hcju/isabelleps huggingface dataset:
extend_eval: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e'' :: "char list exp"   and s'' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"   and "P \<turnstile> \<langle>e'',s''\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
p_sup_p: fixes x :: "'a" shows "dense (x \<squnion> - x)"
integrable_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> c"   and "c \<le> b"   and "f integrable_on {a..c}"   and "f integrable_on {c..b}" shows "f integrable_on {a..b}"
rk2_increment: fixes p :: "real"   and f :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and h :: "real"   and t :: "real"   and x :: "'e" shows "rk2_increment p f h t x = (1 - 1 / (p * 2)) *\<^sub>R f t x + (1 / (p * 2)) *\<^sub>R f (t + h * p) (x + (h * p) *\<^sub>R f t x)"
mat_det_mult_sm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det (k *\<^sub>s\<^sub>m A) = k * k * mat_det A"
hiding_def: fixes P :: "'a process"   and A :: "'a set" shows "P \ A \<equiv> Abs_process ({(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}, div_hide P A)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_member: fixes l :: "'d strand_label"   and x :: "('e, 'f) stateful_strand_step"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" assumes "(l, x) \<in> set A"   and "\<not> stateful_strand_step.is_Receive x"   and "\<not> stateful_strand_step.is_Send x" shows "(l, x) \<in> set (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A)"
U': shows "openin X {x \<in> topspace X. f x \<in> U'} = openin X' U'"
"1_2": fixes i :: "nat" assumes "i \<le> n" shows "to_fract x ^ i / to_fract y ^ (i - 1) \<in> range to_fract"
assms_1: fixes cc :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "terminationPseudo 1 cc Q"
deflation_below_comp2: fixes f :: "'a \<rightarrow> 'a"   and g :: "'a \<rightarrow> 'a"   and x :: "'a" assumes "deflation f"   and "deflation g"   and "f \<sqsubseteq> g" shows "g\<cdot>(f\<cdot>x) = f\<cdot>x"
induced_automorphism_chamber_list_image_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and Cs :: "'b set list" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "order.greater_eq (f \<turnstile> ChamberComplex.chamber_system X) (set Cs)" shows "OpposedThinChamberComplexFoldings.induced_automorphism X f g \<Turnstile> Cs = g \<Turnstile> Cs"
Nleast_def: fixes A :: "nat set" shows "Nleast A = (THE a. a \<in> A \<and> (\<forall>x\<in>A. a \<le> x))"
alpha_sym: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm"   and X' :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "X #= X'" shows "X' #= X"
set_iterator_genord_I: fixes l0 :: "'e list"   and S :: "'e set"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and iti :: "('f \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" assumes "distinct l0"   and "S = set l0"   and "sorted_wrt R l0"   and "iti = foldli l0" shows "set_iterator_genord iti S R"
Int_subset_iff: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set" shows "(C \<subseteq> A \<inter> B) = (C \<subseteq> A \<and> C \<subseteq> B)"
GSMP_def: fixes P :: "('fun, 'var) Term.term set" shows "GSMP P \<equiv> {t \<in> SMP P. fv t = {}}"
goal3_1: fixes x :: "nat"   and y :: "nat \<times> nat"   and xa :: "nat"   and ya :: "nat" assumes "(x, y) = pm_pth (dpm bdd) n"   and "(xa, ya) = y"   and "pointermap_sane (dpm bdd)"   and "pointermap_getmk (v, ni1, ni2) (dpm bdd) = (x1, dpm s')"   and "Rmi_g xa t bdd" shows "Rmi_g xa t s'"
cong_solve_poly: fixes a :: "'c poly"   and n :: "'c poly" assumes "a \<noteq> 0" shows "\<exists>x. [a * x = gcd a n] (mod n)"
partUnfold: fixes P :: "pi"   and Q :: "pi"   and Rel :: "(pi \<times> pi) set"   and \<sigma> :: "(name \<times> name) list" assumes "(P, Q) \<in> substClosed Rel" shows "(P[<\<sigma>>], Q[<\<sigma>>]) \<in> substClosed Rel"
simps: 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 a1 :: "ty"   and a2 :: "ty" shows "P \<turnstile> a1 \<le> a2 = ((\<exists>T. a1 = T \<and> a2 = T) \<or> (\<exists>C D. a1 = Class C \<and> a2 = Class D \<and> P \<turnstile> Path C to D unique) \<or> (\<exists>C. a1 = NT \<and> a2 = Class C))"
obtain_x': shows "x' \<in> PLS (fst p) (snd p) \<and> x' \<le> 2 ^ card (prob_dom PROB) - 1"
assoc'_simps\<^sub>A\<^sub>W\<^sub>C_3: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<noteq> null"   and "g \<star> h \<noteq> null" shows "cod (local.inv \<a>[f, g, h]) = (f \<star> g) \<star> h"
T_on_rand_as_sum: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and s0 :: "'c list"   and rs :: "'c list" shows "T\<^sub>p_on_rand' A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) rs = sum (Partial_Cost_Model.T_on_rand'_n A (fst A s0 \<bind> (\<lambda>is. return_pmf (s0, is))) rs) {..<length rs}"
nonempty_imp_ex_extremal: fixes Q :: "'a set" assumes "Q \<subseteq> A"   and "Q \<noteq> {}" shows "\<exists>z\<in>Q. \<forall>y\<in>Q. \<not> y \<sqsubset> z"
balanced_net_Conv: fixes m :: "(nat \<times> nat) convnet"   and A :: "nat \<times> nat" assumes "balanced_net m" shows "balanced_net (Conv A m)"
coboundedI1: fixes c :: "'d"   and a :: "'d"   and b :: "'d" assumes "c \<le> a" shows "c \<le> sup a b"
r_prenum_def: shows "r_prenum \<equiv> Cn 2 r_ifless [recf.Id 2 1, Cn 2 r_length [recf.Id 2 0], Cn 2 r_nth [recf.Id 2 0, recf.Id 2 1], r_constn 1 0]"
special_coset_below_in_subset_ordering_iso: fixes T :: "'w set"   and w :: "'w" shows "OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater (dual_order.below_in (Pow S) T) ((+o) w \<circ> genby)"
length_vec_smult: fixes \<alpha> :: "'b"   and v :: "'b list" shows "length (vec_smult \<alpha> v) = length v"
Impossible\<^sub>0_def: shows "Impossible \<equiv> \<lambda>p. \<^bold>\<box>\<^bold>\<not>p"
interval_union_mono1: fixes a :: "'a interval"   and A :: "'a interval" shows "set_of a \<subseteq> set_of (sup a A)"
assms: shows "\<forall>(d, e, f)\<in>set leq. \<exists>y'>r. \<forall>x\<in>{r<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
unit_not_snd: fixes P :: "'b trm" shows "Unit \<noteq> Snd P"
cntI: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"   and "\<And>m. m open\<^bsub>T\<^esub> \<Longrightarrow> carrier\<^bsub>S\<^esub> \<inter> f -` m open\<^bsub>S\<^esub>" shows "f \<in> cnt S T"
fps_shift_rev_shift_1: fixes m :: "nat"   and n :: "nat"   and f :: "'a fps" assumes "m \<le> n" shows "fps_shift n (Abs_fps (\<lambda>k. if k < m then 0::'a else fps_nth f (k - m))) = fps_shift (n - m) f"
covif: shows "ov\<inverse> O f \<subseteq> ov\<inverse>"
errMOD_igSubstIGVar2STR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igVarIPresIGWls Delta MOD"   and "FixSyn.igSubstIGVar2 Delta MOD" shows "FixSyn.igSubstIGVar2STR Delta (FixSyn.errMOD Delta MOD)"
preserves_ide: fixes a :: "'a" assumes "ide a" shows "ide (trg a)"
abs_summable_on_comparison_test: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'e" assumes "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> norm (g x)" shows "f abs_summable_on A"
prime_elem_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.prime_elem times one zero p = (p \<noteq> zero \<and> \<not> dvd.dvd times p one \<and> (\<forall>a b. dvd.dvd times p (times a b) \<longrightarrow> dvd.dvd times p a \<or> dvd.dvd times p b))"
combine_add: fixes R :: "('a, 'b) ring_scheme"   and Ks :: "'a list"   and Us :: "'a list"   and Ks' :: "'a list" assumes "ring R"   and "length Ks = length Us"   and "length Ks' = length Us"   and "set Ks \<subseteq> carrier R"   and "set Ks' \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "ring.combine R Ks Us \<oplus>\<^bsub>R\<^esub> ring.combine R Ks' Us = ring.combine R (map2 (\<oplus>\<^bsub>R\<^esub>) Ks Ks') Us"
fin_inf_cases: fixes r :: "'b llist" obtains "r \<in> UNIV\<^sup>\<star>"   | "r \<in> UNIV\<^sup>\<omega>"
spmf_map_outside: fixes x :: "'c"   and f :: "'d \<Rightarrow> 'c"   and M :: "'d spmf" assumes "x \<notin> f ` set_spmf M" shows "spmf (map_spmf f M) x = 0"
r_e2tail_def: shows "r_e2tail \<equiv> Cn 1 r_tl [r_e2stack]"
carrier_direct_lim_def: shows "dlY.carrier_direct_lim \<equiv> Sigma (target.neighborhoods (f x)) \<O>\<^sub>Y / {(x, y). dlY.rel x y}"
elims: fixes x :: "('Var, 'AExp, 'BExp) Stmt"   and xa :: "'Var ModeUpd list"   and y :: "('Var, 'AExp, 'BExp) Stmt" assumes "x \<otimes> xa = y" obtains c :: "('Var, 'AExp, 'BExp) Stmt" where "x = cb__" and "xa = []" and "y = cb__"   | c :: "('Var, 'AExp, 'BExp) Stmt" and a :: "'Var ModeUpd" and as :: "'Var ModeUpd list" where "x = c" and "xa = a # as" and "y = c \<otimes> as@[a]"
permute_u_def: fixes p :: "perm"   and x :: "'a\<^sub>\<bottom>" shows "p \<bullet> x = (case x of up\<cdot>x \<Rightarrow> up\<cdot>(p \<bullet> x))"
sel_3: fixes x1 :: "'c extended"   and x2 :: "'c"   and x3 :: "'c extended"   and x4 :: "'d" shows "S (InsertMessage x1 x2 x3 x4) = x3"
mat_of_cblinfun_Proj_ccspan: fixes S :: "'a list" shows "mat_of_cblinfun (Proj (ccspan (set S))) = (let d = length canonical_basis in mk_projector_orthog d (gram_schmidt0 d (map vec_of_basis_enum S)))"
dvd_power_le: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a"   and y :: "'a"   and n :: "nat"   and m :: "nat" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times x y"   and "n \<le> m" shows "dvd.dvd times (power.power one times x n) (power.power one times y m)"
subset_of_opens: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "U \<in> I" shows "is_open U"
asciis_of_literal_def: shows "String.asciis_of_literal \<equiv> map_fun literal.explode id (map of_char)"
tree_rank_height: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) SkewBinomialTree list" assumes "tree_invar (Node e a r ts)" shows "height_tree (Node e a r ts) = r"
closed_lift2_sup: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c err"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'd \<Rightarrow> 'd err" assumes "closed (err A) (lift2 f)"   and "closed (err B) (lift2 g)" shows "closed (err (A \<times> B)) (lift2 (Product.sup f g))"
sum_proj_k: fixes m :: "nat" assumes "m \<le> K" shows "matrix_sum K proj_k m = proj_fst_k m"
disc_13: fixes x31 :: "'b local.term"   and x32 :: "'b local.term" shows "is_Tensor (x31 \<^bold>\<otimes> x32)"
exhaust: fixes y :: "('d, 'c) trie" obtains x1 :: "'c option" and x2 :: "('d \<times> ('d, 'c) trie) list" where "y = Trie x1 x2"
eq_to_ineqs: fixes x :: "nat"   and y :: "nat" assumes "x \<equiv> y" shows "x \<le> y \<and> y \<le> x"
Says_imp_analz: fixes A' :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list"   and A :: "agent" assumes "Says A' B X \<in> set evs" shows "X \<in> analz (knows A evs)"
unfold: shows "lrev_body2 = (\<Lambda> r v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>ID) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(list2H\<cdot>(wrapH\<cdot>r\<cdot>xs) oo list2H\<cdot>(x :@ lnil)))))"
prm_set_distributes_union: fixes \<pi> :: "'b prm"   and S :: "'b set"   and T :: "'b set" shows "\<pi> {$} (S \<union> T) = \<pi> {$} S \<union> \<pi> {$} T"
is_singletonI: fixes x :: "'f" shows "is_singleton {x}"
that: fixes B :: "'a set" assumes "(0::'a) \<notin> B"   and "B \<subseteq> S"   and "pairwise orthogonal B"   and "independent B"   and "card B = dim S"   and "span B = S" shows "thesis"
arg2_le_c_pair: fixes y :: "nat"   and x :: "nat" shows "y \<le> c_pair x y"
inf_sup_ord_3: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "less_eq x (sup x y)"
lunit_char_3: fixes f :: "'a" assumes "ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : L f \<Rightarrow> f\<guillemotright> \<and> L \<mu> = (\<i>[trg f] \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[trg f, trg f, f]"
dist_on_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and V :: "nat set" shows "dist_on G m n V = path_weight G (SOME p. is_shortest_path_on G m n p V)"
in_own_span: fixes S :: "'c set" assumes "S \<subseteq> carrier M" shows "S \<subseteq> local.span S"
Pow_insert: fixes a :: "'g"   and A :: "'g set" shows "Pow (insert a A) = Pow A \<union> insert a ` Pow A"
that: shows "closed_segment X (X + H) \<subseteq> S"
zero_neutral: fixes x :: "'a"   and y :: "'a" shows "x + y + zero = x + y"
compositionality: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and h :: "'i \<Rightarrow> 'g"   and i :: "'j \<Rightarrow> 'h"   and sum :: "'i + 'j" shows "map_sum f g (map_sum h i sum) = map_sum (f \<circ> h) (g \<circ> i) sum"
surj_of_rat_nat_to_rat_surj: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and r :: "'a" assumes "class.field_char_0 divide inverse times one plus zero minus uminus"   and "r \<in> field_char_0.Rats divide one plus zero minus" shows "\<exists>n. r = field_char_0.of_rat divide one plus zero minus (nat_to_rat_surj n)"
induct: fixes x1 :: "'a list"   and x2 :: "'a list"   and x3 :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "interleavings x1 x2 x3"   and "\<And>ys. P [] ys ys"   and "\<And>xs. P xs [] xs"   and "\<And>xs ys zs x. \<lbrakk>interleavings xs ys zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) ys (x # zs)"   and "\<And>xs ys zs y. \<lbrakk>interleavings xs ys zs; P xs ys zs\<rbrakk> \<Longrightarrow> P xs (y # ys) (y # zs)" shows "P x1 x2 x3"
convergence_G_def: fixes s :: "'s"   and t :: "'s"   and z :: "'b" shows "convergence_G s t z = summable (\<lambda>n. p s t n * norm z ^ n)"
lang_rderiv: fixes n :: "nat"   and r :: "'b rexp"   and as :: "'a" assumes "local.wf n r"   and "as \<in> \<Sigma> n" shows "lang n (rderiv as r) = rQuot as (lang n r)"
simps_2: fixes v :: "'a" shows "Infty + Num v = Infty"
rell_G_compp: fixes L1 :: "'n \<Rightarrow> 'w \<Rightarrow> bool"   and L1' :: "'w \<Rightarrow> 'u \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'x \<Rightarrow> bool"   and L2' :: "'x \<Rightarrow> 'v \<Rightarrow> bool" shows "rell_G (L1 OO L1') (L2 OO L2') = rell_G L1 L2 OO rell_G L1' L2'"
simps: fixes a1 :: "nat"   and a2 :: "nat \<Rightarrow>f value option"   and a3 :: "((nat \<times> nat) \<times> transition) fset"   and a4 :: "nat"   and a5 :: "nat \<Rightarrow>f value option"   and a6 :: "(String.literal \<times> value list) list" shows "obtains a1 a2 a3 a4 a5 a6 = ((\<exists>s r e. a1 = s \<and> a2 = r \<and> a3 = e \<and> a4 = s \<and> a5 = r \<and> a6 = []) \<or> (\<exists>e s' r' l i s r t. a1 = s \<and> a2 = r \<and> a3 = e \<and> a4 = s' \<and> a5 = r' \<and> a6 = (l, i) # t \<and> (\<exists>(s'', T)|\<in>|possible_steps e s' r' l i. obtains s r e s'' (evaluate_updates T i r') t)))"
B_invariant: fixes a :: "'b"   and b :: "'b" assumes "PB a"   and "B a b" shows "QB b"
sets_return: fixes M :: "'d measure"   and x :: "'d" shows "sets (return M x) = sets M"
arr_char: fixes f :: "'a \<times> 'a" shows "VoV.arr f = (fst f \<star> snd f \<noteq> null)"
H_2: shows "(x, b) \<in> closed_segment (x0, y0) (x1, y1)"
Fvars_exu: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "Fvars (exu x \<phi>) = Fvars \<phi> - {x}"
distinct_length_le: fixes s :: "'c set"   and xs :: "'c list" assumes "finite s"   and "set xs \<subseteq> s"   and "distinct xs" shows "length xs \<le> card s"
comparator_def: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order" shows "comparator comp \<equiv> (\<forall>x y. invert_order (comp x y) = comp y x) \<and> (\<forall>x y. comp x y = Eq \<longrightarrow> x = y) \<and> (\<forall>x y z. comp x y = Lt \<longrightarrow> comp y z = Lt \<longrightarrow> comp x z = Lt)"
ext: fixes g :: "'b"   and f :: "'b" assumes "g \<cdot>\<^sub>B f \<noteq> B.null" shows "B.seq g f"
col: fixes nr :: "nat"   and nc :: "nat"   and m :: "'b list list"   and i :: "nat" assumes "mat nr nc m"   and "i < nc" shows "vec nr (col m i)"
13: shows "\<forall>qf\<in>Filters.filter.Rep_filter (stone_phi pf). Rep_lifted_distrib_lattice_top (dl_iso qf) f = get_dense qf f"
INF_eq_bot_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "(Inf (f ` A) = bot) = (\<forall>x. less bot x \<longrightarrow> (\<exists>i\<in>A. less (f i) x))"
nonneg_arcs: fixes e :: "'b" assumes "e \<in> arcs G" shows "0 \<le> c e"
Tree_axioms_def: fixes G :: "('e, 'f) Graph_scheme" shows "Tree_axioms G \<equiv> (\<forall>v w. v \<in> V\<^bsub>G\<^esub> \<longrightarrow> w \<in> V\<^bsub>G\<^esub> \<longrightarrow> Graph.connected G v w) \<and> (\<forall>xs. \<not> Graph.cycle G xs)"
distinct_8: fixes list' :: "char list"   and var' :: "NominalSequents.var"   and form' :: "form"   and list :: "char list"   and form_list :: "form_list" shows "list' \<nabla> [var'].form' \<noteq> Cpd0 list form_list"
map_endofun'_def: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b"   and e :: "'b \<Rightarrow> 'b" shows "map_endofun' f g e = (g ---> f) e"
Lyndon_Lyndon_rec: fixes w :: "'a list"   and n :: "nat" assumes "Lyndon w"   and "0 < n"   and "n < \<^bold>|w\<^bold>|" shows "Lyndon_rec w n"
emeasure_PiM: fixes I :: "bool set"   and A :: "bool \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (case i of True \<Rightarrow> M1 | False \<Rightarrow> M2)" shows "emeasure (Pi\<^sub>M I (case_bool M1 M2)) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (case i of True \<Rightarrow> M1 | False \<Rightarrow> M2) (A i))"
auto_chop_def: fixes A :: "'d \<times> ('c \<Rightarrow> 'd \<Rightarrow> 'd) \<times> ('d \<Rightarrow> bool)" shows "auto_chop A = chop (\<lambda>xs. auto_split A (start A) ([], xs) [] xs)"
polynomial_incl: fixes K :: "'c set"   and p :: "'c list" assumes "polynomial\<^bsub>A\<^esub> K p" shows "set p \<subseteq> K"
compact_frontier_bounded: fixes S :: "'a set" assumes "bounded S" shows "compact (frontier S)"
domain_compose_chart: fixes p :: "'c \<Rightarrow> 'c"   and p' :: "'c \<Rightarrow> 'c"   and c :: "('d, 'c) chart" assumes "homeomorphism UNIV UNIV p p'" shows "domain (compose_chart p p' c) = domain c"
elims: fixes x :: "'a"   and xa :: "'a option"   and y :: "'a" assumes "the_default x xa = y" obtains a :: "'a" where "x = a" and "xa = None" and "y = a"   | uu_ :: "'a" and b :: "'a" where "x = uu_" and "xa = Some b" and "y = b"
FrameStatImp_def: fixes F :: "'b frame"   and G :: "'b frame" shows "F \<hookrightarrow>\<^sub>F G \<equiv> \<forall>\<phi>. F \<turnstile>\<^sub>F \<phi> \<longrightarrow> G \<turnstile>\<^sub>F \<phi>"
g: fixes s :: "complex" assumes "s \<in> A"   and "c < Re s" shows "g s = eval_fds f s"
starlike_imp_contractible: fixes S :: "'a set" assumes "starlike S" shows "contractible S"
negatively_invariant_def: fixes M :: "'a set" shows "rev.negatively_invariant M = (\<forall>x\<in>M. rev.trapped_backward x M)"
0: shows "reflp (\<lambda>a\<^sub>1 a\<^sub>2. g a\<^sub>1 \<le> g a\<^sub>2)"
SeqC: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c1' :: "('test, 'atom) com"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>c (c1', s')" shows "(c1 ;; c2, s) \<rightarrow>c (c1' ;; c2, s')"
bexp_assign_eval: fixes mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and e :: "'AExp" shows "ev\<^sub>B mem (bexp_assign x e) = (mem x = ev\<^sub>A mem e)"
deltafun_is_aezfun: fixes b :: "'d"   and a :: "'c" shows "b \<delta> a \<in> aezfun_set"
relcomppI: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e" assumes "r a b"   and "s b c" shows "(r OO s) a c"
restrict_indets_subst_def: fixes x :: "'b option" shows "restrict_indets_subst x = (case x of None \<Rightarrow> 0 | Some y \<Rightarrow> monomial 1 y)"
simps_1: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "headconst (CN c n p) = headconst p"
card_equiv_k_classes_eq_card_partitions_k_parts: fixes A :: "'b set"   and k :: "nat" shows "card {R. equiv A R \<and> card (A // R) = k} = card {P. partition_on A P \<and> card P = k}"
simps_3: fixes m :: "nat"   and n :: "nat" shows "ack (Suc m) (Suc n) = ack m (ack (Suc m) n)"
assms_2: fixes x :: "'a" assumes "Q x" shows "P x"
CongruenceI: fixes K :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c"   and k :: "'c" assumes "normal_subgroup K G composition unit"   and "a = composition b k"   and "a \<in> G"   and "b \<in> G"   and "k \<in> K" shows "(a, b) \<in> normal_subgroup.Congruence K G composition unit"
inter_ivl_clw_aux: fixes A :: "('c \<times> 'd) set"   and intr :: "'c \<times> 'c \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c" assumes "single_valued A"   and "(intr, (\<inter>)) \<in> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel \<rightarrow> \<langle>A\<rangle>ivl_rel" shows "(\<lambda>xs y. map (intr y) xs, (\<inter>)) \<in> clw_rel (\<langle>A\<rangle>ivl_rel) \<rightarrow> \<langle>A\<rangle>ivl_rel \<rightarrow> clw_rel (\<langle>A\<rangle>ivl_rel)"
indep_var_def: fixes Ma :: "'b measure"   and A :: "'a \<Rightarrow> 'b"   and Mb :: "'b measure"   and B :: "'a \<Rightarrow> 'b" shows "indep_var Ma A Mb B = indep_vars (case_bool Ma Mb) (case_bool A B) UNIV"