xs ! i = [] ! (i - length xs)"
con_comp_failures_def: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" shows "con_comp_failures P Q p q \<equiv> {(xs, X \<union> Y \<union> Z) |xs X Y Z. set xs \<subseteq> range p \<union> range q \<and> X \<subseteq> range p \<and> Y \<subseteq> range q \<and> Z \<subseteq> - (range p \<union> range q) \<and> (map (inv p) (filter (\<lambda>x. x \<in> range p) xs), inv p ` X) \<in> failures P \<and> (map (inv q) (filter (\<lambda>x. x \<in> range q) xs), inv q ` Y) \<in> failures Q} \<union> {(xs, X). xs \<in> con_comp_divergences P Q p q}"
intro: fixes r :: "V" assumes "vpairs r = r" shows "vbrelation r"
fmrel_iff: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap" shows "fmrel R m n = (\<forall>x. rel_option R (fmlookup m x) (fmlookup n x))"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>" shows "\<guillemotleft>L f : L a \<Rightarrow>\<^sub>S L b\<guillemotright>"
pred_set: fixes P :: "'g \<Rightarrow> bool" shows "list_all P = (\<lambda>x. Ball (set x) P)"
circ_isolate_mult: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<circ> * y = x\<^sup>\<circ> * bot \<squnion> x\<^sup>\<star> * y"
\<Delta>_simps_4: shows "local.dom ((\<mu> \<star> r\<^sub>0) \<cdot> \<rho>) = r\<^sub>1"
of_nat_dvd_iff: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "dvd.dvd times (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m dvd n)"
exhaust: fixes y :: "com" obtains "y = SKIP"   | x21 :: "char list" and x22 :: "aexp" where "y = x21 ::= x22"   | x31 :: "com" and x32 :: "com" where "y = x31;; x32"   | x41 :: "bexp" and x42 :: "com" and x43 :: "com" where "y = IF x41 THEN x42 ELSE x43"   | x51 :: "bexp" and x52 :: "com" where "y = WHILE x51 DO x52"
Celarent: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A E B"   and "B Q C" shows "A E C"
runStateT_bindU: fixes m :: "udom\<cdot>('b, 'a) stateT"   and k :: "udom \<rightarrow> udom\<cdot>('b, 'a) stateT"   and s :: "'a" shows "runStateT\<cdot>(bindU\<cdot>m\<cdot>k)\<cdot>s = runStateT\<cdot>m\<cdot>s \<bind> (\<Lambda> (x, s'). runStateT\<cdot>(k\<cdot>x)\<cdot>s')"
IH: fixes e :: "exp" shows "(eta_expand n e)[x::=y] = eta_expand n e[x::=y]"
partition_on_msetD2: fixes A :: "'b multiset"   and P :: "'b multiset multiset" assumes "partition_on_mset A P" shows "{#} \<notin># P"
real_interval_min_interval: fixes a :: "float interval"   and b :: "float interval" shows "real_interval (min_interval a b) = min_interval (real_interval a) (real_interval b)"
1: shows "{x. poly p x = 0 \<and> poly q x = 0} = {}"
chambersubcomplex_image: shows "folding_g.ChamberSubcomplex (g \<turnstile> X)"
widens_refl: fixes P :: "'b prog"   and xs :: "ty list" shows "P \<turnstile> xs [\<le>] xs"
simps_4: fixes x :: "'c event"   and s :: "'c event list"   and X :: "'c event set"   and y :: "'c event"   and t :: "'c event list" shows "setinterleaving (x # s, X, y # t) = (if x \<in> X then if y \<in> X then if x = y then {x # u |u. u setinterleaves ((s, t), X)} else {} else {y # u |u. u setinterleaves ((x # s, t), X)} else if y \<notin> X then {x # u |u. u setinterleaves ((s, y # t), X)} \<union> {y # u |u. u setinterleaves ((x # s, t), X)} else {x # u |u. u setinterleaves ((s, y # t), X)})"
not_in_arcs_H: shows "{uv, rev_I uv} \<inter> arcs H = {}"
eucl_of_list_list_of_eucl: fixes x :: "'b" shows "eucl_of_list (list_of_eucl x) = x"
interior_eq: fixes S :: "'b set" shows "(interior S = S) = open S"
basis_exists: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "local.independent B" and "V \<subseteq> local.span B" and "card B = local.dim V"
strict_suffix_to_prefix: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs <s ys) = (rev xs <p rev ys)"
lincomb_replicate0_left: fixes ms :: "'m list"   and k :: "nat" assumes "set ms \<subseteq> M" shows "replicate k (0::'r) \<bullet>\<cdot> ms = (0::'m)"
step: fixes t :: "'t" assumes "t \<in> X" shows "(\<exists>x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x, shr s\<rangle> \<wrong> \<and> wset s t = None \<and> (\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (X \<union> deadlocked s \<union> final_threads s)))) \<or> (\<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> all_final_except s (X \<union> deadlocked s) \<and> waiting (wset s t)) \<or> (\<exists>x l t' ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> waiting (wset s t) \<and> 0 < ln $ l \<and> has_lock (locks s $ l) t' \<and> t' \<noteq> t \<and> ((t' \<in> X \<or> t' \<in> deadlocked s) \<or> final_thread s t'))"
scale_cancel_left: fixes a :: "complex"   and x :: "'d"   and y :: "'d" shows "(a *\<^sub>C x = a *\<^sub>C y) = (x = y \<or> a = 0)"
get_elements_by_class_name_result_in_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and name :: "char list"   and results :: "(_) element_ptr list"   and to :: "(_) object_ptr list"   and element_ptr :: "(_) element_ptr" assumes "h \<turnstile> get_elements_by_class_name ptr name \<rightarrow>\<^sub>r results"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to"   and "element_ptr \<in> set results" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<in> set to"
assms_1: shows "lambda C -quasi_isometry_on {a..b} f"
mult_scaleC_left: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "complex"   and x :: "'d"   and y :: "'d" assumes "class.complex_algebra scaleR scaleC plus zero minus uminus times" shows "times (scaleC a x) y = scaleC a (times x y)"
freshInputAction: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and c :: "name" assumes "P \<longmapsto> a<b> \<prec> P'"   and "c \<sharp> P" shows "c \<noteq> a"
loop_super_orientation_diagonal: fixes x :: "'a"   and y :: "'a" assumes "loop_super_orientation x y" shows "x \<sqinter> (1::'a) \<le> y \<sqinter> y\<^sup>T"
indspaceD_range: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and V :: "'v set" assumes "f \<in> indspace G FH V" shows "range f \<subseteq> V"
remdups_rev_code_ps_seteq: fixes ps1 :: "'a list"   and ps2 :: "'a list"   and rsa :: "'a list" assumes "set ps1 = set ps2" shows "remdups_rev_code ps1 rsa = remdups_rev_code ps2 rsa"
less_LiminfD: fixes y :: "'b"   and F :: "'d filter"   and f :: "'d \<Rightarrow> 'b" assumes "y < Liminf F f" shows "\<forall>\<^sub>F x in F. y < f x"
simps_1: fixes x1 :: "'e"   and x2 :: "'f"   and y1 :: "'e"   and y2 :: "'f" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
integrable_spike: fixes f :: "'c \<Rightarrow> 'd"   and T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "f integrable_on T"   and "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g integrable_on T"
cSup_union_distrib: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq A"   and "B \<noteq> {}"   and "preorder.bdd_above less_eq B" shows "Sup (A \<union> B) = sup (Sup A) (Sup B)"
Abs_bytes_inverse: fixes y :: "uint8 list" assumes "y \<in> UNIV" shows "Rep_bytes (Abs_bytes y) = y"
wf_lex_prod: fixes ra :: "('b \<times> 'b) set"   and rb :: "('c \<times> 'c) set" assumes "wf ra"   and "wf rb" shows "wf (ra <*lex*> rb)"
sig_of_spair: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "sig_of_pair (Inl (p, q)) = lt (spair p q)"
embed_Field: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "embed r r' f" shows "f ` Field r \<subseteq> Field r'"
single_numeral: fixes n :: "num" shows "Poly_Mapping.single (0::'a) (numeral n) = numeral n"