hcju/isabelleps huggingface dataset:
simple_function_cong: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t" shows "simple_function M f = simple_function M g"
numeral_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "num"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (m + n) = plus (numeral.numeral one plus m) (numeral.numeral one plus n)"
lConsts_def: shows "kbo'.lConsts = {(f, 0) |f. least f}"
subsumes_subsumes_list: fixes Ls :: "('c, 'd) Term.term literal list"   and Ks :: "('c, 'd) Term.term literal list" shows "subsumes (mset Ls) (mset Ks) = subsumes_list Ls Ks Map.empty"
fix_eq: fixes F :: "'d \<rightarrow> 'd" shows "fix\<cdot>F = F\<cdot>(fix\<cdot>F)"
assms_1: shows "orient_lit_inst L1 t s polarity \<sigma>"
IH_3: fixes u' :: "('node, ('var \<times> 'node \<times> Def) set) mapping"   and p' :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping"   and n :: "'node"   and v :: "'var" assumes "set ns \<subseteq> set (\<alpha>n g)"   and "distinct ns"   and "foldr (aux_2 g) ns (Mapping.empty, Mapping.empty) = (u', p')" shows "(if \<exists>n\<in>set ns. n \<in> phiDefNodes_aux g v (\<alpha>n g) n \<and> v \<in> uses g n then Some (map (\<lambda>m. lookupDef g m v) (predecessors g n)) else None) = Mapping.lookup p' (n, v)"
Ex_EH_1: fixes A :: "fm"   and H :: "fm set"   and B :: "fm"   and i :: "name" assumes "insert A H \<turnstile> B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "insert (SyntaxN.Ex i A) H \<turnstile> B"
rel_mset_free: fixes rel :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "'d multiset"   and Y :: "'e multiset"   and xs :: "'d list" assumes "rel_mset rel X Y"   and "mset xs = X" shows "\<exists>ys. mset ys = Y \<and> list_all2 rel xs ys"
pid_system_automaton_simps_3: shows "pid.V0 = {pid_init_gc}"
prod_0: fixes h :: "(char list \<Rightarrow> int option) \<times> nat" assumes "case h of (s, c) \<Rightarrow> s = Map.empty \<and> c = 0" shows "h = 0"
discrete_topology_unique: fixes U :: "'a set"   and X :: "'a topology" shows "(discrete_topology U = X) = (topspace X = U \<and> (\<forall>x\<in>U. openin X {x}))"
rbt_sorted_entries: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "sorted (map fst (entries t))"
Write\<^sub>s\<^sub>b: fixes \<O>' :: "nat set"   and volatile :: "bool"   and \<O> :: "nat set"   and A :: "nat set"   and R :: "nat set"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and W :: "nat set"   and L :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and rs :: "'c memref list" assumes "\<O>' = (if volatile then \<O> \<union> A - R else \<O>)"   and "\<S>' = (if volatile then \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L else \<S>)"   and "\<R>' = (if volatile then Map.empty else \<R>)" shows "(m, Write\<^sub>s\<^sub>b volatile a sop v A L R W # rs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (m(a := v), rs, \<O>', \<R>', \<S>')"
index_mat_2: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'b vec" assumes "i < nr"   and "j < nc" shows "mat\<^sub>r nr nc g $$ (i, j) = g i $ j"
elims_2: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'd" assumes "mat_diagonal x" obtains A :: "'a" and B :: "'b" and C :: "'c" and D :: "'d" where "x = (A, B, C, D)" and "B = (0::'b) \<and> C = (0::'c)"
L_nths: fixes S :: "nat set"   and init :: "'b list" assumes "S \<subseteq> {..<length init}" shows "map_pmf (\<lambda>l. nths l S) (bv (length init)) = bv (length (nths init S))"
n_add_lubI: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "n x \<sqsubseteq> n z"   and "n y \<sqsubseteq> n z" shows "n x \<oplus> n y \<sqsubseteq> n z"
ineq_about_squares: fixes x :: "real"   and y :: "real"   and c :: "real" assumes "x \<le> y" shows "x \<le> c\<^sup>2 + y"
memory_write_low_equal_none: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and bm :: "word4"   and data :: "word32" assumes "low_equal s1 s2"   and "memory_write 10 addr bm data s1 = None" shows "memory_write 10 addr bm data s2 = None"
sel_6: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt" shows "rsub (Node x21 x22 x23 x24 x25) = x25"
cInf_eq: fixes X :: "'a set"   and a :: "'a" assumes "\<And>x. x \<in> X \<Longrightarrow> a \<le> x"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a" shows "Inf X = a"
permutations_of_setD_2: fixes xs :: "'b list"   and A :: "'b set" assumes "xs \<in> permutations_of_set A" shows "distinct xs"
bit_set_bit_iff: fixes a :: "'b"   and m :: "nat"   and b :: "bool"   and n :: "nat" shows "bit (set_bit_class.set_bit a m b) n = ((if m = n then b else bit a n) \<and> (2::'b) ^ n \<noteq> (0::'b))"
simps_1: fixes t1 :: "Formula.trm"   and t2 :: "Formula.trm" shows "formula_of_constraint (t1, True, MEq, t2) = formula.Eq t1 t2"
events_of_def: fixes P :: "'b process" shows "events_of P \<equiv> \<Union>t\<in>T P. {a. ev a \<in> set t}"
intros_8: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<parallel> Q) \<parallel> R \<equiv>\<^sub>s P \<parallel> Q \<parallel> R"
set_iterator_foldli_conv: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'d set" shows "set_iterator iti S = (\<exists>l0. distinct l0 \<and> S = set l0 \<and> iti = foldli l0)"
linorder_neqE_linordered_idom: 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"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "x \<noteq> y" obtains "less x y"   | "less y x"
submatrix_singleton_index: fixes 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 A :: "'c mat"   and n :: "nat"   and m :: "nat"   and a :: "nat"   and b :: "nat" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "A \<in> carrier_mat n m"   and "a < n"   and "b < m" shows "submatrix A {a} {b} $$ (0, 0) = A $$ (a, b)"
sat_And: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (MFOTL.And \<phi> \<psi>) = (MFOTL.sat \<sigma> v i \<phi> \<and> MFOTL.sat \<sigma> v i \<psi>)"
wle: shows "wgreater (fst ([t2]<>v)) (snd ([t1]<>v))"
iffs: fixes c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e"   and c_records' :: "'d zmultiset"   and c_temp' :: "'c \<Rightarrow> 'd zmultiset"   and c_msg' :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob' :: "'c \<Rightarrow> 'd zmultiset"   and more' :: "'e" shows "(\<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = \<lparr>c_records = c_records', c_temp = c_temp', c_msg = c_msg', c_glob = c_glob', \<dots> = more'\<rparr>) = (c_records = c_records' \<and> c_temp = c_temp' \<and> c_msg = c_msg' \<and> c_glob = c_glob' \<and> more = more')"
tranclp_induct2: fixes r :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and aa :: "'a"   and ab :: "'b"   and ba :: "'a"   and bb :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ (aa, ab) (ba, bb)"   and "\<And>a b. r (aa, ab) (a, b) \<Longrightarrow> P a b"   and "\<And>a b ac ba. \<lbrakk>r\<^sup>+\<^sup>+ (aa, ab) (a, b); r (a, b) (ac, ba); P a b\<rbrakk> \<Longrightarrow> P ac ba" shows "P ba bb"
mod_ring_0__integer: shows "mod_ring_rel_integer 0 0"
vimage_subsetD: fixes f :: "'b \<Rightarrow> 'a"   and B :: "'a set"   and A :: "'b set" assumes "surj f"   and "f -` B \<subseteq> A" shows "B \<subseteq> f ` A"
noS: shows "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> (\<lambda>(c, t). - Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) ` set (uset p)"
is_sentence_cons: fixes N :: "symbol"   and s :: "symbol list" shows "is_sentence (N # s) = (is_symbol N \<and> is_sentence s)"
closedin_continuous_map_preimage: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and C :: "'d set" assumes "continuous_map X Y f"   and "closedin Y C" shows "closedin X {x \<in> topspace X. f x \<in> C}"
border_conjug_eq: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "w\<^sup><\<inverse>x  \<cdot> w = w \<cdot> x\<inverse>\<^sup>>w"
bind_distr: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and X :: "'b measure"   and N :: "'b \<Rightarrow> 'd measure"   and K :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M X"   and "N \<in> X \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}" shows "distr M X f \<bind> N = M \<bind> (\<lambda>x. N (f x))"
LC: shows "while brc_cond (brc_step (build_rqrm \<delta>)) (brc_initial \<delta>) = loop"
simps: fixes v :: "nat \<Rightarrow> 'b"   and b :: "(nat \<Rightarrow> 'b option) \<times> (nat \<Rightarrow> 'b option)" shows "v \<Turnstile>\<^sub>b b = (\<forall>x. in_bounds x v b)"
wf_sees_clinit: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and a :: "'b class" assumes "wf_prog wf_md P"   and "class P C = \<lfloor>a\<rfloor>" shows "\<exists>m. P \<turnstile> C sees clinit, Static :  []\<rightarrow>Void = m in C"
indRelRSTPO_to_SRel_and_TRel_3: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> Q" shows "\<forall>TP SQ. TP \<in>T P \<and> SQ \<in>S Q \<longrightarrow> False"
r'_sub: shows "r' \<subseteq> c \<times> c"
semigroup_mult_hom_axioms: shows "semigroup_mult_hom hom"
akra_bazzi_terms_def: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list" shows "akra_bazzi_terms x\<^sub>0 x\<^sub>1 bs ts = (\<forall>i<length bs. akra_bazzi_term x\<^sub>0 x\<^sub>1 (bs ! i) (ts ! i))"
IH_1: shows "chain_boundary p (chain_map p g a) = chain_map (p - Suc 0) g (chain_boundary p a)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg e' e (local.inv \<psi>') (local.inv \<phi>)"
smallest_dist_is_ortho: fixes M :: "'b set"   and h :: "'b"   and k :: "'b" assumes "closed_csubspace M" shows "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k = (h - k \<in> orthogonal_complement M \<and> k \<in> M)"
P2: shows "Bet A DD C \<and> (DD = B \<or> B Out DD D)"
B_2: fixes a :: "'b set" assumes "a \<in> B" shows "snd x \<in> a"
regular_conc: fixes S :: "'b list set"   and T :: "'b list set" assumes "regular S"   and "regular T" shows "regular (S @@ T)"
finite_field_factorization: fixes f :: "'a mod_ring poly"   and c :: "'a mod_ring"   and us :: "'a mod_ring poly list" assumes "square_free f"   and "finite_field_factorization f = (c, us)" shows "unique_factorization Irr_Mon f (c, mset us)"
add_ant_def: fixes z :: "ant"   and w :: "ant" shows "z + w \<equiv> Abs_Ainteg (zag_pl (Rep_Ainteg z) (Rep_Ainteg w))"
assms_2: shows "length xs = length ys + 1 + length ws + 1 + length zs"
subrelI: fixes r :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
remdups_mset_singleton: fixes a :: "'a" shows "remdups_mset {#a#} = {#a#}"
ide_in_hom_1: fixes f :: "'d" assumes "D.ide f" shows "\<guillemotleft>f : src\<^sub>D f \<rightarrow>\<^sub>D trg\<^sub>D f\<guillemotright>"
multiplicity_is_1: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "multiplicity bl = 1"
simps_8: fixes v :: "char list"   and va :: "aexp"   and vb :: "'b"   and vc :: "bexp"   and vd :: "'b acom"   and ve :: "'b acom"   and vf :: "'b" shows "(v ::= va {vb} \<sqsubseteq> IF vc THEN vd ELSE ve {vf}) = False"
diff_minus_eq_add: fixes minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q"   and b :: "'q" assumes "class.group_add minus plus zero uminus" shows "minus a (uminus b) = plus a b"
one_more_dom: fixes l :: "'g"   and f :: "'g \<Rightarrow> 'h option" assumes "l \<in> dom f" shows "\<exists>f'. f = f'(l \<mapsto> the (f l)) \<and> l \<notin> dom f'"
Span_subgroup_props_3: fixes Us :: "'a list"   and v1 :: "'a"   and v2 :: "'a" assumes "set Us \<subseteq> carrier R"   and "v1 \<in> Span K Us"   and "v2 \<in> Span K Us" shows "v1 \<oplus> v2 \<in> Span K Us"
differentiable_eqI: fixes g :: "'b \<Rightarrow> 'c"   and x :: "'b"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "g differentiable at x"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x"   and "x \<in> X"   and "open X" shows "f differentiable at x"
dependent_imp_non_trivial_combine: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "R.dependent K Us" obtains Ks :: "'a list" where "length Ks = length Us" and "R.combine Ks Us = \<zero>" and "set Ks \<subseteq> K" and "set Ks \<noteq> {\<zero>}"
monoidal_functor_axioms: shows "monoidal_functor (\<cdot>) \<F>C.T\<^sub>F\<^sub>M\<^sub>C \<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C \<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C (\<cdot>\<^sub>S) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<F>C.D D.\<phi>"
wf_ts_regex_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat"   and r :: "Formula.formula Regex.regex"   and ts :: "nat list" shows "wf_ts_regex \<sigma> P j r ts = list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [progress_regex \<sigma> P r j..<j] ts"
real_resource_wiring: shows "macode.res (rnd \<eta>) (mac \<eta>) = RES (\<dagger>\<dagger>insec_channel.insec_oracle \<oplus>\<^sub>O local.rorc_channel_send \<oplus>\<^sub>O local.rorc_channel_recv) ((False, ()), Map.empty, Void)"
leg0_in_hom_1: shows "\<guillemotleft>F f : src\<^sub>D (F f) \<rightarrow>\<^sub>D src\<^sub>D (F r)\<guillemotright>"
count_single: fixes b :: "'c"   and a :: "'c" shows "count {#b#} a = (if b = a then 1 else 0)"
are_all_operators_applicable_set: fixes s :: "'b \<Rightarrow> bool option"   and ops :: "'b strips_operator list" shows "are_all_operators_applicable s ops = (\<forall>op\<in>set ops. \<forall>v\<in>set (precondition_of op). s v = Some True)"
length_zip: fixes xs :: "'d list"   and ys :: "'e list" shows "length (xs || ys) = min (length xs) (length ys)"
precondition_after_set_error_code: shows "\<forall>s d a a'. AS_precondition s d a \<and> aborting s (current s) a' \<longrightarrow> AS_precondition (set_error_code s a') d a"
more_permissive_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and q :: "'c \<Rightarrow> 'd decision option" shows "p \<sqsubseteq>\<^sub>A q = (\<forall>x. case q x of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> \<exists>z. p x = \<lfloor>allow z\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> True)"
card_map_to_set: fixes m :: "'a \<Rightarrow> 'b option" shows "card (map_to_set m) = card (dom m)"
tp_cdiv: fixes f :: "real \<Rightarrow> real"   and c :: "real" assumes "trigpoly f" shows "trigpoly (\<lambda>x. f x / c)"
zero_im_sheaf_def: shows "zero_im_sheaf \<equiv> \<lambda>V. zero_str (f \<^sup>\<inverse> X V)"
rep_eq: fixes x :: "nat \<Rightarrow> 'b" shows "llist_of_stream (of_seq x) = inf_llist x"
simps_1: fixes acc :: "'a list" shows "rev_it' acc [] = acc"
l11: assumes "\<not> QuadRes (int p) a" shows "card local.S2 = (p - 1) div 2"
alpha_tAct: fixes f1 :: "'e"   and \<alpha>1 :: "'d"   and t1 :: "('b, 'c, 'd, 'e) Tree"   and f2 :: "'e"   and \<alpha>2 :: "'d"   and t2 :: "('b, 'c, 'd, 'e) Tree" shows "(tAct f1 \<alpha>1 t1 =\<^sub>\<alpha> tAct f2 \<alpha>2 t2) = (f1 = f2 \<and> (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)))"
simps: fixes a :: "'b"   and b :: "bool"   and r :: "(bool \<times> 'b) rexp" shows "delta_a a (b, r) = (False, read a (follow b r))"
mgu_img_consts: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and c :: "'b" assumes "mgu s t = Some \<delta>"   and "Fun c [] \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t subst_range \<delta>" shows "Fun c [] \<in> subterms s \<union> subterms t"
preserving: fixes u :: "complex_homo"   and a :: "complex" assumes "u \<in> unit_disc"   and "cmod a < 1"   and "P (moebius_pt (blaschke a) u)" shows "P u"
csets_def: shows "csets \<equiv> map_fun id ((`) acset) (\<lambda>A. {B. B \<subseteq> A \<and> countable B})"
gso_int_carrier_vec: fixes i :: "nat"   and l :: "nat" assumes "i < length fs"   and "l \<le> i" shows "gso_int i l \<in> carrier_vec n"
valid_selectorD_2: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and selector :: "nat \<times> ('d \<times> nat \<times> nat) set \<Rightarrow> ((('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) \<times> ('e \<times> nat) set) option"   and G :: "nat \<times> ('d \<times> nat \<times> nat) set"   and R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph"   and f :: "('e \<times> nat) set" assumes "valid_selector Rs selector"   and "selector G = Some (R, f)" shows "valid_selection Rs G R f"
nn_integral_fst: fixes f :: "'a \<times> 'b \<Rightarrow> ennreal"   and M1 :: "'a measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M lborel)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>lborel \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M lborel) f"
Sup_le_iff: fixes Inf :: "'k set \<Rightarrow> 'k"   and Sup :: "'k set \<Rightarrow> 'k"   and inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bot :: "'k"   and top :: "'k"   and A :: "'k set"   and b :: "'k" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
ndec_seqn1: fixes a :: "nat"   and A :: "nat set"   and n :: "nat" assumes "a \<in> A"   and "\<nexists>m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)" shows "ndec_seq A a (Suc n) \<le> ndec_seq A a n - 1"
proj2_abs_mult_rep: fixes c :: "real"   and x :: "proj2" assumes "c \<noteq> 0" shows "proj2_abs (c *\<^sub>R proj2_rep x) = x"
mod_mult_cong: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and a' :: "'b"   and b :: "'b"   and b' :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (times a b) c = modulo (times a' b') c"
\<alpha>_limit_set_in_compact_compact: shows "compact (\<alpha>_limit_set x)"
fps_compose_assoc: fixes c :: "'a fps"   and b :: "'a fps"   and a :: "'a fps" assumes "fps_nth c 0 = (0::'a)"   and "fps_nth b 0 = (0::'a)" shows "a oo (b oo c) = a oo b oo c"
LeastI_ex: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"
Collect_iff: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "a \<in> {x. P x} \<equiv> P a"
add_pos_pos: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less zero b" shows "less zero (plus a b)"
idem: fixes a :: "'c DBMEntry" shows "linordered_monoid.min a a = a"
simps_40: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stop"
mono_Sup: 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 bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "order.mono less_eq f" shows "\<Squnion> (f ` A) \<le> f (Sup A)"
to_hma_from_hma\<^sub>v: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec" shows "HMA_Connect.to_hma\<^sub>v (HMA_Connect.from_hma\<^sub>v v) = v"
invc2_baldL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invc2 l"   and "invc r" shows "invc2 (baldL l a r)"
is_factorization_of_def: fixes fctrs :: "'b \<times> ('b \<times> nat) list"   and p :: "'b poly" shows "is_factorization_of fctrs p = (interp_factorization fctrs = p \<and> distinct (map fst (snd fctrs)))"
injf: fixes i :: "'a" assumes "i \<in> I" shows "inj_on (f i) (carrier (G i))"
A_def: shows "A = (\<lambda>m. box (complex_of_real (- (real m + 1)) - complex_of_real (real (2 * m + 2) * pi) * \<i>) (complex_of_real (real m + 1) + complex_of_real (real (2 * m + 2) * pi) * \<i>))"
prime_elem_dvd_prod_mset: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a"   and A :: "'a multiset" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p"   and "dvd.dvd times p (comm_monoid_mult.prod_mset times one A)" shows "\<exists>a\<in>#A. dvd.dvd times p a"
hom_grading_fun_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and d :: "'a \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.monoid_add plus zero" shows "monoid_add.hom_grading_fun plus d f = (\<forall>n. (\<forall>s t. f n (plus s t) = plus (f n s) (f n t)) \<and> (\<forall>t. d (f n t) \<le> n \<and> (d t \<le> n \<longrightarrow> f n t = t)))"
Red_preserves_bconf: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"   and "iconf (shp s) e"   and "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>" shows "P,shp s' \<turnstile>\<^sub>b (e',b') \<surd>"