hcju/isabelleps huggingface dataset:
borel_rev_eq: shows "sigma_sets UNIV {S. open S} = sets borel"
list_all2_split: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and Q :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "list_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> \<exists>z. Q x z \<and> R z y" shows "\<exists>zs. list_all2 Q xs zs \<and> list_all2 R zs ys"
distinct_3: fixes name1 :: "name"   and name2 :: "name" shows "name1<name2> \<noteq> \<tau>"
f: shows "q.eq_m f (Polynomial.smult (lead_coeff f) F)"
d_OUT_def: fixes g :: "'b \<times> 'b \<Rightarrow> ennreal"   and x :: "'b" shows "d_OUT g x = (\<Sum>\<^sup>+ y. g (x, y))"
HInv4c_def: fixes s :: "state"   and p :: "Proc" shows "HInv4c s p = (phase s p \<in> {2, 3} \<longrightarrow> (\<exists>D\<in>MajoritySet. \<forall>d\<in>D. mbal (disk s d p) = bal (dblock s p)))"
esize_mono: fixes esize :: "'b \<Rightarrow> enat"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.esize_order esize less_eq less"   and "less_eq x y" shows "esize x \<le> esize y"
FixVars_def: shows "FixVars TYPE('d) TYPE('e) \<equiv> var_infinite undefined \<and> var_regular undefined \<and> varSort_lt_var undefined undefined"
ring_ops_def: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "ring_ops ops R \<equiv> ((bi_unique R \<and> right_total R) \<and> R (arith_ops_record.zero ops) (0::'c) \<and> R (arith_ops_record.one ops) (1::'c) \<and> (R ===> R ===> R) (arith_ops_record.plus ops) (+)) \<and> ((R ===> R ===> R) (arith_ops_record.minus ops) (-) \<and> (R ===> R) (arith_ops_record.uminus ops) uminus) \<and> (R ===> R ===> R) (arith_ops_record.times ops) (*) \<and> (R ===> R ===> (=)) (=) (=) \<and> Domainp R = arith_ops_record.DP ops"
hyps_6: fixes k :: "('fun, 'var) Term.term" assumes "k \<in> set K" shows "S \<union> ik\<^sub>e\<^sub>s\<^sub>t B \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c k \<cdot> \<I>"
convex_hull_sum: fixes A :: "'b vec set"   and n :: "nat"   and B :: "'b vec set" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> carrier_vec n" shows "gram_schmidt.convex_hull n (A + B) = gram_schmidt.convex_hull n A + gram_schmidt.convex_hull n B"
rep_eq: fixes x :: "'a sctn"   and xa :: "'a pdevs" shows "pdevs_apply (project_coord_pdevs x xa) = (\<lambda>i. project_coord (pdevs_apply xa i) (normal x) 0)"
prems: shows "\<forall>y<Suc z. eval f (y # xs) \<notin> {None, Some 0}"
assms: shows "Poly_Mapping.keys x \<subseteq> S"
transfer: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (list_R1.rel_list_R1 R) (list_all2 R) list_R1.list_R1_to_list list_R1.list_R1_to_list"
iso_inv_iso: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.iso (VV.inv f)"
statImpTauDerivative: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'" shows "insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P') \<Psi>"
weight_spmf_of_set: fixes A :: "'b set" shows "weight_spmf (spmf_of_set A) = (if finite' A then 1 else 0)"
hnz: fixes w :: "complex" assumes "w \<in> ball z0 r" shows "h w \<noteq> 0"
is_gcd_some_gcd: fixes a :: "'a"   and b :: "'a" shows "is_gcd (some_gcd a b) a b"
assms_3: fixes g :: "'a" assumes "f \<noteq> g" shows "F2 g = F1 g"
i_contract_mono: fixes f :: "'a \<Rightarrow> 'c"   and J1 :: "'c set"   and J2 :: "'c set" assumes "f \<in> rHom R R'"   and "ideal R' J1"   and "ideal R' J2"   and "J1 \<subseteq> J2" shows "i_contract f R R' J1 \<subseteq> i_contract f R R' J2"
pop_closed: shows "(\<plusminus>) \<in> carrier R \<rightarrow> carrier R \<rightarrow> carrier R"
Key_unique_lemma: fixes evs :: "event list"   and K :: "nat"   and S :: "agent"   and R :: "agent"   and m :: "nat"   and AO :: "nat"   and cleartext :: "nat"   and q :: "nat"   and hs :: "msg"   and S' :: "agent"   and R' :: "agent"   and m' :: "nat"   and AO' :: "nat"   and cleartext' :: "nat"   and q' :: "nat"   and hs' :: "msg" assumes "evs \<in> certified_mail"   and "Key K \<notin> analz (knows Spy evs)"   and "Says S R \<lbrace>Agent S, Agent TTP, Crypt K (Number m), Number AO, Number cleartext, Nonce q, Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace>\<rbrace> \<in> set evs"   and "Says S' R' \<lbrace>Agent S', Agent TTP, Crypt K (Number m'), Number AO', Number cleartext', Nonce q', Crypt (pubK TTP) \<lbrace>Agent S', Number AO', Key K, Agent R', hs'\<rbrace>\<rbrace> \<in> set evs" shows "R' = R \<and> S' = S \<and> AO' = AO \<and> hs' = hs"
mono: fixes x :: "('h, 'i, 'j) gpv"   and f :: "'h \<Rightarrow> ennreal"   and \<I> :: "('i, 'j) \<I>"   and fail :: "ennreal" shows "lfp.mono_body (\<lambda>expectation_gpv. \<integral>\<^sup>+ generat. (case generat of Pure x \<Rightarrow> f x | IO out c \<Rightarrow> \<Sqinter>r\<in>responses_\<I> \<I> out. expectation_gpv (c r)) \<partial>measure_spmf (the_gpv x) + fail * ennreal (pmf (the_gpv x) None))"
img_def: fixes f :: "'s" shows "img f = mkIde (Img f)"
card_suc: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "card {i. i \<le> Suc n \<and> P i} = card {i. i \<le> n \<and> P i} + card {i. i = Suc n \<and> P i}"
preserves_reflects_arr: fixes f :: "'j" shows "Cop_S.arr ((local.map \<circ> \<chi>) f) = J.arr f"
yD_2: shows "sgn (rai_lb (p2, l2, r2)) = sgn (rai_ub (p2, l2, r2))"
def_sel: shows "\<tau> \<Turnstile> \<delta> select_object_any\<^sub>S\<^sub>e\<^sub>q f s_set"
pf: shows "pf \<in> nnfis (pp f) M"
"1_2": shows "y \<in> {0..<2 * pi}"
STCal_divergent_1: fixes S :: "'a"   and Source :: "'a processCalculus"   and Target :: "'b processCalculus" shows "SourceTerm S \<longmapsto>STCal Source Target\<omega> = S \<longmapsto>Source\<omega>"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'d tree \<Rightarrow> 'c"   and f3 :: "'d tree \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and up2 :: "'d up2" shows "h (case up2 of Same2 \<Rightarrow> f1 | Bal2 x \<Rightarrow> f2 x | Unbal2 x xa xb \<Rightarrow> f3 x xa xb) = (case up2 of Same2 \<Rightarrow> h f1 | Bal2 x \<Rightarrow> h (f2 x) | Unbal2 x1 x2 x3 \<Rightarrow> h (f3 x1 x2 x3))"
nz: shows "(0::'a) \<notin> fst ` set cs"
m_s_def: fixes S :: "'a st"   and X :: "char list set" shows "m\<^sub>s S X = (\<Sum>x\<in>X. m (fun S x))"
state_change_on_either_2: fixes i :: "nat"   and t :: "nat" assumes "gap i (Suc t) \<noteq> gap i t" shows "gap i (Suc t) = e_length (block i t)"
B: shows "B = fst (gauss_jordan (mat_of_rows n G)\<^sup>T (1\<^sub>m n))"
C_bound: fixes b :: "real"   and x :: "real" assumes "b \<in> set bs"   and "x\<^sub>1 \<le> x" shows "C * x \<le> b * x - hb * x / ln x powr (1 + e)"
Func_def: fixes A :: "'c set"   and B :: "'d set" shows "Func A B = {f. (\<forall>a\<in>A. f a \<in> B) \<and> (\<forall>a. a \<notin> A \<longrightarrow> f a = undefined)}"
g2nonzero: shows "(gcd c d)\<^sup>2 \<noteq> 0"
Rep_action_inverse: fixes x :: "'d action" shows "Abs_action (Rep_action x) = x"
cos_converges: fixes x :: "'b" shows "(\<lambda>n. cos_coeff n *\<^sub>R x ^ n) sums cos x"
surjective: fixes r :: "('a, 'b, 'c, 'd, 'e) galcon_scheme" shows "r = \<lparr>orderA = \<X>\<^bsub>r\<^esub>, orderB = \<Y>\<^bsub>r\<^esub>, lower = \<pi>\<^sup>*\<^bsub>r\<^esub>, upper = \<pi>\<^sub>*\<^bsub>r\<^esub>, \<dots> = galcon.more r\<rparr>"
isomorphic_implies_hpar_4: fixes f :: "'d"   and f' :: "'d" assumes "D.isomorphic f f'" shows "trg\<^sub>D f = trg\<^sub>D f'"
empty: fixes g :: "'e \<Rightarrow> 'd" shows "sum g {} = (0::'d)"
ideD_2: fixes a :: "'a" assumes "B.ide a" shows "B.dom a = a"
Le: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "dfa_of_pf n (Le ks l) = ineq_dfa n ks l"
Leaf_in_space_tree_sigma: fixes M :: "'b measure" shows "\<langle>\<rangle> \<in> space (tree_sigma M)"
ex_terminal_arrow: fixes y :: "'a" assumes "ide y" shows "\<exists>x e. terminal_arrow_from_functor C C (\<lambda>x. local.prod x b) x y e"
callee_invariant: fixes y :: "'b"   and s' :: "'s1"   and s :: "'s1"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee1 s x)"   and "I1 s"   and "x \<in> outs_\<I> \<I>" shows "I1 s'"
apply_guards_subset: fixes g' :: "vname gexp list"   and g :: "vname gexp list"   and c :: "vname \<Rightarrow> value option" assumes "set g' \<subseteq> set g" shows "apply_guards g c \<longrightarrow> apply_guards g' c"
sigma_finite_measure_axioms: shows "sigma_finite_measure (lborel \<Otimes>\<^sub>M M)"
f: fixes n :: "nat" shows "flow \<Psi> (f n)"
lderivs_PLUS: fixes xs :: "'a list"   and ys :: "'b rexp list" shows "lderivs xs (PLUS ys) = PLUS (map (lderivs xs) ys)"
src_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.hseq \<mu> \<nu>" shows "src (\<mu> \<star> \<nu>) = src \<nu>"
val_inf_n_val_inf: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "(v x = \<infinity>) = (n_val K v x = \<infinity>)"
cases_scheme: fixes r :: "'a state_scheme" obtains heap :: "heap" and locals :: "locals" and more :: "'a" where "r = \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr>"
step_eq_in: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and as :: "nat"   and ins :: "abc_inst"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "abc_fetch as ap = Some ins"   and "step (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) = (s', l', r')"   and "s' \<noteq> 0" shows "step0 (s, l, r) tp = (s', l', r')"
mult_L_circ: fixes x :: "'a" shows "(x * L)\<^sup>\<circ> = Rf x L"
intros_29: fixes \<C> :: "t_context"   and n :: "nat"   and a :: "nat"   and tp_sx :: "(tp \<times> sx) option"   and t :: "t"   and off :: "nat" assumes "memory \<C> = Some n"   and "load_store_t_bounds a (option_projl tp_sx) t" shows "\<C> \<turnstile> [Load t tp_sx a off] : [T_i32] _> [t]"
leftmost_exists: fixes a :: "symbol list"   and k :: "nat" assumes "is_sentence a"   and "k < length a"   and "is_nonterminal (a ! k)" shows "\<exists>i. leftmost i a \<and> i \<le> k"
wordinterval_element_set_eq: fixes el :: "'a word"   and rg :: "'a wordinterval" shows "wordinterval_element el rg = (el \<in> wordinterval_to_set rg)"
obj_simps_2: fixes a :: "'a" assumes "B.obj a" shows "src a = a"
to_nat_mono: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less a b" shows "mod_type.to_nat Rep a < mod_type.to_nat Rep b"
term_poly_list_relD_2: fixes xs :: "char list list"   and ys :: "char list multiset" assumes "(xs, ys) \<in> term_poly_list_rel" shows "ys = mset xs"
IH: shows "concat m = concat (map (\<lambda>i. map ((!) (m ! i)) [0..<nr]) [0..<nc])"
terminal_def: fixes b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "terminal b \<equiv> ide b \<and> (\<forall>a. ide a \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow> b\<guillemotright>))"
mon_c_empty: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme" shows "mon_c fg {#} = {}"
inverse_add_Infinitesimal_approx: fixes x :: "'a star"   and h :: "'a star" assumes "x \<in> HFinite - Infinitesimal"   and "h \<in> Infinitesimal" shows "inverse (x + h) \<approx> inverse x"
simps_2: fixes p :: "'b fm"   and ps :: "'b fm list"   and q :: "'b fm" shows "imply (p # ps) q = (p \<^bold>\<longrightarrow> imply ps q)"
card_double: shows "2 * card {{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} |i. i < n} = card (\<Union> {{dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} |i. i < n})"
rcResF: fixes P' :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "F (<\<nu>x>P')"
igSubstAllIPresIGWlsAll_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gSubstAllPresGWlsAll MOD" shows "igSubstAllIPresIGWlsAll (fromMOD MOD)"
value_mprod_expTr: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v" shows "(\<forall>j\<le>n. e j \<in> Zset) \<and> (\<forall>j\<le>n. f j \<in> carrier K - {\<zero>}) \<longrightarrow> v (mprod_exp K e f n) = ASum (\<lambda>j. e j *\<^sub>a v (f j)) n"
bottomI: fixes z :: "'b"   and P :: "'b set" assumes "z \<in> P"   and "\<And>y. y \<in> P \<Longrightarrow> order.greater_eq y z" shows "z = order.bottom P"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
beukers_integral1_different: fixes s :: "nat"   and r :: "nat" assumes "s \<le> r"   and "s < r" shows "beukers_integral1 r s = (\<Sum>k\<in>{s<..r}. 1 / real (k\<^sup>2)) / real (r - s)"
ideD_2: fixes a :: "'b \<times> 'b" assumes "B.VV.ide a" shows "B.VV.dom a = a"
UPFDefs_24: fixes p :: "'b" shows "D\<^sub>U p = \<lfloor>deny ()\<rfloor>"
intros_4: fixes v :: "Syntax.term"   and x :: "Syntax.var"   and \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and uv :: "mode" assumes "value v"   and "atom x \<sharp> (v, \<pi>)" shows "\<lless>\<pi>, App (Rec x e) v\<ggreater> uv\<rightarrow> \<lless>\<pi>, App (e[Rec x e / x]) v\<ggreater>"
fr1: shows "(q \<bullet> set as) \<sharp>* (x, c, f as x c, f bs y c)"
inner_prod_with_itself_Im: fixes u :: "complex vec" shows "Im \<langle>u|u\<rangle> = 0"
enum_in: fixes i :: "nat" assumes "i \<le> n" shows "t.enum i \<in> t"
simps_36: fixes vd :: "'b acom"   and ve :: "'b acom"   and v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b" shows "(vd;; ve \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
\<iota>_in_hom': shows "\<guillemotleft>\<iota>\<^sub>C : C.cod \<iota>\<^sub>C \<otimes>\<^sub>C C.cod \<iota>\<^sub>C \<rightarrow>\<^sub>C C.cod \<iota>\<^sub>C\<guillemotright>"
convex_cball: fixes x :: "'b"   and e :: "real" shows "convex (cball x e)"
theFTOne_length_finished: fixes n :: "nat"   and cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "n \<in> theFTOne cl dl" shows "n < length cl \<and> finished (cl ! n) \<or> n < length dl \<and> finished (dl ! n)"
fact_num_eq_if: fixes m :: "nat" shows "fact m = (if m = 0 then 1::'a else of_nat m * fact (m - 1))"
power_diff_sumr2: fixes x :: "'b"   and n :: "nat"   and y :: "'b" shows "x ^ n - y ^ n = (x - y) * (\<Sum>i<n. y ^ (n - Suc i) * x ^ i)"
supremum_unique: fixes a :: "'a"   and A :: "'a set"   and b :: "'a" assumes "a \<in> supremum A"   and "b \<in> supremum A" shows "a = b"
isodefl_DEFL_imp_ID: fixes d :: "'a \<rightarrow> 'a" assumes "isodefl d DEFL('a)" shows "d = ID"
d1_def: shows "stm2.d1 = prod_list stm2.dims1"
irreducibleD: fixes C :: "'s set"   and a :: "'s"   and b :: "'s" assumes "C \<in> UNIV // communicating"   and "a \<in> C"   and "b \<in> C" shows "(a, b) \<in> communicating"
mat_hom_one: fixes hom :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "semiring_hom hom" shows "map_mat hom (1\<^sub>m n) = 1\<^sub>m n"
sigma_finite_measure_axioms: shows "sigma_finite_measure (distr M S X)"
chine_assoc'_props_2: shows "Prj\<^sub>1\<^sub>1 \<cdot> chine_assoc' = Prj\<^sub>1"
Phase_partitioning_general: fixes x0 :: "nat"   and y0 :: "nat"   and c :: "real"   and \<sigma> :: "nat list"   and P :: "(nat list \<times> 'a) pmf \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool"   and A :: "(nat list \<Rightarrow> 'a pmf) \<times> (nat list \<times> 'a \<Rightarrow> nat \<Rightarrow> ((nat \<times> nat list) \<times> 'a) pmf)" assumes "x0 \<noteq> y0"   and "0 \<le> c"   and "set \<sigma> \<subseteq> {x0, y0}"   and "P (map_pmf (Pair [x0, y0]) (fst A [x0, y0])) x0 [x0, y0]"   and "\<And>a b \<sigma> s. \<lbrakk>\<sigma> \<in> Lxx a b; a \<noteq> b; {a, b} = {x0, y0}; P s a [x0, y0]; set \<sigma> \<subseteq> {a, b}\<rbrakk> \<Longrightarrow> T\<^sub>p_on_rand' A s \<sigma> \<le> c * real (T\<^sub>p [a, b] \<sigma> (OPT2 \<sigma> [a, b])) \<and> P (Partial_Cost_Model.config'_rand A s \<sigma>) (last \<sigma>) [x0, y0]" shows "T\<^sub>p_on_rand' A (fst A [x0, y0] \<bind> (\<lambda>is. return_pmf ([x0, y0], is))) \<sigma> \<le> c * real (T\<^sub>p_opt [x0, y0] \<sigma>) + c"
prefI: fixes A :: "'b \<Rightarrow> bool" assumes "\<And>w. A w" shows "|~ A"
orbit_altdef_self_in: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "s \<in> orbit f s" shows "orbit f s = {(f ^^ n) s |n. True}"
list_all_conv_same_lab_h: fixes ls :: "t list list"   and ts :: "t list"   and ils :: "nat list" assumes "list_all (\<lambda>i. i < length ls \<and> ls ! i = ts) ils" shows "same_lab_h ils ls ts = Some ts"
ParamDefs_return_target_length: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ins :: "'var list"   and outs :: "'var list" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "(p, ins, outs) \<in> set procs" shows "length (ParamDefs (targetnode a)) = length outs"
abs_eq: shows "1\<^sub>h = abs_complex_homo 1\<^sub>h\<^sub>c"
chine_hcomp_ide_ide: fixes g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide g"   and "Span.ide f"   and "Span.src g = Span.trg f" shows "Span.chine_hcomp g f = Maps.pbdom (Leg0 (Dom g)) (Leg1 (Dom f))"
cut_ge_Max_eq: fixes I :: "'a set"   and t :: "'a" assumes "finite I"   and "I \<down>\<ge> t \<noteq> {}" shows "Max (I \<down>\<ge> t) = Max I"
hyps: fixes \<S>' :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat set" shows "dom (release sb \<S>' \<R>) = dom (release sb \<S> \<R>)"
cINF_greatest: fixes A :: "'k set"   and m :: "'l multiset"   and f :: "'k \<Rightarrow> 'l multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "m \<subseteq># Inf (f ` A)"
God_starNecExists: shows "universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v = xa x v)))"
Field_rel_restrict: fixes R :: "('a \<times> 'a) set"   and A :: "'a set" shows "Field (rel_restrict R A) \<subseteq> Field R - A"
span_mono: fixes A :: "('t \<Rightarrow>\<^sub>0 'f) set"   and B :: "('t \<Rightarrow>\<^sub>0 'f) set" assumes "A \<subseteq> B" shows "pmdl A \<subseteq> pmdl B"
sup_mono: fixes a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "a \<le> c"   and "b \<le> d" shows "sup a b \<le> sup c d"
to_fun_Fun_add: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "x \<in> carrier (UP Zp)"   and "y \<in> carrier (UP Zp)" shows "to_fun (x \<oplus>\<^bsub>UP Zp\<^esub> y) = to_fun x \<oplus>\<^bsub>function_ring (carrier Zp) Zp\<^esub> to_fun y"
set_iterator_rev_linord_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and it :: "('b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and S0 :: "'a set" assumes "class.linorder less_eq less" shows "linorder.set_iterator_rev_linord less_eq it S0 \<equiv> set_iterator_genord it S0 (\<lambda>x y. less_eq y x)"
ang_vec_bounded: fixes z1 :: "complex"   and z2 :: "complex" shows "- pi < \<angle> z1 z2 \<and> \<angle> z1 z2 \<le> pi"
sup_ge1: fixes x :: "'b"   and y :: "'b" shows "x \<le> x + y"
simps_5: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "(case OK x2 of ERR \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f2 x2"
order_widen: fixes m :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "wf_prog m P" shows "order (subtype P)"
bisim1LAss2: fixes P :: "'a prog"   and h :: "'heap"   and V :: "nat"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,V:=e,h \<turnstile> (unit, xs) \<leftrightarrow> ([], xs, Suc (length (compE2 e)), None)"
onorm_id: shows "onorm (\<lambda>x. x) = 1"
proper_interval_set_aux: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "proper_intrvl.proper_interval_set_aux less proper_interval xs ys = (\<exists>A. ord.set_less_aux less_eq (set xs) A \<and> ord.set_less_aux less_eq A (set ys))"
push_down_state_rank_tokens: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and i :: "nat" assumes "\<MM>.state_rank q n = Some i" shows "\<MM>.configuration q n \<noteq> {}"
simps_3: fixes a :: "'a poly"   and b :: "'a poly" shows "num_params (Add a b) = max (num_params a) (num_params b)"
apps_append: fixes s :: "('a, 'b) tm"   and ss :: "('a, 'b) tm list"   and ts :: "('a, 'b) tm list" shows "apps s (ss @ ts) = apps (apps s ss) ts"
g_add_dj_def: fixes m1 :: "'s"   and m2 :: "'s" shows "g_add_dj m1 m2 \<equiv> iterate m2 (\<lambda>(x, y). update_dj x y) m1"
borel_def: fixes "open" :: "'d set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.borel open = sigma UNIV {S. open S}"
simps_13: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
hcomp_simps_3: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "D.hseq \<nu> \<mu>" shows "D.dom (\<nu> \<star>\<^sub>D \<mu>) = D.dom \<nu> \<star>\<^sub>D D.dom \<mu>"
freq_mergeSibling: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "sibling t a \<noteq> a" shows "freq (mergeSibling t a) = (\<lambda>c. if c = a then freq t a + freq t (sibling t a) else if c = sibling t a then 0 else freq t c)"
ind_hyp: fixes l' :: "'a list" shows "(delete_first P l = Some l') = (\<exists>l1 x l2. P x \<and> (\<forall>x\<in>set l1. \<not> P x) \<and> l = l1 @ x # l2 \<and> l' = l1 @ l2)"
four: shows "4\<^sub>\<nat> = ZFC_in_HOL.set {[]\<^sub>\<circ>, 1\<^sub>\<nat>, 2\<^sub>\<nat>, 3\<^sub>\<nat>}"
abs_eq: fixes x :: "'b \<times> 'b" assumes "fractrel x x" shows "quot_of_fract (abs_fract x) = normalize_quot x"
ball_conj_distrib: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>A. P x \<and> Q x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>A. Q x))"
extensinoal0_mult: fixes S :: "'c set"   and x :: "'c \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'd" assumes "extensional0 S x"   and "extensional0 S y" shows "extensional0 S (x * y)"
sum: fixes a :: "real"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "a *\<^sub>R sum g A = (\<Sum>x\<in>A. a *\<^sub>R g x)"
exhaust: fixes y :: "'a list" obtains "y = []"   | x21 :: "'a" and x22 :: "'a list" where "y = Cons x21 x22"
poly_div_radical_dvd_pderiv: fixes p :: "'b poly" shows "p div radical p dvd pderiv p"
assms: shows "(m, l') = foldl f (m', t') l"
us_facts_2: shows "set us \<inter> FvarsT r = {}"