hcju/isabelleps huggingface dataset:
wf_equallength: fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and \<Gamma>' :: "binding list" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "map is_TVarB \<Gamma>' = map is_TVarB \<Gamma>" shows "\<Gamma>' \<turnstile>\<^sub>w\<^sub>f T"
eq: fixes z :: "complex" assumes "z \<in> S" shows "1 - (f z)\<^sup>2 = (g z)\<^sup>2"
bot_set_code: shows "\<bottom> = Seq (\<lambda>u. Empty)"
regular_star: fixes S :: "'b list set" assumes "regular S" shows "regular (star S)"
inner_eq_zero_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "(inner x x = 0) = (x = zero)"
divide_le_eq_numeral1_1: fixes b :: "'c"   and w :: "num"   and a :: "'c" shows "(b / numeral w \<le> a) = (b \<le> a * numeral w)"
mat1_rt_mult: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "int poly fract list list" assumes "mat nr nc m1" shows "rat_poly.matrix_mult m1 (mat1 nc) = m1"
assms: shows "r \<subseteq>\<^sub>\<circ> A \<times>\<^sub>\<circ> B"
closed_vimage_Int: fixes s :: "'a set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "closed s"   and "continuous_on t f"   and "closed t" shows "closed (f -` s \<inter> t)"
fold_A_chamber_system_image_fixespointwise': fixes F :: "'a set" assumes "F \<in> {F \<in> ChamberComplex.chamber_system A. face_distance z F = chamber_distance C F}" shows "fixespointwise fold_A F"
proj_psi_mat: shows "proj_psi = mat N N (\<lambda>x. complex_of_real (1 / real N))"
C'_def: shows "C' \<equiv> - f (real_of_int a) + F (real_of_int a) + C + (\<Sum>k<N. (bernoulli' (Suc k) / fact (Suc k)) *\<^sub>R fs k (real_of_int a))"
snd_sylow_def: fixes G :: "('c, 'd) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" shows "snd_sylow G p a m \<equiv> sylow G p a m \<and> snd_sylow_axioms p m"
set_val_ok: fixes h :: "(_) heap"   and character_data_ptr :: "(_) character_data_ptr"   and tag :: "char list" assumes "DocumentClass.type_wf h"   and "character_data_ptr |\<in>| character_data_ptr_kinds h" shows "h \<turnstile> ok set_val character_data_ptr tag"
pp: shows "pderiv p = (\<Prod>(a, i)\<in>as. a ^ i) * (\<Sum>(a, i)\<in>as. (\<Prod>(b, j)\<in>as - {(a, i)}. b) * smult (of_nat (Suc i)) (pderiv a))"
pref_len': fixes u :: "'b list"   and z :: "'b list" shows "\<^bold>|u\<^bold>| \<le> \<^bold>|u \<cdot> z\<^bold>|"
G\<Psi>: fixes m :: "nat"   and M :: "nat set"   and l :: "nat" assumes "\<Phi> m M" shows "case G l m M of (n, N, j) \<Rightarrow> \<Psi> (Suc l) m n M N j"
assms_7: shows "\<forall>q\<in>ps\<^sub>R. l \<le> fst q"
coeff_liftPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'a mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (liftPoly b i mp) x = (if \<exists>j\<in>{b..<b + i}. 0 < lookup x j then 0::'a else MPoly_Type.coeff mp (higherPowers b i x))"
nv: shows "set (normal_vectors ws) \<subseteq> carrier_vec n"
continuous_within: fixes x :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous (at x within s) f = (f \<longlongrightarrow> f x) (at x within s)"
poly_mult_zero_1: fixes p :: "'a list" assumes "set p \<subseteq> carrier R" shows "poly_mult [] p = []"
subst_pp_def: fixes f :: "'f \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e"   and t :: "'f \<Rightarrow>\<^sub>0 nat" shows "subst_pp f t = (\<Prod>x\<in>keys t. f x ^ lookup t x)"
mset_insert_wrt: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "mset (insert_wrt R x xs) = add_mset x (mset xs)"
simps_2: fixes x :: "Preliminaries.var"   and T1 :: "'fsym trm"   and T2 :: "'fsym trm" shows "isGuard x (Neg (Eq T1 T2)) = (T1 = Var x \<and> (\<exists>f Tl. T2 = Fn f Tl) \<or> T2 = Var x \<and> (\<exists>f Tl. T1 = Fn f Tl))"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> T = vcard T"
cap_simps_3: fixes x :: "'v"   and y :: "'v" shows "cap (\<langle>x\<rangle>, \<langle>y\<rangle>) = (if edge \<Gamma> x y \<and> x \<noteq> a then f (x, y) else if edge \<Gamma> y x then max (weight \<Gamma> x) (weight \<Gamma> y) + 1 else 0)"
qTermQSwappedLess_def: shows "qTermQSwappedLess = qTermLess_modulo qSwapped"
set_rem_condlessD: fixes x :: "('a, 'b) sas_plus_operator"   and s :: "'a \<Rightarrow> 'b option"   and ops :: "('a, 'b) sas_plus_operator list" assumes "x \<in> set (rem_condless_ops s ops)" shows "x \<in> set ops"
filterlim_sup: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and F1 :: "'a filter"   and F2 :: "'a filter" assumes "filterlim f F F1"   and "filterlim f F F2" shows "filterlim f F (sup F1 F2)"
cases: fixes x :: "'a acom" obtains P :: "'a" where "x = SKIP {P__}"   | xa :: "char list" and e :: "aexp" and P :: "'a" where "x = xa ::= e {Pa__}"   | c1 :: "'a acom" and c2 :: "'a acom" where "x = c1__;; c2__"   | b :: "bexp" and c1 :: "'a acom" and c2 :: "'a acom" and P :: "'a" where "x = IF b__ THEN c1 ELSE c2 {Pb__}"   | Inv :: "'a" and b :: "bexp" and c :: "'a acom" and P :: "'a" where "x = {Inv} WHILE b DO c {P}"
assms: shows "vs \<subseteq> prob_dom (prob_proj PROB (prob_dom PROB - vs'))"
is_monic_setI: fixes B :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>b. \<lbrakk>b \<in> B; b \<noteq> 0\<rbrakk> \<Longrightarrow> punit.lc b = (1::'c)" shows "punit.is_monic_set B"
select_convs_2: fixes I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "M \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = M"
empty_Collect_eq: fixes P :: "'c \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
cardinalityIntersectionEquality: fixes X :: "'a set"   and Y :: "'a set" assumes "finite X"   and "finite Y"   and "card X = card Y" shows "(card (X \<inter> Y) = card X) = (X = Y)"
exhaust: fixes y :: "pseqp" obtains "y = PAodv"   | "y = PNewPkt"   | "y = PPkt"   | "y = PRreq"   | "y = PRrep"   | "y = PRerr"
and_and_mask_simple_not: fixes y :: "'b"   and n :: "nat"   and x :: "'b" assumes "y AND mask n = (0::'b)" shows "(x AND y) AND mask n = (0::'b)"
derf': fixes t :: "real" assumes "t \<in> {0..1} - {t. (1 - t) *\<^sub>R c + t *\<^sub>R x \<in> K}" shows "((\<lambda>u. f ((1 - u) *\<^sub>R c + u *\<^sub>R x)) has_derivative (\<lambda>h. 0::'b)) (at t within {0..1})"
card_of_Un_infinite_simps_1: fixes A :: "'a set"   and B :: "'a set" assumes "infinite A"   and "|B| \<le>o |A|" shows "|A \<union> B| =o |A|"
match_list_to_match_expr_disjunction: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and ms :: "'a match_expr list"   and a :: "action"   and p :: "'b" shows "match_list \<gamma> ms a p = matches \<gamma> (match_list_to_match_expr ms) a p"
mono_def: fixes f :: "'a" shows "local.mono f = (arr f \<and> inj_on ((\<cdot>) f) {g. seq f g})"
mk_v_concrete: fixes I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE"   and \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and sol :: "(real, 'd) vec" shows "mk_v I ODE \<nu> sol = (\<chi>i. (if Inl i \<in> semBV I ODE then sol else fst \<nu>) $ i, \<chi>i. (if Inr i \<in> semBV I ODE then ODE_sem I ODE sol else snd \<nu>) $ i)"
simps: fixes A :: "('b \<times> bal) tree"   and a :: "'b"   and ba :: "bal"   and BC :: "('b \<times> bal) tree" shows "balR A a ba BC = (case ba of Lh \<Rightarrow> \<langle>A, (a, Bal), BC\<rangle> | Bal \<Rightarrow> \<langle>A, (a, Rh), BC\<rangle> | Rh \<Rightarrow> case BC of \<langle>B, (c, Lh), C\<rangle> \<Rightarrow> rot2 A a B c C | \<langle>B, (c, Bal), C\<rangle> \<Rightarrow> \<langle>\<langle>A, (a, Rh), B\<rangle>, (c, Lh), C\<rangle> | \<langle>B, (c, Rh), C\<rangle> \<Rightarrow> \<langle>\<langle>A, (a, Bal), B\<rangle>, (c, Bal), C\<rangle>)"
int_pow_1: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> 1 = x"
shorten_nondistinct: shows "\<forall>p. path M p q1 \<and> target p q1 = q2 \<and> \<not> distinct (q1 # states p q1) \<longrightarrow> (\<exists>p'. path M p' q1 \<and> target p' q1 = q2 \<and> length p' < length p)"
simps_1: fixes s :: "'b" shows "last_state (s, []) = s"
A_lb_finite: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and x :: "nat" assumes "lb T A j"   and "x \<in> {1..m}" shows "finite (A x)"
sorted_wrt_insert_wrt: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" assumes "linorder_on A R"   and "set (x # xs) \<subseteq> A"   and "Linorder_Relations.sorted_wrt R xs" shows "Linorder_Relations.sorted_wrt R (insert_wrt R x xs)"
monom_eq_0: fixes n :: "nat" shows "monom (0::'b) n = 0"
eq_elem_cong_r: fixes A :: "'a set"   and A' :: "'a set"   and a :: "'a" assumes "A \<subseteq> carrier S"   and "A' \<subseteq> carrier S"   and "A {.=} A'"   and "a \<in> carrier S"   and "a .\<in> A" shows "a .\<in> A'"
app\<^sub>2: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" shows "(\<Gamma>, Lam [y]. e, Arg x # S) \<Rightarrow> (\<Gamma>, e[y::=x], S)"
\<omega>_limit_set_def: fixes x :: "'a" shows "\<omega>_limit_set x = {p. \<omega>_limit_point x p}"
laterInPrefixRetainsPrecedes: fixes p :: "'a list"   and l :: "'a list"   and a :: "'a"   and b :: "'a" assumes "isPrefix p l"   and "precedes a b l"   and "b \<in> set p" shows "precedes a b p"
dual_iff: fixes x :: "'a"   and y :: "'a" shows "(- x = y) = (x = - y)"
simps_3: fixes x :: "'a list"   and xs :: "'a list list"   and y :: "'a list"   and ys :: "'a list list" shows "interact (x # xs) (y # ys) = x @ y @ interact xs ys"
less_eq_Assertion_def: fixes p :: "'b Assertion"   and q :: "'b Assertion" shows "(p \<le> q) = ({\<cdot> p } \<le> {\<cdot> q })"
assms_4: shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> FW M n"
coerce_eq_emb: shows "COERCE('a, udom) = EMB('a)"
has_density_emeasure_space': fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_density M N f" shows "emeasure (density N f) (space (density N f)) = integral\<^sup>N N f"
mult_eq_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
a_1: fixes g :: "'a"   and P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list" assumes "g \<in> set (P f)" shows "set (map (Fun g) (U P T)) \<subseteq> set (term_variants P (Fun f T))"
matrix_right_invertible_span_columns: fixes A :: "(('b, 'd) vec, 'c) vec" shows "(\<exists>B. A ** B = mat (1::'b)) = (vec.span (columns A) = UNIV)"
integrable_on_null: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "content (cbox a b) = 0" shows "f integrable_on cbox a b"
skl1_step3_refines_a0i_commit_skip_i: fixes A :: "agent"   and B :: "agent"   and Ra :: "fid_t"   and Nr :: "msg"   and gny :: "msg" shows "{R0sk1iai \<inter> UNIV \<times> skl1_inv3} (a0i_commit A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>)\<^sup>=, skl1_step3 Ra A B Nr gny {> R0sk1iai}"
"4_3": fixes i :: "nat" assumes "i < length (x # xs)"   and "i < length (y # ys)"   and "snd (P ((x # xs) ! i) ((y # ys) ! i))" shows "snd (P' ((x # xs) ! i) ((y # ys) ! i))"
alpha_Tree_eqvt: fixes t1 :: "('b, 'c, 'd, 'e) Tree"   and t2 :: "('b, 'c, 'd, 'e) Tree"   and p :: "perm" assumes "t1 =\<^sub>\<alpha> t2" shows "p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2"
lunit_eqI: fixes f :: "'a"   and \<mu> :: "'a" assumes "ide f"   and "\<guillemotleft>\<mu> : trg f \<star> f \<Rightarrow> f\<guillemotright>"   and "trg f \<star> \<mu> = (\<i>' (trg f) \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]" shows "\<mu> = PB.lunit f"
poly_pos_greater: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x>a. 0 < poly p x) = (p \<noteq> 0 \<and> poly_inf p = 1 \<and> count_roots_above p a = 0)"
trd3_def: shows "trd3 \<equiv> snd \<circ> snd"
crypts_used: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> crypts H"
subspace_sum: fixes A :: "'a set"   and f :: "'c \<Rightarrow> 'a"   and B :: "'c set" assumes "A \<subseteq> S"   and "\<forall>x. f x \<in> S"   and "subspace A"   and "\<And>x. x \<in> B \<Longrightarrow> f x \<in> A" shows "sum f B \<in> A"
odd_one: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) one"
simps_5: fixes vb :: "'a local.term"   and vc :: "'a local.term"   and va :: "'a local.term" shows "Diag ((vb \<^bold>\<otimes> vc) \<^bold>\<otimes> va) = False"
image_mset_const_eq: fixes c :: "'b"   and M :: "'c multiset" shows "{#c. a \<in># M#} = replicate_mset (size M) c"
multiset_inter_commute: fixes A :: "'a multiset"   and B :: "'a multiset" shows "A \<inter># B = B \<inter># A"
height_gt_0_imp_mergeSibling_left: fixes t\<^sub>1 :: "'a tree"   and w :: "nat"   and t\<^sub>2 :: "'a tree"   and a :: "'a" assumes "0 < height t\<^sub>1" shows "mergeSibling (Node w t\<^sub>1 t\<^sub>2) a = Node w (mergeSibling t\<^sub>1 a) (mergeSibling t\<^sub>2 a)"
induct: fixes x :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite x"   and "P {}"   and "\<And>A a. \<lbrakk>finite A; P A\<rbrakk> \<Longrightarrow> P (insert a A)" shows "P x"
IH: assumes "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>s\<^sub>t \<theta>"
ih: fixes C :: "'c literal multiset" assumes "poss AA \<subseteq># map_clause f C" shows "\<exists>AA0. poss AA0 \<subseteq># C \<and> AA = image_mset f AA0"
size_list_length_diff1: fixes xs :: "'a list list" assumes "xs \<noteq> []"   and "[] \<notin> set xs" shows "size_list (\<lambda>xs. length xs - Suc 0) xs < size_list length xs"
add_mset_subseteq_single_iff: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b" shows "(add_mset a M \<subseteq># {#b#}) = (M = {#} \<and> a = b)"
intro: fixes block_collection :: "'b set multiset" assumes "int (size block_collection) \<noteq> 0" shows "proper_design_axioms block_collection"
Max_image_cong_simp: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "S = T"   and "\<And>x. x \<in> T =simp=> f x = g x" shows "Max (f ` S) = Max (g ` T)"
simps_1: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.getMinTree [t] = t"
approx_add: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star"   and d :: "'a star" assumes "a \<approx> b"   and "c \<approx> d" shows "a + c \<approx> b + d"
r_constn_prim: fixes n :: "nat"   and c :: "nat" shows "prim_recfn (Suc n) (r_constn n c)"
mult_of_nat_commute: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and x :: "nat"   and y :: "'d" assumes "class.semiring_1 one times plus zero" shows "times (semiring_1.of_nat one plus zero x) y = times y (semiring_1.of_nat one plus zero x)"
infinite_growing: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "class.linorder less_eq less"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. less x y" shows "infinite X"
Abs_mod_ring_inverse: fixes y :: "int" assumes "y \<in> {0..<int CARD('b)}" shows "Rep_mod_ring (Abs_mod_ring y) = y"
comp_left_dist_sum: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.idempotent_semiring one times sup less_eq less bot" shows "comm_monoid_add.sum sup bot (\<lambda>k. times x (f k)) {k. True} = times x (comm_monoid_add.sum sup bot f {k. True})"
degree_rebase_poly_eq: fixes p :: "'p mod_ring poly" shows "degree (#p) = degree p"
\<alpha>_def: shows "\<alpha> = exp (- 2 * M - 1)"
PhiDef: assumes "v \<notin> defs g n"   and "length (predecessors g n) \<noteq> 1"   and "lookupDef g n v = (v, n, PhiDef)" shows "thesis"
filter_join_def: fixes pos :: "bool"   and X :: "'a set"   and m :: "('a, 'b) mapping" shows "filter_join pos X m = Mapping.filter (join_filter_cond pos X) m"
prime_factors_gauss_int_of_nat_correct: fixes n :: "nat" shows "prime_factors_gauss_int_of_nat n = prime_factors (of_nat n)"
flift2_def: fixes f :: "'b \<Rightarrow> 'c" shows "flift2 f = (\<Lambda> (Def x). Def (f x))"
power_int_mult: fixes x :: "'a"   and m :: "int"   and n :: "int" shows "x powi (m * n) = (x powi m) powi n"
simps_2: shows "noDenyAll1 [] = noDenyAll []"
RS: fixes W :: "'a \<Rightarrow> 'a nres"   and s :: "'a"   and s' :: "'b" assumes "\<And>s s'. (s, s') \<in> R \<Longrightarrow> W s \<le> M s'"   and "(s, s') \<in> R" shows "(if b s then f s \<bind> W else RETURN s) \<le> M s'"
seqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "S.VV.arr f"   and "S.VV.arr g"   and "S.VV.dom g = S.VV.cod f" shows "S.VV.seq g f"
in_homI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and a :: "'b"   and b :: "'b" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a"   and "partial_magma.cod C f = b" shows "partial_magma.in_hom C f a b"
E'_def: shows "E' \<equiv> edges T - {(x, y), (y, x)}"
inf_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<squnion> y = x"
integral_eq_nn_integral: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x" shows "integral\<^sup>L M f = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>M)"
absorb_iff1: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" shows "(a \<preceq> b) = (linordered_monoid.min a b = a)"
that: fixes g :: "nat \<Rightarrow> 'a" assumes "[g[a .. b ..  c]X] \<or> [g[c .. b ..  a]X]" shows "thesis"
rSim: fixes P :: "ccs"   and Q :: "ccs" assumes "(P, Q) \<in> X" shows "P \<leadsto>\<^sup>^<(weakBisimulation O X O bisim)> Q"
semiring_norm_6: fixes m :: "num"   and n :: "num" shows "num.Bit0 m + num.Bit0 n = num.Bit0 (m + n)"
integrable_density: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integrable (density M (\<lambda>x. ennreal (g x))) f = integrable M (\<lambda>x. g x *\<^sub>R f x)"
rbd_bot_pres: shows "bot_pres bd\<^sub>\<R>"
triangle_F': fixes b :: "'d" assumes "D.ide b" shows "C.inverse_arrows (F (\<eta> b)) (\<epsilon> (F b))"
scale_left_commute: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and b :: "'d"   and x :: "'e" assumes "module scale" shows "scale a (scale b x) = scale b (scale a x)"
assms: fixes k :: "'a\<cdot>'e error\<cdot>'f" shows "P (ErrorT\<cdot>k)"
\<omega>_simps_4: shows "S.dom \<omega> = w \<star> w"
one: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "\<one>\<^bsub>G\<^esub> \<in> generate G H"
distinct_61: shows "NotEq \<noteq> ShiftRightSigned"
mst_extends_new_tree: fixes t :: "'a"   and w :: "'a"   and v :: "'a"   and e :: "'a" assumes "t \<le> w"   and "t \<le> v \<sqinter> id v"   and "v \<sqinter> top = v" shows "t \<squnion> e \<le> w \<sqinter> - (w \<sqinter> - id v \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> id (w \<sqinter> (- v \<sqinter> - id v) \<sqinter> (top \<sqinter> e \<sqinter> top)) \<squnion> e"
leofD: fixes m :: "'a nres"   and m' :: "'a nres" assumes "nofail m"   and "m \<le>\<^sub>n m'" shows "m \<le> m'"
weak_duality_aux: fixes i :: "rat"   and j :: "rat" assumes "i \<in> {c \<bullet> x |x. x \<in> sat_primal A b}"   and "j \<in> {b \<bullet> y |y. y \<in> sat_dual A c}" shows "i \<le> j"
sum_set_upt_eq_sum_list: fixes f :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "sum f {m..<n} = sum_list (map f [m..<n])"
simps_2: fixes f :: "'a" shows "Diag \<^bold>\<langle>f\<^bold>\<rangle> = C.arr f"
eq_fold: fixes M :: "'k multiset" shows "\<Sum>\<^sub># M = fold_mset (+) (0::'k) M"
distinct_1: fixes x1 :: "'a"   and x2 :: "'a" shows "Pos x1 \<noteq> Neg x2"
sep_add_zero_sym: fixes x :: "'a" shows "(0::'a) + x = x"
xX_simp2: shows "backflows stateful - (backflows (X - {x}) - E) = backflows stateful"
dep: shows "\<forall>a\<in>set as. depends\<^sub>d\<^sub>l\<^sub>o a"
le: shows "int_of_uint64 x * int_of_uint64 x' < 18446744073709551616"
ereal_inverse_eq_0: fixes x :: "ereal" shows "(inverse x = 0) = (x = \<infinity> \<or> x = - \<infinity>)"
cont: fixes Y :: "'a set" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "\<And>x. x \<in> Y \<Longrightarrow> \<not> x \<le> bound" shows "f (\<Squnion> Y) = \<Or> (f ` Y)"
separation_by_component_open_aux: fixes S :: "'a set"   and T :: "'a set" assumes "closed S"   and "closed T"   and "S \<inter> T = {}"   and "S \<noteq> {}"   and "T \<noteq> {}" obtains C :: "'a set" where "C \<in> components (- (S \<union> T))" and "C \<noteq> {}" and "frontier C \<inter> S \<noteq> {}" and "frontier C \<inter> T \<noteq> {}"
length_remdups_card_conv: fixes xs :: "'d list" shows "length (remdups xs) = card (set xs)"
max_fun_diff_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "t.max_fun_diff f g \<equiv> t.maxim {a \<in> Field t. f a \<noteq> g a}"
set_bset_bsingleton: fixes x :: "'a" shows "set_bset (bsingleton x) = {x}"
h2: fixes y :: "'m"   and z :: "'n" assumes "y \<in> rel_interior {t. f t \<noteq> {}}" shows "((y, z) \<in> rel_interior S) = (z \<in> rel_interior (f y))"
inject_4: fixes x41 :: "'b"   and x42 :: "value list"   and y41 :: "'b"   and y42 :: "value list" shows "(In x41 x42 = In y41 y42) = (x41 = y41 \<and> x42 = y42)"
pj_csrp_mem_coeff: fixes P :: "'a set"   and n :: "nat"   and c' :: "nat \<Rightarrow> 'a set" assumes "ideal S P"   and "pol_coeff (S /\<^sub>r P) (n, c')" shows "\<forall>j\<le>n. pj S P (csrp_fn S P (c' j)) = c' j"
simps_2: fixes vb :: "'c \<times> ('d, 'e) rbt"   and vc :: "('c \<times> ('d, 'e) rbt) list"   and va :: "('d, 'e) rbt" shows "rbt_has_next (vb # vc, va) = True"
assms: fixes t :: "('a, 'b) Term.term" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
isAction_isChoice: fixes p :: "'a process" shows "isAction p \<or> isChoice p"
embedding_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "embedding_map X Y f \<equiv> homeomorphic_map X (subtopology Y (f ` topspace X)) f"
assms_1: shows "SAMS A' B' C' D' E' F'"
primes_coprime: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and q :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "algebraic_semidom.coprime times one p q"