hcju/isabelleps huggingface dataset:
nn_integral_set_ennreal: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "\<integral>\<^sup>+x\<in>A. ennreal (f x)\<partial>M = \<integral>\<^sup>+ x. ennreal (f x * indicat_real A x) \<partial>M"
top_healthy: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel" assumes "utp_theory_lattice hcond" shows "\<top>\<^bsub>utp_order hcond\<^esub> is hcond"
simps: fixes p :: "('a, 'b) form" shows "dest_Neg (Neg p) = p"
assms_2: shows "nprv (insert \<phi>1 F) \<psi>"
Sources_DSources: fixes i :: "AbstrLevelsID"   and C :: "CSet" shows "DSources i C \<subseteq> Sources i C"
nn_integral_swap: fixes f :: "'a \<times> 'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (lborel \<Otimes>\<^sub>M lborel)" shows "integral\<^sup>N (lborel \<Otimes>\<^sub>M lborel) f = \<integral>\<^sup>+ (y, x). f (x, y) \<partial>lborel \<Otimes>\<^sub>M lborel"
assms_4: shows "\<forall>y\<in>ball x e. f x \<le> f y"
set_of_eq: fixes x :: "'a interval" shows "set_of x = {lower x..upper x}"
Collect_neg_eq: fixes P :: "'e \<Rightarrow> bool" shows "{x. \<not> P x} = - {x. P x}"
h_q2_len: shows "1 < length qs \<longrightarrow> 0 < length (drop (length qs div 2) qs)"
fdisjnt_alt_def: fixes M :: "'a fset"   and N :: "'a fset" shows "fdisjnt M N = (M |\<inter>| N = {||})"
deriv_c1: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c\<^sub>1 Q,A"
list_augment_twice: fixes xs :: "'a list"   and i :: "nat"   and u :: "'a"   and j :: "nat"   and v :: "'a" shows "list_augment (list_augment xs i u) j v = (list_pad_out xs (max i j))[i := u, j := v]"
exhaust: fixes y :: "'b event" obtains x1 :: "'b" where "y = ev x1"   | "y = tick"
simps_7: fixes V :: "'a set"   and uw :: "('b, 'a) Term.term"   and ux :: "('b, 'a) Term.term"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (delete\<langle>uw,ux\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t V S"
Restrp_irreflexive: shows "irreflexive UNIV ((\<sqsubset>) \<restriction> A \<restriction> UNIV)"
cases: fixes a1 :: "'b list"   and a2 :: "'c sequent" assumes "(a1, a2) \<in> upRules" obtains c :: "'c sequent" and R :: "'c" and Fs :: "'c form list" and ps :: "'b list" where "a1 = ps" and "a2 = c" and "SRCTransforms.mset c \<equiv> \<LM> Compound R Fs  \<RM>" and "ps \<noteq> []"
Suffixes_rev: fixes A :: "'a list set" shows "Suffixes (rev ` A) = rev ` Prefixes A"
left_ideal_minus: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "left_ideal I"   and "a \<in> I"   and "b \<in> I" shows "a - b \<in> I"
memb_correct: fixes s :: "'s"   and x :: "natural" assumes "invar s" shows "memb x s = (x \<in> \<alpha> s)"
power_Suc: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
Nil: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "P,E \<turnstile> \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>[],s\<rangle>"
refine: fixes ai :: "'b list"   and X :: "'c set" assumes "(ai, X) \<in> appr_rel" shows "(RETURN (ivl_rep_of_set_impl ai), op_ivl_rep_of_set X) \<in> \<langle>lv_rel \<times>\<^sub>r lv_rel\<rangle>nres_rel"
Plus_rotate: fixes r :: "'a language"   and s :: "'a language"   and t :: "'a language" shows "Plus r (Plus s t) = Plus s (Plus r t)"
intro: fixes x :: "'c \<Longrightarrow> 'b" assumes "\<And>\<sigma> \<rho>. put\<^bsub>x\<^esub> \<sigma> (get\<^bsub>x\<^esub> \<rho>) = \<rho>" shows "bij_lens_axioms x"
the_min_min: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and the_min :: "'c"   and x :: "'c" assumes "class.the_min less_eq less the_min" shows "less_eq the_min x"
Done_def: shows "Done \<equiv> \<Lambda> a. resT_abs\<cdot>(sinl\<cdot>(up\<cdot>a))"
sum_two_rw: fixes f :: "nat \<Rightarrow> 'b" shows "sum f {0..<2} = sum f {0, 1}"
coinduct_strong: fixes R :: "('d, 'e, 'f) gpv \<Rightarrow> ('d, 'e, 'f) gpv \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and gpv' :: "('d, 'e, 'f) gpv" assumes "R gpv gpv'"   and "\<And>gpv gpv'. R gpv gpv' \<Longrightarrow> rel_spmf (rel_generat (=) (=) (rel_fun (=) (\<lambda>gpv gpv'. R gpv gpv' \<or> gpv = gpv'))) (the_gpv gpv) (the_gpv gpv')" shows "gpv = gpv'"
iT_Plus_neg_inext: fixes k :: "nat"   and n :: "nat"   and I :: "nat set" assumes "k \<le> n" shows "inext (n - k) (I \<oplus>- k) = inext n I - k"
intros_7: fixes C :: "char list"   and Cs :: "char list list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and a :: "nat"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "C \<notin> set Cs"   and "(last Cs, C) \<notin> (subcls1 P)\<^sup>*" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>ref (a, Cs),s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt ClassCast, [ClassCast]),s\<rangle>"
normalize_const_poly: fixes c :: "'a" shows "normalize [:c:] = [:normalize c:]"
add: shows "local.add a p\<^sub>1 p\<^sub>2 = p\<^sub>2"
not_dvd_pderiv: fixes p :: "'b poly" assumes "Polynomial.degree p \<noteq> 0" shows "\<not> p dvd pderiv p"
rel_FGco_def: fixes L1 :: "'y \<Rightarrow> 'al \<Rightarrow> bool"   and Co1 :: "'z \<Rightarrow> 'am \<Rightarrow> bool"   and Co2 :: "'ab \<Rightarrow> 'ao \<Rightarrow> bool"   and Co3 :: "'ac \<Rightarrow> 'ap \<Rightarrow> bool"   and Co4 :: "'ad \<Rightarrow> 'aq \<Rightarrow> bool"   and Co5 :: "'aa \<Rightarrow> 'an \<Rightarrow> bool"   and Co6 :: "'ah \<Rightarrow> 'at \<Rightarrow> bool"   and Contra1 :: "'ae \<Rightarrow> 'ar \<Rightarrow> bool"   and Contra2 :: "'af \<Rightarrow> 'as \<Rightarrow> bool"   and Contra3 :: "'ai \<Rightarrow> 'au \<Rightarrow> bool"   and Contra4 :: "'aj \<Rightarrow> 'av \<Rightarrow> bool" shows "rel_FGco L1 Co1 Co2 Co3 Co4 Co5 Co6 Contra1 Contra2 Contra3 Contra4 = rel_F L1 Co1 Co5 (rel_G Co1 Co2 Co3 Co4 Contra1 Contra2) Co3 Co6 Contra1 Contra3 Contra4"
GroupByPresentationInducedFun_axioms: shows "GroupByPresentationInducedFun S P f"
R: fixes S :: "'a set"   and T :: "'a set" assumes "compactin X S \<and> compactin X T \<and> disjnt S T" shows "\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V"
distinct_1: fixes x1 :: "bool"   and x2 :: "'a" shows "Bool x1 \<noteq> Value x2"
coeff_keys: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'a mpoly" shows "(m \<in> keys (mapping_of p)) = (coeff p m \<noteq> (0::'a))"
opI: fixes a :: "'a" assumes "a \<in> S" shows "\<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. f x = f a)"
is_Var_def: fixes "term" :: "('c, 'd) Term.term" shows "is_Var term = (\<exists>x1. term = Var x1)"
mulextp_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and m :: "'b multiset"   and n :: "'b multiset" shows "mulextp f m n = (smulextp f m n, nsmulextp f m n)"
IH: assumes "S \<noteq> {}"   and "S \<subseteq> carrier (union_ring C)" shows "\<exists>R\<in>C. S \<subseteq> carrier R"
first_summand_disjointness: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "\<not> O z (x \<oplus> y)" shows "\<not> O z x"
WT_gpv_inline': fixes \<I> :: "('b, 'c) \<I>"   and \<I>' :: "('d, 'e) \<I>"   and callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a, 'd, 'e) gpv"   and gpv :: "('f, 'b, 'c) gpv"   and s :: "'a" assumes "\<And>s x. x \<in> outs_\<I> \<I> \<Longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> UNIV"   and "\<And>x s. x \<in> outs_\<I> \<I> \<Longrightarrow> \<I>' \<turnstile>g callee s x \<surd>"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<I>' \<turnstile>g inline callee gpv s \<surd>"
mult_less_cancel_left2: 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 c :: "'c"   and a :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (times c a) c = ((less_eq zero c \<longrightarrow> less a one) \<and> (less_eq c zero \<longrightarrow> less one a))"
imp_conjR: fixes P :: "bool"   and Q :: "bool"   and R :: "bool" shows "(P \<longrightarrow> Q \<and> R) = ((P \<longrightarrow> Q) \<and> (P \<longrightarrow> R))"
divisor_sigma_of_nat: fixes x :: "nat"   and n :: "nat" shows "divisor_sigma (of_nat x) n = of_nat (divisor_sigma x n)"
bij_betw_ide_Obj_1: shows "J_C.MkIde \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>)) \<rightarrow> Collect J_C.ide"
g: shows "g \<in> M \<rightarrow>\<^sub>M M2"
extend_constrains: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "(extend h F \<in> extend_set h A co extend_set h B) = (F \<in> A co B)"
tab_def: shows "Tgh.\<rho>\<sigma>.tab \<equiv> \<a>\<^sup>-\<^sup>1[g, h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> (g \<star> \<a>[h, tab\<^sub>0 h, g\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> (g \<star> h.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> (g \<star> g\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[g, tab\<^sub>0 g, g\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (g.tab \<star> g\<^sub>0h\<^sub>1.p\<^sub>1)"
fl3: fixes p :: "('b, 'c) Combinators list"   and x :: "('b, 'c) Combinators" assumes "NetsCollected p"   and "\<And>a. \<lbrakk>first_bothNet x \<noteq> firstList p; a \<in> set p\<rbrakk> \<Longrightarrow> first_bothNet x \<noteq> first_bothNet a" shows "NetsCollected (x # p)"
Sigma_Int_distrib2: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'd set" shows "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> Sigma I B"
children_eq_h: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
continuous_on_winding_number: fixes \<gamma> :: "real \<Rightarrow> complex" assumes "path \<gamma>" shows "continuous_on (- path_image \<gamma>) (winding_number \<gamma>)"
sum_case_def: fixes f :: "hf \<Rightarrow> 'b"   and g :: "hf \<Rightarrow> 'b"   and a :: "hf" shows "sum_case f g a \<equiv> THE z. (\<forall>x. a = HF.Inl x \<longrightarrow> z = f x) \<and> (\<forall>y. a = HF.Inr y \<longrightarrow> z = g y) \<and> (\<not> is_hsum a \<longrightarrow> z = undefined)"
Rep_bit1_def: fixes x :: "'b bit1" shows "Rep x = Rep_bit1 x"
or_propag_on_create_graph: fixes \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>nd\<in>nds. \<mu> or\<^sub>r \<eta> \<in> old nd \<longrightarrow> {\<mu>, \<eta>} \<inter> old nd \<noteq> {})"
alphaAbs_qAbs_iff: fixes A :: "('f, 'g, 'h, 'i, 'j) qAbs"   and xs' :: "'h"   and x' :: "'i"   and X' :: "('f, 'g, 'h, 'i, 'j) qTerm" shows "(A $= qAbs xs' x' X') = (\<exists>x y X. A = qAbs xs' x X \<and> y \<notin> {x, x'} \<and> qAFresh xs' y X \<and> qAFresh xs' y X' \<and> X #[[y \<and> x]]_xs' #= X' #[[y \<and> x']]_xs')"
at_to_infinity: shows "at (0::'c) = filtermap inverse at_infinity"
induct: fixes P :: "fm \<Rightarrow> bool"   and fm :: "fm" assumes "P T"   and "P F"   and "\<And>x. P (Lt x)"   and "\<And>x. P (Le x)"   and "\<And>x. P (Gt x)"   and "\<And>x. P (Ge x)"   and "\<And>x. P (Eq x)"   and "\<And>x. P (NEq x)"   and "\<And>x1 x2. P (Dvd x1 x2)"   and "\<And>x1 x2. P (NDvd x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (NOT x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (And x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Or x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Imp x1 x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (Iff x1 x2)"   and "\<And>x. P x \<Longrightarrow> P (E x)"   and "\<And>x. P x \<Longrightarrow> P (A x)"   and "\<And>x. P (Closed x)"   and "\<And>x. P (NClosed x)" shows "P fm"
while_elim: fixes e :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>While e c, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = mds \<and> mem' = mem"
inject_1: fixes x1 :: "i32"   and y1 :: "i32" shows "(ConstInt32 x1 = ConstInt32 y1) = (x1 = y1)"
simps_5: fixes x :: "nat"   and v :: "exp"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp" shows "subst x v (EPrim f e1 e2) = EPrim f (subst x v e1) (subst x v e2)"
S_intersection2_rep_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection2_rep p q \<equiv> S_intersection_coeff2 p q *\<^sub>R p + q"
surj_iff_all: fixes f :: "'e \<Rightarrow> 'd" shows "surj f = (\<forall>x. f (inv f x) = x)"
prj0_ext: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "\<not> (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g)" shows "prj0 f g = partial_magma.null C"
square_le_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less_eq (uminus one) x"   and "less_eq x one" shows "less_eq (power.power one times x 2) one"
coprime_imp_no_common_roots: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" assumes "algebraic_semidom_class.coprime p q" shows "\<not> (poly p x = (0::'a) \<and> poly q x = (0::'a))"
induct: fixes P :: "'a stream \<Rightarrow> bool"   and x1 :: "nat"   and x2 :: "'a stream"   and Pa :: "nat \<Rightarrow> 'a stream \<Rightarrow> bool" assumes "ev_at P x1 x2"   and "\<And>\<omega>. P \<omega> \<Longrightarrow> Pa 0 \<omega>"   and "\<And>\<omega> n. \<lbrakk>\<not> P \<omega>; ev_at P n (stl \<omega>); Pa n (stl \<omega>)\<rbrakk> \<Longrightarrow> Pa (Suc n) \<omega>" shows "Pa x1 x2"
binrelchain_overlap_join: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and ys :: "'b list" assumes "binrelchain P (xs @ [x])"   and "binrelchain P (x # ys)" shows "binrelchain P (xs @ x # ys)"
groupWIs2_def: fixes c :: "parts_connection"   and rs :: "'a simple_rule list" shows "groupWIs2 c rs = (let P = getParts rs; W = map getOneIp P; filterW = filter (\<lambda>r. simple_conn_matches (match_sel r) c) rs; f = \<lambda>wi. (map (\<lambda>d. runFw (getOneIp wi) d c filterW) W, map (\<lambda>s. runFw s (getOneIp wi) c filterW) W) in map (map fst) (groupF snd (map (\<lambda>x. (x, f x)) P)))"
order_eq_one_iff: fixes f :: "'a perm"   and a :: "'a" shows "(order f a = Suc 0) = (a \<notin> affected f)"
sup_top_right: fixes 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 top :: "'b"   and x :: "'b" assumes "class.bounded_lattice_top inf less_eq less sup top" shows "sup x top = top"
f: shows "\<forall>\<sigma>. dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k \<longrightarrow> (cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), subst \<sigma> mem\<^sub>h\<rangle>"
trancl_mono_set: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set" assumes "r \<subseteq> s" shows "r\<^sup>+ \<subseteq> s\<^sup>+"
ccLinear_join: fixes S :: "Vars.var set"   and G :: "CoCalls"   and G' :: "CoCalls" shows "ccLinear S (G \<squnion> G') = (ccLinear S G \<and> ccLinear S G')"
admissible_leI: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "mcont luba orda Sup (\<le>) f"   and "mcont luba orda Sup (\<le>) g" shows "ccpo.admissible luba orda (\<lambda>x. f x \<le> g x)"
gallery_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and Cs :: "'b set list" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.maxsimpchain X Cs" shows "SimplicialComplex.maxsimpchain Y (f \<Turnstile> Cs)"
info_real_alg_2: fixes x :: "real_alg"   and q :: "rat" assumes "info_real_alg x = Inl q" shows "real_of x = real_of_rat q"
comp_null_2: fixes f :: "'c" shows "C f C.null = C.null"
Some_le: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b option" shows "(Some x \<sqsubseteq>\<^bsub>Opt.le r\<^esub> z) = (\<exists>y. z = Some y \<and> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
lass_val_of_nsub_RI: fixes e :: "'c exp"   and a :: "'c \<times> val" assumes "lass_val_of e = \<lfloor>a\<rfloor>" shows "\<not> sub_RI e"
totalI: fixes f :: "recf" assumes "\<And>xs. length xs = arity f \<Longrightarrow> eval f xs \<down>" shows "Partial_Recursive.total f"
count_roots_interval_sf_def: fixes p :: "real poly" shows "count_roots_interval_sf p = (let ps = sturm_squarefree p in (\<lambda>a b. sign_changes ps a - sign_changes ps b + (if poly p a = 0 then 1 else 0), \<lambda>a. sign_changes_neg_inf ps - sign_changes ps a))"
CFG_SSA_Transformed_code_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and "uses" :: "'a \<Rightarrow> ('b, 'c set) mapping" shows "CFG_SSA_Transformed_code_axioms \<alpha>n uses \<equiv> \<forall>g. Mapping.keys (uses g) \<subseteq> set (\<alpha>n g)"
a_rcosetsI: fixes G :: "('e, 'f) ring_scheme"   and H :: "'e set"   and x :: "'e" assumes "abelian_group G"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H +>\<^bsub>G\<^esub> x \<in> a_rcosets\<^bsub>G\<^esub> H"
nn_integral_def: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "integral\<^sup>N M f = \<Squnion> (integral\<^sup>S M ` {g. simple_function M g \<and> g \<le> f})"
mu_const: fixes P :: "'d" shows "(\<mu> X \<bullet> P) = P"
gcd_mult_unit1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a one" shows "gcd (times b a) c = gcd b c"
axioms_1: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "diagram J C D" shows "category C"
prems_4: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"
is_column: fixes n :: "nat"   and v :: "complex Matrix.mat" assumes "state n v" shows "dim_col v = 1"
takeWhile_replicate: fixes P :: "'b \<Rightarrow> bool"   and n :: "nat"   and x :: "'b" shows "takeWhile P (replicate n x) = (if P x then replicate n x else [])"
split_root: fixes P :: "real \<Rightarrow> bool"   and n :: "nat"   and x :: "real" shows "P (root n x) = ((n = 0 \<longrightarrow> P 0) \<and> (0 < n \<longrightarrow> (\<forall>y. sgn y * \<bar>y\<bar> ^ n = x \<longrightarrow> P y)))"
sylvester_mat_dim_2: fixes p :: "'b poly"   and q :: "'b poly" shows "dim_col (sylvester_mat p q) = degree p + degree q"
until_def: fixes \<gamma>' :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> bool" shows "dynamic_component.until \<gamma>' \<gamma> \<equiv> \<lambda>t n. \<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')"
sorted_insort_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (linorder.insort_insert_key less_eq (\<lambda>x. x) x xs)"
theorem3_part1: assumes "\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'a}" shows "\<forall>A. admits_triangular_reduction A"
right_adjoint_simps_4: fixes f :: "'a" assumes "is_left_adjoint f" shows "local.dom f\<^sup>* = f\<^sup>*"
SC_RS1: fixes p :: "('b set set, 'c) Combinators list" assumes "singleCombinators p"   and "allNetsDistinct p" shows "singleCombinators (removeShadowRules1 p)"
h_is_map: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and h :: "'b"   and k :: "'b" assumes "cospan_of_maps_in_bicategory_of_spans V H \<a> \<i> src trg h k" shows "\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg h g"
all_maps_def: fixes G :: "'b pair_pre_digraph" shows "all_maps G \<equiv> to_map (arcs (with_proj G)) ` {f. f permutes arcs (with_proj G) \<and> (\<forall>v\<in>verts (with_proj G). out_arcs (with_proj G) v \<noteq> {} \<longrightarrow> cyclic_on f (out_arcs (with_proj G) v))}"
stake_smap: fixes n :: "nat"   and f :: "'d \<Rightarrow> 'c"   and s :: "'d stream" shows "stake n (smap f s) = map f (stake n s)"
t_def: shows "t \<equiv> THE x'. x' ~ state x \<and> x' \<in> set_pmf (action cfg)"
assms_1: shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
v: shows "v = (punit.lt (rep_list (monom_mult (1::'b) (pp_of_term u - lp a) a)) - punit.lt (rep_list b)) \<oplus> lt b"
bit_horner_sum_uint_exp_iff: fixes ws :: "'b word list"   and n :: "nat" shows "bit (horner_sum uint (2 ^ LENGTH('b)) ws) n = (n div LENGTH('b) < length ws \<and> bit (ws ! (n div LENGTH('b))) (n mod LENGTH('b)))"
simps_4: fixes dims :: "nat list"   and n :: "nat"   and M :: "nat \<Rightarrow> complex mat"   and S :: "com list" shows "state_sig.well_com dims (Measure n M S) = (measurement (state_sig.d dims) n M \<and> length S = n \<and> list_all (state_sig.well_com dims) S)"
lprefix_refl: fixes xs :: "'a llist" shows "lprefix xs xs"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set1_rose_tree\<^sub>m x"   and "b \<in> set1_rose_tree\<^sub>m y" shows "m a b = m' a b"