--- a/NEWS Mon Sep 19 17:37:22 2016 +0200
+++ b/NEWS Tue Sep 20 11:35:10 2016 +0200
@@ -635,10 +635,20 @@
msetsum ~> sum_mset
msetprod ~> prod_mset
+* The symbols for intersection and union of multisets have been changed:
+ #\<inter> ~> \<inter>#
+ #\<union> ~> \<union>#
+INCOMPATIBILITY.
+
* The lemma one_step_implies_mult_aux on multisets has been removed, use
one_step_implies_mult instead.
INCOMPATIBILITY.
+* The following theorems have been renamed:
+ setsum_left_distrib ~> setsum_distrib_right
+ setsum_right_distrib ~> setsum_distrib_left
+INCOMPATIBILITY.
+
* Compound constants INFIMUM and SUPREMUM are mere abbreviations now.
INCOMPATIBILITY.
@@ -785,6 +795,9 @@
nn_integral :: 'a measure => ('a => ennreal) => ennreal
INCOMPATIBILITY.
+* Renamed HOL/Quotient_Examples/FSet.thy to
+HOL/Quotient_Examples/Quotient_FSet.thy
+INCOMPATIBILITY.
*** ML ***
--- a/src/HOL/Algebra/Divisibility.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Algebra/Divisibility.thy Tue Sep 20 11:35:10 2016 +0200
@@ -2306,10 +2306,10 @@
by (fast elim: wfactorsE)
have "\<exists>c cs. c \<in> carrier G \<and> set cs \<subseteq> carrier G \<and> wfactors G cs c \<and>
- fmset G cs = fmset G as #\<inter> fmset G bs"
+ fmset G cs = fmset G as \<inter># fmset G bs"
proof (intro mset_wfactorsEx)
fix X
- assume "X \<in># fmset G as #\<inter> fmset G bs"
+ assume "X \<in># fmset G as \<inter># fmset G bs"
then have "X \<in># fmset G as" by simp
then have "X \<in> set (map (assocs G) as)"
by (simp add: fmset_def)
@@ -2328,7 +2328,7 @@
where ccarr: "c \<in> carrier G"
and cscarr: "set cs \<subseteq> carrier G"
and csirr: "wfactors G cs c"
- and csmset: "fmset G cs = fmset G as #\<inter> fmset G bs"
+ and csmset: "fmset G cs = fmset G as \<inter># fmset G bs"
by auto
have "c gcdof a b"
--- a/src/HOL/Analysis/Bounded_Linear_Function.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Bounded_Linear_Function.thy Tue Sep 20 11:35:10 2016 +0200
@@ -352,7 +352,7 @@
apply (rule norm_blinfun_bound)
apply (simp add: setsum_nonneg)
apply (subst euclidean_representation[symmetric, where 'a='a])
- apply (simp only: blinfun.bilinear_simps setsum_left_distrib)
+ apply (simp only: blinfun.bilinear_simps setsum_distrib_right)
apply (rule order.trans[OF norm_setsum setsum_mono])
apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
done
@@ -406,7 +406,7 @@
"norm (blinfun_of_matrix a) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. \<bar>a i j\<bar>)"
apply (rule norm_blinfun_bound)
apply (simp add: setsum_nonneg)
- apply (simp only: blinfun_of_matrix_apply setsum_left_distrib)
+ apply (simp only: blinfun_of_matrix_apply setsum_distrib_right)
apply (rule order_trans[OF norm_setsum setsum_mono])
apply (rule order_trans[OF norm_setsum setsum_mono])
apply (simp add: abs_mult mult_right_mono ac_simps Basis_le_norm)
--- a/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Cartesian_Euclidean_Space.thy Tue Sep 20 11:35:10 2016 +0200
@@ -119,8 +119,8 @@
val ss1 =
simpset_of (put_simpset HOL_basic_ss @{context}
addsimps [@{thm setsum.distrib} RS sym,
- @{thm setsum_subtractf} RS sym, @{thm setsum_right_distrib},
- @{thm setsum_left_distrib}, @{thm setsum_negf} RS sym])
+ @{thm setsum_subtractf} RS sym, @{thm setsum_distrib_left},
+ @{thm setsum_distrib_right}, @{thm setsum_negf} RS sym])
val ss2 =
simpset_of (@{context} addsimps
[@{thm plus_vec_def}, @{thm times_vec_def},
@@ -326,7 +326,7 @@
lemma setsum_cmul:
fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
- by (simp add: vec_eq_iff setsum_right_distrib)
+ by (simp add: vec_eq_iff setsum_distrib_left)
lemma setsum_norm_allsubsets_bound_cart:
fixes f:: "'a \<Rightarrow> real ^'n"
@@ -517,14 +517,14 @@
done
lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
- apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult.assoc)
+ apply (vector matrix_matrix_mult_def setsum_distrib_left setsum_distrib_right mult.assoc)
apply (subst setsum.commute)
apply simp
done
lemma matrix_vector_mul_assoc: "A *v (B *v x) = (A ** B) *v x"
apply (vector matrix_matrix_mult_def matrix_vector_mult_def
- setsum_right_distrib setsum_left_distrib mult.assoc)
+ setsum_distrib_left setsum_distrib_right mult.assoc)
apply (subst setsum.commute)
apply simp
done
@@ -555,7 +555,7 @@
by (simp add: matrix_vector_mult_def inner_vec_def)
lemma dot_lmul_matrix: "((x::real ^_) v* A) \<bullet> y = x \<bullet> (A *v y)"
- apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib ac_simps)
+ apply (simp add: inner_vec_def matrix_vector_mult_def vector_matrix_mult_def setsum_distrib_right setsum_distrib_left ac_simps)
apply (subst setsum.commute)
apply simp
done
@@ -630,7 +630,7 @@
lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::real ^ _))"
by (simp add: linear_iff matrix_vector_mult_def vec_eq_iff
- field_simps setsum_right_distrib setsum.distrib)
+ field_simps setsum_distrib_left setsum.distrib)
lemma matrix_works:
assumes lf: "linear f"
@@ -660,7 +660,7 @@
lemma adjoint_matrix: "adjoint(\<lambda>x. (A::real^'n^'m) *v x) = (\<lambda>x. transpose A *v x)"
apply (rule adjoint_unique)
apply (simp add: transpose_def inner_vec_def matrix_vector_mult_def
- setsum_left_distrib setsum_right_distrib)
+ setsum_distrib_right setsum_distrib_left)
apply (subst setsum.commute)
apply (auto simp add: ac_simps)
done
--- a/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Cauchy_Integral_Theorem.thy Tue Sep 20 11:35:10 2016 +0200
@@ -6215,7 +6215,7 @@
by (metis (no_types) dist_norm divide_eq_1_iff less_irrefl mem_ball norm_minus_commute r w)
have "norm ((\<Sum>k<N. (w - z) ^ k * f u / (u - z) ^ Suc k) * (u - w) - f u)
= norm ((\<Sum>k<N. (((w - z) / (u - z)) ^ k)) * f u * (u - w) / (u - z) - f u)"
- unfolding setsum_left_distrib setsum_divide_distrib power_divide by (simp add: algebra_simps)
+ unfolding setsum_distrib_right setsum_divide_distrib power_divide by (simp add: algebra_simps)
also have "... = norm ((((w - z) / (u - z)) ^ N - 1) * (u - w) / (((w - z) / (u - z) - 1) * (u - z)) - 1) * norm (f u)"
using \<open>0 < B\<close>
apply (auto simp: geometric_sum [OF wzu_not1])
--- a/src/HOL/Analysis/Complex_Analysis_Basics.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Complex_Analysis_Basics.thy Tue Sep 20 11:35:10 2016 +0200
@@ -951,7 +951,7 @@
then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
- by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
+ by (simp add: norm_mult [symmetric] field_simps setsum_distrib_left)
qed
} note ** = this
show ?thesis
--- a/src/HOL/Analysis/Complex_Transcendental.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Complex_Transcendental.thy Tue Sep 20 11:35:10 2016 +0200
@@ -597,7 +597,7 @@
text\<open>32-bit Approximation to e\<close>
lemma e_approx_32: "\<bar>exp(1) - 5837465777 / 2147483648\<bar> \<le> (inverse(2 ^ 32)::real)"
using Taylor_exp [of 1 14] exp_le
- apply (simp add: setsum_left_distrib in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
+ apply (simp add: setsum_distrib_right in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
apply (simp only: pos_le_divide_eq [symmetric], linarith)
done
--- a/src/HOL/Analysis/Conformal_Mappings.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Conformal_Mappings.thy Tue Sep 20 11:35:10 2016 +0200
@@ -2846,7 +2846,7 @@
qed
qed
also have "... = c * (\<Sum>p\<in>pts. winding_number g p * residue f p)"
- by (simp add: setsum_right_distrib algebra_simps)
+ by (simp add: setsum_distrib_left algebra_simps)
finally show ?thesis unfolding c_def .
qed
@@ -3459,7 +3459,7 @@
qed
qed
then have "(\<Sum>p\<in>zeros. w p * cont p) = c * (\<Sum>p\<in>zeros. w p * h p * zorder f p)"
- apply (subst setsum_right_distrib)
+ apply (subst setsum_distrib_left)
by (simp add:algebra_simps)
moreover have "(\<Sum>p\<in>poles. w p * cont p) = (\<Sum>p\<in>poles. - c * w p * h p * porder f p)"
proof (rule setsum.cong[of poles poles,simplified])
@@ -3479,7 +3479,7 @@
qed
qed
then have "(\<Sum>p\<in>poles. w p * cont p) = - c * (\<Sum>p\<in>poles. w p * h p * porder f p)"
- apply (subst setsum_right_distrib)
+ apply (subst setsum_distrib_left)
by (simp add:algebra_simps)
ultimately show ?thesis by (simp add: right_diff_distrib)
qed
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy Tue Sep 20 11:35:10 2016 +0200
@@ -523,7 +523,7 @@
qed auto
then show ?thesis
apply (rule_tac IA[of "s - {x}" "\<lambda>y. (inverse (1 - u x) * u y)"])
- unfolding setsum_right_distrib[symmetric]
+ unfolding setsum_distrib_left[symmetric]
using as and *** and True
apply auto
done
@@ -536,7 +536,7 @@
then show ?thesis
using as(3)[THEN bspec[where x=a], THEN bspec[where x=b]]
using *** *(2) and \<open>s \<subseteq> V\<close>
- unfolding setsum_right_distrib
+ unfolding setsum_distrib_left
by (auto simp add: setsum_clauses(2))
qed
then have "u x + (1 - u x) = 1 \<Longrightarrow>
@@ -619,7 +619,7 @@
unfolding scaleR_left_distrib setsum.distrib if_smult scaleR_zero_left
** setsum.inter_restrict[OF xy, symmetric]
unfolding scaleR_scaleR[symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric]
- and setsum_right_distrib[symmetric]
+ and setsum_distrib_left[symmetric]
unfolding x y
using x(1-3) y(1-3) uv
apply simp
@@ -1323,7 +1323,7 @@
apply (rule_tac x="s - {v}" in exI)
apply (rule_tac x="\<lambda>x. - (1 / u v) * u x" in exI)
unfolding scaleR_scaleR[symmetric] and scaleR_right.setsum [symmetric]
- unfolding setsum_right_distrib[symmetric] and setsum_diff1[OF as(1)]
+ unfolding setsum_distrib_left[symmetric] and setsum_diff1[OF as(1)]
using as
apply auto
done
@@ -1793,7 +1793,7 @@
apply rule
unfolding * and setsum.If_cases[OF finite_atLeastAtMost[of 1 "k1 + k2"]] and
setsum.reindex[OF inj] and o_def Collect_mem_eq
- unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_right_distrib[symmetric]
+ unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] setsum_distrib_left[symmetric]
proof -
fix i
assume i: "i \<in> {1..k1+k2}"
@@ -1844,7 +1844,7 @@
}
moreover
have "(\<Sum>x\<in>s. u * ux x + v * uy x) = 1"
- unfolding setsum.distrib and setsum_right_distrib[symmetric] and ux(2) uy(2)
+ unfolding setsum.distrib and setsum_distrib_left[symmetric] and ux(2) uy(2)
using uv(3) by auto
moreover
have "(\<Sum>x\<in>s. (u * ux x + v * uy x) *\<^sub>R x) = u *\<^sub>R (\<Sum>x\<in>s. ux x *\<^sub>R x) + v *\<^sub>R (\<Sum>x\<in>s. uy x *\<^sub>R x)"
@@ -3306,7 +3306,7 @@
have *: "\<And>f. setsum f (s - {a}) = setsum f s - ((f a)::'b::ab_group_add)"
unfolding setsum.remove[OF obt(1) \<open>a\<in>s\<close>] by auto
have "(\<Sum>v\<in>s. u v + t * w v) = 1"
- unfolding setsum.distrib wv(1) setsum_right_distrib[symmetric] obt(5) by auto
+ unfolding setsum.distrib wv(1) setsum_distrib_left[symmetric] obt(5) by auto
moreover have "(\<Sum>v\<in>s. u v *\<^sub>R v + (t * w v) *\<^sub>R v) - (u a *\<^sub>R a + (t * w a) *\<^sub>R a) = y"
unfolding setsum.distrib obt(6) scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] wv(4)
using a(2) [THEN eq_neg_iff_add_eq_0 [THEN iffD2]] by simp
@@ -5279,7 +5279,7 @@
apply (rule_tac x="{v \<in> c. u v < 0}" in exI)
apply (rule_tac x="\<lambda>y. inverse (setsum u {x\<in>c. u x > 0}) * - u y" in exI)
using assms(1) unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
- apply (auto simp add: setsum_negf setsum_right_distrib[symmetric])
+ apply (auto simp add: setsum_negf setsum_distrib_left[symmetric])
done
moreover have "\<forall>x\<in>{v \<in> c. 0 < u v}. 0 \<le> inverse (setsum u {x \<in> c. 0 < u x}) * u x"
apply rule
@@ -5294,7 +5294,7 @@
using assms(1)
unfolding scaleR_scaleR[symmetric] scaleR_right.setsum [symmetric] and z_def
using *
- apply (auto simp add: setsum_negf setsum_right_distrib[symmetric])
+ apply (auto simp add: setsum_negf setsum_distrib_left[symmetric])
done
ultimately show ?thesis
apply (rule_tac x="{v\<in>c. u v \<le> 0}" in exI)
@@ -5683,7 +5683,7 @@
unfolding convex_hull_indexed mem_Collect_eq by auto
have "(\<Sum>i = 1..k. u i * f (v i)) \<le> b"
using setsum_mono[of "{1..k}" "\<lambda>i. u i * f (v i)" "\<lambda>i. u i * b"]
- unfolding setsum_left_distrib[symmetric] obt(2) mult_1
+ unfolding setsum_distrib_right[symmetric] obt(2) mult_1
apply (drule_tac meta_mp)
apply (rule mult_left_mono)
using assms(2) obt(1)
@@ -9200,9 +9200,9 @@
have ge0: "\<forall>i\<in>I. e i \<ge> 0"
using e_def xc yc uv by simp
have "setsum (\<lambda>i. u * c i) I = u * setsum c I"
- by (simp add: setsum_right_distrib)
+ by (simp add: setsum_distrib_left)
moreover have "setsum (\<lambda>i. v * d i) I = v * setsum d I"
- by (simp add: setsum_right_distrib)
+ by (simp add: setsum_distrib_left)
ultimately have sum1: "setsum e I = 1"
using e_def xc yc uv by (simp add: setsum.distrib)
define q where "q i = (if e i = 0 then p i else (u * c i / e i) *\<^sub>R s i + (v * d i / e i) *\<^sub>R t i)"
@@ -11857,7 +11857,7 @@
have sum_dd0: "setsum dd S = 0"
unfolding dd_def using S
by (simp add: sumSS' comm_monoid_add_class.setsum.distrib setsum_subtractf
- algebra_simps setsum_left_distrib [symmetric] b1)
+ algebra_simps setsum_distrib_right [symmetric] b1)
have "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R (\<Sum>v\<in>S. b v *\<^sub>R v)" for x
by (simp add: pth_5 real_vector.scale_setsum_right mult.commute)
then have *: "(\<Sum>v\<in>S. (b v * x) *\<^sub>R v) = x *\<^sub>R a" for x
--- a/src/HOL/Analysis/Derivative.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Derivative.thy Tue Sep 20 11:35:10 2016 +0200
@@ -2244,7 +2244,7 @@
{
fix n :: nat and x h :: 'a assume nx: "n \<ge> N" "x \<in> s"
have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) = norm ((\<Sum>i<n. f' i x) - g' x) * norm h"
- by (simp add: norm_mult [symmetric] ring_distribs setsum_left_distrib)
+ by (simp add: norm_mult [symmetric] ring_distribs setsum_distrib_right)
also from N[OF nx] have "norm ((\<Sum>i<n. f' i x) - g' x) \<le> e" by simp
hence "norm ((\<Sum>i<n. f' i x) - g' x) * norm h \<le> e * norm h"
by (intro mult_right_mono) simp_all
--- a/src/HOL/Analysis/Determinants.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Determinants.thy Tue Sep 20 11:35:10 2016 +0200
@@ -226,7 +226,7 @@
fixes A :: "'a::comm_ring_1^'n^'n"
assumes p: "p permutes (UNIV :: 'n::finite set)"
shows "det (\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
- apply (simp add: det_def setsum_right_distrib mult.assoc[symmetric])
+ apply (simp add: det_def setsum_distrib_left mult.assoc[symmetric])
apply (subst sum_permutations_compose_right[OF p])
proof (rule setsum.cong)
let ?U = "UNIV :: 'n set"
@@ -372,7 +372,7 @@
fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
- unfolding det_def vec_lambda_beta setsum_right_distrib
+ unfolding det_def vec_lambda_beta setsum_distrib_left
proof (rule setsum.cong)
let ?U = "UNIV :: 'n set"
let ?pU = "{p. p permutes ?U}"
@@ -645,7 +645,7 @@
lemma det_rows_mul:
"det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
-proof (simp add: det_def setsum_right_distrib cong add: setprod.cong, rule setsum.cong)
+proof (simp add: det_def setsum_distrib_left cong add: setprod.cong, rule setsum.cong)
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
fix p
--- a/src/HOL/Analysis/Finite_Cartesian_Product.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Finite_Cartesian_Product.thy Tue Sep 20 11:35:10 2016 +0200
@@ -461,7 +461,7 @@
by (simp add: inner_add_left setsum.distrib)
show "inner (scaleR r x) y = r * inner x y"
unfolding inner_vec_def
- by (simp add: setsum_right_distrib)
+ by (simp add: setsum_distrib_left)
show "0 \<le> inner x x"
unfolding inner_vec_def
by (simp add: setsum_nonneg)
--- a/src/HOL/Analysis/Gamma_Function.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Gamma_Function.thy Tue Sep 20 11:35:10 2016 +0200
@@ -450,7 +450,7 @@
(\<Sum>k=Suc m..n. 2 * (norm t + (norm t)\<^sup>2) / (real_of_nat k)\<^sup>2)" using t_nz N(2) mn norm_t
by (intro order.trans[OF norm_setsum setsum_mono[OF ln_Gamma_series_complex_converges_aux]]) simp_all
also have "... \<le> 2 * (norm t + norm t^2) * (\<Sum>k=Suc m..n. 1 / (of_nat k)\<^sup>2)"
- by (simp add: setsum_right_distrib)
+ by (simp add: setsum_distrib_left)
also have "... < 2 * (norm t + norm t^2) * e'" using mn z t_nz
by (intro mult_strict_left_mono N mult_pos_pos add_pos_pos) simp_all
also from e''_pos have "... = e * ((cmod t + (cmod t)\<^sup>2) / e'')"
@@ -543,7 +543,7 @@
by (subst atLeast0LessThan [symmetric], subst setsum_shift_bounds_Suc_ivl [symmetric],
subst atLeastLessThanSuc_atLeastAtMost) simp_all
also have "\<dots> = z * of_real (harm n) - (\<Sum>k=1..n. ln (1 + z / of_nat k))"
- by (simp add: harm_def setsum_subtractf setsum_right_distrib divide_inverse)
+ by (simp add: harm_def setsum_subtractf setsum_distrib_left divide_inverse)
also from n have "\<dots> - ?g n = 0"
by (simp add: ln_Gamma_series_def setsum_subtractf algebra_simps Ln_of_nat)
finally show "(\<Sum>k<n. ?f k) - ?g n = 0" .
@@ -944,7 +944,7 @@
using z' n by (intro uniformly_convergent_mult Polygamma_converges) (simp_all add: n'_def)
thus "uniformly_convergent_on (ball z d)
(\<lambda>k z. \<Sum>i<k. - of_nat n' * inverse ((z + of_nat i :: 'a) ^ (n'+1)))"
- by (subst (asm) setsum_right_distrib) simp
+ by (subst (asm) setsum_distrib_left) simp
qed (insert Polygamma_converges'[OF z' n'] d, simp_all)
also have "(\<Sum>k. - of_nat n' * inverse ((z + of_nat k) ^ (n' + 1))) =
(- of_nat n') * (\<Sum>k. inverse ((z + of_nat k) ^ (n' + 1)))"
@@ -2573,7 +2573,7 @@
proof -
have "(\<Prod>k=1..n. exp (z * of_real (ln (1 + 1 / of_nat k)))) =
exp (z * of_real (\<Sum>k = 1..n. ln (1 + 1 / real_of_nat k)))"
- by (subst exp_setsum [symmetric]) (simp_all add: setsum_right_distrib)
+ by (subst exp_setsum [symmetric]) (simp_all add: setsum_distrib_left)
also have "(\<Sum>k=1..n. ln (1 + 1 / of_nat k) :: real) = ln (\<Prod>k=1..n. 1 + 1 / real_of_nat k)"
by (subst ln_setprod [symmetric]) (auto intro!: add_pos_nonneg)
also have "(\<Prod>k=1..n. 1 + 1 / of_nat k :: real) = (\<Prod>k=1..n. (of_nat k + 1) / of_nat k)"
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy Tue Sep 20 11:35:10 2016 +0200
@@ -3756,7 +3756,7 @@
have "norm (setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
using norm_setsum by blast
also have "... \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
- by (simp add: setsum_right_distrib[symmetric] mult.commute assms(2) mult_right_mono setsum_nonneg)
+ by (simp add: setsum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono setsum_nonneg)
also have "... \<le> e * content (cbox a b)"
apply (rule mult_left_mono [OF _ e])
apply (simp add: sumeq)
@@ -3792,7 +3792,7 @@
apply (rule order_trans[OF setsum_mono])
apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
apply (metis norm)
- unfolding setsum_left_distrib[symmetric]
+ unfolding setsum_distrib_right[symmetric]
using con setsum_le
apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
done
@@ -4697,7 +4697,7 @@
done
have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *
norm (setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x :: real) (q i))) {..N+1}"
- unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
+ unfolding real_norm_def setsum_distrib_left abs_of_nonneg[OF *] diff_0_right
apply (rule order_trans)
apply (rule norm_setsum)
apply (subst sum_sum_product)
@@ -4775,7 +4775,7 @@
done
qed
also have "\<dots> < e * inverse 2 * 2"
- unfolding divide_inverse setsum_right_distrib[symmetric]
+ unfolding divide_inverse setsum_distrib_left[symmetric]
apply (rule mult_strict_left_mono)
unfolding power_inverse [symmetric] lessThan_Suc_atMost[symmetric]
apply (subst geometric_sum)
@@ -5313,7 +5313,7 @@
show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
unfolding content_real[OF assms(1), simplified box_real[symmetric]] additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of f,symmetric]
unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "\<lambda>x. x",symmetric]
- unfolding setsum_right_distrib
+ unfolding setsum_distrib_left
defer
unfolding setsum_subtractf[symmetric]
proof (rule setsum_norm_le,safe)
@@ -6479,7 +6479,7 @@
by arith
show ?case
unfolding content_real[OF assms(1)] and *[of "\<lambda>x. x"] *[of f] setsum_subtractf[symmetric] split_minus
- unfolding setsum_right_distrib
+ unfolding setsum_distrib_left
apply (subst(2) pA)
apply (subst pA)
unfolding setsum.union_disjoint[OF pA(2-)]
@@ -6548,7 +6548,7 @@
apply (unfold split_paired_all split_conv)
defer
unfolding setsum.union_disjoint[OF pA(2-),symmetric] pA(1)[symmetric]
- unfolding setsum_right_distrib[symmetric]
+ unfolding setsum_distrib_left[symmetric]
apply (subst additive_tagged_division_1[OF _ as(1)])
apply (rule assms)
proof -
@@ -8856,7 +8856,7 @@
apply rule
apply (drule qq)
defer
- unfolding divide_inverse setsum_left_distrib[symmetric]
+ unfolding divide_inverse setsum_distrib_right[symmetric]
unfolding divide_inverse[symmetric]
using * apply (auto simp add: field_simps)
done
@@ -8976,7 +8976,7 @@
done
have th: "op ^ x \<circ> op + m = (\<lambda>i. x^m * x^i)"
by (rule ext) (simp add: power_add power_mult)
- from setsum.reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
+ from setsum.reindex[OF i, of "op ^ x", unfolded f th setsum_distrib_left[symmetric]]
have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})"
by simp
then show ?thesis
@@ -9164,7 +9164,7 @@
apply (rule norm_setsum)
apply (rule setsum_mono)
unfolding split_paired_all split_conv
- unfolding split_def setsum_left_distrib[symmetric] scaleR_diff_right[symmetric]
+ unfolding split_def setsum_distrib_right[symmetric] scaleR_diff_right[symmetric]
unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
proof -
fix x k
@@ -9201,7 +9201,7 @@
proof
show "(\<Sum>i = 0..s. e / 2 ^ (i + 2)) < e / 2"
unfolding power_add divide_inverse inverse_mult_distrib
- unfolding setsum_right_distrib[symmetric] setsum_left_distrib[symmetric]
+ unfolding setsum_distrib_left[symmetric] setsum_distrib_right[symmetric]
unfolding power_inverse [symmetric] sum_gp
apply(rule mult_strict_left_mono[OF _ e])
unfolding power2_eq_square
@@ -10350,7 +10350,7 @@
by auto
qed
finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
- unfolding setsum_left_distrib[symmetric] real_scaleR_def
+ unfolding setsum_distrib_right[symmetric] real_scaleR_def
apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
using xl(2)[unfolded uv]
unfolding uv
@@ -10614,7 +10614,7 @@
proof goal_cases
case prems: (2 d)
have "(\<Sum>k\<in>d. norm (integral k (h \<circ> f))) \<le> setsum (\<lambda>k. norm(integral k f)) d * b"
- unfolding setsum_left_distrib
+ unfolding setsum_distrib_right
apply (rule setsum_mono)
proof goal_cases
case (1 k)
--- a/src/HOL/Analysis/Homeomorphism.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Homeomorphism.thy Tue Sep 20 11:35:10 2016 +0200
@@ -1007,7 +1007,7 @@
have gf[simp]: "g (f x) = x" for x
apply (rule euclidean_eqI)
apply (simp add: f_def g_def inner_setsum_left scaleR_setsum_left algebra_simps)
- apply (simp add: Groups_Big.setsum_right_distrib [symmetric] *)
+ apply (simp add: Groups_Big.setsum_distrib_left [symmetric] *)
done
then have "inj f" by (metis injI)
have gfU: "g ` f ` U = U"
--- a/src/HOL/Analysis/L2_Norm.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/L2_Norm.thy Tue Sep 20 11:35:10 2016 +0200
@@ -58,7 +58,7 @@
"0 \<le> r \<Longrightarrow> r * setL2 f A = setL2 (\<lambda>x. r * f x) A"
unfolding setL2_def
apply (simp add: power_mult_distrib)
- apply (simp add: setsum_right_distrib [symmetric])
+ apply (simp add: setsum_distrib_left [symmetric])
apply (simp add: real_sqrt_mult setsum_nonneg)
done
@@ -66,7 +66,7 @@
"0 \<le> r \<Longrightarrow> setL2 f A * r = setL2 (\<lambda>x. f x * r) A"
unfolding setL2_def
apply (simp add: power_mult_distrib)
- apply (simp add: setsum_left_distrib [symmetric])
+ apply (simp add: setsum_distrib_right [symmetric])
apply (simp add: real_sqrt_mult setsum_nonneg)
done
--- a/src/HOL/Analysis/Lebesgue_Measure.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Lebesgue_Measure.thy Tue Sep 20 11:35:10 2016 +0200
@@ -235,7 +235,7 @@
by auto
also have "... = (\<Sum>i \<in> S. F(r i) - F(l i)) +
(epsilon / 4) * (\<Sum>i \<in> S. (1 / 2)^i)" (is "_ = ?t + _")
- by (subst setsum.distrib) (simp add: field_simps setsum_right_distrib)
+ by (subst setsum.distrib) (simp add: field_simps setsum_distrib_left)
also have "... \<le> ?t + (epsilon / 4) * (\<Sum> i < Suc n. (1 / 2)^i)"
apply (rule add_left_mono)
apply (rule mult_left_mono)
--- a/src/HOL/Analysis/Linear_Algebra.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Linear_Algebra.thy Tue Sep 20 11:35:10 2016 +0200
@@ -1960,7 +1960,7 @@
qed
from setsum_norm_le[of _ ?g, OF th]
show "norm (f x) \<le> ?B * norm x"
- unfolding th0 setsum_left_distrib by metis
+ unfolding th0 setsum_distrib_right by metis
qed
qed
@@ -2021,7 +2021,7 @@
unfolding bilinear_setsum[OF bh finite_Basis finite_Basis] ..
finally have th: "norm (h x y) = \<dots>" .
show "norm (h x y) \<le> (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. norm (h i j)) * norm x * norm y"
- apply (auto simp add: setsum_left_distrib th setsum.cartesian_product)
+ apply (auto simp add: setsum_distrib_right th setsum.cartesian_product)
apply (rule setsum_norm_le)
apply simp
apply (auto simp add: bilinear_rmul[OF bh] bilinear_lmul[OF bh]
--- a/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Nonnegative_Lebesgue_Integration.thy Tue Sep 20 11:35:10 2016 +0200
@@ -550,7 +550,7 @@
qed
also have "\<dots> = (\<Sum>y\<in>f`space M. (\<Sum>z\<in>g`space M.
if \<exists>x\<in>space M. y = f x \<and> z = g x then y * emeasure M {x\<in>space M. g x = z} else 0))"
- by (auto intro!: setsum.cong simp: setsum_right_distrib)
+ by (auto intro!: setsum.cong simp: setsum_distrib_left)
also have "\<dots> = ?r"
by (subst setsum.commute)
(auto intro!: setsum.cong simp: setsum.If_cases scaleR_setsum_right[symmetric] eq)
@@ -592,7 +592,7 @@
using f by (intro simple_function_partition) auto
also have "\<dots> = c * integral\<^sup>S M f"
using f unfolding simple_integral_def
- by (subst setsum_right_distrib) (auto simp: mult.assoc Int_def conj_commute)
+ by (subst setsum_distrib_left) (auto simp: mult.assoc Int_def conj_commute)
finally show ?thesis .
qed
--- a/src/HOL/Analysis/Poly_Roots.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Poly_Roots.thy Tue Sep 20 11:35:10 2016 +0200
@@ -24,7 +24,7 @@
lemma setsum_power_add:
fixes x :: "'a::{comm_ring,monoid_mult}"
shows "(\<Sum>i\<in>I. x^(m+i)) = x^m * (\<Sum>i\<in>I. x^i)"
- by (simp add: setsum_right_distrib power_add)
+ by (simp add: setsum_distrib_left power_add)
lemma setsum_power_shift:
fixes x :: "'a::{comm_ring,monoid_mult}"
@@ -32,7 +32,7 @@
shows "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i\<le>n-m. x^i)"
proof -
have "(\<Sum>i=m..n. x^i) = x^m * (\<Sum>i=m..n. x^(i-m))"
- by (simp add: setsum_right_distrib power_add [symmetric])
+ by (simp add: setsum_distrib_left power_add [symmetric])
also have "(\<Sum>i=m..n. x^(i-m)) = (\<Sum>i\<le>n-m. x^i)"
using \<open>m \<le> n\<close> by (intro setsum.reindex_bij_witness[where j="\<lambda>i. i - m" and i="\<lambda>i. i + m"]) auto
finally show ?thesis .
@@ -88,7 +88,7 @@
also have "... = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
by (simp add: power_diff_sumr2 ac_simps)
also have "... = (x - y) * (\<Sum>i\<le>n. (\<Sum>j<i. a i * y^(i - Suc j) * x^j))"
- by (simp add: setsum_right_distrib ac_simps)
+ by (simp add: setsum_distrib_left ac_simps)
also have "... = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - Suc j) * x^j))"
by (simp add: nested_setsum_swap')
finally show ?thesis .
@@ -115,7 +115,7 @@
{ fix z
have "(\<Sum>i\<le>n. c i * z^i) - (\<Sum>i\<le>n. c i * a^i) =
(z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)"
- by (simp add: sub_polyfun setsum_left_distrib)
+ by (simp add: sub_polyfun setsum_distrib_right)
then have "(\<Sum>i\<le>n. c i * z^i) =
(z - a) * (\<Sum>j<n. (\<Sum>k = Suc j..n. c k * a^(k - Suc j)) * z^j)
+ (\<Sum>i\<le>n. c i * a^i)"
--- a/src/HOL/Analysis/Polytope.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Polytope.thy Tue Sep 20 11:35:10 2016 +0200
@@ -303,7 +303,7 @@
have cge0: "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> c i"
using a b u01 by (simp add: c_def)
have sumc1: "setsum c S = 1"
- by (simp add: c_def setsum.distrib setsum_right_distrib [symmetric] asum bsum)
+ by (simp add: c_def setsum.distrib setsum_distrib_left [symmetric] asum bsum)
have sumci_xy: "(\<Sum>i\<in>S. c i *\<^sub>R i) = (1 - u) *\<^sub>R x + u *\<^sub>R y"
apply (simp add: c_def setsum.distrib scaleR_left_distrib)
by (simp only: scaleR_scaleR [symmetric] Real_Vector_Spaces.scaleR_right.setsum [symmetric] aeqx beqy)
@@ -357,7 +357,7 @@
apply (rule_tac x="\<lambda>i. inverse (1-k) * c i" in exI)
apply auto
apply (metis sumcf cge0 inverse_nonnegative_iff_nonnegative mult_nonneg_nonneg S(2) setsum_nonneg subsetCE)
- apply (metis False mult.commute right_inverse right_minus_eq setsum_right_distrib sumcf)
+ apply (metis False mult.commute right_inverse right_minus_eq setsum_distrib_left sumcf)
by (metis (mono_tags, lifting) scaleR_right.setsum scaleR_scaleR setsum.cong)
with \<open>0 < k\<close> have "inverse(k) *\<^sub>R (w - setsum (\<lambda>i. c i *\<^sub>R i) T) \<in> affine hull T"
by (simp add: affine_diff_divide [OF affine_affine_hull] False waff convex_hull_subset_affine_hull [THEN subsetD])
@@ -365,7 +365,7 @@
apply (simp add: weq_sumsum convex_hull_finite fin)
apply (rule_tac x="\<lambda>i. inverse k * c i" in exI)
using \<open>k > 0\<close> cge0
- apply (auto simp: scaleR_right.setsum setsum_right_distrib [symmetric] k_def [symmetric])
+ apply (auto simp: scaleR_right.setsum setsum_distrib_left [symmetric] k_def [symmetric])
done
ultimately show ?thesis
using disj by blast
--- a/src/HOL/Analysis/Summation_Tests.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Summation_Tests.thy Tue Sep 20 11:35:10 2016 +0200
@@ -288,7 +288,7 @@
qed
from this and A have "Bseq (\<lambda>n. \<Sum>k<n. 2^k * f (2^Suc k))" by (rule Bseq_eventually_mono)
from Bseq_mult[OF Bfun_const[of 2] this] have "Bseq (\<lambda>n. \<Sum>k<n. 2^Suc k * f (2^Suc k))"
- by (simp add: setsum_right_distrib setsum_left_distrib mult_ac)
+ by (simp add: setsum_distrib_left setsum_distrib_right mult_ac)
hence "Bseq (\<lambda>n. (\<Sum>k=Suc 0..<Suc n. 2^k * f (2^k)) + f 1)"
by (intro Bseq_add, subst setsum_shift_bounds_Suc_ivl) (simp add: atLeast0LessThan)
hence "Bseq (\<lambda>n. (\<Sum>k=0..<Suc n. 2^k * f (2^k)))"
@@ -424,7 +424,7 @@
have n: "n > m" by (simp add: n_def)
from r have "r * norm (\<Sum>k\<le>n. f k) = norm (\<Sum>k\<le>n. r * f k)"
- by (simp add: setsum_right_distrib[symmetric] abs_mult)
+ by (simp add: setsum_distrib_left[symmetric] abs_mult)
also from n have "{..n} = {..m} \<union> {Suc m..n}" by auto
hence "(\<Sum>k\<le>n. r * f k) = (\<Sum>k\<in>{..m} \<union> {Suc m..n}. r * f k)" by (simp only:)
also have "\<dots> = (\<Sum>k\<le>m. r * f k) + (\<Sum>k=Suc m..n. r * f k)"
--- a/src/HOL/Analysis/Weierstrass_Theorems.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/Weierstrass_Theorems.thy Tue Sep 20 11:35:10 2016 +0200
@@ -38,7 +38,7 @@
lemma sum_k_Bernstein [simp]: "(\<Sum>k = 0..n. real k * Bernstein n k x) = of_nat n * x"
apply (subst binomial_deriv1 [of n x "1-x", simplified, symmetric])
- apply (simp add: setsum_left_distrib)
+ apply (simp add: setsum_distrib_right)
apply (auto simp: Bernstein_def algebra_simps realpow_num_eq_if intro!: setsum.cong)
done
@@ -46,7 +46,7 @@
proof -
have "(\<Sum> k = 0..n. real k * (real k - 1) * Bernstein n k x) = real_of_nat n * real_of_nat (n - Suc 0) * x\<^sup>2"
apply (subst binomial_deriv2 [of n x "1-x", simplified, symmetric])
- apply (simp add: setsum_left_distrib)
+ apply (simp add: setsum_distrib_right)
apply (rule setsum.cong [OF refl])
apply (simp add: Bernstein_def power2_eq_square algebra_simps)
apply (rename_tac k)
@@ -98,7 +98,7 @@
by (simp add: algebra_simps power2_eq_square)
have "(\<Sum> k = 0..n. (k - n * x)\<^sup>2 * Bernstein n k x) = n * x * (1 - x)"
apply (simp add: * setsum.distrib)
- apply (simp add: setsum_right_distrib [symmetric] mult.assoc)
+ apply (simp add: setsum_distrib_left [symmetric] mult.assoc)
apply (simp add: algebra_simps power2_eq_square)
done
then have "(\<Sum> k = 0..n. (k - n * x)\<^sup>2 * Bernstein n k x)/n^2 = x * (1 - x) / n"
@@ -138,14 +138,14 @@
qed
} note * = this
have "\<bar>f x - (\<Sum> k = 0..n. f(k / n) * Bernstein n k x)\<bar> \<le> \<bar>\<Sum> k = 0..n. (f x - f(k / n)) * Bernstein n k x\<bar>"
- by (simp add: setsum_subtractf setsum_right_distrib [symmetric] algebra_simps)
+ by (simp add: setsum_subtractf setsum_distrib_left [symmetric] algebra_simps)
also have "... \<le> (\<Sum> k = 0..n. (e/2 + (2 * M / d\<^sup>2) * (x - k / n)\<^sup>2) * Bernstein n k x)"
apply (rule order_trans [OF setsum_abs setsum_mono])
using *
apply (simp add: abs_mult Bernstein_nonneg x mult_right_mono)
done
also have "... \<le> e/2 + (2 * M) / (d\<^sup>2 * n)"
- apply (simp only: setsum.distrib Rings.semiring_class.distrib_right setsum_right_distrib [symmetric] mult.assoc sum_bern)
+ apply (simp only: setsum.distrib Rings.semiring_class.distrib_right setsum_distrib_left [symmetric] mult.assoc sum_bern)
using \<open>d>0\<close> x
apply (simp add: divide_simps Mge0 mult_le_one mult_left_le)
done
--- a/src/HOL/Analysis/ex/Approximations.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Analysis/ex/Approximations.thy Tue Sep 20 11:35:10 2016 +0200
@@ -30,7 +30,7 @@
by (subst atLeast0LessThan [symmetric], subst setsum_head_upt_Suc) simp_all
also have "(\<Sum>k=Suc 0..<Suc m. f k * x^k) = (\<Sum>k<m. f (k+1) * x^k) * x"
by (subst setsum_shift_bounds_Suc_ivl)
- (simp add: setsum_left_distrib algebra_simps atLeast0LessThan power_commutes)
+ (simp add: setsum_distrib_right algebra_simps atLeast0LessThan power_commutes)
finally have "(\<Sum>k<Suc m. f k * x ^ k) = f 0 + (\<Sum>k<m. f (k + 1) * x ^ k) * x" .
}
from this[of "pred_numeral n"]
@@ -199,7 +199,7 @@
lemma euler_approx_aux_Suc:
"euler_approx_aux (Suc m) = 1 + Suc m * euler_approx_aux m"
unfolding euler_approx_aux_def
- by (subst setsum_right_distrib) (simp add: atLeastAtMostSuc_conv)
+ by (subst setsum_distrib_left) (simp add: atLeastAtMostSuc_conv)
lemma eval_euler_approx_aux:
"euler_approx_aux 0 = 1"
@@ -209,7 +209,7 @@
proof -
have A: "euler_approx_aux (Suc m) = 1 + Suc m * euler_approx_aux m" for m :: nat
unfolding euler_approx_aux_def
- by (subst setsum_right_distrib) (simp add: atLeastAtMostSuc_conv)
+ by (subst setsum_distrib_left) (simp add: atLeastAtMostSuc_conv)
show ?th by (subst numeral_eq_Suc, subst A, subst numeral_eq_Suc [symmetric]) simp
qed (simp_all add: euler_approx_aux_def)
@@ -281,7 +281,7 @@
y_def [symmetric] d_def [symmetric])
also have "2 * y * (\<Sum>k<n. inverse (real (2 * k + 1)) * y\<^sup>2 ^ k) =
(\<Sum>k<n. 2 * y^(2*k+1) / (real (2 * k + 1)))"
- by (subst setsum_right_distrib, simp, subst power_mult)
+ by (subst setsum_distrib_left, simp, subst power_mult)
(simp_all add: divide_simps mult_ac power_mult)
finally show ?case by (simp only: d_def y_def approx_def)
qed
@@ -380,7 +380,7 @@
from sums_split_initial_segment[OF this, of n]
have "(\<lambda>i. x * ((- x\<^sup>2) ^ (i + n) / real (2 * (i + n) + 1))) sums
(arctan x - arctan_approx n x)"
- by (simp add: arctan_approx_def setsum_right_distrib)
+ by (simp add: arctan_approx_def setsum_distrib_left)
from sums_group[OF this, of 2] assms
have sums: "(\<lambda>k. x * (x\<^sup>2)^n * (x^4)^k * c k) sums (arctan x - arctan_approx n x)"
by (simp add: algebra_simps power_add power_mult [symmetric] c_def)
@@ -423,7 +423,7 @@
by (subst atLeast0LessThan [symmetric], subst setsum_head_upt_Suc) simp_all
also have "(\<Sum>k=Suc 0..<Suc m. inverse (f k * x^k)) = (\<Sum>k<m. inverse (f (k+1) * x^k)) / x"
by (subst setsum_shift_bounds_Suc_ivl)
- (simp add: setsum_right_distrib divide_inverse algebra_simps
+ (simp add: setsum_distrib_left divide_inverse algebra_simps
atLeast0LessThan power_commutes)
finally have "(\<Sum>k<Suc m. inverse (f k) * inverse (x ^ k)) =
inverse (f 0) + (\<Sum>k<m. inverse (f (k + 1)) * inverse (x ^ k)) / x" by simp
--- a/src/HOL/Binomial.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Binomial.thy Tue Sep 20 11:35:10 2016 +0200
@@ -341,7 +341,7 @@
by (rule distrib_right)
also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^(k+1) * b^(n-k)) +
(\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n - k + 1))"
- by (auto simp add: setsum_right_distrib ac_simps)
+ by (auto simp add: setsum_distrib_left ac_simps)
also have "\<dots> = (\<Sum>k=0..n. of_nat (n choose k) * a^k * b^(n + 1 - k)) +
(\<Sum>k=1..n+1. of_nat (n choose (k - 1)) * a^k * b^(n + 1 - k))"
by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le field_simps del: setsum_cl_ivl_Suc)
@@ -463,7 +463,7 @@
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) + (-1) ^ i * of_nat (n choose i))"
by (simp add: setsum.distrib)
also have "\<dots> = 2 * (\<Sum>i\<le>n. if even i then of_nat (n choose i) else 0)"
- by (subst setsum_right_distrib, intro setsum.cong) simp_all
+ by (subst setsum_distrib_left, intro setsum.cong) simp_all
finally show ?thesis ..
qed
@@ -477,7 +477,7 @@
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) - (-1) ^ i * of_nat (n choose i))"
by (simp add: setsum_subtractf)
also have "\<dots> = 2 * (\<Sum>i\<le>n. if odd i then of_nat (n choose i) else 0)"
- by (subst setsum_right_distrib, intro setsum.cong) simp_all
+ by (subst setsum_distrib_left, intro setsum.cong) simp_all
finally show ?thesis ..
qed
@@ -914,7 +914,7 @@
have "(\<Sum>i\<le>n. i * (n choose i)) = (\<Sum>i\<le>Suc m. i * (Suc m choose i))"
by (simp add: Suc)
also have "\<dots> = Suc m * 2 ^ m"
- by (simp only: setsum_atMost_Suc_shift Suc_times_binomial setsum_right_distrib[symmetric])
+ by (simp only: setsum_atMost_Suc_shift Suc_times_binomial setsum_distrib_left[symmetric])
(simp add: choose_row_sum')
finally show ?thesis
using Suc by simp
@@ -934,9 +934,9 @@
(\<Sum>i\<le>Suc m. (-1) ^ i * of_nat i * of_nat (Suc m choose i))"
by (simp add: Suc)
also have "\<dots> = (\<Sum>i\<le>m. (-1) ^ (Suc i) * of_nat (Suc i * (Suc m choose Suc i)))"
- by (simp only: setsum_atMost_Suc_shift setsum_right_distrib[symmetric] mult_ac of_nat_mult) simp
+ by (simp only: setsum_atMost_Suc_shift setsum_distrib_left[symmetric] mult_ac of_nat_mult) simp
also have "\<dots> = - of_nat (Suc m) * (\<Sum>i\<le>m. (-1) ^ i * of_nat (m choose i))"
- by (subst setsum_right_distrib, rule setsum.cong[OF refl], subst Suc_times_binomial)
+ by (subst setsum_distrib_left, rule setsum.cong[OF refl], subst Suc_times_binomial)
(simp add: algebra_simps)
also have "(\<Sum>i\<le>m. (-1 :: 'a) ^ i * of_nat ((m choose i))) = 0"
using choose_alternating_sum[OF \<open>m > 0\<close>] by simp
@@ -978,7 +978,7 @@
by (subst setsum_atMost_Suc_shift) (simp add: ring_distribs setsum.distrib)
also have "(\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a (Suc i) * pochhammer b (n - i)) =
a * pochhammer ((a + 1) + b) n"
- by (subst Suc) (simp add: setsum_right_distrib pochhammer_rec mult_ac)
+ by (subst Suc) (simp add: setsum_distrib_left pochhammer_rec mult_ac)
also have "(\<Sum>i\<le>n. of_nat (n choose Suc i) * pochhammer a (Suc i) * pochhammer b (n - i)) +
pochhammer b (Suc n) =
(\<Sum>i=0..Suc n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc n - i))"
@@ -992,7 +992,7 @@
also have "\<dots> = (\<Sum>i\<le>n. of_nat (n choose i) * pochhammer a i * pochhammer b (Suc (n - i)))"
by (intro setsum.cong) (simp_all add: Suc_diff_le)
also have "\<dots> = b * pochhammer (a + (b + 1)) n"
- by (subst Suc) (simp add: setsum_right_distrib mult_ac pochhammer_rec)
+ by (subst Suc) (simp add: setsum_distrib_left mult_ac pochhammer_rec)
also have "a * pochhammer ((a + 1) + b) n + b * pochhammer (a + (b + 1)) n =
pochhammer (a + b) (Suc n)"
by (simp add: pochhammer_rec algebra_simps)
@@ -1263,9 +1263,9 @@
by (simp only:)
qed
also have "\<dots> + ?B = y * (\<Sum>k=1..mm. (G mm k)) + (of_nat mm + r gchoose (Suc mm)) * x^(Suc mm)"
- unfolding G_def by (subst setsum_right_distrib) (simp add: algebra_simps)
+ unfolding G_def by (subst setsum_distrib_left) (simp add: algebra_simps)
also have "(\<Sum>k=0..mm. (of_nat mm + r gchoose k) * x^(Suc k) * y^(mm - k)) = x * (S mm)"
- unfolding S_def by (subst setsum_right_distrib) (simp add: atLeast0AtMost algebra_simps)
+ unfolding S_def by (subst setsum_distrib_left) (simp add: atLeast0AtMost algebra_simps)
also have "(G (Suc mm) 0) = y * (G mm 0)"
by (simp add: G_def)
finally have "S (Suc mm) =
@@ -1283,7 +1283,7 @@
also have "(x + y) * S mm + \<dots> = (x + y) * ?rhs mm + (- r gchoose (Suc mm)) * (- x)^Suc mm"
unfolding S_def by (subst Suc.IH) simp
also have "(x + y) * ?rhs mm = (\<Sum>n\<le>mm. ((- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n)))"
- by (subst setsum_right_distrib, rule setsum.cong) (simp_all add: Suc_diff_le)
+ by (subst setsum_distrib_left, rule setsum.cong) (simp_all add: Suc_diff_le)
also have "\<dots> + (-r gchoose (Suc mm)) * (-x)^Suc mm =
(\<Sum>n\<le>Suc mm. (- r gchoose n) * (- x) ^ n * (x + y) ^ (Suc mm - n))"
by simp
@@ -1345,7 +1345,7 @@
using gbinomial_partial_sum_poly_xpos[where x="1" and y="1" and r="of_nat m + 1" and m="m"]
by (simp add: add_ac)
also have "\<dots> = 2 ^ m * (\<Sum>k\<le>m. (of_nat (m + k) gchoose k) / 2 ^ k)"
- by (subst setsum_right_distrib) (simp add: algebra_simps power_diff)
+ by (subst setsum_distrib_left) (simp add: algebra_simps power_diff)
finally show ?thesis
by (subst (asm) mult_left_cancel) simp_all
qed
@@ -1444,7 +1444,7 @@
by simp
also have "\<dots> = nat (\<Sum>I | I \<subseteq> A \<and> I \<noteq> {}. (\<Sum>_\<in>\<Inter>I. (- 1) ^ (card I + 1)))"
(is "_ = nat ?rhs")
- by (subst setsum_right_distrib) simp
+ by (subst setsum_distrib_left) simp
also have "?rhs = (\<Sum>(I, _)\<in>Sigma {I. I \<subseteq> A \<and> I \<noteq> {}} Inter. (- 1) ^ (card I + 1))"
using assms by (subst setsum.Sigma) auto
also have "\<dots> = (\<Sum>(x, I)\<in>(SIGMA x:UNIV. {I. I \<subseteq> A \<and> I \<noteq> {} \<and> x \<in> \<Inter>I}). (- 1) ^ (card I + 1))"
@@ -1474,7 +1474,7 @@
also have "\<dots> = (\<Sum>i=1..card A. (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. (- 1) ^ (i + 1)))"
using assms by (subst setsum.Sigma) auto
also have "\<dots> = (\<Sum>i=1..card A. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> A \<and> card I = i \<and> x \<in> \<Inter>I. 1))"
- by (subst setsum_right_distrib) simp
+ by (subst setsum_distrib_left) simp
also have "\<dots> = (\<Sum>i=1..card K. (- 1) ^ (i + 1) * (\<Sum>I|I \<subseteq> K \<and> card I = i. 1))"
(is "_ = ?rhs")
proof (rule setsum.mono_neutral_cong_right[rule_format])
@@ -1508,7 +1508,7 @@
also have "\<dots> = (\<Sum>i = 0..card K. (- 1) * ((- 1) ^ i * int (card K choose i))) + 1"
using K by (subst n_subsets[symmetric]) simp_all
also have "\<dots> = - (\<Sum>i = 0..card K. (- 1) ^ i * int (card K choose i)) + 1"
- by (subst setsum_right_distrib[symmetric]) simp
+ by (subst setsum_distrib_left[symmetric]) simp
also have "\<dots> = - ((-1 + 1) ^ card K) + 1"
by (subst binomial_ring) (simp add: ac_simps)
also have "\<dots> = 1"
--- a/src/HOL/Decision_Procs/Approximation.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Decision_Procs/Approximation.thy Tue Sep 20 11:35:10 2016 +0200
@@ -31,7 +31,7 @@
have shift_pow: "\<And>i. - (x * ((-1)^i * a (Suc i) * x ^ i)) = (-1)^(Suc i) * a (Suc i) * x ^ (Suc i)"
by auto
show ?thesis
- unfolding setsum_right_distrib shift_pow uminus_add_conv_diff [symmetric] setsum_negf[symmetric]
+ unfolding setsum_distrib_left shift_pow uminus_add_conv_diff [symmetric] setsum_negf[symmetric]
setsum_head_upt_Suc[OF zero_less_Suc]
setsum.reindex[OF inj_Suc, unfolded comp_def, symmetric, of "\<lambda> n. (-1)^n *a n * x^n"] by auto
qed
@@ -514,7 +514,7 @@
proof -
have "(sqrt y * lb_arctan_horner prec n 1 y) \<le> ?S n"
using bounds(1) \<open>0 \<le> sqrt y\<close>
- apply (simp only: power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric])
+ apply (simp only: power_add power_one_right mult.assoc[symmetric] setsum_distrib_right[symmetric])
apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
apply (auto intro!: mult_left_mono)
done
@@ -527,7 +527,7 @@
have "arctan (sqrt y) \<le> ?S (Suc n)" using arctan_bounds ..
also have "\<dots> \<le> (sqrt y * ub_arctan_horner prec (Suc n) 1 y)"
using bounds(2)[of "Suc n"] \<open>0 \<le> sqrt y\<close>
- apply (simp only: power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric])
+ apply (simp only: power_add power_one_right mult.assoc[symmetric] setsum_distrib_right[symmetric])
apply (simp only: mult.commute[where 'a=real] mult.commute[of _ "2::nat"] power_mult)
apply (auto intro!: mult_left_mono)
done
@@ -1212,7 +1212,7 @@
from horner_bounds[where lb="lb_sin_cos_aux prec" and ub="ub_sin_cos_aux prec" and j'=0,
OF \<open>0 \<le> real_of_float (x * x)\<close> f_eq lb_sin_cos_aux.simps ub_sin_cos_aux.simps]
show "?lb" and "?ub" using \<open>0 \<le> real_of_float x\<close>
- apply (simp_all only: power_add power_one_right mult.assoc[symmetric] setsum_left_distrib[symmetric])
+ apply (simp_all only: power_add power_one_right mult.assoc[symmetric] setsum_distrib_right[symmetric])
apply (simp_all only: mult.commute[where 'a=real] of_nat_fact)
apply (auto intro!: mult_left_mono simp add: power_mult power2_eq_square[of "real_of_float x"])
done
@@ -2193,7 +2193,7 @@
let "?s n" = "(- 1) ^ n * (1 / real (1 + n)) * (real_of_float x)^(Suc n)"
have "?lb \<le> setsum ?s {0 ..< 2 * ev}"
- unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric]
+ unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_distrib_right[symmetric]
unfolding mult.commute[of "real_of_float x"] ev
using horner_bounds(1)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x"
and lb="\<lambda>n i k x. lb_ln_horner prec n k x"
@@ -2208,7 +2208,7 @@
have "?ln \<le> setsum ?s {0 ..< 2 * od + 1}"
using ln_bounds(2)[OF \<open>0 \<le> real_of_float x\<close> \<open>real_of_float x < 1\<close>] by auto
also have "\<dots> \<le> ?ub"
- unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_left_distrib[symmetric]
+ unfolding power_Suc2 mult.assoc[symmetric] times_float.rep_eq setsum_distrib_right[symmetric]
unfolding mult.commute[of "real_of_float x"] od
using horner_bounds(2)[where G="\<lambda> i k. Suc k" and F="\<lambda>x. x" and f="\<lambda>x. x" and lb="\<lambda>n i k x. lb_ln_horner prec n k x" and ub="\<lambda>n i k x. ub_ln_horner prec n k x" and j'=1 and n="2*od+1",
OF \<open>0 \<le> real_of_float x\<close> refl lb_ln_horner.simps ub_ln_horner.simps] \<open>0 \<le> real_of_float x\<close>
@@ -4019,7 +4019,7 @@
inverse (real (\<Prod> j \<in> {k..<k+Suc n}. j)) * ?f (Suc n) t * (xs!x - c)^Suc n" (is "_ = ?T")
unfolding funpow_Suc C_def[symmetric] setsum_move0 setprod_head_Suc
by (auto simp add: algebra_simps)
- (simp only: mult.left_commute [of _ "inverse (real k)"] setsum_right_distrib [symmetric])
+ (simp only: mult.left_commute [of _ "inverse (real k)"] setsum_distrib_left [symmetric])
finally have "?T \<in> {l .. u}" .
}
thus ?thesis using DERIV by blast
--- a/src/HOL/Deriv.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Deriv.thy Tue Sep 20 11:35:10 2016 +0200
@@ -368,7 +368,7 @@
using insert by (intro has_derivative_mult) auto
also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
using insert(1,2)
- by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum.cong)
+ by (auto simp add: setsum_distrib_left insert_Diff_if intro!: ext setsum.cong)
finally show ?case
using insert by simp
qed
@@ -845,7 +845,7 @@
"(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow>
((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
by (rule has_derivative_imp_has_field_derivative [OF has_derivative_setsum])
- (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
+ (auto simp: setsum_distrib_left mult_commute_abs dest: has_field_derivative_imp_has_derivative)
lemma DERIV_inverse'[derivative_intros]:
assumes "(f has_field_derivative D) (at x within s)"
--- a/src/HOL/Groups_Big.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Groups_Big.thy Tue Sep 20 11:35:10 2016 +0200
@@ -729,7 +729,7 @@
"finite F \<Longrightarrow> (setsum f F = 0) = (\<forall>a\<in>F. f a = 0)"
by (intro ballI setsum_nonneg_eq_0_iff zero_le)
-lemma setsum_right_distrib: "r * setsum f A = setsum (\<lambda>n. r * f n) A"
+lemma setsum_distrib_left: "r * setsum f A = setsum (\<lambda>n. r * f n) A"
for f :: "'a \<Rightarrow> 'b::semiring_0"
proof (induct A rule: infinite_finite_induct)
case infinite
@@ -742,7 +742,7 @@
then show ?case by (simp add: distrib_left)
qed
-lemma setsum_left_distrib: "setsum f A * r = (\<Sum>n\<in>A. f n * r)"
+lemma setsum_distrib_right: "setsum f A * r = (\<Sum>n\<in>A. f n * r)"
for r :: "'a::semiring_0"
proof (induct A rule: infinite_finite_induct)
case infinite
@@ -811,7 +811,7 @@
lemma setsum_product:
fixes f :: "'a \<Rightarrow> 'b::semiring_0"
shows "setsum f A * setsum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
- by (simp add: setsum_right_distrib setsum_left_distrib) (rule setsum.commute)
+ by (simp add: setsum_distrib_left setsum_distrib_right) (rule setsum.commute)
lemma setsum_mult_setsum_if_inj:
fixes f :: "'a \<Rightarrow> 'b::semiring_0"
--- a/src/HOL/Inequalities.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Inequalities.thy Tue Sep 20 11:35:10 2016 +0200
@@ -59,7 +59,7 @@
let ?S = "(\<Sum>j=0..<n. (\<Sum>k=0..<n. (a j - a k) * (b j - b k)))"
have "2 * (of_nat n * (\<Sum>j=0..<n. (a j * b j)) - (\<Sum>j=0..<n. b j) * (\<Sum>k=0..<n. a k)) = ?S"
by (simp only: one_add_one[symmetric] algebra_simps)
- (simp add: algebra_simps setsum_subtractf setsum.distrib setsum.commute[of "\<lambda>i j. a i * b j"] setsum_right_distrib)
+ (simp add: algebra_simps setsum_subtractf setsum.distrib setsum.commute[of "\<lambda>i j. a i * b j"] setsum_distrib_left)
also
{ fix i j::nat assume "i<n" "j<n"
hence "a i - a j \<le> 0 \<and> b i - b j \<ge> 0 \<or> a i - a j \<ge> 0 \<and> b i - b j \<le> 0"
--- a/src/HOL/Library/BigO.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/BigO.thy Tue Sep 20 11:35:10 2016 +0200
@@ -556,7 +556,7 @@
apply (subst abs_of_nonneg) back back
apply (rule setsum_nonneg)
apply force
- apply (subst setsum_right_distrib)
+ apply (subst setsum_distrib_left)
apply (rule allI)
apply (rule order_trans)
apply (rule setsum_abs)
--- a/src/HOL/Library/Convex.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Convex.thy Tue Sep 20 11:35:10 2016 +0200
@@ -633,7 +633,7 @@
OF asum_le less_imp_le[OF i0]], of "a i * f (y i)"]
by simp
also have "\<dots> = (\<Sum> j \<in> s. (1 - a i) * ?a j * f (y j)) + a i * f (y i)"
- unfolding setsum_right_distrib[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
+ unfolding setsum_distrib_left[of "1 - a i" "\<lambda> j. ?a j * f (y j)"]
using i0 by auto
also have "\<dots> = (\<Sum> j \<in> s. a j * f (y j)) + a i * f (y i)"
using i0 by auto
--- a/src/HOL/Library/Extended_Real.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Extended_Real.thy Tue Sep 20 11:35:10 2016 +0200
@@ -1098,7 +1098,7 @@
"(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> setsum f A * r = (\<Sum>n\<in>A. f n * r :: ereal)"
using setsum_ereal_right_distrib[of A f r] by (simp add: mult_ac)
-lemma setsum_left_distrib_ereal:
+lemma setsum_distrib_right_ereal:
"c \<ge> 0 \<Longrightarrow> setsum f A * ereal c = (\<Sum>x\<in>A. f x * c :: ereal)"
by(subst setsum_comp_morphism[where h="\<lambda>x. x * ereal c", symmetric])(simp_all add: distrib_left_ereal_nn)
--- a/src/HOL/Library/Formal_Power_Series.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy Tue Sep 20 11:35:10 2016 +0200
@@ -141,7 +141,7 @@
(\<Sum>j=0..n. \<Sum>i=0..n - j. a$j * b$i * c$(n - j - i))"
by (rule fps_mult_assoc_lemma)
then show "((a * b) * c) $ n = (a * (b * c)) $ n"
- by (simp add: fps_mult_nth setsum_right_distrib setsum_left_distrib mult.assoc)
+ by (simp add: fps_mult_nth setsum_distrib_left setsum_distrib_right mult.assoc)
qed
qed
@@ -1562,7 +1562,7 @@
also have "\<dots> = (fps_deriv (f * g)) $ n"
apply (simp only: fps_deriv_nth fps_mult_nth setsum.distrib)
unfolding s0 s1
- unfolding setsum.distrib[symmetric] setsum_right_distrib
+ unfolding setsum.distrib[symmetric] setsum_distrib_left
apply (rule setsum.cong)
apply (auto simp add: of_nat_diff field_simps)
done
@@ -2320,7 +2320,7 @@
apply auto
apply (rule setsum.cong)
apply (rule refl)
- unfolding setsum_left_distrib
+ unfolding setsum_distrib_right
apply (rule sym)
apply (rule_tac l = "\<lambda>xs. xs @ [n - x]" in setsum.reindex_cong)
apply (simp add: inj_on_def)
@@ -3082,7 +3082,7 @@
have "(fps_deriv (a oo b))$n = (((fps_deriv a) oo b) * (fps_deriv b)) $n" for n
proof -
have "(fps_deriv (a oo b))$n = setsum (\<lambda>i. a $ i * (fps_deriv (b^i))$n) {0.. Suc n}"
- by (simp add: fps_compose_def field_simps setsum_right_distrib del: of_nat_Suc)
+ by (simp add: fps_compose_def field_simps setsum_distrib_left del: of_nat_Suc)
also have "\<dots> = setsum (\<lambda>i. a$i * ((fps_const (of_nat i)) * (fps_deriv b * (b^(i - 1))))$n) {0.. Suc n}"
by (simp add: field_simps fps_deriv_power del: fps_mult_left_const_nth of_nat_Suc)
also have "\<dots> = setsum (\<lambda>i. of_nat i * a$i * (((b^(i - 1)) * fps_deriv b))$n) {0.. Suc n}"
@@ -3102,7 +3102,7 @@
have "(((fps_deriv a) oo b) * (fps_deriv b))$n = setsum (\<lambda>i. (fps_deriv b)$ (n - i) * ((fps_deriv a) oo b)$i) {0..n}"
unfolding fps_mult_nth by (simp add: ac_simps)
also have "\<dots> = setsum (\<lambda>i. setsum (\<lambda>j. of_nat (n - i +1) * b$(n - i + 1) * of_nat (j + 1) * a$(j+1) * (b^j)$i) {0..n}) {0..n}"
- unfolding fps_deriv_nth fps_compose_nth setsum_right_distrib mult.assoc
+ unfolding fps_deriv_nth fps_compose_nth setsum_distrib_left mult.assoc
apply (rule setsum.cong)
apply (rule refl)
apply (rule setsum.mono_neutral_left)
@@ -3114,7 +3114,7 @@
apply simp
done
also have "\<dots> = setsum (\<lambda>i. of_nat (i + 1) * a$(i+1) * (setsum (\<lambda>j. (b^ i)$j * of_nat (n - j + 1) * b$(n - j + 1)) {0..n})) {0.. n}"
- unfolding setsum_right_distrib
+ unfolding setsum_distrib_left
apply (subst setsum.commute)
apply (rule setsum.cong, rule refl)+
apply simp
@@ -3295,7 +3295,7 @@
apply auto
done
have "?r = setsum (\<lambda>i. setsum (\<lambda>(k,m). a$k * (c^k)$i * b$m * (d^m) $ (n - i)) {(k,m). k + m \<le> n}) {0..n}"
- apply (simp add: fps_mult_nth setsum_right_distrib)
+ apply (simp add: fps_mult_nth setsum_distrib_left)
apply (subst setsum.commute)
apply (rule setsum.cong)
apply (auto simp add: field_simps)
@@ -3377,7 +3377,7 @@
assumes c0: "c $ 0 = (0::'a::idom)"
shows "(a * b) oo c = (a oo c) * (b oo c)"
apply (simp add: fps_eq_iff fps_compose_mult_distrib_lemma [OF c0])
- apply (simp add: fps_compose_nth fps_mult_nth setsum_left_distrib)
+ apply (simp add: fps_compose_nth fps_mult_nth setsum_distrib_right)
done
lemma fps_compose_setprod_distrib:
@@ -3498,7 +3498,7 @@
qed
lemma fps_const_mult_apply_left: "fps_const c * (a oo b) = (fps_const c * a) oo b"
- by (simp add: fps_eq_iff fps_compose_nth setsum_right_distrib mult.assoc)
+ by (simp add: fps_eq_iff fps_compose_nth setsum_distrib_left mult.assoc)
lemma fps_const_mult_apply_right:
"(a oo b) * fps_const (c::'a::comm_semiring_1) = (fps_const c * a) oo b"
@@ -3513,11 +3513,11 @@
proof -
have "?l$n = (setsum (\<lambda>i. (fps_const (a$i) * b^i) oo c) {0..n})$n"
by (simp add: fps_compose_nth fps_compose_power[OF c0] fps_const_mult_apply_left
- setsum_right_distrib mult.assoc fps_setsum_nth)
+ setsum_distrib_left mult.assoc fps_setsum_nth)
also have "\<dots> = ((setsum (\<lambda>i. fps_const (a$i) * b^i) {0..n}) oo c)$n"
by (simp add: fps_compose_setsum_distrib)
also have "\<dots> = ?r$n"
- apply (simp add: fps_compose_nth fps_setsum_nth setsum_left_distrib mult.assoc)
+ apply (simp add: fps_compose_nth fps_setsum_nth setsum_distrib_right mult.assoc)
apply (rule setsum.cong)
apply (rule refl)
apply (rule setsum.mono_neutral_right)
@@ -4224,7 +4224,7 @@
apply (simp add: th00)
unfolding gbinomial_pochhammer
using bn0
- apply (simp add: setsum_left_distrib setsum_right_distrib field_simps)
+ apply (simp add: setsum_distrib_right setsum_distrib_left field_simps)
done
finally show ?thesis by simp
qed
@@ -4253,7 +4253,7 @@
by (simp add: pochhammer_eq_0_iff)
from Vandermonde_pochhammer_lemma[where a = "?a" and b="?b" and n=n, OF h, unfolded th0 th1]
show ?thesis
- using nz by (simp add: field_simps setsum_right_distrib)
+ using nz by (simp add: field_simps setsum_distrib_left)
qed
--- a/src/HOL/Library/Groups_Big_Fun.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Groups_Big_Fun.thy Tue Sep 20 11:35:10 2016 +0200
@@ -240,7 +240,7 @@
note assms
moreover have "{a. g a * r \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
ultimately show ?thesis
- by (simp add: setsum_left_distrib Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
+ by (simp add: setsum_distrib_right Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
qed
lemma Sum_any_right_distrib:
@@ -251,7 +251,7 @@
note assms
moreover have "{a. r * g a \<noteq> 0} \<subseteq> {a. g a \<noteq> 0}" by auto
ultimately show ?thesis
- by (simp add: setsum_right_distrib Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
+ by (simp add: setsum_distrib_left Sum_any.expand_superset [of "{a. g a \<noteq> 0}"])
qed
lemma Sum_any_product:
--- a/src/HOL/Library/Multiset.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Multiset.thy Tue Sep 20 11:35:10 2016 +0200
@@ -656,40 +656,40 @@
subsubsection \<open>Intersection\<close>
-definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where
+definition inf_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "\<inter>#" 70) where
multiset_inter_def: "inf_subset_mset A B = A - (A - B)"
interpretation subset_mset: semilattice_inf inf_subset_mset "op \<subseteq>#" "op \<subset>#"
proof -
have [simp]: "m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" for m n q :: nat
by arith
- show "class.semilattice_inf op #\<inter> op \<subseteq># op \<subset>#"
+ show "class.semilattice_inf op \<inter># op \<subseteq># op \<subset>#"
by standard (auto simp add: multiset_inter_def subseteq_mset_def)
qed
\<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
lemma multiset_inter_count [simp]:
fixes A B :: "'a multiset"
- shows "count (A #\<inter> B) x = min (count A x) (count B x)"
+ shows "count (A \<inter># B) x = min (count A x) (count B x)"
by (simp add: multiset_inter_def)
lemma set_mset_inter [simp]:
- "set_mset (A #\<inter> B) = set_mset A \<inter> set_mset B"
+ "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] multiset_inter_count) simp
lemma diff_intersect_left_idem [simp]:
- "M - M #\<inter> N = M - N"
+ "M - M \<inter># N = M - N"
by (simp add: multiset_eq_iff min_def)
lemma diff_intersect_right_idem [simp]:
- "M - N #\<inter> M = M - N"
+ "M - N \<inter># M = M - N"
by (simp add: multiset_eq_iff min_def)
-lemma multiset_inter_single[simp]: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
+lemma multiset_inter_single[simp]: "a \<noteq> b \<Longrightarrow> {#a#} \<inter># {#b#} = {#}"
by (rule multiset_eqI) auto
lemma multiset_union_diff_commute:
- assumes "B #\<inter> C = {#}"
+ assumes "B \<inter># C = {#}"
shows "A + B - C = A - C + B"
proof (rule multiset_eqI)
fix x
@@ -702,7 +702,7 @@
qed
lemma disjunct_not_in:
- "A #\<inter> B = {#} \<longleftrightarrow> (\<forall>a. a \<notin># A \<or> a \<notin># B)" (is "?P \<longleftrightarrow> ?Q")
+ "A \<inter># B = {#} \<longleftrightarrow> (\<forall>a. a \<notin># A \<or> a \<notin># B)" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P
show ?Q
@@ -722,46 +722,46 @@
fix a
from \<open>?Q\<close> have "count A a = 0 \<or> count B a = 0"
by (auto simp add: not_in_iff)
- then show "count (A #\<inter> B) a = count {#} a"
+ then show "count (A \<inter># B) a = count {#} a"
by auto
qed
qed
lemma add_mset_inter_add_mset[simp]:
- "add_mset a A #\<inter> add_mset a B = add_mset a (A #\<inter> B)"
+ "add_mset a A \<inter># add_mset a B = add_mset a (A \<inter># B)"
by (metis add_mset_add_single add_mset_diff_bothsides diff_subset_eq_self multiset_inter_def
subset_mset.diff_add_assoc2)
lemma add_mset_disjoint [simp]:
- "add_mset a A #\<inter> B = {#} \<longleftrightarrow> a \<notin># B \<and> A #\<inter> B = {#}"
- "{#} = add_mset a A #\<inter> B \<longleftrightarrow> a \<notin># B \<and> {#} = A #\<inter> B"
+ "add_mset a A \<inter># B = {#} \<longleftrightarrow> a \<notin># B \<and> A \<inter># B = {#}"
+ "{#} = add_mset a A \<inter># B \<longleftrightarrow> a \<notin># B \<and> {#} = A \<inter># B"
by (auto simp: disjunct_not_in)
lemma disjoint_add_mset [simp]:
- "B #\<inter> add_mset a A = {#} \<longleftrightarrow> a \<notin># B \<and> B #\<inter> A = {#}"
- "{#} = A #\<inter> add_mset b B \<longleftrightarrow> b \<notin># A \<and> {#} = A #\<inter> B"
+ "B \<inter># add_mset a A = {#} \<longleftrightarrow> a \<notin># B \<and> B \<inter># A = {#}"
+ "{#} = A \<inter># add_mset b B \<longleftrightarrow> b \<notin># A \<and> {#} = A \<inter># B"
by (auto simp: disjunct_not_in)
-lemma empty_inter[simp]: "{#} #\<inter> M = {#}"
+lemma empty_inter[simp]: "{#} \<inter># M = {#}"
by (simp add: multiset_eq_iff)
-lemma inter_empty[simp]: "M #\<inter> {#} = {#}"
+lemma inter_empty[simp]: "M \<inter># {#} = {#}"
by (simp add: multiset_eq_iff)
-lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) #\<inter> N = M #\<inter> N"
+lemma inter_add_left1: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) \<inter># N = M \<inter># N"
by (simp add: multiset_eq_iff not_in_iff)
-lemma inter_add_left2: "x \<in># N \<Longrightarrow> (add_mset x M) #\<inter> N = add_mset x (M #\<inter> (N - {#x#}))"
+lemma inter_add_left2: "x \<in># N \<Longrightarrow> (add_mset x M) \<inter># N = add_mset x (M \<inter># (N - {#x#}))"
by (auto simp add: multiset_eq_iff elim: mset_add)
-lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N #\<inter> (add_mset x M) = N #\<inter> M"
+lemma inter_add_right1: "\<not> x \<in># N \<Longrightarrow> N \<inter># (add_mset x M) = N \<inter># M"
by (simp add: multiset_eq_iff not_in_iff)
-lemma inter_add_right2: "x \<in># N \<Longrightarrow> N #\<inter> (add_mset x M) = add_mset x ((N - {#x#}) #\<inter> M)"
+lemma inter_add_right2: "x \<in># N \<Longrightarrow> N \<inter># (add_mset x M) = add_mset x ((N - {#x#}) \<inter># M)"
by (auto simp add: multiset_eq_iff elim: mset_add)
lemma disjunct_set_mset_diff:
- assumes "M #\<inter> N = {#}"
+ assumes "M \<inter># N = {#}"
shows "set_mset (M - N) = set_mset M"
proof (rule set_eqI)
fix a
@@ -787,85 +787,85 @@
qed
lemma inter_iff:
- "a \<in># A #\<inter> B \<longleftrightarrow> a \<in># A \<and> a \<in># B"
+ "a \<in># A \<inter># B \<longleftrightarrow> a \<in># A \<and> a \<in># B"
by simp
lemma inter_union_distrib_left:
- "A #\<inter> B + C = (A + C) #\<inter> (B + C)"
+ "A \<inter># B + C = (A + C) \<inter># (B + C)"
by (simp add: multiset_eq_iff min_add_distrib_left)
lemma inter_union_distrib_right:
- "C + A #\<inter> B = (C + A) #\<inter> (C + B)"
+ "C + A \<inter># B = (C + A) \<inter># (C + B)"
using inter_union_distrib_left [of A B C] by (simp add: ac_simps)
lemma inter_subset_eq_union:
- "A #\<inter> B \<subseteq># A + B"
+ "A \<inter># B \<subseteq># A + B"
by (auto simp add: subseteq_mset_def)
subsubsection \<open>Bounded union\<close>
-definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "#\<union>" 70)
+definition sup_subset_mset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"(infixl "\<union>#" 70)
where "sup_subset_mset A B = A + (B - A)" \<comment> \<open>FIXME irregular fact name\<close>
interpretation subset_mset: semilattice_sup sup_subset_mset "op \<subseteq>#" "op \<subset>#"
proof -
have [simp]: "m \<le> n \<Longrightarrow> q \<le> n \<Longrightarrow> m + (q - m) \<le> n" for m n q :: nat
by arith
- show "class.semilattice_sup op #\<union> op \<subseteq># op \<subset>#"
+ show "class.semilattice_sup op \<union># op \<subseteq># op \<subset>#"
by standard (auto simp add: sup_subset_mset_def subseteq_mset_def)
qed
\<comment> \<open>FIXME: avoid junk stemming from type class interpretation\<close>
-interpretation subset_mset: bounded_lattice_bot "op #\<inter>" "op \<subseteq>#" "op \<subset>#"
- "op #\<union>" "{#}"
+interpretation subset_mset: bounded_lattice_bot "op \<inter>#" "op \<subseteq>#" "op \<subset>#"
+ "op \<union>#" "{#}"
by standard auto
lemma sup_subset_mset_count [simp]: \<comment> \<open>FIXME irregular fact name\<close>
- "count (A #\<union> B) x = max (count A x) (count B x)"
+ "count (A \<union># B) x = max (count A x) (count B x)"
by (simp add: sup_subset_mset_def)
lemma set_mset_sup [simp]:
- "set_mset (A #\<union> B) = set_mset A \<union> set_mset B"
+ "set_mset (A \<union># B) = set_mset A \<union> set_mset B"
by (simp only: set_eq_iff count_greater_zero_iff [symmetric] sup_subset_mset_count)
(auto simp add: not_in_iff elim: mset_add)
-lemma empty_sup: "{#} #\<union> M = M"
+lemma empty_sup: "{#} \<union># M = M"
by (fact subset_mset.sup_bot.left_neutral)
-lemma sup_empty: "M #\<union> {#} = M"
+lemma sup_empty: "M \<union># {#} = M"
by (fact subset_mset.sup_bot.right_neutral)
-lemma sup_union_left1 [simp]: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) #\<union> N = add_mset x (M #\<union> N)"
+lemma sup_union_left1 [simp]: "\<not> x \<in># N \<Longrightarrow> (add_mset x M) \<union># N = add_mset x (M \<union># N)"
by (simp add: multiset_eq_iff not_in_iff)
-lemma sup_union_left2: "x \<in># N \<Longrightarrow> (add_mset x M) #\<union> N = add_mset x (M #\<union> (N - {#x#}))"
+lemma sup_union_left2: "x \<in># N \<Longrightarrow> (add_mset x M) \<union># N = add_mset x (M \<union># (N - {#x#}))"
by (simp add: multiset_eq_iff)
-lemma sup_union_right1 [simp]: "\<not> x \<in># N \<Longrightarrow> N #\<union> (add_mset x M) = add_mset x (N #\<union> M)"
+lemma sup_union_right1 [simp]: "\<not> x \<in># N \<Longrightarrow> N \<union># (add_mset x M) = add_mset x (N \<union># M)"
by (simp add: multiset_eq_iff not_in_iff)
-lemma sup_union_right2: "x \<in># N \<Longrightarrow> N #\<union> (add_mset x M) = add_mset x ((N - {#x#}) #\<union> M)"
+lemma sup_union_right2: "x \<in># N \<Longrightarrow> N \<union># (add_mset x M) = add_mset x ((N - {#x#}) \<union># M)"
by (simp add: multiset_eq_iff)
lemma sup_union_distrib_left:
- "A #\<union> B + C = (A + C) #\<union> (B + C)"
+ "A \<union># B + C = (A + C) \<union># (B + C)"
by (simp add: multiset_eq_iff max_add_distrib_left)
lemma union_sup_distrib_right:
- "C + A #\<union> B = (C + A) #\<union> (C + B)"
+ "C + A \<union># B = (C + A) \<union># (C + B)"
using sup_union_distrib_left [of A B C] by (simp add: ac_simps)
lemma union_diff_inter_eq_sup:
- "A + B - A #\<inter> B = A #\<union> B"
+ "A + B - A \<inter># B = A \<union># B"
by (auto simp add: multiset_eq_iff)
lemma union_diff_sup_eq_inter:
- "A + B - A #\<union> B = A #\<inter> B"
+ "A + B - A \<union># B = A \<inter># B"
by (auto simp add: multiset_eq_iff)
lemma add_mset_union:
- \<open>add_mset a A #\<union> add_mset a B = add_mset a (A #\<union> B)\<close>
+ \<open>add_mset a A \<union># add_mset a B = add_mset a (A \<union># B)\<close>
by (auto simp: multiset_eq_iff max_def)
@@ -1096,7 +1096,7 @@
using assms by (simp add: Sup_multiset_def Abs_multiset_inverse Sup_multiset_in_multiset)
-interpretation subset_mset: conditionally_complete_lattice Inf Sup "op #\<inter>" "op \<subseteq>#" "op \<subset>#" "op #\<union>"
+interpretation subset_mset: conditionally_complete_lattice Inf Sup "op \<inter>#" "op \<subseteq>#" "op \<subset>#" "op \<union>#"
proof
fix X :: "'a multiset" and A
assume "X \<in> A"
@@ -1220,10 +1220,10 @@
with assms show ?thesis by (simp add: in_Sup_multiset_iff)
qed
-interpretation subset_mset: distrib_lattice "op #\<inter>" "op \<subseteq>#" "op \<subset>#" "op #\<union>"
+interpretation subset_mset: distrib_lattice "op \<inter>#" "op \<subseteq>#" "op \<subset>#" "op \<union>#"
proof
fix A B C :: "'a multiset"
- show "A #\<union> (B #\<inter> C) = A #\<union> B #\<inter> (A #\<union> C)"
+ show "A \<union># (B \<inter># C) = A \<union># B \<inter># (A \<union># C)"
by (intro multiset_eqI) simp_all
qed
@@ -1263,10 +1263,10 @@
lemma filter_diff_mset [simp]: "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
by (rule multiset_eqI) simp
-lemma filter_inter_mset [simp]: "filter_mset P (M #\<inter> N) = filter_mset P M #\<inter> filter_mset P N"
+lemma filter_inter_mset [simp]: "filter_mset P (M \<inter># N) = filter_mset P M \<inter># filter_mset P N"
by (rule multiset_eqI) simp
-lemma filter_sup_mset[simp]: "filter_mset P (A #\<union> B) = filter_mset P A #\<union> filter_mset P B"
+lemma filter_sup_mset[simp]: "filter_mset P (A \<union># B) = filter_mset P A \<union># filter_mset P B"
by (rule multiset_eqI) simp
lemma filter_mset_add_mset [simp]:
@@ -2702,10 +2702,10 @@
lemma mult_cancel_max:
assumes "trans s" and "irrefl s"
- shows "(X, Y) \<in> mult s \<longleftrightarrow> (X - X #\<inter> Y, Y - X #\<inter> Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
+ shows "(X, Y) \<in> mult s \<longleftrightarrow> (X - X \<inter># Y, Y - X \<inter># Y) \<in> mult s" (is "?L \<longleftrightarrow> ?R")
proof -
- have "X - X #\<inter> Y + X #\<inter> Y = X" "Y - X #\<inter> Y + X #\<inter> Y = Y" by (auto simp: count_inject[symmetric])
- thus ?thesis using mult_cancel[OF assms, of "X - X #\<inter> Y" "X #\<inter> Y" "Y - X #\<inter> Y"] by auto
+ have "X - X \<inter># Y + X \<inter># Y = X" "Y - X \<inter># Y + X \<inter># Y = Y" by (auto simp: count_inject[symmetric])
+ thus ?thesis using mult_cancel[OF assms, of "X - X \<inter># Y" "X \<inter># Y" "Y - X \<inter># Y"] by auto
qed
@@ -2714,37 +2714,37 @@
text \<open>
Predicate variants of \<open>mult\<close> and the reflexive closure of \<open>mult\<close>, which are
executable whenever the given predicate \<open>P\<close> is. Together with the standard
- code equations for \<open>op #\<inter>\<close> and \<open>op -\<close> this should yield quadratic
+ code equations for \<open>op \<inter>#\<close> and \<open>op -\<close> this should yield quadratic
(with respect to calls to \<open>P\<close>) implementations of \<open>multp\<close> and \<open>multeqp\<close>.
\<close>
definition multp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
"multp P N M =
- (let Z = M #\<inter> N; X = M - Z in
+ (let Z = M \<inter># N; X = M - Z in
X \<noteq> {#} \<and> (let Y = N - Z in (\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x)))"
definition multeqp :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where
"multeqp P N M =
- (let Z = M #\<inter> N; X = M - Z; Y = N - Z in
+ (let Z = M \<inter># N; X = M - Z; Y = N - Z in
(\<forall>y \<in> set_mset Y. \<exists>x \<in> set_mset X. P y x))"
lemma multp_iff:
assumes "irrefl R" and "trans R" and [simp]: "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
shows "multp P N M \<longleftrightarrow> (N, M) \<in> mult R" (is "?L \<longleftrightarrow> ?R")
proof -
- have *: "M #\<inter> N + (N - M #\<inter> N) = N" "M #\<inter> N + (M - M #\<inter> N) = M"
- "(M - M #\<inter> N) #\<inter> (N - M #\<inter> N) = {#}" by (auto simp: count_inject[symmetric])
+ have *: "M \<inter># N + (N - M \<inter># N) = N" "M \<inter># N + (M - M \<inter># N) = M"
+ "(M - M \<inter># N) \<inter># (N - M \<inter># N) = {#}" by (auto simp: count_inject[symmetric])
show ?thesis
proof
assume ?L thus ?R
- using one_step_implies_mult[of "M - M #\<inter> N" "N - M #\<inter> N" R "M #\<inter> N"] *
+ using one_step_implies_mult[of "M - M \<inter># N" "N - M \<inter># N" R "M \<inter># N"] *
by (auto simp: multp_def Let_def)
next
- { fix I J K :: "'a multiset" assume "(I + J) #\<inter> (I + K) = {#}"
+ { fix I J K :: "'a multiset" assume "(I + J) \<inter># (I + K) = {#}"
then have "I = {#}" by (metis inter_union_distrib_right union_eq_empty)
} note [dest!] = this
assume ?R thus ?L
- using mult_implies_one_step[OF assms(2), of "N - M #\<inter> N" "M - M #\<inter> N"]
+ using mult_implies_one_step[OF assms(2), of "N - M \<inter># N" "M - M \<inter># N"]
mult_cancel_max[OF assms(2,1), of "N" "M"] * by (auto simp: multp_def)
qed
qed
@@ -2753,9 +2753,9 @@
assumes "irrefl R" and "trans R" and "\<And>x y. P x y \<longleftrightarrow> (x, y) \<in> R"
shows "multeqp P N M \<longleftrightarrow> (N, M) \<in> (mult R)\<^sup>="
proof -
- { assume "N \<noteq> M" "M - M #\<inter> N = {#}"
+ { assume "N \<noteq> M" "M - M \<inter># N = {#}"
then obtain y where "count N y \<noteq> count M y" by (auto simp: count_inject[symmetric])
- then have "\<exists>y. count M y < count N y" using `M - M #\<inter> N = {#}`
+ then have "\<exists>y. count M y < count N y" using `M - M \<inter># N = {#}`
by (auto simp: count_inject[symmetric] dest!: le_neq_implies_less fun_cong[of _ _ y])
}
then have "multeqp P N M \<longleftrightarrow> multp P N M \<or> N = M"
@@ -3035,13 +3035,13 @@
lemma mset_subset_trans: "(M::'a multiset) \<subset># K \<Longrightarrow> K \<subset># N \<Longrightarrow> M \<subset># N"
by (fact subset_mset.less_trans)
-lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
+lemma multiset_inter_commute: "A \<inter># B = B \<inter># A"
by (fact subset_mset.inf.commute)
-lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
+lemma multiset_inter_assoc: "A \<inter># (B \<inter># C) = A \<inter># B \<inter># C"
by (fact subset_mset.inf.assoc [symmetric])
-lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
+lemma multiset_inter_left_commute: "A \<inter># (B \<inter># C) = B \<inter># (A \<inter># C)"
by (fact subset_mset.inf.left_commute)
lemmas multiset_inter_ac =
@@ -3112,24 +3112,24 @@
by (rule sym, induct ys arbitrary: xs) (simp_all add: diff_add diff_right_commute diff_diff_add)
lemma [code]:
- "mset xs #\<inter> mset ys =
+ "mset xs \<inter># mset ys =
mset (snd (fold (\<lambda>x (ys, zs).
if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, [])))"
proof -
have "\<And>zs. mset (snd (fold (\<lambda>x (ys, zs).
if x \<in> set ys then (remove1 x ys, x # zs) else (ys, zs)) xs (ys, zs))) =
- (mset xs #\<inter> mset ys) + mset zs"
+ (mset xs \<inter># mset ys) + mset zs"
by (induct xs arbitrary: ys)
(auto simp add: inter_add_right1 inter_add_right2 ac_simps)
then show ?thesis by simp
qed
lemma [code]:
- "mset xs #\<union> mset ys =
+ "mset xs \<union># mset ys =
mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, [])))"
proof -
have "\<And>zs. mset (case_prod append (fold (\<lambda>x (ys, zs). (remove1 x ys, x # zs)) xs (ys, zs))) =
- (mset xs #\<union> mset ys) + mset zs"
+ (mset xs \<union># mset ys) + mset zs"
by (induct xs arbitrary: ys) (simp_all add: multiset_eq_iff)
then show ?thesis by simp
qed
--- a/src/HOL/Library/Polynomial.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Polynomial.thy Tue Sep 20 11:35:10 2016 +0200
@@ -507,7 +507,7 @@
also note pCons.IH
also have "a + x * (\<Sum>i\<le>degree p. coeff p i * x ^ i) =
coeff ?p' 0 * x^0 + (\<Sum>i\<le>degree p. coeff ?p' (Suc i) * x^Suc i)"
- by (simp add: field_simps setsum_right_distrib coeff_pCons)
+ by (simp add: field_simps setsum_distrib_left coeff_pCons)
also note setsum_atMost_Suc_shift[symmetric]
also note degree_pCons_eq[OF \<open>p \<noteq> 0\<close>, of a, symmetric]
finally show ?thesis .
@@ -3412,7 +3412,7 @@
by (subst setprod.insert, insert insert, auto)
} note id2 = this
show ?case
- unfolding id pderiv_mult insert(3) setsum_right_distrib
+ unfolding id pderiv_mult insert(3) setsum_distrib_left
by (auto simp add: ac_simps id2 intro!: setsum.cong)
qed auto
--- a/src/HOL/Library/Stirling.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Library/Stirling.thy Tue Sep 20 11:35:10 2016 +0200
@@ -110,7 +110,7 @@
also have "\<dots> = Suc n * (\<Sum>k=1..n. fact n div k) + Suc n * fact n div Suc n"
by (metis nat.distinct(1) nonzero_mult_divide_cancel_left)
also have "\<dots> = (\<Sum>k=1..n. fact (Suc n) div k) + fact (Suc n) div Suc n"
- by (simp add: setsum_right_distrib div_mult_swap dvd_fact)
+ by (simp add: setsum_distrib_left div_mult_swap dvd_fact)
also have "\<dots> = (\<Sum>k=1..Suc n. fact (Suc n) div k)"
by simp
finally show ?thesis .
@@ -162,7 +162,7 @@
also have "\<dots> = (\<Sum>k\<le>n. n * stirling n (Suc k) + stirling n k)"
by simp
also have "\<dots> = n * (\<Sum>k\<le>n. stirling n (Suc k)) + (\<Sum>k\<le>n. stirling n k)"
- by (simp add: setsum.distrib setsum_right_distrib)
+ by (simp add: setsum.distrib setsum_distrib_left)
also have "\<dots> = n * fact n + fact n"
proof -
have "n * (\<Sum>k\<le>n. stirling n (Suc k)) = n * ((\<Sum>k\<le>Suc n. stirling n k) - stirling n 0)"
@@ -195,7 +195,7 @@
by (subst setsum_atMost_Suc_shift) (simp add: setsum.distrib ring_distribs)
also have "\<dots> = pochhammer x (Suc n)"
by (subst setsum_atMost_Suc_shift [symmetric])
- (simp add: algebra_simps setsum.distrib setsum_right_distrib pochhammer_Suc Suc [symmetric])
+ (simp add: algebra_simps setsum.distrib setsum_distrib_left pochhammer_Suc Suc [symmetric])
finally show ?case .
qed
--- a/src/HOL/Metis_Examples/Big_O.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Metis_Examples/Big_O.thy Tue Sep 20 11:35:10 2016 +0200
@@ -499,7 +499,7 @@
apply (subst abs_of_nonneg) back back
apply (rule setsum_nonneg)
apply force
-apply (subst setsum_right_distrib)
+apply (subst setsum_distrib_left)
apply (rule allI)
apply (rule order_trans)
apply (rule setsum_abs)
--- a/src/HOL/Nonstandard_Analysis/HSeries.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Nonstandard_Analysis/HSeries.thy Tue Sep 20 11:35:10 2016 +0200
@@ -59,7 +59,7 @@
lemma sumhr_mult:
"!!m n. hypreal_of_real r * sumhr(m,n,f) = sumhr(m,n,%n. r * f n)"
-unfolding sumhr_app by transfer (rule setsum_right_distrib)
+unfolding sumhr_app by transfer (rule setsum_distrib_left)
lemma sumhr_split_add:
"!!n p. n < p ==> sumhr(0,n,f) + sumhr(n,p,f) = sumhr(0,p,f)"
--- a/src/HOL/Number_Theory/Factorial_Ring.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Number_Theory/Factorial_Ring.thy Tue Sep 20 11:35:10 2016 +0200
@@ -541,7 +541,7 @@
by simp
with A B C have subset: "A \<subseteq># B + C"
by (subst (asm) prod_mset_dvd_prod_mset_primes_iff) auto
- define A1 and A2 where "A1 = A #\<inter> B" and "A2 = A - A1"
+ define A1 and A2 where "A1 = A \<inter># B" and "A2 = A - A1"
have "A = A1 + A2" unfolding A1_def A2_def
by (rule sym, intro subset_mset.add_diff_inverse) simp_all
from subset have "A1 \<subseteq># B" "A2 \<subseteq># C"
@@ -1440,10 +1440,10 @@
definition "gcd_factorial a b = (if a = 0 then normalize b
else if b = 0 then normalize a
- else prod_mset (prime_factorization a #\<inter> prime_factorization b))"
+ else prod_mset (prime_factorization a \<inter># prime_factorization b))"
definition "lcm_factorial a b = (if a = 0 \<or> b = 0 then 0
- else prod_mset (prime_factorization a #\<union> prime_factorization b))"
+ else prod_mset (prime_factorization a \<union># prime_factorization b))"
definition "Gcd_factorial A =
(if A \<subseteq> {0} then 0 else prod_mset (Inf (prime_factorization ` (A - {0}))))"
@@ -1457,24 +1457,24 @@
lemma prime_factorization_gcd_factorial:
assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
- shows "prime_factorization (gcd_factorial a b) = prime_factorization a #\<inter> prime_factorization b"
+ shows "prime_factorization (gcd_factorial a b) = prime_factorization a \<inter># prime_factorization b"
proof -
have "prime_factorization (gcd_factorial a b) =
- prime_factorization (prod_mset (prime_factorization a #\<inter> prime_factorization b))"
+ prime_factorization (prod_mset (prime_factorization a \<inter># prime_factorization b))"
by (simp add: gcd_factorial_def)
- also have "\<dots> = prime_factorization a #\<inter> prime_factorization b"
+ also have "\<dots> = prime_factorization a \<inter># prime_factorization b"
by (subst prime_factorization_prod_mset_primes) auto
finally show ?thesis .
qed
lemma prime_factorization_lcm_factorial:
assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
- shows "prime_factorization (lcm_factorial a b) = prime_factorization a #\<union> prime_factorization b"
+ shows "prime_factorization (lcm_factorial a b) = prime_factorization a \<union># prime_factorization b"
proof -
have "prime_factorization (lcm_factorial a b) =
- prime_factorization (prod_mset (prime_factorization a #\<union> prime_factorization b))"
+ prime_factorization (prod_mset (prime_factorization a \<union># prime_factorization b))"
by (simp add: lcm_factorial_def)
- also have "\<dots> = prime_factorization a #\<union> prime_factorization b"
+ also have "\<dots> = prime_factorization a \<union># prime_factorization b"
by (subst prime_factorization_prod_mset_primes) auto
finally show ?thesis .
qed
@@ -1527,8 +1527,8 @@
lemma normalize_gcd_factorial: "normalize (gcd_factorial a b) = gcd_factorial a b"
proof -
- have "normalize (prod_mset (prime_factorization a #\<inter> prime_factorization b)) =
- prod_mset (prime_factorization a #\<inter> prime_factorization b)"
+ have "normalize (prod_mset (prime_factorization a \<inter># prime_factorization b)) =
+ prod_mset (prime_factorization a \<inter># prime_factorization b)"
by (intro normalize_prod_mset_primes) auto
thus ?thesis by (simp add: gcd_factorial_def)
qed
@@ -1541,7 +1541,7 @@
from that False have "?p c \<subseteq># ?p a" "?p c \<subseteq># ?p b"
by (simp_all add: prime_factorization_subset_iff_dvd)
hence "prime_factorization c \<subseteq>#
- prime_factorization (prod_mset (prime_factorization a #\<inter> prime_factorization b))"
+ prime_factorization (prod_mset (prime_factorization a \<inter># prime_factorization b))"
using False by (subst prime_factorization_prod_mset_primes) auto
with False show ?thesis
by (auto simp: gcd_factorial_def prime_factorization_subset_iff_dvd [symmetric])
@@ -1553,12 +1553,12 @@
case False
let ?p = "prime_factorization"
from False have "prod_mset (?p (a * b)) div gcd_factorial a b =
- prod_mset (?p a + ?p b - ?p a #\<inter> ?p b)"
+ prod_mset (?p a + ?p b - ?p a \<inter># ?p b)"
by (subst prod_mset_diff) (auto simp: lcm_factorial_def gcd_factorial_def
prime_factorization_mult subset_mset.le_infI1)
also from False have "prod_mset (?p (a * b)) = normalize (a * b)"
by (intro prod_mset_prime_factorization) simp_all
- also from False have "prod_mset (?p a + ?p b - ?p a #\<inter> ?p b) = lcm_factorial a b"
+ also from False have "prod_mset (?p a + ?p b - ?p a \<inter># ?p b) = lcm_factorial a b"
by (simp add: union_diff_inter_eq_sup lcm_factorial_def)
finally show ?thesis ..
qed (auto simp: lcm_factorial_def)
@@ -1750,12 +1750,12 @@
lemma prime_factorization_gcd:
assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
- shows "prime_factorization (gcd a b) = prime_factorization a #\<inter> prime_factorization b"
+ shows "prime_factorization (gcd a b) = prime_factorization a \<inter># prime_factorization b"
by (simp add: gcd_eq_gcd_factorial prime_factorization_gcd_factorial)
lemma prime_factorization_lcm:
assumes [simp]: "a \<noteq> 0" "b \<noteq> 0"
- shows "prime_factorization (lcm a b) = prime_factorization a #\<union> prime_factorization b"
+ shows "prime_factorization (lcm a b) = prime_factorization a \<union># prime_factorization b"
by (simp add: lcm_eq_lcm_factorial prime_factorization_lcm_factorial)
lemma prime_factorization_Gcd:
--- a/src/HOL/Probability/Distributions.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Probability/Distributions.thy Tue Sep 20 11:35:10 2016 +0200
@@ -149,7 +149,7 @@
assume "0 \<le> x"
have "(\<Sum>n\<le>k. (l * x) ^ n * exp (- (l * x)) / fact n) =
exp (- (l * x)) * (\<Sum>n\<le>k. (l * x) ^ n / fact n)"
- unfolding setsum_right_distrib by (intro setsum.cong) (auto simp: field_simps)
+ unfolding setsum_distrib_left by (intro setsum.cong) (auto simp: field_simps)
also have "\<dots> = (\<Sum>n\<le>k. (l * x) ^ n / fact n) / exp (l * x)"
by (simp add: exp_minus field_simps)
also have "\<dots> \<le> 1"
--- a/src/HOL/Probability/Probability_Mass_Function.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Probability/Probability_Mass_Function.thy Tue Sep 20 11:35:10 2016 +0200
@@ -1610,7 +1610,7 @@
lemma nn_integral_pmf_of_set: "nn_integral (measure_pmf pmf_of_set) f = setsum f S / card S"
by (subst nn_integral_measure_pmf_finite)
- (simp_all add: setsum_left_distrib[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def
+ (simp_all add: setsum_distrib_right[symmetric] card_gt_0_iff S_not_empty S_finite divide_ennreal_def
divide_ennreal[symmetric] ennreal_of_nat_eq_real_of_nat[symmetric] ennreal_times_divide)
lemma integral_pmf_of_set: "integral\<^sup>L (measure_pmf pmf_of_set) f = setsum f S / card S"
--- a/src/HOL/Probability/Projective_Limit.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Probability/Projective_Limit.thy Tue Sep 20 11:35:10 2016 +0200
@@ -290,7 +290,7 @@
finally show "\<mu>G (Z i - Z' i) \<le> (2 powr - real i) * ?a" .
qed
also have "\<dots> = ennreal ((\<Sum> i\<in>{1..n}. (2 powr -enn2real i)) * enn2real ?a)"
- using r by (simp add: setsum_left_distrib ennreal_mult[symmetric])
+ using r by (simp add: setsum_distrib_right ennreal_mult[symmetric])
also have "\<dots> < ennreal (1 * enn2real ?a)"
proof (intro ennreal_lessI mult_strict_right_mono)
have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)"
--- a/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Probability/ex/Koepf_Duermuth_Countermeasure.thy Tue Sep 20 11:35:10 2016 +0200
@@ -100,7 +100,7 @@
show ?case unfolding *
using Suc[of "\<lambda>i. P (Suc i)"]
by (simp add: setsum.reindex split_conv setsum_cartesian_product'
- lessThan_Suc_eq_insert_0 setprod.reindex setsum_left_distrib[symmetric] setsum_right_distrib[symmetric])
+ lessThan_Suc_eq_insert_0 setprod.reindex setsum_distrib_right[symmetric] setsum_distrib_left[symmetric])
qed
declare space_point_measure[simp]
@@ -158,8 +158,8 @@
have [simp]: "\<And>A. inj_on (\<lambda>(xs, n). n # xs) A" by (force intro!: inj_onI)
- note setsum_right_distrib[symmetric, simp]
- note setsum_left_distrib[symmetric, simp]
+ note setsum_distrib_left[symmetric, simp]
+ note setsum_distrib_right[symmetric, simp]
note setsum_cartesian_product'[simp]
have "(\<Sum>ms | set ms \<subseteq> messages \<and> length ms = n. \<Prod>x<length ms. M (ms ! x)) = 1"
@@ -256,7 +256,7 @@
apply (simp add: * P_def)
apply (simp add: setsum_cartesian_product')
using setprod_setsum_distrib_lists[OF M.finite_space, of M n "\<lambda>x x. True"] \<open>k \<in> keys\<close>
- by (auto simp add: setsum_right_distrib[symmetric] subset_eq setprod.neutral_const)
+ by (auto simp add: setsum_distrib_left[symmetric] subset_eq setprod.neutral_const)
qed
lemma fst_image_msgs[simp]: "fst`msgs = keys"
@@ -323,7 +323,7 @@
also have "\<dots> = - (\<Sum>y\<in>Y`msgs. (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))))"
by (subst setsum.commute) rule
also have "\<dots> = -(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) * log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))"
- by (auto simp add: setsum_right_distrib vimage_def intro!: setsum.cong prob_conj_imp1)
+ by (auto simp add: setsum_distrib_left vimage_def intro!: setsum.cong prob_conj_imp1)
finally show "- (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` msgs. (\<P>(X ; Y) {(x, y)}) * log b ((\<P>(X ; Y) {(x, y)}) / (\<P>(Y) {y}))) =
-(\<Sum>y\<in>Y`msgs. (\<P>(Y) {y}) * (\<Sum>x\<in>X`msgs. (\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}) * log b ((\<P>(X ; Y) {(x,y)}) / (\<P>(Y) {y}))))" .
qed simp_all
@@ -402,7 +402,7 @@
also have "\<P>(t\<circ>OB | fst) {(t obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(t\<circ>OB|fst) {(t obs, k')} * \<P>(fst) {k'}) =
\<P>(OB | fst) {(obs, k)} * \<P>(fst) {k} / (\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'})"
using CP_t_K[OF \<open>k\<in>keys\<close> obs] CP_t_K[OF _ obs] \<open>real (card ?S) \<noteq> 0\<close>
- by (simp only: setsum_right_distrib[symmetric] ac_simps
+ by (simp only: setsum_distrib_left[symmetric] ac_simps
mult_divide_mult_cancel_left[OF \<open>real (card ?S) \<noteq> 0\<close>]
cong: setsum.cong)
also have "(\<Sum>k'\<in>keys. \<P>(OB|fst) {(obs, k')} * \<P>(fst) {k'}) = \<P>(OB) {obs}"
@@ -450,7 +450,7 @@
apply (safe intro!: setsum.cong)
using P_t_sum_P_O
by (simp add: setsum_divide_distrib[symmetric] field_simps **
- setsum_right_distrib[symmetric] setsum_left_distrib[symmetric])
+ setsum_distrib_left[symmetric] setsum_distrib_right[symmetric])
also have "\<dots> = \<H>(fst | t\<circ>OB)"
unfolding conditional_entropy_eq_ce_with_hypothesis
by (simp add: comp_def image_image[symmetric])
--- a/src/HOL/Quotient_Examples/FSet.thy Mon Sep 19 17:37:22 2016 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1163 +0,0 @@
-(* Title: HOL/Quotient_Examples/FSet.thy
- Author: Cezary Kaliszyk, TU Munich
- Author: Christian Urban, TU Munich
-
-Type of finite sets.
-*)
-
-(********************************************************************
- WARNING: There is a formalization of 'a fset as a subtype of sets in
- HOL/Library/FSet.thy using Lifting/Transfer. The user should use
- that file rather than this file unless there are some very specific
- reasons.
-*********************************************************************)
-
-theory FSet
-imports "~~/src/HOL/Library/Multiset" "~~/src/HOL/Library/Quotient_List"
-begin
-
-text \<open>
- The type of finite sets is created by a quotient construction
- over lists. The definition of the equivalence:
-\<close>
-
-definition
- list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
-where
- [simp]: "xs \<approx> ys \<longleftrightarrow> set xs = set ys"
-
-lemma list_eq_reflp:
- "reflp list_eq"
- by (auto intro: reflpI)
-
-lemma list_eq_symp:
- "symp list_eq"
- by (auto intro: sympI)
-
-lemma list_eq_transp:
- "transp list_eq"
- by (auto intro: transpI)
-
-lemma list_eq_equivp:
- "equivp list_eq"
- by (auto intro: equivpI list_eq_reflp list_eq_symp list_eq_transp)
-
-text \<open>The \<open>fset\<close> type\<close>
-
-quotient_type
- 'a fset = "'a list" / "list_eq"
- by (rule list_eq_equivp)
-
-text \<open>
- Definitions for sublist, cardinality,
- intersection, difference and respectful fold over
- lists.
-\<close>
-
-declare List.member_def [simp]
-
-definition
- sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
-where
- [simp]: "sub_list xs ys \<longleftrightarrow> set xs \<subseteq> set ys"
-
-definition
- card_list :: "'a list \<Rightarrow> nat"
-where
- [simp]: "card_list xs = card (set xs)"
-
-definition
- inter_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- [simp]: "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"
-
-definition
- diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
-where
- [simp]: "diff_list xs ys = [x \<leftarrow> xs. x \<notin> set ys]"
-
-definition
- rsp_fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
-where
- "rsp_fold f \<longleftrightarrow> (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
-
-lemma rsp_foldI:
- "(\<And>u v. f u \<circ> f v = f v \<circ> f u) \<Longrightarrow> rsp_fold f"
- by (simp add: rsp_fold_def)
-
-lemma rsp_foldE:
- assumes "rsp_fold f"
- obtains "f u \<circ> f v = f v \<circ> f u"
- using assms by (simp add: rsp_fold_def)
-
-definition
- fold_once :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
-where
- "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
-
-lemma fold_once_default [simp]:
- "\<not> rsp_fold f \<Longrightarrow> fold_once f xs = id"
- by (simp add: fold_once_def)
-
-lemma fold_once_fold_remdups:
- "rsp_fold f \<Longrightarrow> fold_once f xs = fold f (remdups xs)"
- by (simp add: fold_once_def)
-
-
-section \<open>Quotient composition lemmas\<close>
-
-lemma list_all2_refl':
- assumes q: "equivp R"
- shows "(list_all2 R) r r"
- by (rule list_all2_refl) (metis equivp_def q)
-
-lemma compose_list_refl:
- assumes q: "equivp R"
- shows "(list_all2 R OOO op \<approx>) r r"
-proof
- have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
- show "list_all2 R r r" by (rule list_all2_refl'[OF q])
- with * show "(op \<approx> OO list_all2 R) r r" ..
-qed
-
-lemma map_list_eq_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
- by (simp only: list_eq_def set_map)
-
-lemma quotient_compose_list_g:
- assumes q: "Quotient3 R Abs Rep"
- and e: "equivp R"
- shows "Quotient3 ((list_all2 R) OOO (op \<approx>))
- (abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
- unfolding Quotient3_def comp_def
-proof (intro conjI allI)
- fix a r s
- show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
- by (simp add: abs_o_rep[OF q] Quotient3_abs_rep[OF Quotient3_fset] List.map.id)
- have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
- by (rule list_all2_refl'[OF e])
- have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
- by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
- show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
- by (rule, rule list_all2_refl'[OF e]) (rule c)
- show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
- (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
- proof (intro iffI conjI)
- show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl[OF e])
- show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl[OF e])
- next
- assume a: "(list_all2 R OOO op \<approx>) r s"
- then have b: "map Abs r \<approx> map Abs s"
- proof (elim relcomppE)
- fix b ba
- assume c: "list_all2 R r b"
- assume d: "b \<approx> ba"
- assume e: "list_all2 R ba s"
- have f: "map Abs r = map Abs b"
- using Quotient3_rel[OF list_quotient3[OF q]] c by blast
- have "map Abs ba = map Abs s"
- using Quotient3_rel[OF list_quotient3[OF q]] e by blast
- then have g: "map Abs s = map Abs ba" by simp
- then show "map Abs r \<approx> map Abs s" using d f map_list_eq_cong by simp
- qed
- then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
- using Quotient3_rel[OF Quotient3_fset] by blast
- next
- assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
- \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
- then have s: "(list_all2 R OOO op \<approx>) s s" by simp
- have d: "map Abs r \<approx> map Abs s"
- by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
- have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"
- by (rule map_list_eq_cong[OF d])
- have y: "list_all2 R (map Rep (map Abs s)) s"
- by (fact rep_abs_rsp_left[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of s]])
- have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"
- by (rule relcomppI) (rule b, rule y)
- have z: "list_all2 R r (map Rep (map Abs r))"
- by (fact rep_abs_rsp[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of r]])
- then show "(list_all2 R OOO op \<approx>) r s"
- using a c relcomppI by simp
- qed
-qed
-
-lemma quotient_compose_list[quot_thm]:
- shows "Quotient3 ((list_all2 op \<approx>) OOO (op \<approx>))
- (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
- by (rule quotient_compose_list_g, rule Quotient3_fset, rule list_eq_equivp)
-
-
-section \<open>Quotient definitions for fsets\<close>
-
-
-subsection \<open>Finite sets are a bounded, distributive lattice with minus\<close>
-
-instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
-begin
-
-quotient_definition
- "bot :: 'a fset"
- is "Nil :: 'a list" done
-
-abbreviation
- empty_fset ("{||}")
-where
- "{||} \<equiv> bot :: 'a fset"
-
-quotient_definition
- "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
- is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)" by simp
-
-abbreviation
- subset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
-where
- "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
-
-definition
- less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
-where
- "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
-
-abbreviation
- psubset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
-where
- "xs |\<subset>| ys \<equiv> xs < ys"
-
-quotient_definition
- "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
-
-abbreviation
- union_fset (infixl "|\<union>|" 65)
-where
- "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
-
-quotient_definition
- "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
-
-abbreviation
- inter_fset (infixl "|\<inter>|" 65)
-where
- "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
-
-quotient_definition
- "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by fastforce
-
-instance
-proof
- fix x y z :: "'a fset"
- show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
- by (unfold less_fset_def, descending) auto
- show "x |\<subseteq>| x" by (descending) (simp)
- show "{||} |\<subseteq>| x" by (descending) (simp)
- show "x |\<subseteq>| x |\<union>| y" by (descending) (simp)
- show "y |\<subseteq>| x |\<union>| y" by (descending) (simp)
- show "x |\<inter>| y |\<subseteq>| x" by (descending) (auto)
- show "x |\<inter>| y |\<subseteq>| y" by (descending) (auto)
- show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)"
- by (descending) (auto)
-next
- fix x y z :: "'a fset"
- assume a: "x |\<subseteq>| y"
- assume b: "y |\<subseteq>| z"
- show "x |\<subseteq>| z" using a b by (descending) (simp)
-next
- fix x y :: "'a fset"
- assume a: "x |\<subseteq>| y"
- assume b: "y |\<subseteq>| x"
- show "x = y" using a b by (descending) (auto)
-next
- fix x y z :: "'a fset"
- assume a: "y |\<subseteq>| x"
- assume b: "z |\<subseteq>| x"
- show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp)
-next
- fix x y z :: "'a fset"
- assume a: "x |\<subseteq>| y"
- assume b: "x |\<subseteq>| z"
- show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (auto)
-qed
-
-end
-
-
-subsection \<open>Other constants for fsets\<close>
-
-quotient_definition
- "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is "Cons" by auto
-
-syntax
- "_insert_fset" :: "args => 'a fset" ("{|(_)|}")
-
-translations
- "{|x, xs|}" == "CONST insert_fset x {|xs|}"
- "{|x|}" == "CONST insert_fset x {||}"
-
-quotient_definition
- fset_member
-where
- "fset_member :: 'a fset \<Rightarrow> 'a \<Rightarrow> bool" is "List.member" by fastforce
-
-abbreviation
- in_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50)
-where
- "x |\<in>| S \<equiv> fset_member S x"
-
-abbreviation
- notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
-where
- "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
-
-
-subsection \<open>Other constants on the Quotient Type\<close>
-
-quotient_definition
- "card_fset :: 'a fset \<Rightarrow> nat"
- is card_list by simp
-
-quotient_definition
- "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
- is map by simp
-
-quotient_definition
- "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is removeAll by simp
-
-quotient_definition
- "fset :: 'a fset \<Rightarrow> 'a set"
- is "set" by simp
-
-lemma fold_once_set_equiv:
- assumes "xs \<approx> ys"
- shows "fold_once f xs = fold_once f ys"
-proof (cases "rsp_fold f")
- case False then show ?thesis by simp
-next
- case True
- then have "\<And>x y. x \<in> set (remdups xs) \<Longrightarrow> y \<in> set (remdups xs) \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
- by (rule rsp_foldE)
- moreover from assms have "mset (remdups xs) = mset (remdups ys)"
- by (simp add: set_eq_iff_mset_remdups_eq)
- ultimately have "fold f (remdups xs) = fold f (remdups ys)"
- by (rule fold_multiset_equiv)
- with True show ?thesis by (simp add: fold_once_fold_remdups)
-qed
-
-quotient_definition
- "fold_fset :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b \<Rightarrow> 'b"
- is fold_once by (rule fold_once_set_equiv)
-
-lemma concat_rsp_pre:
- assumes a: "list_all2 op \<approx> x x'"
- and b: "x' \<approx> y'"
- and c: "list_all2 op \<approx> y' y"
- and d: "\<exists>x\<in>set x. xa \<in> set x"
- shows "\<exists>x\<in>set y. xa \<in> set x"
-proof -
- obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
- have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
- then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
- have "ya \<in> set y'" using b h by simp
- then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
- then show ?thesis using f i by auto
-qed
-
-quotient_definition
- "concat_fset :: ('a fset) fset \<Rightarrow> 'a fset"
- is concat
-proof (elim relcomppE)
-fix a b ba bb
- assume a: "list_all2 op \<approx> a ba"
- with list_symp [OF list_eq_symp] have a': "list_all2 op \<approx> ba a" by (rule sympE)
- assume b: "ba \<approx> bb"
- with list_eq_symp have b': "bb \<approx> ba" by (rule sympE)
- assume c: "list_all2 op \<approx> bb b"
- with list_symp [OF list_eq_symp] have c': "list_all2 op \<approx> b bb" by (rule sympE)
- have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
- proof
- fix x
- show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
- proof
- assume d: "\<exists>xa\<in>set a. x \<in> set xa"
- show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
- next
- assume e: "\<exists>xa\<in>set b. x \<in> set xa"
- show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
- qed
- qed
- then show "concat a \<approx> concat b" by auto
-qed
-
-quotient_definition
- "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
- is filter by force
-
-
-subsection \<open>Compositional respectfulness and preservation lemmas\<close>
-
-lemma Nil_rsp2 [quot_respect]:
- shows "(list_all2 op \<approx> OOO op \<approx>) Nil Nil"
- by (rule compose_list_refl, rule list_eq_equivp)
-
-lemma Cons_rsp2 [quot_respect]:
- shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
- apply (auto intro!: rel_funI)
- apply (rule_tac b="x # b" in relcomppI)
- apply auto
- apply (rule_tac b="x # ba" in relcomppI)
- apply auto
- done
-
-lemma Nil_prs2 [quot_preserve]:
- assumes "Quotient3 R Abs Rep"
- shows "(Abs \<circ> map f) [] = Abs []"
- by simp
-
-lemma Cons_prs2 [quot_preserve]:
- assumes q: "Quotient3 R1 Abs1 Rep1"
- and r: "Quotient3 R2 Abs2 Rep2"
- shows "(Rep1 ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
- by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
-
-lemma append_prs2 [quot_preserve]:
- assumes q: "Quotient3 R1 Abs1 Rep1"
- and r: "Quotient3 R2 Abs2 Rep2"
- shows "((map Rep1 \<circ> Rep2) ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) op @ =
- (Rep2 ---> Rep2 ---> Abs2) op @"
- by (simp add: fun_eq_iff abs_o_rep[OF q] List.map.id)
-
-lemma list_all2_app_l:
- assumes a: "reflp R"
- and b: "list_all2 R l r"
- shows "list_all2 R (z @ l) (z @ r)"
- using a b by (induct z) (auto elim: reflpE)
-
-lemma append_rsp2_pre0:
- assumes a:"list_all2 op \<approx> x x'"
- shows "list_all2 op \<approx> (x @ z) (x' @ z)"
- using a apply (induct x x' rule: list_induct2')
- by simp_all (rule list_all2_refl'[OF list_eq_equivp])
-
-lemma append_rsp2_pre1:
- assumes a:"list_all2 op \<approx> x x'"
- shows "list_all2 op \<approx> (z @ x) (z @ x')"
- using a apply (induct x x' arbitrary: z rule: list_induct2')
- apply (rule list_all2_refl'[OF list_eq_equivp])
- apply (simp_all del: list_eq_def)
- apply (rule list_all2_app_l)
- apply (simp_all add: reflpI)
- done
-
-lemma append_rsp2_pre:
- assumes "list_all2 op \<approx> x x'"
- and "list_all2 op \<approx> z z'"
- shows "list_all2 op \<approx> (x @ z) (x' @ z')"
- using assms by (rule list_all2_appendI)
-
-lemma compositional_rsp3:
- assumes "(R1 ===> R2 ===> R3) C C" and "(R4 ===> R5 ===> R6) C C"
- shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
- by (auto intro!: rel_funI)
- (metis (full_types) assms rel_funE relcomppI)
-
-lemma append_rsp2 [quot_respect]:
- "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) append append"
- by (intro compositional_rsp3)
- (auto intro!: rel_funI simp add: append_rsp2_pre)
-
-lemma map_rsp2 [quot_respect]:
- "((op \<approx> ===> op \<approx>) ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) map map"
-proof (auto intro!: rel_funI)
- fix f f' :: "'a list \<Rightarrow> 'b list"
- fix xa ya x y :: "'a list list"
- assume fs: "(op \<approx> ===> op \<approx>) f f'" and x: "list_all2 op \<approx> xa x" and xy: "set x = set y" and y: "list_all2 op \<approx> y ya"
- have a: "(list_all2 op \<approx>) (map f xa) (map f x)"
- using x
- by (induct xa x rule: list_induct2')
- (simp_all, metis fs rel_funE list_eq_def)
- have b: "set (map f x) = set (map f y)"
- using xy fs
- by (induct x y rule: list_induct2')
- (simp_all, metis image_insert)
- have c: "(list_all2 op \<approx>) (map f y) (map f' ya)"
- using y fs
- by (induct y ya rule: list_induct2')
- (simp_all, metis apply_rsp' list_eq_def)
- show "(list_all2 op \<approx> OOO op \<approx>) (map f xa) (map f' ya)"
- by (metis a b c list_eq_def relcomppI)
-qed
-
-lemma map_prs2 [quot_preserve]:
- shows "((abs_fset ---> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> abs_fset \<circ> map abs_fset) map = (id ---> rep_fset ---> abs_fset) map"
- by (auto simp add: fun_eq_iff)
- (simp only: map_map[symmetric] map_prs_aux[OF Quotient3_fset Quotient3_fset])
-
-section \<open>Lifted theorems\<close>
-
-subsection \<open>fset\<close>
-
-lemma fset_simps [simp]:
- shows "fset {||} = {}"
- and "fset (insert_fset x S) = insert x (fset S)"
- by (descending, simp)+
-
-lemma finite_fset [simp]:
- shows "finite (fset S)"
- by (descending) (simp)
-
-lemma fset_cong:
- shows "fset S = fset T \<longleftrightarrow> S = T"
- by (descending) (simp)
-
-lemma filter_fset [simp]:
- shows "fset (filter_fset P xs) = Collect P \<inter> fset xs"
- by (descending) (auto)
-
-lemma remove_fset [simp]:
- shows "fset (remove_fset x xs) = fset xs - {x}"
- by (descending) (simp)
-
-lemma inter_fset [simp]:
- shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
- by (descending) (auto)
-
-lemma union_fset [simp]:
- shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
- by (lifting set_append)
-
-lemma minus_fset [simp]:
- shows "fset (xs - ys) = fset xs - fset ys"
- by (descending) (auto)
-
-
-subsection \<open>in_fset\<close>
-
-lemma in_fset:
- shows "x |\<in>| S \<longleftrightarrow> x \<in> fset S"
- by descending simp
-
-lemma notin_fset:
- shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"
- by (simp add: in_fset)
-
-lemma notin_empty_fset:
- shows "x |\<notin>| {||}"
- by (simp add: in_fset)
-
-lemma fset_eq_iff:
- shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
- by descending auto
-
-lemma none_in_empty_fset:
- shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
- by descending simp
-
-
-subsection \<open>insert_fset\<close>
-
-lemma in_insert_fset_iff [simp]:
- shows "x |\<in>| insert_fset y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
- by descending simp
-
-lemma
- shows insert_fsetI1: "x |\<in>| insert_fset x S"
- and insert_fsetI2: "x |\<in>| S \<Longrightarrow> x |\<in>| insert_fset y S"
- by simp_all
-
-lemma insert_absorb_fset [simp]:
- shows "x |\<in>| S \<Longrightarrow> insert_fset x S = S"
- by (descending) (auto)
-
-lemma empty_not_insert_fset[simp]:
- shows "{||} \<noteq> insert_fset x S"
- and "insert_fset x S \<noteq> {||}"
- by (descending, simp)+
-
-lemma insert_fset_left_comm:
- shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
- by (descending) (auto)
-
-lemma insert_fset_left_idem:
- shows "insert_fset x (insert_fset x S) = insert_fset x S"
- by (descending) (auto)
-
-lemma singleton_fset_eq[simp]:
- shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
- by (descending) (auto)
-
-lemma in_fset_mdef:
- shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> F = insert_fset x (F - {|x|})"
- by (descending) (auto)
-
-
-subsection \<open>union_fset\<close>
-
-lemmas [simp] =
- sup_bot_left[where 'a="'a fset"]
- sup_bot_right[where 'a="'a fset"]
-
-lemma union_insert_fset [simp]:
- shows "insert_fset x S |\<union>| T = insert_fset x (S |\<union>| T)"
- by (lifting append.simps(2))
-
-lemma singleton_union_fset_left:
- shows "{|a|} |\<union>| S = insert_fset a S"
- by simp
-
-lemma singleton_union_fset_right:
- shows "S |\<union>| {|a|} = insert_fset a S"
- by (subst sup.commute) simp
-
-lemma in_union_fset:
- shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
- by (descending) (simp)
-
-
-subsection \<open>minus_fset\<close>
-
-lemma minus_in_fset:
- shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
- by (descending) (simp)
-
-lemma minus_insert_fset:
- shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
- by (descending) (auto)
-
-lemma minus_insert_in_fset[simp]:
- shows "x |\<in>| ys \<Longrightarrow> insert_fset x xs - ys = xs - ys"
- by (simp add: minus_insert_fset)
-
-lemma minus_insert_notin_fset[simp]:
- shows "x |\<notin>| ys \<Longrightarrow> insert_fset x xs - ys = insert_fset x (xs - ys)"
- by (simp add: minus_insert_fset)
-
-lemma in_minus_fset:
- shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
- unfolding in_fset minus_fset
- by blast
-
-lemma notin_minus_fset:
- shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
- unfolding in_fset minus_fset
- by blast
-
-
-subsection \<open>remove_fset\<close>
-
-lemma in_remove_fset:
- shows "x |\<in>| remove_fset y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
- by (descending) (simp)
-
-lemma notin_remove_fset:
- shows "x |\<notin>| remove_fset x S"
- by (descending) (simp)
-
-lemma notin_remove_ident_fset:
- shows "x |\<notin>| S \<Longrightarrow> remove_fset x S = S"
- by (descending) (simp)
-
-lemma remove_fset_cases:
- shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = insert_fset x (remove_fset x S))"
- by (descending) (auto simp add: insert_absorb)
-
-
-subsection \<open>inter_fset\<close>
-
-lemma inter_empty_fset_l:
- shows "{||} |\<inter>| S = {||}"
- by simp
-
-lemma inter_empty_fset_r:
- shows "S |\<inter>| {||} = {||}"
- by simp
-
-lemma inter_insert_fset:
- shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
- by (descending) (auto)
-
-lemma in_inter_fset:
- shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
- by (descending) (simp)
-
-
-subsection \<open>subset_fset and psubset_fset\<close>
-
-lemma subset_fset:
- shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
- by (descending) (simp)
-
-lemma psubset_fset:
- shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
- unfolding less_fset_def
- by (descending) (auto)
-
-lemma subset_insert_fset:
- shows "(insert_fset x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
- by (descending) (simp)
-
-lemma subset_in_fset:
- shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
- by (descending) (auto)
-
-lemma subset_empty_fset:
- shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
- by (descending) (simp)
-
-lemma not_psubset_empty_fset:
- shows "\<not> xs |\<subset>| {||}"
- by (metis fset_simps(1) psubset_fset not_psubset_empty)
-
-
-subsection \<open>map_fset\<close>
-
-lemma map_fset_simps [simp]:
- shows "map_fset f {||} = {||}"
- and "map_fset f (insert_fset x S) = insert_fset (f x) (map_fset f S)"
- by (descending, simp)+
-
-lemma map_fset_image [simp]:
- shows "fset (map_fset f S) = f ` (fset S)"
- by (descending) (simp)
-
-lemma inj_map_fset_cong:
- shows "inj f \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T"
- by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
-
-lemma map_union_fset:
- shows "map_fset f (S |\<union>| T) = map_fset f S |\<union>| map_fset f T"
- by (descending) (simp)
-
-lemma in_fset_map_fset[simp]: "a |\<in>| map_fset f X = (\<exists>b. b |\<in>| X \<and> a = f b)"
- by descending auto
-
-
-subsection \<open>card_fset\<close>
-
-lemma card_fset:
- shows "card_fset xs = card (fset xs)"
- by (descending) (simp)
-
-lemma card_insert_fset_iff [simp]:
- shows "card_fset (insert_fset x S) = (if x |\<in>| S then card_fset S else Suc (card_fset S))"
- by (descending) (simp add: insert_absorb)
-
-lemma card_fset_0[simp]:
- shows "card_fset S = 0 \<longleftrightarrow> S = {||}"
- by (descending) (simp)
-
-lemma card_empty_fset[simp]:
- shows "card_fset {||} = 0"
- by (simp add: card_fset)
-
-lemma card_fset_1:
- shows "card_fset S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
- by (descending) (auto simp add: card_Suc_eq)
-
-lemma card_fset_gt_0:
- shows "x \<in> fset S \<Longrightarrow> 0 < card_fset S"
- by (descending) (auto simp add: card_gt_0_iff)
-
-lemma card_notin_fset:
- shows "(x |\<notin>| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
- by simp
-
-lemma card_fset_Suc:
- shows "card_fset S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> card_fset T = n"
- apply(descending)
- apply(auto dest!: card_eq_SucD)
- by (metis Diff_insert_absorb set_removeAll)
-
-lemma card_remove_fset_iff [simp]:
- shows "card_fset (remove_fset y S) = (if y |\<in>| S then card_fset S - 1 else card_fset S)"
- by (descending) (simp)
-
-lemma card_Suc_exists_in_fset:
- shows "card_fset S = Suc n \<Longrightarrow> \<exists>a. a |\<in>| S"
- by (drule card_fset_Suc) (auto)
-
-lemma in_card_fset_not_0:
- shows "a |\<in>| A \<Longrightarrow> card_fset A \<noteq> 0"
- by (descending) (auto)
-
-lemma card_fset_mono:
- shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset xs \<le> card_fset ys"
- unfolding card_fset psubset_fset
- by (simp add: card_mono subset_fset)
-
-lemma card_subset_fset_eq:
- shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset ys \<le> card_fset xs \<Longrightarrow> xs = ys"
- unfolding card_fset subset_fset
- by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)
-
-lemma psubset_card_fset_mono:
- shows "xs |\<subset>| ys \<Longrightarrow> card_fset xs < card_fset ys"
- unfolding card_fset subset_fset
- by (metis finite_fset psubset_fset psubset_card_mono)
-
-lemma card_union_inter_fset:
- shows "card_fset xs + card_fset ys = card_fset (xs |\<union>| ys) + card_fset (xs |\<inter>| ys)"
- unfolding card_fset union_fset inter_fset
- by (rule card_Un_Int[OF finite_fset finite_fset])
-
-lemma card_union_disjoint_fset:
- shows "xs |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| ys) = card_fset xs + card_fset ys"
- unfolding card_fset union_fset
- apply (rule card_Un_disjoint[OF finite_fset finite_fset])
- by (metis inter_fset fset_simps(1))
-
-lemma card_remove_fset_less1:
- shows "x |\<in>| xs \<Longrightarrow> card_fset (remove_fset x xs) < card_fset xs"
- unfolding card_fset in_fset remove_fset
- by (rule card_Diff1_less[OF finite_fset])
-
-lemma card_remove_fset_less2:
- shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> card_fset (remove_fset y (remove_fset x xs)) < card_fset xs"
- unfolding card_fset remove_fset in_fset
- by (rule card_Diff2_less[OF finite_fset])
-
-lemma card_remove_fset_le1:
- shows "card_fset (remove_fset x xs) \<le> card_fset xs"
- unfolding remove_fset card_fset
- by (rule card_Diff1_le[OF finite_fset])
-
-lemma card_psubset_fset:
- shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| xs"
- unfolding card_fset psubset_fset subset_fset
- by (rule card_psubset[OF finite_fset])
-
-lemma card_map_fset_le:
- shows "card_fset (map_fset f xs) \<le> card_fset xs"
- unfolding card_fset map_fset_image
- by (rule card_image_le[OF finite_fset])
-
-lemma card_minus_insert_fset[simp]:
- assumes "a |\<in>| A" and "a |\<notin>| B"
- shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
- using assms
- unfolding in_fset card_fset minus_fset
- by (simp add: card_Diff_insert[OF finite_fset])
-
-lemma card_minus_subset_fset:
- assumes "B |\<subseteq>| A"
- shows "card_fset (A - B) = card_fset A - card_fset B"
- using assms
- unfolding subset_fset card_fset minus_fset
- by (rule card_Diff_subset[OF finite_fset])
-
-lemma card_minus_fset:
- shows "card_fset (A - B) = card_fset A - card_fset (A |\<inter>| B)"
- unfolding inter_fset card_fset minus_fset
- by (rule card_Diff_subset_Int) (simp)
-
-
-subsection \<open>concat_fset\<close>
-
-lemma concat_empty_fset [simp]:
- shows "concat_fset {||} = {||}"
- by descending simp
-
-lemma concat_insert_fset [simp]:
- shows "concat_fset (insert_fset x S) = x |\<union>| concat_fset S"
- by descending simp
-
-lemma concat_union_fset [simp]:
- shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| concat_fset ys"
- by descending simp
-
-lemma map_concat_fset:
- shows "map_fset f (concat_fset xs) = concat_fset (map_fset (map_fset f) xs)"
- by (lifting map_concat)
-
-subsection \<open>filter_fset\<close>
-
-lemma subset_filter_fset:
- "filter_fset P xs |\<subseteq>| filter_fset Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
- by descending auto
-
-lemma eq_filter_fset:
- "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
- by descending auto
-
-lemma psubset_filter_fset:
- "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow>
- filter_fset P xs |\<subset>| filter_fset Q xs"
- unfolding less_fset_def by (auto simp add: subset_filter_fset eq_filter_fset)
-
-
-subsection \<open>fold_fset\<close>
-
-lemma fold_empty_fset:
- "fold_fset f {||} = id"
- by descending (simp add: fold_once_def)
-
-lemma fold_insert_fset: "fold_fset f (insert_fset a A) =
- (if rsp_fold f then if a |\<in>| A then fold_fset f A else fold_fset f A \<circ> f a else id)"
- by descending (simp add: fold_once_fold_remdups)
-
-lemma remdups_removeAll:
- "remdups (removeAll x xs) = remove1 x (remdups xs)"
- by (induct xs) auto
-
-lemma member_commute_fold_once:
- assumes "rsp_fold f"
- and "x \<in> set xs"
- shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
-proof -
- from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \<circ> f x"
- by (auto intro!: fold_remove1_split elim: rsp_foldE)
- then show ?thesis using \<open>rsp_fold f\<close> by (simp add: fold_once_fold_remdups remdups_removeAll)
-qed
-
-lemma in_commute_fold_fset:
- "rsp_fold f \<Longrightarrow> h |\<in>| b \<Longrightarrow> fold_fset f b = fold_fset f (remove_fset h b) \<circ> f h"
- by descending (simp add: member_commute_fold_once)
-
-
-subsection \<open>Choice in fsets\<close>
-
-lemma fset_choice:
- assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
- shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
- using a
- apply(descending)
- using finite_set_choice
- by (auto simp add: Ball_def)
-
-
-section \<open>Induction and Cases rules for fsets\<close>
-
-lemma fset_exhaust [case_names empty insert, cases type: fset]:
- assumes empty_fset_case: "S = {||} \<Longrightarrow> P"
- and insert_fset_case: "\<And>x S'. S = insert_fset x S' \<Longrightarrow> P"
- shows "P"
- using assms by (lifting list.exhaust)
-
-lemma fset_induct [case_names empty insert]:
- assumes empty_fset_case: "P {||}"
- and insert_fset_case: "\<And>x S. P S \<Longrightarrow> P (insert_fset x S)"
- shows "P S"
- using assms
- by (descending) (blast intro: list.induct)
-
-lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
- assumes empty_fset_case: "P {||}"
- and insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (insert_fset x S)"
- shows "P S"
-proof(induct S rule: fset_induct)
- case empty
- show "P {||}" using empty_fset_case by simp
-next
- case (insert x S)
- have "P S" by fact
- then show "P (insert_fset x S)" using insert_fset_case
- by (cases "x |\<in>| S") (simp_all)
-qed
-
-lemma fset_card_induct:
- assumes empty_fset_case: "P {||}"
- and card_fset_Suc_case: "\<And>S T. Suc (card_fset S) = (card_fset T) \<Longrightarrow> P S \<Longrightarrow> P T"
- shows "P S"
-proof (induct S)
- case empty
- show "P {||}" by (rule empty_fset_case)
-next
- case (insert x S)
- have h: "P S" by fact
- have "x |\<notin>| S" by fact
- then have "Suc (card_fset S) = card_fset (insert_fset x S)"
- using card_fset_Suc by auto
- then show "P (insert_fset x S)"
- using h card_fset_Suc_case by simp
-qed
-
-lemma fset_raw_strong_cases:
- obtains "xs = []"
- | ys x where "\<not> List.member ys x" and "xs \<approx> x # ys"
-proof (induct xs)
- case Nil
- then show thesis by simp
-next
- case (Cons a xs)
- have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
- by (rule Cons(1))
- have b: "\<And>x' ys'. \<lbrakk>\<not> List.member ys' x'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
- have c: "xs = [] \<Longrightarrow> thesis" using b
- apply(simp)
- by (metis list.set(1) emptyE empty_subsetI)
- have "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
- proof -
- fix x :: 'a
- fix ys :: "'a list"
- assume d:"\<not> List.member ys x"
- assume e:"xs \<approx> x # ys"
- show thesis
- proof (cases "x = a")
- assume h: "x = a"
- then have f: "\<not> List.member ys a" using d by simp
- have g: "a # xs \<approx> a # ys" using e h by auto
- show thesis using b f g by simp
- next
- assume h: "x \<noteq> a"
- then have f: "\<not> List.member (a # ys) x" using d by auto
- have g: "a # xs \<approx> x # (a # ys)" using e h by auto
- show thesis using b f g by (simp del: List.member_def)
- qed
- qed
- then show thesis using a c by blast
-qed
-
-
-lemma fset_strong_cases:
- obtains "xs = {||}"
- | ys x where "x |\<notin>| ys" and "xs = insert_fset x ys"
- by (lifting fset_raw_strong_cases)
-
-
-lemma fset_induct2:
- "P {||} {||} \<Longrightarrow>
- (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (insert_fset x xs) {||}) \<Longrightarrow>
- (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (insert_fset y ys)) \<Longrightarrow>
- (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (insert_fset x xs) (insert_fset y ys)) \<Longrightarrow>
- P xsa ysa"
- apply (induct xsa arbitrary: ysa)
- apply (induct_tac x rule: fset_induct_stronger)
- apply simp_all
- apply (induct_tac xa rule: fset_induct_stronger)
- apply simp_all
- done
-
-text \<open>Extensionality\<close>
-
-lemma fset_eqI:
- assumes "\<And>x. x \<in> fset A \<longleftrightarrow> x \<in> fset B"
- shows "A = B"
-using assms proof (induct A arbitrary: B)
- case empty then show ?case
- by (auto simp add: in_fset none_in_empty_fset [symmetric] sym)
-next
- case (insert x A)
- from insert.prems insert.hyps(1) have "\<And>z. z \<in> fset A \<longleftrightarrow> z \<in> fset (B - {|x|})"
- by (auto simp add: in_fset)
- then have A: "A = B - {|x|}" by (rule insert.hyps(2))
- with insert.prems [symmetric, of x] have "x |\<in>| B" by (simp add: in_fset)
- with A show ?case by (metis in_fset_mdef)
-qed
-
-subsection \<open>alternate formulation with a different decomposition principle
- and a proof of equivalence\<close>
-
-inductive
- list_eq2 :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>2 _")
-where
- "(a # b # xs) \<approx>2 (b # a # xs)"
-| "[] \<approx>2 []"
-| "xs \<approx>2 ys \<Longrightarrow> ys \<approx>2 xs"
-| "(a # a # xs) \<approx>2 (a # xs)"
-| "xs \<approx>2 ys \<Longrightarrow> (a # xs) \<approx>2 (a # ys)"
-| "xs1 \<approx>2 xs2 \<Longrightarrow> xs2 \<approx>2 xs3 \<Longrightarrow> xs1 \<approx>2 xs3"
-
-lemma list_eq2_refl:
- shows "xs \<approx>2 xs"
- by (induct xs) (auto intro: list_eq2.intros)
-
-lemma cons_delete_list_eq2:
- shows "(a # (removeAll a A)) \<approx>2 (if List.member A a then A else a # A)"
- apply (induct A)
- apply (simp add: list_eq2_refl)
- apply (case_tac "List.member (aa # A) a")
- apply (simp_all)
- apply (case_tac [!] "a = aa")
- apply (simp_all)
- apply (case_tac "List.member A a")
- apply (auto)[2]
- apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
- apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
- apply (auto simp add: list_eq2_refl)
- done
-
-lemma member_delete_list_eq2:
- assumes a: "List.member r e"
- shows "(e # removeAll e r) \<approx>2 r"
- using a cons_delete_list_eq2[of e r]
- by simp
-
-lemma list_eq2_equiv:
- "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
-proof
- show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
-next
- {
- fix n
- assume a: "card_list l = n" and b: "l \<approx> r"
- have "l \<approx>2 r"
- using a b
- proof (induct n arbitrary: l r)
- case 0
- have "card_list l = 0" by fact
- then have "\<forall>x. \<not> List.member l x" by auto
- then have z: "l = []" by auto
- then have "r = []" using \<open>l \<approx> r\<close> by simp
- then show ?case using z list_eq2_refl by simp
- next
- case (Suc m)
- have b: "l \<approx> r" by fact
- have d: "card_list l = Suc m" by fact
- then have "\<exists>a. List.member l a"
- apply(simp)
- apply(drule card_eq_SucD)
- apply(blast)
- done
- then obtain a where e: "List.member l a" by auto
- then have e': "List.member r a" using list_eq_def [simplified List.member_def [symmetric], of l r] b
- by auto
- have f: "card_list (removeAll a l) = m" using e d by (simp)
- have g: "removeAll a l \<approx> removeAll a r" using remove_fset.rsp b by simp
- have "(removeAll a l) \<approx>2 (removeAll a r)" by (rule Suc.hyps[OF f g])
- then have h: "(a # removeAll a l) \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(5))
- have i: "l \<approx>2 (a # removeAll a l)"
- by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
- have "l \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
- then show ?case using list_eq2.intros(6)[OF _ member_delete_list_eq2[OF e']] by simp
- qed
- }
- then show "l \<approx> r \<Longrightarrow> l \<approx>2 r" by blast
-qed
-
-
-(* We cannot write it as "assumes .. shows" since Isabelle changes
- the quantifiers to schematic variables and reintroduces them in
- a different order *)
-lemma fset_eq_cases:
- "\<lbrakk>a1 = a2;
- \<And>a b xs. \<lbrakk>a1 = insert_fset a (insert_fset b xs); a2 = insert_fset b (insert_fset a xs)\<rbrakk> \<Longrightarrow> P;
- \<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
- \<And>a xs. \<lbrakk>a1 = insert_fset a (insert_fset a xs); a2 = insert_fset a xs\<rbrakk> \<Longrightarrow> P;
- \<And>xs ys a. \<lbrakk>a1 = insert_fset a xs; a2 = insert_fset a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
- \<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
- \<Longrightarrow> P"
- by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
-
-lemma fset_eq_induct:
- assumes "x1 = x2"
- and "\<And>a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
- and "P {||} {||}"
- and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
- and "\<And>a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
- and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (insert_fset a xs) (insert_fset a ys)"
- and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"
- shows "P x1 x2"
- using assms
- by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
-
-ML \<open>
-fun dest_fsetT (Type (@{type_name fset}, [T])) = T
- | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
-\<close>
-
-no_notation
- list_eq (infix "\<approx>" 50) and
- list_eq2 (infix "\<approx>2" 50)
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Quotient_Examples/Quotient_FSet.thy Tue Sep 20 11:35:10 2016 +0200
@@ -0,0 +1,1163 @@
+(* Title: HOL/Quotient_Examples/Quotient_FSet.thy
+ Author: Cezary Kaliszyk, TU Munich
+ Author: Christian Urban, TU Munich
+
+Type of finite sets.
+*)
+
+(********************************************************************
+ WARNING: There is a formalization of 'a fset as a subtype of sets in
+ HOL/Library/FSet.thy using Lifting/Transfer. The user should use
+ that file rather than this file unless there are some very specific
+ reasons.
+*********************************************************************)
+
+theory Quotient_FSet
+imports "~~/src/HOL/Library/Multiset" "~~/src/HOL/Library/Quotient_List"
+begin
+
+text \<open>
+ The type of finite sets is created by a quotient construction
+ over lists. The definition of the equivalence:
+\<close>
+
+definition
+ list_eq :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infix "\<approx>" 50)
+where
+ [simp]: "xs \<approx> ys \<longleftrightarrow> set xs = set ys"
+
+lemma list_eq_reflp:
+ "reflp list_eq"
+ by (auto intro: reflpI)
+
+lemma list_eq_symp:
+ "symp list_eq"
+ by (auto intro: sympI)
+
+lemma list_eq_transp:
+ "transp list_eq"
+ by (auto intro: transpI)
+
+lemma list_eq_equivp:
+ "equivp list_eq"
+ by (auto intro: equivpI list_eq_reflp list_eq_symp list_eq_transp)
+
+text \<open>The \<open>fset\<close> type\<close>
+
+quotient_type
+ 'a fset = "'a list" / "list_eq"
+ by (rule list_eq_equivp)
+
+text \<open>
+ Definitions for sublist, cardinality,
+ intersection, difference and respectful fold over
+ lists.
+\<close>
+
+declare List.member_def [simp]
+
+definition
+ sub_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"
+where
+ [simp]: "sub_list xs ys \<longleftrightarrow> set xs \<subseteq> set ys"
+
+definition
+ card_list :: "'a list \<Rightarrow> nat"
+where
+ [simp]: "card_list xs = card (set xs)"
+
+definition
+ inter_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ [simp]: "inter_list xs ys = [x \<leftarrow> xs. x \<in> set xs \<and> x \<in> set ys]"
+
+definition
+ diff_list :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list"
+where
+ [simp]: "diff_list xs ys = [x \<leftarrow> xs. x \<notin> set ys]"
+
+definition
+ rsp_fold :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
+where
+ "rsp_fold f \<longleftrightarrow> (\<forall>u v. f u \<circ> f v = f v \<circ> f u)"
+
+lemma rsp_foldI:
+ "(\<And>u v. f u \<circ> f v = f v \<circ> f u) \<Longrightarrow> rsp_fold f"
+ by (simp add: rsp_fold_def)
+
+lemma rsp_foldE:
+ assumes "rsp_fold f"
+ obtains "f u \<circ> f v = f v \<circ> f u"
+ using assms by (simp add: rsp_fold_def)
+
+definition
+ fold_once :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a list \<Rightarrow> 'b \<Rightarrow> 'b"
+where
+ "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
+
+lemma fold_once_default [simp]:
+ "\<not> rsp_fold f \<Longrightarrow> fold_once f xs = id"
+ by (simp add: fold_once_def)
+
+lemma fold_once_fold_remdups:
+ "rsp_fold f \<Longrightarrow> fold_once f xs = fold f (remdups xs)"
+ by (simp add: fold_once_def)
+
+
+section \<open>Quotient composition lemmas\<close>
+
+lemma list_all2_refl':
+ assumes q: "equivp R"
+ shows "(list_all2 R) r r"
+ by (rule list_all2_refl) (metis equivp_def q)
+
+lemma compose_list_refl:
+ assumes q: "equivp R"
+ shows "(list_all2 R OOO op \<approx>) r r"
+proof
+ have *: "r \<approx> r" by (rule equivp_reflp[OF fset_equivp])
+ show "list_all2 R r r" by (rule list_all2_refl'[OF q])
+ with * show "(op \<approx> OO list_all2 R) r r" ..
+qed
+
+lemma map_list_eq_cong: "b \<approx> ba \<Longrightarrow> map f b \<approx> map f ba"
+ by (simp only: list_eq_def set_map)
+
+lemma quotient_compose_list_g:
+ assumes q: "Quotient3 R Abs Rep"
+ and e: "equivp R"
+ shows "Quotient3 ((list_all2 R) OOO (op \<approx>))
+ (abs_fset \<circ> (map Abs)) ((map Rep) \<circ> rep_fset)"
+ unfolding Quotient3_def comp_def
+proof (intro conjI allI)
+ fix a r s
+ show "abs_fset (map Abs (map Rep (rep_fset a))) = a"
+ by (simp add: abs_o_rep[OF q] Quotient3_abs_rep[OF Quotient3_fset] List.map.id)
+ have b: "list_all2 R (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+ by (rule list_all2_refl'[OF e])
+ have c: "(op \<approx> OO list_all2 R) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+ by (rule, rule equivp_reflp[OF fset_equivp]) (rule b)
+ show "(list_all2 R OOO op \<approx>) (map Rep (rep_fset a)) (map Rep (rep_fset a))"
+ by (rule, rule list_all2_refl'[OF e]) (rule c)
+ show "(list_all2 R OOO op \<approx>) r s = ((list_all2 R OOO op \<approx>) r r \<and>
+ (list_all2 R OOO op \<approx>) s s \<and> abs_fset (map Abs r) = abs_fset (map Abs s))"
+ proof (intro iffI conjI)
+ show "(list_all2 R OOO op \<approx>) r r" by (rule compose_list_refl[OF e])
+ show "(list_all2 R OOO op \<approx>) s s" by (rule compose_list_refl[OF e])
+ next
+ assume a: "(list_all2 R OOO op \<approx>) r s"
+ then have b: "map Abs r \<approx> map Abs s"
+ proof (elim relcomppE)
+ fix b ba
+ assume c: "list_all2 R r b"
+ assume d: "b \<approx> ba"
+ assume e: "list_all2 R ba s"
+ have f: "map Abs r = map Abs b"
+ using Quotient3_rel[OF list_quotient3[OF q]] c by blast
+ have "map Abs ba = map Abs s"
+ using Quotient3_rel[OF list_quotient3[OF q]] e by blast
+ then have g: "map Abs s = map Abs ba" by simp
+ then show "map Abs r \<approx> map Abs s" using d f map_list_eq_cong by simp
+ qed
+ then show "abs_fset (map Abs r) = abs_fset (map Abs s)"
+ using Quotient3_rel[OF Quotient3_fset] by blast
+ next
+ assume a: "(list_all2 R OOO op \<approx>) r r \<and> (list_all2 R OOO op \<approx>) s s
+ \<and> abs_fset (map Abs r) = abs_fset (map Abs s)"
+ then have s: "(list_all2 R OOO op \<approx>) s s" by simp
+ have d: "map Abs r \<approx> map Abs s"
+ by (subst Quotient3_rel [OF Quotient3_fset, symmetric]) (simp add: a)
+ have b: "map Rep (map Abs r) \<approx> map Rep (map Abs s)"
+ by (rule map_list_eq_cong[OF d])
+ have y: "list_all2 R (map Rep (map Abs s)) s"
+ by (fact rep_abs_rsp_left[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of s]])
+ have c: "(op \<approx> OO list_all2 R) (map Rep (map Abs r)) s"
+ by (rule relcomppI) (rule b, rule y)
+ have z: "list_all2 R r (map Rep (map Abs r))"
+ by (fact rep_abs_rsp[OF list_quotient3[OF q], OF list_all2_refl'[OF e, of r]])
+ then show "(list_all2 R OOO op \<approx>) r s"
+ using a c relcomppI by simp
+ qed
+qed
+
+lemma quotient_compose_list[quot_thm]:
+ shows "Quotient3 ((list_all2 op \<approx>) OOO (op \<approx>))
+ (abs_fset \<circ> (map abs_fset)) ((map rep_fset) \<circ> rep_fset)"
+ by (rule quotient_compose_list_g, rule Quotient3_fset, rule list_eq_equivp)
+
+
+section \<open>Quotient definitions for fsets\<close>
+
+
+subsection \<open>Finite sets are a bounded, distributive lattice with minus\<close>
+
+instantiation fset :: (type) "{bounded_lattice_bot, distrib_lattice, minus}"
+begin
+
+quotient_definition
+ "bot :: 'a fset"
+ is "Nil :: 'a list" done
+
+abbreviation
+ empty_fset ("{||}")
+where
+ "{||} \<equiv> bot :: 'a fset"
+
+quotient_definition
+ "less_eq_fset :: ('a fset \<Rightarrow> 'a fset \<Rightarrow> bool)"
+ is "sub_list :: ('a list \<Rightarrow> 'a list \<Rightarrow> bool)" by simp
+
+abbreviation
+ subset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subseteq>|" 50)
+where
+ "xs |\<subseteq>| ys \<equiv> xs \<le> ys"
+
+definition
+ less_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool"
+where
+ "xs < ys \<equiv> xs \<le> ys \<and> xs \<noteq> (ys::'a fset)"
+
+abbreviation
+ psubset_fset :: "'a fset \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<subset>|" 50)
+where
+ "xs |\<subset>| ys \<equiv> xs < ys"
+
+quotient_definition
+ "sup :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "append :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
+
+abbreviation
+ union_fset (infixl "|\<union>|" 65)
+where
+ "xs |\<union>| ys \<equiv> sup xs (ys::'a fset)"
+
+quotient_definition
+ "inf :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "inter_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by simp
+
+abbreviation
+ inter_fset (infixl "|\<inter>|" 65)
+where
+ "xs |\<inter>| ys \<equiv> inf xs (ys::'a fset)"
+
+quotient_definition
+ "minus :: 'a fset \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "diff_list :: 'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" by fastforce
+
+instance
+proof
+ fix x y z :: "'a fset"
+ show "x |\<subset>| y \<longleftrightarrow> x |\<subseteq>| y \<and> \<not> y |\<subseteq>| x"
+ by (unfold less_fset_def, descending) auto
+ show "x |\<subseteq>| x" by (descending) (simp)
+ show "{||} |\<subseteq>| x" by (descending) (simp)
+ show "x |\<subseteq>| x |\<union>| y" by (descending) (simp)
+ show "y |\<subseteq>| x |\<union>| y" by (descending) (simp)
+ show "x |\<inter>| y |\<subseteq>| x" by (descending) (auto)
+ show "x |\<inter>| y |\<subseteq>| y" by (descending) (auto)
+ show "x |\<union>| (y |\<inter>| z) = x |\<union>| y |\<inter>| (x |\<union>| z)"
+ by (descending) (auto)
+next
+ fix x y z :: "'a fset"
+ assume a: "x |\<subseteq>| y"
+ assume b: "y |\<subseteq>| z"
+ show "x |\<subseteq>| z" using a b by (descending) (simp)
+next
+ fix x y :: "'a fset"
+ assume a: "x |\<subseteq>| y"
+ assume b: "y |\<subseteq>| x"
+ show "x = y" using a b by (descending) (auto)
+next
+ fix x y z :: "'a fset"
+ assume a: "y |\<subseteq>| x"
+ assume b: "z |\<subseteq>| x"
+ show "y |\<union>| z |\<subseteq>| x" using a b by (descending) (simp)
+next
+ fix x y z :: "'a fset"
+ assume a: "x |\<subseteq>| y"
+ assume b: "x |\<subseteq>| z"
+ show "x |\<subseteq>| y |\<inter>| z" using a b by (descending) (auto)
+qed
+
+end
+
+
+subsection \<open>Other constants for fsets\<close>
+
+quotient_definition
+ "insert_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is "Cons" by auto
+
+syntax
+ "_insert_fset" :: "args => 'a fset" ("{|(_)|}")
+
+translations
+ "{|x, xs|}" == "CONST insert_fset x {|xs|}"
+ "{|x|}" == "CONST insert_fset x {||}"
+
+quotient_definition
+ fset_member
+where
+ "fset_member :: 'a fset \<Rightarrow> 'a \<Rightarrow> bool" is "List.member" by fastforce
+
+abbreviation
+ in_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<in>|" 50)
+where
+ "x |\<in>| S \<equiv> fset_member S x"
+
+abbreviation
+ notin_fset :: "'a \<Rightarrow> 'a fset \<Rightarrow> bool" (infix "|\<notin>|" 50)
+where
+ "x |\<notin>| S \<equiv> \<not> (x |\<in>| S)"
+
+
+subsection \<open>Other constants on the Quotient Type\<close>
+
+quotient_definition
+ "card_fset :: 'a fset \<Rightarrow> nat"
+ is card_list by simp
+
+quotient_definition
+ "map_fset :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b fset"
+ is map by simp
+
+quotient_definition
+ "remove_fset :: 'a \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is removeAll by simp
+
+quotient_definition
+ "fset :: 'a fset \<Rightarrow> 'a set"
+ is "set" by simp
+
+lemma fold_once_set_equiv:
+ assumes "xs \<approx> ys"
+ shows "fold_once f xs = fold_once f ys"
+proof (cases "rsp_fold f")
+ case False then show ?thesis by simp
+next
+ case True
+ then have "\<And>x y. x \<in> set (remdups xs) \<Longrightarrow> y \<in> set (remdups xs) \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"
+ by (rule rsp_foldE)
+ moreover from assms have "mset (remdups xs) = mset (remdups ys)"
+ by (simp add: set_eq_iff_mset_remdups_eq)
+ ultimately have "fold f (remdups xs) = fold f (remdups ys)"
+ by (rule fold_multiset_equiv)
+ with True show ?thesis by (simp add: fold_once_fold_remdups)
+qed
+
+quotient_definition
+ "fold_fset :: ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'a fset \<Rightarrow> 'b \<Rightarrow> 'b"
+ is fold_once by (rule fold_once_set_equiv)
+
+lemma concat_rsp_pre:
+ assumes a: "list_all2 op \<approx> x x'"
+ and b: "x' \<approx> y'"
+ and c: "list_all2 op \<approx> y' y"
+ and d: "\<exists>x\<in>set x. xa \<in> set x"
+ shows "\<exists>x\<in>set y. xa \<in> set x"
+proof -
+ obtain xb where e: "xb \<in> set x" and f: "xa \<in> set xb" using d by auto
+ have "\<exists>y. y \<in> set x' \<and> xb \<approx> y" by (rule list_all2_find_element[OF e a])
+ then obtain ya where h: "ya \<in> set x'" and i: "xb \<approx> ya" by auto
+ have "ya \<in> set y'" using b h by simp
+ then have "\<exists>yb. yb \<in> set y \<and> ya \<approx> yb" using c by (rule list_all2_find_element)
+ then show ?thesis using f i by auto
+qed
+
+quotient_definition
+ "concat_fset :: ('a fset) fset \<Rightarrow> 'a fset"
+ is concat
+proof (elim relcomppE)
+fix a b ba bb
+ assume a: "list_all2 op \<approx> a ba"
+ with list_symp [OF list_eq_symp] have a': "list_all2 op \<approx> ba a" by (rule sympE)
+ assume b: "ba \<approx> bb"
+ with list_eq_symp have b': "bb \<approx> ba" by (rule sympE)
+ assume c: "list_all2 op \<approx> bb b"
+ with list_symp [OF list_eq_symp] have c': "list_all2 op \<approx> b bb" by (rule sympE)
+ have "\<forall>x. (\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
+ proof
+ fix x
+ show "(\<exists>xa\<in>set a. x \<in> set xa) = (\<exists>xa\<in>set b. x \<in> set xa)"
+ proof
+ assume d: "\<exists>xa\<in>set a. x \<in> set xa"
+ show "\<exists>xa\<in>set b. x \<in> set xa" by (rule concat_rsp_pre[OF a b c d])
+ next
+ assume e: "\<exists>xa\<in>set b. x \<in> set xa"
+ show "\<exists>xa\<in>set a. x \<in> set xa" by (rule concat_rsp_pre[OF c' b' a' e])
+ qed
+ qed
+ then show "concat a \<approx> concat b" by auto
+qed
+
+quotient_definition
+ "filter_fset :: ('a \<Rightarrow> bool) \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
+ is filter by force
+
+
+subsection \<open>Compositional respectfulness and preservation lemmas\<close>
+
+lemma Nil_rsp2 [quot_respect]:
+ shows "(list_all2 op \<approx> OOO op \<approx>) Nil Nil"
+ by (rule compose_list_refl, rule list_eq_equivp)
+
+lemma Cons_rsp2 [quot_respect]:
+ shows "(op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) Cons Cons"
+ apply (auto intro!: rel_funI)
+ apply (rule_tac b="x # b" in relcomppI)
+ apply auto
+ apply (rule_tac b="x # ba" in relcomppI)
+ apply auto
+ done
+
+lemma Nil_prs2 [quot_preserve]:
+ assumes "Quotient3 R Abs Rep"
+ shows "(Abs \<circ> map f) [] = Abs []"
+ by simp
+
+lemma Cons_prs2 [quot_preserve]:
+ assumes q: "Quotient3 R1 Abs1 Rep1"
+ and r: "Quotient3 R2 Abs2 Rep2"
+ shows "(Rep1 ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) (op #) = (id ---> Rep2 ---> Abs2) (op #)"
+ by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
+
+lemma append_prs2 [quot_preserve]:
+ assumes q: "Quotient3 R1 Abs1 Rep1"
+ and r: "Quotient3 R2 Abs2 Rep2"
+ shows "((map Rep1 \<circ> Rep2) ---> (map Rep1 \<circ> Rep2) ---> (Abs2 \<circ> map Abs1)) op @ =
+ (Rep2 ---> Rep2 ---> Abs2) op @"
+ by (simp add: fun_eq_iff abs_o_rep[OF q] List.map.id)
+
+lemma list_all2_app_l:
+ assumes a: "reflp R"
+ and b: "list_all2 R l r"
+ shows "list_all2 R (z @ l) (z @ r)"
+ using a b by (induct z) (auto elim: reflpE)
+
+lemma append_rsp2_pre0:
+ assumes a:"list_all2 op \<approx> x x'"
+ shows "list_all2 op \<approx> (x @ z) (x' @ z)"
+ using a apply (induct x x' rule: list_induct2')
+ by simp_all (rule list_all2_refl'[OF list_eq_equivp])
+
+lemma append_rsp2_pre1:
+ assumes a:"list_all2 op \<approx> x x'"
+ shows "list_all2 op \<approx> (z @ x) (z @ x')"
+ using a apply (induct x x' arbitrary: z rule: list_induct2')
+ apply (rule list_all2_refl'[OF list_eq_equivp])
+ apply (simp_all del: list_eq_def)
+ apply (rule list_all2_app_l)
+ apply (simp_all add: reflpI)
+ done
+
+lemma append_rsp2_pre:
+ assumes "list_all2 op \<approx> x x'"
+ and "list_all2 op \<approx> z z'"
+ shows "list_all2 op \<approx> (x @ z) (x' @ z')"
+ using assms by (rule list_all2_appendI)
+
+lemma compositional_rsp3:
+ assumes "(R1 ===> R2 ===> R3) C C" and "(R4 ===> R5 ===> R6) C C"
+ shows "(R1 OOO R4 ===> R2 OOO R5 ===> R3 OOO R6) C C"
+ by (auto intro!: rel_funI)
+ (metis (full_types) assms rel_funE relcomppI)
+
+lemma append_rsp2 [quot_respect]:
+ "(list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) append append"
+ by (intro compositional_rsp3)
+ (auto intro!: rel_funI simp add: append_rsp2_pre)
+
+lemma map_rsp2 [quot_respect]:
+ "((op \<approx> ===> op \<approx>) ===> list_all2 op \<approx> OOO op \<approx> ===> list_all2 op \<approx> OOO op \<approx>) map map"
+proof (auto intro!: rel_funI)
+ fix f f' :: "'a list \<Rightarrow> 'b list"
+ fix xa ya x y :: "'a list list"
+ assume fs: "(op \<approx> ===> op \<approx>) f f'" and x: "list_all2 op \<approx> xa x" and xy: "set x = set y" and y: "list_all2 op \<approx> y ya"
+ have a: "(list_all2 op \<approx>) (map f xa) (map f x)"
+ using x
+ by (induct xa x rule: list_induct2')
+ (simp_all, metis fs rel_funE list_eq_def)
+ have b: "set (map f x) = set (map f y)"
+ using xy fs
+ by (induct x y rule: list_induct2')
+ (simp_all, metis image_insert)
+ have c: "(list_all2 op \<approx>) (map f y) (map f' ya)"
+ using y fs
+ by (induct y ya rule: list_induct2')
+ (simp_all, metis apply_rsp' list_eq_def)
+ show "(list_all2 op \<approx> OOO op \<approx>) (map f xa) (map f' ya)"
+ by (metis a b c list_eq_def relcomppI)
+qed
+
+lemma map_prs2 [quot_preserve]:
+ shows "((abs_fset ---> rep_fset) ---> (map rep_fset \<circ> rep_fset) ---> abs_fset \<circ> map abs_fset) map = (id ---> rep_fset ---> abs_fset) map"
+ by (auto simp add: fun_eq_iff)
+ (simp only: map_map[symmetric] map_prs_aux[OF Quotient3_fset Quotient3_fset])
+
+section \<open>Lifted theorems\<close>
+
+subsection \<open>fset\<close>
+
+lemma fset_simps [simp]:
+ shows "fset {||} = {}"
+ and "fset (insert_fset x S) = insert x (fset S)"
+ by (descending, simp)+
+
+lemma finite_fset [simp]:
+ shows "finite (fset S)"
+ by (descending) (simp)
+
+lemma fset_cong:
+ shows "fset S = fset T \<longleftrightarrow> S = T"
+ by (descending) (simp)
+
+lemma filter_fset [simp]:
+ shows "fset (filter_fset P xs) = Collect P \<inter> fset xs"
+ by (descending) (auto)
+
+lemma remove_fset [simp]:
+ shows "fset (remove_fset x xs) = fset xs - {x}"
+ by (descending) (simp)
+
+lemma inter_fset [simp]:
+ shows "fset (xs |\<inter>| ys) = fset xs \<inter> fset ys"
+ by (descending) (auto)
+
+lemma union_fset [simp]:
+ shows "fset (xs |\<union>| ys) = fset xs \<union> fset ys"
+ by (lifting set_append)
+
+lemma minus_fset [simp]:
+ shows "fset (xs - ys) = fset xs - fset ys"
+ by (descending) (auto)
+
+
+subsection \<open>in_fset\<close>
+
+lemma in_fset:
+ shows "x |\<in>| S \<longleftrightarrow> x \<in> fset S"
+ by descending simp
+
+lemma notin_fset:
+ shows "x |\<notin>| S \<longleftrightarrow> x \<notin> fset S"
+ by (simp add: in_fset)
+
+lemma notin_empty_fset:
+ shows "x |\<notin>| {||}"
+ by (simp add: in_fset)
+
+lemma fset_eq_iff:
+ shows "S = T \<longleftrightarrow> (\<forall>x. (x |\<in>| S) = (x |\<in>| T))"
+ by descending auto
+
+lemma none_in_empty_fset:
+ shows "(\<forall>x. x |\<notin>| S) \<longleftrightarrow> S = {||}"
+ by descending simp
+
+
+subsection \<open>insert_fset\<close>
+
+lemma in_insert_fset_iff [simp]:
+ shows "x |\<in>| insert_fset y S \<longleftrightarrow> x = y \<or> x |\<in>| S"
+ by descending simp
+
+lemma
+ shows insert_fsetI1: "x |\<in>| insert_fset x S"
+ and insert_fsetI2: "x |\<in>| S \<Longrightarrow> x |\<in>| insert_fset y S"
+ by simp_all
+
+lemma insert_absorb_fset [simp]:
+ shows "x |\<in>| S \<Longrightarrow> insert_fset x S = S"
+ by (descending) (auto)
+
+lemma empty_not_insert_fset[simp]:
+ shows "{||} \<noteq> insert_fset x S"
+ and "insert_fset x S \<noteq> {||}"
+ by (descending, simp)+
+
+lemma insert_fset_left_comm:
+ shows "insert_fset x (insert_fset y S) = insert_fset y (insert_fset x S)"
+ by (descending) (auto)
+
+lemma insert_fset_left_idem:
+ shows "insert_fset x (insert_fset x S) = insert_fset x S"
+ by (descending) (auto)
+
+lemma singleton_fset_eq[simp]:
+ shows "{|x|} = {|y|} \<longleftrightarrow> x = y"
+ by (descending) (auto)
+
+lemma in_fset_mdef:
+ shows "x |\<in>| F \<longleftrightarrow> x |\<notin>| (F - {|x|}) \<and> F = insert_fset x (F - {|x|})"
+ by (descending) (auto)
+
+
+subsection \<open>union_fset\<close>
+
+lemmas [simp] =
+ sup_bot_left[where 'a="'a fset"]
+ sup_bot_right[where 'a="'a fset"]
+
+lemma union_insert_fset [simp]:
+ shows "insert_fset x S |\<union>| T = insert_fset x (S |\<union>| T)"
+ by (lifting append.simps(2))
+
+lemma singleton_union_fset_left:
+ shows "{|a|} |\<union>| S = insert_fset a S"
+ by simp
+
+lemma singleton_union_fset_right:
+ shows "S |\<union>| {|a|} = insert_fset a S"
+ by (subst sup.commute) simp
+
+lemma in_union_fset:
+ shows "x |\<in>| S |\<union>| T \<longleftrightarrow> x |\<in>| S \<or> x |\<in>| T"
+ by (descending) (simp)
+
+
+subsection \<open>minus_fset\<close>
+
+lemma minus_in_fset:
+ shows "x |\<in>| (xs - ys) \<longleftrightarrow> x |\<in>| xs \<and> x |\<notin>| ys"
+ by (descending) (simp)
+
+lemma minus_insert_fset:
+ shows "insert_fset x xs - ys = (if x |\<in>| ys then xs - ys else insert_fset x (xs - ys))"
+ by (descending) (auto)
+
+lemma minus_insert_in_fset[simp]:
+ shows "x |\<in>| ys \<Longrightarrow> insert_fset x xs - ys = xs - ys"
+ by (simp add: minus_insert_fset)
+
+lemma minus_insert_notin_fset[simp]:
+ shows "x |\<notin>| ys \<Longrightarrow> insert_fset x xs - ys = insert_fset x (xs - ys)"
+ by (simp add: minus_insert_fset)
+
+lemma in_minus_fset:
+ shows "x |\<in>| F - S \<Longrightarrow> x |\<notin>| S"
+ unfolding in_fset minus_fset
+ by blast
+
+lemma notin_minus_fset:
+ shows "x |\<in>| S \<Longrightarrow> x |\<notin>| F - S"
+ unfolding in_fset minus_fset
+ by blast
+
+
+subsection \<open>remove_fset\<close>
+
+lemma in_remove_fset:
+ shows "x |\<in>| remove_fset y S \<longleftrightarrow> x |\<in>| S \<and> x \<noteq> y"
+ by (descending) (simp)
+
+lemma notin_remove_fset:
+ shows "x |\<notin>| remove_fset x S"
+ by (descending) (simp)
+
+lemma notin_remove_ident_fset:
+ shows "x |\<notin>| S \<Longrightarrow> remove_fset x S = S"
+ by (descending) (simp)
+
+lemma remove_fset_cases:
+ shows "S = {||} \<or> (\<exists>x. x |\<in>| S \<and> S = insert_fset x (remove_fset x S))"
+ by (descending) (auto simp add: insert_absorb)
+
+
+subsection \<open>inter_fset\<close>
+
+lemma inter_empty_fset_l:
+ shows "{||} |\<inter>| S = {||}"
+ by simp
+
+lemma inter_empty_fset_r:
+ shows "S |\<inter>| {||} = {||}"
+ by simp
+
+lemma inter_insert_fset:
+ shows "insert_fset x S |\<inter>| T = (if x |\<in>| T then insert_fset x (S |\<inter>| T) else S |\<inter>| T)"
+ by (descending) (auto)
+
+lemma in_inter_fset:
+ shows "x |\<in>| (S |\<inter>| T) \<longleftrightarrow> x |\<in>| S \<and> x |\<in>| T"
+ by (descending) (simp)
+
+
+subsection \<open>subset_fset and psubset_fset\<close>
+
+lemma subset_fset:
+ shows "xs |\<subseteq>| ys \<longleftrightarrow> fset xs \<subseteq> fset ys"
+ by (descending) (simp)
+
+lemma psubset_fset:
+ shows "xs |\<subset>| ys \<longleftrightarrow> fset xs \<subset> fset ys"
+ unfolding less_fset_def
+ by (descending) (auto)
+
+lemma subset_insert_fset:
+ shows "(insert_fset x xs) |\<subseteq>| ys \<longleftrightarrow> x |\<in>| ys \<and> xs |\<subseteq>| ys"
+ by (descending) (simp)
+
+lemma subset_in_fset:
+ shows "xs |\<subseteq>| ys = (\<forall>x. x |\<in>| xs \<longrightarrow> x |\<in>| ys)"
+ by (descending) (auto)
+
+lemma subset_empty_fset:
+ shows "xs |\<subseteq>| {||} \<longleftrightarrow> xs = {||}"
+ by (descending) (simp)
+
+lemma not_psubset_empty_fset:
+ shows "\<not> xs |\<subset>| {||}"
+ by (metis fset_simps(1) psubset_fset not_psubset_empty)
+
+
+subsection \<open>map_fset\<close>
+
+lemma map_fset_simps [simp]:
+ shows "map_fset f {||} = {||}"
+ and "map_fset f (insert_fset x S) = insert_fset (f x) (map_fset f S)"
+ by (descending, simp)+
+
+lemma map_fset_image [simp]:
+ shows "fset (map_fset f S) = f ` (fset S)"
+ by (descending) (simp)
+
+lemma inj_map_fset_cong:
+ shows "inj f \<Longrightarrow> map_fset f S = map_fset f T \<longleftrightarrow> S = T"
+ by (descending) (metis inj_vimage_image_eq list_eq_def set_map)
+
+lemma map_union_fset:
+ shows "map_fset f (S |\<union>| T) = map_fset f S |\<union>| map_fset f T"
+ by (descending) (simp)
+
+lemma in_fset_map_fset[simp]: "a |\<in>| map_fset f X = (\<exists>b. b |\<in>| X \<and> a = f b)"
+ by descending auto
+
+
+subsection \<open>card_fset\<close>
+
+lemma card_fset:
+ shows "card_fset xs = card (fset xs)"
+ by (descending) (simp)
+
+lemma card_insert_fset_iff [simp]:
+ shows "card_fset (insert_fset x S) = (if x |\<in>| S then card_fset S else Suc (card_fset S))"
+ by (descending) (simp add: insert_absorb)
+
+lemma card_fset_0[simp]:
+ shows "card_fset S = 0 \<longleftrightarrow> S = {||}"
+ by (descending) (simp)
+
+lemma card_empty_fset[simp]:
+ shows "card_fset {||} = 0"
+ by (simp add: card_fset)
+
+lemma card_fset_1:
+ shows "card_fset S = 1 \<longleftrightarrow> (\<exists>x. S = {|x|})"
+ by (descending) (auto simp add: card_Suc_eq)
+
+lemma card_fset_gt_0:
+ shows "x \<in> fset S \<Longrightarrow> 0 < card_fset S"
+ by (descending) (auto simp add: card_gt_0_iff)
+
+lemma card_notin_fset:
+ shows "(x |\<notin>| S) = (card_fset (insert_fset x S) = Suc (card_fset S))"
+ by simp
+
+lemma card_fset_Suc:
+ shows "card_fset S = Suc n \<Longrightarrow> \<exists>x T. x |\<notin>| T \<and> S = insert_fset x T \<and> card_fset T = n"
+ apply(descending)
+ apply(auto dest!: card_eq_SucD)
+ by (metis Diff_insert_absorb set_removeAll)
+
+lemma card_remove_fset_iff [simp]:
+ shows "card_fset (remove_fset y S) = (if y |\<in>| S then card_fset S - 1 else card_fset S)"
+ by (descending) (simp)
+
+lemma card_Suc_exists_in_fset:
+ shows "card_fset S = Suc n \<Longrightarrow> \<exists>a. a |\<in>| S"
+ by (drule card_fset_Suc) (auto)
+
+lemma in_card_fset_not_0:
+ shows "a |\<in>| A \<Longrightarrow> card_fset A \<noteq> 0"
+ by (descending) (auto)
+
+lemma card_fset_mono:
+ shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset xs \<le> card_fset ys"
+ unfolding card_fset psubset_fset
+ by (simp add: card_mono subset_fset)
+
+lemma card_subset_fset_eq:
+ shows "xs |\<subseteq>| ys \<Longrightarrow> card_fset ys \<le> card_fset xs \<Longrightarrow> xs = ys"
+ unfolding card_fset subset_fset
+ by (auto dest: card_seteq[OF finite_fset] simp add: fset_cong)
+
+lemma psubset_card_fset_mono:
+ shows "xs |\<subset>| ys \<Longrightarrow> card_fset xs < card_fset ys"
+ unfolding card_fset subset_fset
+ by (metis finite_fset psubset_fset psubset_card_mono)
+
+lemma card_union_inter_fset:
+ shows "card_fset xs + card_fset ys = card_fset (xs |\<union>| ys) + card_fset (xs |\<inter>| ys)"
+ unfolding card_fset union_fset inter_fset
+ by (rule card_Un_Int[OF finite_fset finite_fset])
+
+lemma card_union_disjoint_fset:
+ shows "xs |\<inter>| ys = {||} \<Longrightarrow> card_fset (xs |\<union>| ys) = card_fset xs + card_fset ys"
+ unfolding card_fset union_fset
+ apply (rule card_Un_disjoint[OF finite_fset finite_fset])
+ by (metis inter_fset fset_simps(1))
+
+lemma card_remove_fset_less1:
+ shows "x |\<in>| xs \<Longrightarrow> card_fset (remove_fset x xs) < card_fset xs"
+ unfolding card_fset in_fset remove_fset
+ by (rule card_Diff1_less[OF finite_fset])
+
+lemma card_remove_fset_less2:
+ shows "x |\<in>| xs \<Longrightarrow> y |\<in>| xs \<Longrightarrow> card_fset (remove_fset y (remove_fset x xs)) < card_fset xs"
+ unfolding card_fset remove_fset in_fset
+ by (rule card_Diff2_less[OF finite_fset])
+
+lemma card_remove_fset_le1:
+ shows "card_fset (remove_fset x xs) \<le> card_fset xs"
+ unfolding remove_fset card_fset
+ by (rule card_Diff1_le[OF finite_fset])
+
+lemma card_psubset_fset:
+ shows "ys |\<subseteq>| xs \<Longrightarrow> card_fset ys < card_fset xs \<Longrightarrow> ys |\<subset>| xs"
+ unfolding card_fset psubset_fset subset_fset
+ by (rule card_psubset[OF finite_fset])
+
+lemma card_map_fset_le:
+ shows "card_fset (map_fset f xs) \<le> card_fset xs"
+ unfolding card_fset map_fset_image
+ by (rule card_image_le[OF finite_fset])
+
+lemma card_minus_insert_fset[simp]:
+ assumes "a |\<in>| A" and "a |\<notin>| B"
+ shows "card_fset (A - insert_fset a B) = card_fset (A - B) - 1"
+ using assms
+ unfolding in_fset card_fset minus_fset
+ by (simp add: card_Diff_insert[OF finite_fset])
+
+lemma card_minus_subset_fset:
+ assumes "B |\<subseteq>| A"
+ shows "card_fset (A - B) = card_fset A - card_fset B"
+ using assms
+ unfolding subset_fset card_fset minus_fset
+ by (rule card_Diff_subset[OF finite_fset])
+
+lemma card_minus_fset:
+ shows "card_fset (A - B) = card_fset A - card_fset (A |\<inter>| B)"
+ unfolding inter_fset card_fset minus_fset
+ by (rule card_Diff_subset_Int) (simp)
+
+
+subsection \<open>concat_fset\<close>
+
+lemma concat_empty_fset [simp]:
+ shows "concat_fset {||} = {||}"
+ by descending simp
+
+lemma concat_insert_fset [simp]:
+ shows "concat_fset (insert_fset x S) = x |\<union>| concat_fset S"
+ by descending simp
+
+lemma concat_union_fset [simp]:
+ shows "concat_fset (xs |\<union>| ys) = concat_fset xs |\<union>| concat_fset ys"
+ by descending simp
+
+lemma map_concat_fset:
+ shows "map_fset f (concat_fset xs) = concat_fset (map_fset (map_fset f) xs)"
+ by (lifting map_concat)
+
+subsection \<open>filter_fset\<close>
+
+lemma subset_filter_fset:
+ "filter_fset P xs |\<subseteq>| filter_fset Q xs = (\<forall> x. x |\<in>| xs \<longrightarrow> P x \<longrightarrow> Q x)"
+ by descending auto
+
+lemma eq_filter_fset:
+ "(filter_fset P xs = filter_fset Q xs) = (\<forall>x. x |\<in>| xs \<longrightarrow> P x = Q x)"
+ by descending auto
+
+lemma psubset_filter_fset:
+ "(\<And>x. x |\<in>| xs \<Longrightarrow> P x \<Longrightarrow> Q x) \<Longrightarrow> (x |\<in>| xs & \<not> P x & Q x) \<Longrightarrow>
+ filter_fset P xs |\<subset>| filter_fset Q xs"
+ unfolding less_fset_def by (auto simp add: subset_filter_fset eq_filter_fset)
+
+
+subsection \<open>fold_fset\<close>
+
+lemma fold_empty_fset:
+ "fold_fset f {||} = id"
+ by descending (simp add: fold_once_def)
+
+lemma fold_insert_fset: "fold_fset f (insert_fset a A) =
+ (if rsp_fold f then if a |\<in>| A then fold_fset f A else fold_fset f A \<circ> f a else id)"
+ by descending (simp add: fold_once_fold_remdups)
+
+lemma remdups_removeAll:
+ "remdups (removeAll x xs) = remove1 x (remdups xs)"
+ by (induct xs) auto
+
+lemma member_commute_fold_once:
+ assumes "rsp_fold f"
+ and "x \<in> set xs"
+ shows "fold_once f xs = fold_once f (removeAll x xs) \<circ> f x"
+proof -
+ from assms have "fold f (remdups xs) = fold f (remove1 x (remdups xs)) \<circ> f x"
+ by (auto intro!: fold_remove1_split elim: rsp_foldE)
+ then show ?thesis using \<open>rsp_fold f\<close> by (simp add: fold_once_fold_remdups remdups_removeAll)
+qed
+
+lemma in_commute_fold_fset:
+ "rsp_fold f \<Longrightarrow> h |\<in>| b \<Longrightarrow> fold_fset f b = fold_fset f (remove_fset h b) \<circ> f h"
+ by descending (simp add: member_commute_fold_once)
+
+
+subsection \<open>Choice in fsets\<close>
+
+lemma fset_choice:
+ assumes a: "\<forall>x. x |\<in>| A \<longrightarrow> (\<exists>y. P x y)"
+ shows "\<exists>f. \<forall>x. x |\<in>| A \<longrightarrow> P x (f x)"
+ using a
+ apply(descending)
+ using finite_set_choice
+ by (auto simp add: Ball_def)
+
+
+section \<open>Induction and Cases rules for fsets\<close>
+
+lemma fset_exhaust [case_names empty insert, cases type: fset]:
+ assumes empty_fset_case: "S = {||} \<Longrightarrow> P"
+ and insert_fset_case: "\<And>x S'. S = insert_fset x S' \<Longrightarrow> P"
+ shows "P"
+ using assms by (lifting list.exhaust)
+
+lemma fset_induct [case_names empty insert]:
+ assumes empty_fset_case: "P {||}"
+ and insert_fset_case: "\<And>x S. P S \<Longrightarrow> P (insert_fset x S)"
+ shows "P S"
+ using assms
+ by (descending) (blast intro: list.induct)
+
+lemma fset_induct_stronger [case_names empty insert, induct type: fset]:
+ assumes empty_fset_case: "P {||}"
+ and insert_fset_case: "\<And>x S. \<lbrakk>x |\<notin>| S; P S\<rbrakk> \<Longrightarrow> P (insert_fset x S)"
+ shows "P S"
+proof(induct S rule: fset_induct)
+ case empty
+ show "P {||}" using empty_fset_case by simp
+next
+ case (insert x S)
+ have "P S" by fact
+ then show "P (insert_fset x S)" using insert_fset_case
+ by (cases "x |\<in>| S") (simp_all)
+qed
+
+lemma fset_card_induct:
+ assumes empty_fset_case: "P {||}"
+ and card_fset_Suc_case: "\<And>S T. Suc (card_fset S) = (card_fset T) \<Longrightarrow> P S \<Longrightarrow> P T"
+ shows "P S"
+proof (induct S)
+ case empty
+ show "P {||}" by (rule empty_fset_case)
+next
+ case (insert x S)
+ have h: "P S" by fact
+ have "x |\<notin>| S" by fact
+ then have "Suc (card_fset S) = card_fset (insert_fset x S)"
+ using card_fset_Suc by auto
+ then show "P (insert_fset x S)"
+ using h card_fset_Suc_case by simp
+qed
+
+lemma fset_raw_strong_cases:
+ obtains "xs = []"
+ | ys x where "\<not> List.member ys x" and "xs \<approx> x # ys"
+proof (induct xs)
+ case Nil
+ then show thesis by simp
+next
+ case (Cons a xs)
+ have a: "\<lbrakk>xs = [] \<Longrightarrow> thesis; \<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis\<rbrakk> \<Longrightarrow> thesis"
+ by (rule Cons(1))
+ have b: "\<And>x' ys'. \<lbrakk>\<not> List.member ys' x'; a # xs \<approx> x' # ys'\<rbrakk> \<Longrightarrow> thesis" by fact
+ have c: "xs = [] \<Longrightarrow> thesis" using b
+ apply(simp)
+ by (metis list.set(1) emptyE empty_subsetI)
+ have "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis"
+ proof -
+ fix x :: 'a
+ fix ys :: "'a list"
+ assume d:"\<not> List.member ys x"
+ assume e:"xs \<approx> x # ys"
+ show thesis
+ proof (cases "x = a")
+ assume h: "x = a"
+ then have f: "\<not> List.member ys a" using d by simp
+ have g: "a # xs \<approx> a # ys" using e h by auto
+ show thesis using b f g by simp
+ next
+ assume h: "x \<noteq> a"
+ then have f: "\<not> List.member (a # ys) x" using d by auto
+ have g: "a # xs \<approx> x # (a # ys)" using e h by auto
+ show thesis using b f g by (simp del: List.member_def)
+ qed
+ qed
+ then show thesis using a c by blast
+qed
+
+
+lemma fset_strong_cases:
+ obtains "xs = {||}"
+ | ys x where "x |\<notin>| ys" and "xs = insert_fset x ys"
+ by (lifting fset_raw_strong_cases)
+
+
+lemma fset_induct2:
+ "P {||} {||} \<Longrightarrow>
+ (\<And>x xs. x |\<notin>| xs \<Longrightarrow> P (insert_fset x xs) {||}) \<Longrightarrow>
+ (\<And>y ys. y |\<notin>| ys \<Longrightarrow> P {||} (insert_fset y ys)) \<Longrightarrow>
+ (\<And>x xs y ys. \<lbrakk>P xs ys; x |\<notin>| xs; y |\<notin>| ys\<rbrakk> \<Longrightarrow> P (insert_fset x xs) (insert_fset y ys)) \<Longrightarrow>
+ P xsa ysa"
+ apply (induct xsa arbitrary: ysa)
+ apply (induct_tac x rule: fset_induct_stronger)
+ apply simp_all
+ apply (induct_tac xa rule: fset_induct_stronger)
+ apply simp_all
+ done
+
+text \<open>Extensionality\<close>
+
+lemma fset_eqI:
+ assumes "\<And>x. x \<in> fset A \<longleftrightarrow> x \<in> fset B"
+ shows "A = B"
+using assms proof (induct A arbitrary: B)
+ case empty then show ?case
+ by (auto simp add: in_fset none_in_empty_fset [symmetric] sym)
+next
+ case (insert x A)
+ from insert.prems insert.hyps(1) have "\<And>z. z \<in> fset A \<longleftrightarrow> z \<in> fset (B - {|x|})"
+ by (auto simp add: in_fset)
+ then have A: "A = B - {|x|}" by (rule insert.hyps(2))
+ with insert.prems [symmetric, of x] have "x |\<in>| B" by (simp add: in_fset)
+ with A show ?case by (metis in_fset_mdef)
+qed
+
+subsection \<open>alternate formulation with a different decomposition principle
+ and a proof of equivalence\<close>
+
+inductive
+ list_eq2 :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" ("_ \<approx>2 _")
+where
+ "(a # b # xs) \<approx>2 (b # a # xs)"
+| "[] \<approx>2 []"
+| "xs \<approx>2 ys \<Longrightarrow> ys \<approx>2 xs"
+| "(a # a # xs) \<approx>2 (a # xs)"
+| "xs \<approx>2 ys \<Longrightarrow> (a # xs) \<approx>2 (a # ys)"
+| "xs1 \<approx>2 xs2 \<Longrightarrow> xs2 \<approx>2 xs3 \<Longrightarrow> xs1 \<approx>2 xs3"
+
+lemma list_eq2_refl:
+ shows "xs \<approx>2 xs"
+ by (induct xs) (auto intro: list_eq2.intros)
+
+lemma cons_delete_list_eq2:
+ shows "(a # (removeAll a A)) \<approx>2 (if List.member A a then A else a # A)"
+ apply (induct A)
+ apply (simp add: list_eq2_refl)
+ apply (case_tac "List.member (aa # A) a")
+ apply (simp_all)
+ apply (case_tac [!] "a = aa")
+ apply (simp_all)
+ apply (case_tac "List.member A a")
+ apply (auto)[2]
+ apply (metis list_eq2.intros(3) list_eq2.intros(4) list_eq2.intros(5) list_eq2.intros(6))
+ apply (metis list_eq2.intros(1) list_eq2.intros(5) list_eq2.intros(6))
+ apply (auto simp add: list_eq2_refl)
+ done
+
+lemma member_delete_list_eq2:
+ assumes a: "List.member r e"
+ shows "(e # removeAll e r) \<approx>2 r"
+ using a cons_delete_list_eq2[of e r]
+ by simp
+
+lemma list_eq2_equiv:
+ "(l \<approx> r) \<longleftrightarrow> (list_eq2 l r)"
+proof
+ show "list_eq2 l r \<Longrightarrow> l \<approx> r" by (induct rule: list_eq2.induct) auto
+next
+ {
+ fix n
+ assume a: "card_list l = n" and b: "l \<approx> r"
+ have "l \<approx>2 r"
+ using a b
+ proof (induct n arbitrary: l r)
+ case 0
+ have "card_list l = 0" by fact
+ then have "\<forall>x. \<not> List.member l x" by auto
+ then have z: "l = []" by auto
+ then have "r = []" using \<open>l \<approx> r\<close> by simp
+ then show ?case using z list_eq2_refl by simp
+ next
+ case (Suc m)
+ have b: "l \<approx> r" by fact
+ have d: "card_list l = Suc m" by fact
+ then have "\<exists>a. List.member l a"
+ apply(simp)
+ apply(drule card_eq_SucD)
+ apply(blast)
+ done
+ then obtain a where e: "List.member l a" by auto
+ then have e': "List.member r a" using list_eq_def [simplified List.member_def [symmetric], of l r] b
+ by auto
+ have f: "card_list (removeAll a l) = m" using e d by (simp)
+ have g: "removeAll a l \<approx> removeAll a r" using remove_fset.rsp b by simp
+ have "(removeAll a l) \<approx>2 (removeAll a r)" by (rule Suc.hyps[OF f g])
+ then have h: "(a # removeAll a l) \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(5))
+ have i: "l \<approx>2 (a # removeAll a l)"
+ by (rule list_eq2.intros(3)[OF member_delete_list_eq2[OF e]])
+ have "l \<approx>2 (a # removeAll a r)" by (rule list_eq2.intros(6)[OF i h])
+ then show ?case using list_eq2.intros(6)[OF _ member_delete_list_eq2[OF e']] by simp
+ qed
+ }
+ then show "l \<approx> r \<Longrightarrow> l \<approx>2 r" by blast
+qed
+
+
+(* We cannot write it as "assumes .. shows" since Isabelle changes
+ the quantifiers to schematic variables and reintroduces them in
+ a different order *)
+lemma fset_eq_cases:
+ "\<lbrakk>a1 = a2;
+ \<And>a b xs. \<lbrakk>a1 = insert_fset a (insert_fset b xs); a2 = insert_fset b (insert_fset a xs)\<rbrakk> \<Longrightarrow> P;
+ \<lbrakk>a1 = {||}; a2 = {||}\<rbrakk> \<Longrightarrow> P; \<And>xs ys. \<lbrakk>a1 = ys; a2 = xs; xs = ys\<rbrakk> \<Longrightarrow> P;
+ \<And>a xs. \<lbrakk>a1 = insert_fset a (insert_fset a xs); a2 = insert_fset a xs\<rbrakk> \<Longrightarrow> P;
+ \<And>xs ys a. \<lbrakk>a1 = insert_fset a xs; a2 = insert_fset a ys; xs = ys\<rbrakk> \<Longrightarrow> P;
+ \<And>xs1 xs2 xs3. \<lbrakk>a1 = xs1; a2 = xs3; xs1 = xs2; xs2 = xs3\<rbrakk> \<Longrightarrow> P\<rbrakk>
+ \<Longrightarrow> P"
+ by (lifting list_eq2.cases[simplified list_eq2_equiv[symmetric]])
+
+lemma fset_eq_induct:
+ assumes "x1 = x2"
+ and "\<And>a b xs. P (insert_fset a (insert_fset b xs)) (insert_fset b (insert_fset a xs))"
+ and "P {||} {||}"
+ and "\<And>xs ys. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P ys xs"
+ and "\<And>a xs. P (insert_fset a (insert_fset a xs)) (insert_fset a xs)"
+ and "\<And>xs ys a. \<lbrakk>xs = ys; P xs ys\<rbrakk> \<Longrightarrow> P (insert_fset a xs) (insert_fset a ys)"
+ and "\<And>xs1 xs2 xs3. \<lbrakk>xs1 = xs2; P xs1 xs2; xs2 = xs3; P xs2 xs3\<rbrakk> \<Longrightarrow> P xs1 xs3"
+ shows "P x1 x2"
+ using assms
+ by (lifting list_eq2.induct[simplified list_eq2_equiv[symmetric]])
+
+ML \<open>
+fun dest_fsetT (Type (@{type_name fset}, [T])) = T
+ | dest_fsetT T = raise TYPE ("dest_fsetT: fset type expected", [T], []);
+\<close>
+
+no_notation
+ list_eq (infix "\<approx>" 50) and
+ list_eq2 (infix "\<approx>2" 50)
+
+end
--- a/src/HOL/ROOT Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/ROOT Tue Sep 20 11:35:10 2016 +0200
@@ -996,7 +996,7 @@
options [document = false]
theories
DList
- FSet
+ Quotient_FSet
Quotient_Int
Quotient_Message
Lift_FSet
--- a/src/HOL/Set_Interval.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Set_Interval.thy Tue Sep 20 11:35:10 2016 +0200
@@ -1889,7 +1889,7 @@
also have "... = (x - y) * (y * (\<Sum>p<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x^n)"
by (simp only: mult.left_commute)
also have "... = (x - y) * (\<Sum>p<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
- by (simp add: field_simps Suc_diff_le setsum_left_distrib setsum_right_distrib)
+ by (simp add: field_simps Suc_diff_le setsum_distrib_right setsum_distrib_left)
finally show ?case .
qed simp
@@ -1944,7 +1944,7 @@
also from ngt1 have "\<dots> = ?n*a + (\<Sum>i\<in>{..<n}. ?I i*d)" by simp
also from ngt1 have "\<dots> = (?n*a + d*(\<Sum>i\<in>{1..<n}. ?I i))"
unfolding One_nat_def
- by (simp add: setsum_right_distrib atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt ac_simps)
+ by (simp add: setsum_distrib_left atLeast0LessThan[symmetric] setsum_shift_lb_Suc0_0_upt ac_simps)
also have "2*\<dots> = 2*?n*a + d*2*(\<Sum>i\<in>{1..<n}. ?I i)"
by (simp add: algebra_simps)
also from ngt1 have "{1..<n} = {1..n - 1}"
--- a/src/HOL/Transcendental.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/Transcendental.thy Tue Sep 20 11:35:10 2016 +0200
@@ -639,11 +639,11 @@
apply (subst lemma_realpow_rev_sumr)
apply (subst sumr_diff_mult_const2)
apply simp
- apply (simp only: lemma_termdiff1 setsum_right_distrib)
+ apply (simp only: lemma_termdiff1 setsum_distrib_left)
apply (rule setsum.cong [OF refl])
apply (simp add: less_iff_Suc_add)
apply clarify
- apply (simp add: setsum_right_distrib diff_power_eq_setsum ac_simps
+ apply (simp add: setsum_distrib_left diff_power_eq_setsum ac_simps
del: setsum_lessThan_Suc power_Suc)
apply (subst mult.assoc [symmetric], subst power_add [symmetric])
apply (simp add: ac_simps)
@@ -1448,7 +1448,7 @@
also have "\<dots> = x * (\<Sum>i\<le>n. S x i * S y (n - i)) + y * (\<Sum>i\<le>n. S x i * S y (n - i))"
by (rule distrib_right)
also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * y * S y (n - i))"
- by (simp add: setsum_right_distrib ac_simps S_comm)
+ by (simp add: setsum_distrib_left ac_simps S_comm)
also have "\<dots> = (\<Sum>i\<le>n. x * S x i * S y (n - i)) + (\<Sum>i\<le>n. S x i * (y * S y (n - i)))"
by (simp add: ac_simps)
also have "\<dots> = (\<Sum>i\<le>n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n - i))) +
@@ -3340,7 +3340,7 @@
(if even p
then of_real ((-1) ^ (p div 2) / (fact p)) * (\<Sum>n\<le>p. (p choose n) *\<^sub>R (x^n) * y^(p-n))
else 0)"
- by (auto simp: setsum_right_distrib field_simps scaleR_conv_of_real nonzero_of_real_divide)
+ by (auto simp: setsum_distrib_left field_simps scaleR_conv_of_real nonzero_of_real_divide)
also have "\<dots> = cos_coeff p *\<^sub>R ((x + y) ^ p)"
by (simp add: cos_coeff_def binomial_ring [of x y] scaleR_conv_of_real atLeast0AtMost)
finally show ?thesis .
@@ -5835,7 +5835,7 @@
by (auto simp: pairs_le_eq_Sigma setsum.Sigma)
also have "\<dots> = (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
apply (subst setsum_triangle_reindex_eq)
- apply (auto simp: algebra_simps setsum_right_distrib intro!: setsum.cong)
+ apply (auto simp: algebra_simps setsum_distrib_left intro!: setsum.cong)
apply (metis le_add_diff_inverse power_add)
done
finally show ?thesis .
@@ -5864,7 +5864,7 @@
also have "\<dots> = (\<Sum>i\<le>n. a i * (x - y) * (\<Sum>j<i. y^(i - Suc j) * x^j))"
by (simp add: power_diff_sumr2 mult.assoc)
also have "\<dots> = (\<Sum>i\<le>n. \<Sum>j<i. a i * (x - y) * (y^(i - Suc j) * x^j))"
- by (simp add: setsum_right_distrib)
+ by (simp add: setsum_distrib_left)
also have "\<dots> = (\<Sum>(i,j) \<in> (SIGMA i : atMost n. lessThan i). a i * (x - y) * (y^(i - Suc j) * x^j))"
by (simp add: setsum.Sigma)
also have "\<dots> = (\<Sum>(j,i) \<in> (SIGMA j : lessThan n. {Suc j..n}). a i * (x - y) * (y^(i - Suc j) * x^j))"
@@ -5872,7 +5872,7 @@
also have "\<dots> = (\<Sum>j<n. \<Sum>i=Suc j..n. a i * (x - y) * (y^(i - Suc j) * x^j))"
by (simp add: setsum.Sigma)
also have "\<dots> = (x - y) * (\<Sum>j<n. (\<Sum>i=Suc j..n. a i * y^(i - j - 1)) * x^j)"
- by (simp add: setsum_right_distrib mult_ac)
+ by (simp add: setsum_distrib_left mult_ac)
finally show ?thesis .
qed
@@ -5894,7 +5894,7 @@
by (auto simp add: setsum.reindex_bij_betw [OF h, symmetric] intro: setsum.strong_cong)
qed
then show ?thesis
- by (simp add: polyfun_diff [OF assms] setsum_left_distrib)
+ by (simp add: polyfun_diff [OF assms] setsum_distrib_right)
qed
lemma polyfun_linear_factor: (*COMPLEX_POLYFUN_LINEAR_FACTOR in HOL Light*)
@@ -5949,7 +5949,7 @@
unfolding Set_Interval.setsum_atMost_Suc_shift
by simp
also have "\<dots> = w * (\<Sum>i\<le>n. c (Suc i) * w^i)"
- by (simp add: setsum_right_distrib ac_simps)
+ by (simp add: setsum_distrib_left ac_simps)
finally show ?thesis .
qed
then have w: "\<And>w. w \<noteq> 0 \<Longrightarrow> (\<Sum>i\<le>n. c (Suc i) * w^i) = 0"
--- a/src/HOL/ex/Sum_of_Powers.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/ex/Sum_of_Powers.thy Tue Sep 20 11:35:10 2016 +0200
@@ -110,7 +110,7 @@
unfolding bernpoly_def by (rule DERIV_cong) (fast intro!: derivative_intros, simp)
moreover have "(\<Sum>k\<le>n. real (Suc n - k) * x ^ (n - k) * (real (Suc n choose k) * bernoulli k)) = (n + 1) * bernpoly n x"
unfolding bernpoly_def
- by (auto intro: setsum.cong simp add: setsum_right_distrib real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 of_nat_diff)
+ by (auto intro: setsum.cong simp add: setsum_distrib_left real_binomial_eq_mult_binomial_Suc[of _ n] Suc_eq_plus1 of_nat_diff)
ultimately show ?thesis by auto
qed
@@ -135,7 +135,7 @@
lemma sum_of_powers: "(\<Sum>k\<le>n::nat. (real k) ^ m) = (bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0) / (m + 1)"
proof -
from diff_bernpoly[of "Suc m", simplified] have "(m + (1::real)) * (\<Sum>k\<le>n. (real k) ^ m) = (\<Sum>k\<le>n. bernpoly (Suc m) (real k + 1) - bernpoly (Suc m) (real k))"
- by (auto simp add: setsum_right_distrib intro!: setsum.cong)
+ by (auto simp add: setsum_distrib_left intro!: setsum.cong)
also have "... = (\<Sum>k\<le>n. bernpoly (Suc m) (real (k + 1)) - bernpoly (Suc m) (real k))"
by simp
also have "... = bernpoly (Suc m) (n + 1) - bernpoly (Suc m) 0"
--- a/src/HOL/ex/ThreeDivides.thy Mon Sep 19 17:37:22 2016 +0200
+++ b/src/HOL/ex/ThreeDivides.thy Tue Sep 20 11:35:10 2016 +0200
@@ -193,7 +193,7 @@
"m = 10*(\<Sum>x<nd. m div 10 div 10^x mod 10 * 10^x) + c" by simp
also have
"\<dots> = (\<Sum>x<nd. m div 10 div 10^x mod 10 * 10^(x+1)) + c"
- by (subst setsum_right_distrib) (simp add: ac_simps)
+ by (subst setsum_distrib_left) (simp add: ac_simps)
also have
"\<dots> = (\<Sum>x<nd. m div 10^(Suc x) mod 10 * 10^(Suc x)) + c"
by (simp add: div_mult2_eq[symmetric])