hcju/isabelleps huggingface dataset:
eval_elim: fixes 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>c, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" obtains ca :: "('Var, 'AExp, 'BExp) Stmt" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "c = cxt_to_stmt E__ ca__" and "(ca__, mem) \<leadsto>\<^sub>s (c'b__, mem')" and "c'a__ = cxt_to_stmt E__ c'b__" and "mds' = mds"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" where "c = c\<^sub>1'__ ;; c\<^sub>2" and "c'a__ = c\<^sub>1' ;; c\<^sub>2" and "\<langle>c\<^sub>1'__, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>w"   | ca :: "('Var, 'AExp, 'BExp) Stmt" and mu :: "'Var ModeUpd" and c' :: "('Var, 'AExp, 'BExp) Stmt" and E :: "('Var, 'AExp, 'BExp) Stmt list" where "c = cxt_to_stmt E (ca@[mu])" and "c'a__ = cxt_to_stmt E c'" and "\<langle>ca, update_modes mu mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"   | b :: "'BExp" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" where "c = Await b c\<^sub>1" and "eval\<^sub>B mem b" and "no_await c\<^sub>1" and "\<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w\<^sup>+ \<langle>c'a__, mds', mem'\<rangle>\<^sub>w" and "is_final c'a__"
conseq': fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"   and "\<forall>s. P s \<le> P' s"   and "\<forall>s. Q' s \<le> Q s" shows "\<turnstile>\<^sub>2\<^sub>' {P'} c {Q'}"
simps_11: shows "pratt2_max \<top>\<^sub>2 \<top>\<^sub>2 = \<top>\<^sub>2"
Int_iff: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
Min_less_iff: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(Min A < x) = (\<exists>a\<in>A. a < x)"
Re_strict_mono: fixes x :: "complex"   and y :: "complex" assumes "x < y" shows "Re x < Re y"
2: shows "discrete_evolution (rk2_increment p (\<lambda>t x. f (t, x))) (t + h) t (x t) - x t - h *\<^sub>R f (t, x t) - (h\<^sup>2 / 2) *\<^sub>R f' (t, x t) $ (1, f (t, x t)) \<in> (*\<^sub>R) (1 / 2) ` closure (convex hull (\<lambda>s. (h ^ 3 * p / 2) *\<^sub>R f'' (t + s * (h * p), x t + (s * (h * p)) *\<^sub>R f (t, x t)) $ (1, f (t, x t)) $ (1, f (t, x t))) ` cbox 0 1)"
simps_1: fixes x1 :: "'c"   and y1 :: "'c" shows "(Unblinded x1 = Unblinded y1) = (x1 = y1)"
Trans: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language"   and z :: "'a language" assumes "Plus_cong R x y"   and "Plus_cong R y z" shows "Plus_cong R x z"
isomorphic_implies_hpar_4: fixes f :: "'c"   and f' :: "'c" assumes "C.isomorphic f f'" shows "trg\<^sub>C f = trg\<^sub>C f'"
can_mkarr_Can: fixes t :: "'c local.term" assumes "Can t" shows "\<F>C.can (\<F>C.mkarr t)"
finsum_smult: fixes c :: "'a"   and g :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "c \<in> carrier R"   and "g \<in> A \<rightarrow> carrier M" shows "c \<odot>\<^bsub>M\<^esub> finsum M g A = (\<Oplus>\<^bsub>M\<^esub>x\<in>A. c \<odot>\<^bsub>M\<^esub> g x)"
product_code: fixes xs :: "'a list"   and ys :: "'b list" shows "Product_Type.product (set xs) (set ys) = set (concat (map (\<lambda>x. map (Pair x) ys) xs))"
wfactors_dividesI: fixes fs :: "'a list"   and a :: "'a"   and f :: "'a" assumes "wfactors G fs a"   and "set fs \<subseteq> carrier G"   and "a \<in> carrier G"   and "f \<in> set fs" shows "f divides a"
proj_ik_append: fixes l :: "'f"   and A :: "('f strand_label \<times> ('d, 'e) strand_step) list"   and B :: "('f strand_label \<times> ('d, 'e) strand_step) list" shows "ik\<^sub>s\<^sub>t (proj_unl l (A @ B)) = ik\<^sub>s\<^sub>t (proj_unl l A) \<union> ik\<^sub>s\<^sub>t (proj_unl l B)"
assms_1: shows "\<forall>t1\<le>t. y t1 = []"
risk_neutral_prob_def: fixes N :: "bool stream measure" shows "risk_neutral_prob N = (prob_space N \<and> (\<forall>asset\<in>stocks Mkt. martingale N G (discounted_value r (prices Mkt asset))))"
ergodic: fixes A :: "'a set" assumes "A \<in> sets Invariants" shows "A \<in> null_sets M \<or> space M - A \<in> null_sets M"
perp_per_2: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A B Perp A C" shows "Per B A C"
hyps_1: fixes P :: "'a set" assumes "x = card P"   and "finite P" shows "finite {p. distinct p \<and> set p \<subseteq> P}"
dg_is_arrE: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b" obtains "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" and "\<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a" and "\<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
supp_map_bset_Rep_formula: fixes xset :: "('e, 'f, 'g) formula set['h]" shows "supp (map_bset Rep_formula xset) = supp xset"
sum_abs: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "\<bar>sum f A\<bar> \<le> (\<Sum>i\<in>A. \<bar>f i\<bar>)"
prv_imp_cnj5: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<chi>1 :: "'c"   and \<chi>2 :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (imp \<phi>1 \<chi>1)"   and "prv (imp \<phi>2 \<chi>2)" shows "prv (imp \<phi>1 (imp \<phi>2 (cnj \<chi>1 \<chi>2)))"
not_Cons_self2: fixes x :: "'f"   and xs :: "'f list" shows "x # xs \<noteq> xs"
prod_topology_subtopology_eu: fixes S :: "'a set"   and T :: "'b set" shows "prod_topology (top_of_set S) (top_of_set T) = top_of_set (S \<times> T)"
mem_restr_dropI: fixes b :: "nat"   and R :: "event_data list set"   and xs :: "event_data option list" assumes "mem_restr (lift_envs' b R) xs" shows "mem_restr R (drop b xs)"
N_\<sigma>: fixes i :: "nat"   and j :: "nat"   and l :: "nat" assumes "i < length fs"   and "j \<le> i"   and "l \<le> j" shows "\<bar>\<sigma> l i j\<bar> \<le> of_nat l * N ^ (2 * l + 2)"
disconnected_nodes_eq_h: fixes doc_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "doc_ptr \<noteq> new_document_ptr" shows "h \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes = h' \<turnstile> get_disconnected_nodes doc_ptr \<rightarrow>\<^sub>r disc_nodes"
Fvars_PPf: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm"   and "xx \<notin> FvarsT t1" shows "Fvars (PPf t1 t2) = FvarsT t1 \<union> FvarsT t2"
Deduction_theorem: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "AX0 \<union> ((F::'b formula) \<triangleright> (\<Gamma>::'b formula set)) \<turnstile>\<^sub>H (G::'b formula)" shows "AX0 \<union> (\<Gamma>::'b formula set) \<turnstile>\<^sub>H (F::'b formula) \<^bold>\<rightarrow> (G::'b formula)"
to_partial_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and f :: "'d \<Rightarrow> 'e" assumes "sifum_security stop eval some_val some_val'" shows "to_partial f = (\<lambda>x. Some (f x))"
frefI: fixes P :: "'e \<Rightarrow> bool"   and R :: "('d \<times> 'e) set"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and S :: "('f \<times> 'g) set" assumes "\<And>x y. \<lbrakk>P y; (x, y) \<in> R\<rbrakk> \<Longrightarrow> (f x, g y) \<in> S" shows "(f, g) \<in> [P]\<^sub>f R \<rightarrow> S"
elims: fixes x :: "dir"   and y :: "int" assumes "Grid_Point.sgn x = y" obtains "x = left" and "y = - 1"   | "x = right" and "y = 1"
while_def: fixes gs :: "('f \<times> 'd set) list"   and b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "while gs b c = guards gs (While b (Seq c (guards gs Skip)))"
valid_sops_stmt_invariant: fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)"   and "valid_sops_stmt t s" shows "valid_sops_stmt t' s'"
lift_f_trivial_P': fixes S :: "'a set"   and P :: "('a \<times> bool) list set"   and f :: "'a \<Rightarrow> 'b"   and p :: "'a freeword" assumes "GroupByPresentationInducedFun S P f"   and "p \<in> Abs_freeword ` P" shows "freeword_funlift f p = (0::'b)"
\<sigma>1': shows "\<sigma>1' \<in> seq ars"
simps: fixes pre :: "object atom formula"   and eff :: "object ast_effect" shows "wf_ground_action (Ground_Action pre eff) = (wf_fmla objT pre \<and> wf_effect objT eff)"
simps_1: fixes s :: "'state_t"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "run_total 0 s execs = s"
integral_minus_sets: fixes c :: "real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b" assumes "c \<le> a"   and "c \<le> b"   and "f integrable_on {c..max a b}" shows "integral {c..a} f - integral {c..b} f = (if a \<le> b then - integral {a..b} f else integral {b..a} f)"
vec_neg: fixes x :: "'b vec"   and n :: "nat" assumes "x \<in> carrier_vec n" shows "\<ominus>\<^bsub>module_vec TYPE('b) n\<^esub> x = - x"
MkIde_Dom': fixes a :: "('a E.term, 'a) arr" assumes "ide a" shows "MkIde (Dom a) = a"
assms_1: shows "A B C D FSC A' B' C' D'"
max_min_same_2: fixes x :: "'c"   and y :: "'c" shows "max (min x y) x = x"
union_impl: fixes a :: "'a"   and b :: "'a" shows "(a l\<rightarrow> b) \<squnion>1 (b l\<rightarrow> a) = (1::'a)"
VarNonOccForm_def: fixes v :: "hf"   and x :: "hf" shows "VarNonOccForm v x \<equiv> Form x \<and> SubstForm v 0 x x"
assms_1: shows "\<forall>w. prices Mkt asset 0 w \<noteq> 0"
UN_E: fixes b :: "'f"   and B :: "'g \<Rightarrow> 'f set"   and A :: "'g set" assumes "b \<in> \<Union> (B ` A)" obtains x :: "'g" where "x \<in> A" and "b \<in> B x"
sccs_def: fixes G :: "('e, 'f) pre_digraph" shows "pre_digraph.sccs G \<equiv> {H. induced_subgraph H G \<and> strongly_connected H \<and> (\<nexists>H'. induced_subgraph H' G \<and> strongly_connected H' \<and> verts H \<subset> verts H')}"
lt_strict_1: fixes compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a" assumes "class.Ord_strict compare" shows "Ord.lt compare\<cdot>\<bottom>\<cdot>x = \<bottom>"
real_is_int_numeral: fixes x :: "num" shows "real_is_int (numeral x)"
Ints_of_real: fixes x :: "real" assumes "x \<in> \<int>" shows "of_real x \<in> \<int>"
constr_rep_1: fixes x2 :: "vrs"   and x1 :: "ty" shows "Rep_binding (VarB x2 x1) = VarB_Rep x2 x1"
meas_2: shows "(\<lambda>x. (fst x, - snd x)) \<in> borel_measurable borel"
value_gt_true_Some: fixes a :: "value option"   and b :: "value option" assumes "value_gt a b = true" shows "(\<exists>x. a = Some x) \<and> (\<exists>y. b = Some y)"
obs_comp_def: fixes a :: "nat \<Rightarrow> 'out option"   and b :: "nat \<Rightarrow> 'out option" shows "a \<approx> b \<equiv> a \<lesssim> b \<or> b \<lesssim> a"
measurable_component_singleton: fixes i :: "'d"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "i \<in> I" shows "(\<lambda>x. x i) \<in> Pi\<^sub>M I M \<rightarrow>\<^sub>M M i"
base: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option" shows "recognises_execution e s r []"
inteq_reflection: fixes x :: "'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'c" assumes "\<turnstile> x = y" shows "x \<equiv> y"
closure_Times: fixes A :: "'a set"   and B :: "'b set" shows "closure (A \<times> B) = closure A \<times> closure B"
ovalidNF_property_def: fixes P :: "'c \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option" shows "ovalidNF_property P x = (\<lambda>s f. x s f \<noteq> None \<and> (\<forall>r. Some r = x s f \<longrightarrow> P r s))"
assms_1: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
not_no_matching_Goto_cases: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list" assumes "\<not> no_matching_Goto \<gamma> p rs"   and "rs \<noteq> []" shows "\<exists>rs1 m chain rs2. rs = rs1 @ Rule m (Goto chain) # rs2 \<and> no_matching_Goto \<gamma> p rs1 \<and> matches \<gamma> m p"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "nat"   and y4 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "(SkewBinomialHeapStruc.Node x1 x2 x3 x4 = SkewBinomialHeapStruc.Node y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
f_join_Nil_conv': fixes xs :: "'b list"   and I :: "nat set" shows "(xs \<Join>\<^sub> I = []) = (\<forall>i<length xs. i \<notin> I)"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "H.VxV.seq g f" obtains "V.seq (fst g) (fst f)" and "V.seq (snd g) (snd f)"
2: shows "ss \<subseteq> {s. fmdom' s = domain}"
is_max: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<aa> :: "'c set" assumes "max_ideal R I addition multiplication zero unit"   and "ideal \<aa> R addition multiplication zero unit"   and "\<aa> \<noteq> R"   and "I \<subseteq> \<aa>" shows "I = \<aa>"
simps_8: fixes P1 :: "pol"   and j :: "nat"   and c :: "int"   and v :: "nat"   and Q :: "pol" shows "isnorm (PX (PX P1 j (Pc c)) (Suc v) Q) = (c \<noteq> 0 \<and> isnorm (PX P1 j (Pc c)) \<and> isnorm Q)"
meld_def: fixes q1 :: "('a, 'b) BinomialHeap"   and q2 :: "('a, 'b) BinomialHeap" shows "BinomialHeap_loc.meld q1 q2 \<equiv> Abs_BinomialHeap (BinomialHeapStruc.meld (Rep_BinomialHeap q1) (Rep_BinomialHeap q2))"
concat_eq_appendD: fixes xss :: "'d list list"   and ys :: "'d list"   and zs :: "'d list" assumes "concat xss = ys @ zs"   and "xss \<noteq> []" shows "\<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2"
cs_join: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "relation.cs r a b = ((a, b) \<in> {(x, y). r x y}\<^sup>\<down>)"
divmod_numeral_simps_17: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m \<le> num.Bit0 n) = (m \<le> n)"
strong_cases: fixes a1 :: "binding list"   and a2 :: "ty"   and a3 :: "ty"   and X :: "tyvrs" assumes "a1\<turnstile>a2<:a3" obtains \<Gamma> :: "binding list" and S :: "ty" where "a1 = \<Gamma>__" and "a2 = S__" and "a3 = Top" and "\<turnstile> \<Gamma>__ ok" and "S__ closed_in \<Gamma>__"   | \<Gamma> :: "binding list" and X :: "tyvrs" where "a1 = \<Gamma>'__" and "a2 = Tvar Xb__" and "a3 = Tvar Xb__" and "\<turnstile> \<Gamma>'__ ok" and "Xb__ \<in> ty_dom \<Gamma>'__"   | X :: "tyvrs" and S :: "ty" and \<Gamma> :: "binding list" and T :: "ty" where "a1 = \<Gamma>''__" and "a2 = Tvar X" and "a3 = T" and "TVarB X S \<in> set \<Gamma>''__" and "\<Gamma>''__\<turnstile>S<:T"   | \<Gamma> :: "binding list" and T\<^sub>1 :: "ty" and S\<^sub>1 :: "ty" and S\<^sub>2 :: "ty" and T\<^sub>2 :: "ty" where "a1 = \<Gamma>'''__" and "a2 = S\<^sub>1__ \<rightarrow> S\<^sub>2__" and "a3 = T\<^sub>1__ \<rightarrow> T\<^sub>2'__" and "\<Gamma>'''__\<turnstile>T\<^sub>1__<:S\<^sub>1__" and "\<Gamma>'''__\<turnstile>S\<^sub>2__<:T\<^sub>2'__"   | \<Gamma> :: "binding list" and T\<^sub>1 :: "ty" and S\<^sub>1 :: "ty" and S\<^sub>2 :: "ty" and T\<^sub>2 :: "ty" where "\<lbrakk>Xa__ \<sharp> a1; Xa__ \<sharp> a2; Xa__ \<sharp> a3\<rbrakk> \<Longrightarrow> a1 = \<Gamma> \<and> a2 = (\<forall>Xa__<:S\<^sub>1. S\<^sub>2) \<and> a3 = (\<forall>Xa__<:T\<^sub>1. T\<^sub>2) \<and> \<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1 \<and> (TVarB Xa__ T\<^sub>1 # \<Gamma>)\<turnstile>S\<^sub>2<:T\<^sub>2"
accI: fixes x :: "'a"   and r :: "('a \<times> 'a) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
rtranclp_tranclp_tranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
functor_ToCT: shows "functor C.CCC.comp (\<cdot>) C.T.ToCT"
in_carrier: shows "[\<one>, inv a \<otimes> b] \<in> carrier (poly_ring R)"
add_independentS_double_sum_conv_append: fixes As :: "'a set list"   and Bs :: "'a set list" assumes "\<forall>X\<in>set As. (0::'a) \<in> X"   and "add_independentS As"   and "add_independentS Bs"   and "add_independentS [\<Sum>X\<leftarrow>As. X, \<Sum>X\<leftarrow>Bs. X]" shows "add_independentS (As @ Bs)"
ct_prefix_def: fixes xs :: "ct list"   and ys :: "ct list" shows "ct_prefix xs ys = (\<exists>as bs. ys = as @ bs \<and> ct_list_eq as xs)"
axioms: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat" assumes "rpo ground_heads_var gt_sym extf arity_sym arity_var" shows "rpo_basis ground_heads_var gt_sym arity_sym arity_var extf"
assms: shows "T \<in> set\<^sub>v (mat_tensorlist_mult A Ts ds)"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) (\<lambda>f. C.tensor f (cod \<iota>))"
p'_in_spg: fixes p' :: "(nat \<times> int) list" assumes "p' \<in> lgrid (base {0..<d} p) {0..<d} lm" shows "p' \<in> sparsegrid dm lm"
psi_2: fixes i :: "nat"   and x :: "nat" shows "eval psi [i, x] = \<psi> i x"
app_invimage_def: fixes r :: "V"   and A :: "V" shows "r -`\<^sub>\<circ> A = r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> A"
SN_imp_wf: fixes r :: "('a \<times> 'a) set" assumes "SN r" shows "wf (r\<inverse>)"
one_def_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c" assumes "class.linear_aggregation_algebra one times uminus inf less_eq less sup bot top plus conv" shows "one = top"
in_minimize_wrtI: fixes x :: "'a"   and xs :: "'a list"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<forall>y\<in>set xs. P y x" shows "x \<in> set (minimize_wrt P xs)"
equivalence_pseudofunctor_axioms: shows "equivalence_pseudofunctor (\<cdot>) (\<star>) \<a> \<i> src trg (\<cdot>\<^sub>B) (\<star>\<^sub>B) \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B DN cmp\<^sub>D\<^sub>N"
is_scc_alt_restrict_path: fixes E :: "('b \<times> 'b) set"   and U :: "'b set" shows "is_scc E U = (U \<noteq> {} \<and> U \<times> U \<subseteq> (Restr E U)\<^sup>* \<and> E\<^sup>* `` (E\<^sup>* `` U - U) \<inter> U = {})"
simps_4: fixes v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Plus v va) Zero = False"
f_const_has_max_value_2: assumes "const 0 \<or> const 1" shows "(cmod (\<Sum>k<2 ^ n. (- 1) ^ (f k + 1)))\<^sup>2 = real (2 ^ (2 * n))"
w0: shows "(of_nat (Suc k) * fps_radical r (Suc k) a ^ k) $ 0 \<noteq> (0::'a)"
jkbpC_represents: shows "Clock.represents Clock.jkbpC"
algebraic_diff_cancel_right: fixes y :: "'a"   and x :: "'a" assumes "algebraic y" shows "algebraic (x - y) = algebraic x"
elims: fixes x :: "atomUni"   and y :: "real \<times> real \<times> real" assumes "getPoly x = y" obtains p :: "real \<times> real \<times> real" where "x = EqUni pa__" and "y = pa__"   | p :: "real \<times> real \<times> real" where "x = LeqUni pb__" and "y = pb__"   | p :: "real \<times> real \<times> real" where "x = NeqUni pc__" and "y = pc__"   | p :: "real \<times> real \<times> real" where "x = LessUni p" and "y = p"
fls_residue_power_series: fixes f :: "'a fls" assumes "0 \<le> fls_subdegree f" shows "fls_residue f = (0::'a)"
finsert_is_funion: fixes a :: "'a"   and A :: "'a fset" shows "finsert a A = {|a|} |\<union>| A"
degree_isnpolyh_Suc: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "Reflected_Multivariate_Polynomial.degree p = 0"
linear_componentwise_iff: fixes f' :: "'a \<Rightarrow> 'b" shows "linear f' = (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))"
it_dfs_SPEC: fixes P :: "('v, 'es) state_scheme \<Rightarrow> bool" assumes "\<And>s. \<lbrakk>DFS_invar G param s; \<not> cond s\<rbrakk> \<Longrightarrow> P s" shows "it_dfs \<le> SPEC P"
galleryD_chamber: fixes xs :: "'a set list"   and x :: "'a set" assumes "folding_g.gallery xs"   and "x \<in> set xs" shows "folding_g.chamber x"
Limsup_ereal_mult_left: fixes F :: "'b filter"   and c :: "real"   and f :: "'b \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "0 \<le> c" shows "Limsup F (\<lambda>n. ereal c * f n) = ereal c * Limsup F f"
wt_elim_cases_21: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (Expr e)\<Colon>x" obtains T :: "ty" where "x = Inl (PrimT Void)" and "E,dt\<Turnstile>e\<Colon>-T"
map_simp_4: shows "local.map J.j1 = f1"
hoare_choice: fixes p :: "'a"   and S :: "'b \<Rightarrow> 'a"   and T :: "'b \<Rightarrow> 'a"   and q :: "'b" shows "\<Turnstile> p{| S \<sqinter> T |}q = (\<Turnstile> p{| S |}q \<and> \<Turnstile> p{| T |}q)"
dis_rews_7: shows "is_scons\<cdot>[::] = FF"
eq_\<I>_gpv_conversep: fixes A :: "'d \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>" shows "eq_\<I>_gpv A\<inverse>\<inverse> \<I> = (eq_\<I>_gpv A \<I>)\<inverse>\<inverse>"
dvd_normalize_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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times a (normalize b) = dvd.dvd times a b"
cinner_same_CBasis: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and CBasis :: "'a set"   and u :: "'a" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "u \<in> CBasis" shows "cinner u u = 1"