src/HOL/Numeral_Simprocs.thy
 author haftmann Mon, 04 Nov 2013 20:10:06 +0100 changeset 54249 ce00f2fef556 parent 48891 c0eafbd55de3 child 54489 03ff4d1e6784 permissions -rw-r--r--
streamlined setup of linear arithmetic
```
(* Author: Various *)

header {* Combination and Cancellation Simprocs for Numeral Expressions *}

theory Numeral_Simprocs
imports Divides
begin

ML_file "~~/src/Provers/Arith/assoc_fold.ML"
ML_file "~~/src/Provers/Arith/cancel_numerals.ML"
ML_file "~~/src/Provers/Arith/combine_numerals.ML"
ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML"
ML_file "~~/src/Provers/Arith/extract_common_term.ML"

lemmas semiring_norm =
Let_def arith_simps diff_nat_numeral rel_simps
if_False if_True
add_0 add_Suc add_numeral_left
add_neg_numeral_left mult_numeral_left
numeral_1_eq_1 [symmetric] Suc_eq_plus1
eq_numeral_iff_iszero not_iszero_Numeral1

declare split_div [of _ _ "numeral k", arith_split] for k
declare split_mod [of _ _ "numeral k", arith_split] for k

text {* For @{text combine_numerals} *}

lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
by (simp add: add_mult_distrib)

text {* For @{text cancel_numerals} *}

lemma nat_diff_add_eq1:
"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
by (simp split add: nat_diff_split add: add_mult_distrib)

lemma nat_diff_add_eq2:
"i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
by (simp split add: nat_diff_split add: add_mult_distrib)

lemma nat_eq_add_iff1:
"j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)

lemma nat_eq_add_iff2:
"i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)

lemma nat_less_add_iff1:
"j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)

lemma nat_less_add_iff2:
"i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)

lemma nat_le_add_iff1:
"j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)

lemma nat_le_add_iff2:
"i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
by (auto split add: nat_diff_split simp add: add_mult_distrib)

text {* For @{text cancel_numeral_factors} *}

lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
by auto

lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
by auto

lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
by auto

lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
by auto

lemma nat_mult_dvd_cancel_disj[simp]:
"(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1)

lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
by(auto)

text {* For @{text cancel_factor} *}

lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
by auto

lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
by auto

lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
by auto

lemma nat_mult_div_cancel_disj[simp]:
"(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
by (simp add: nat_mult_div_cancel1)

ML_file "Tools/numeral_simprocs.ML"

simproc_setup semiring_assoc_fold
("(a::'a::comm_semiring_1_cancel) * b") =
{* fn phi => Numeral_Simprocs.assoc_fold *}

(* TODO: see whether the type class can be generalized further *)
simproc_setup int_combine_numerals
("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") =
{* fn phi => Numeral_Simprocs.combine_numerals *}

simproc_setup field_combine_numerals
("(i::'a::{field_inverse_zero,ring_char_0}) + j"
|"(i::'a::{field_inverse_zero,ring_char_0}) - j") =
{* fn phi => Numeral_Simprocs.field_combine_numerals *}

simproc_setup inteq_cancel_numerals
("(l::'a::comm_ring_1) + m = n"
|"(l::'a::comm_ring_1) = m + n"
|"(l::'a::comm_ring_1) - m = n"
|"(l::'a::comm_ring_1) = m - n"
|"(l::'a::comm_ring_1) * m = n"
|"(l::'a::comm_ring_1) = m * n"
|"- (l::'a::comm_ring_1) = m"
|"(l::'a::comm_ring_1) = - m") =
{* fn phi => Numeral_Simprocs.eq_cancel_numerals *}

simproc_setup intless_cancel_numerals
("(l::'a::linordered_idom) + m < n"
|"(l::'a::linordered_idom) < m + n"
|"(l::'a::linordered_idom) - m < n"
|"(l::'a::linordered_idom) < m - n"
|"(l::'a::linordered_idom) * m < n"
|"(l::'a::linordered_idom) < m * n"
|"- (l::'a::linordered_idom) < m"
|"(l::'a::linordered_idom) < - m") =
{* fn phi => Numeral_Simprocs.less_cancel_numerals *}

simproc_setup intle_cancel_numerals
("(l::'a::linordered_idom) + m \<le> n"
|"(l::'a::linordered_idom) \<le> m + n"
|"(l::'a::linordered_idom) - m \<le> n"
|"(l::'a::linordered_idom) \<le> m - n"
|"(l::'a::linordered_idom) * m \<le> n"
|"(l::'a::linordered_idom) \<le> m * n"
|"- (l::'a::linordered_idom) \<le> m"
|"(l::'a::linordered_idom) \<le> - m") =
{* fn phi => Numeral_Simprocs.le_cancel_numerals *}

simproc_setup ring_eq_cancel_numeral_factor
("(l::'a::{idom,ring_char_0}) * m = n"
|"(l::'a::{idom,ring_char_0}) = m * n") =
{* fn phi => Numeral_Simprocs.eq_cancel_numeral_factor *}

simproc_setup ring_less_cancel_numeral_factor
("(l::'a::linordered_idom) * m < n"
|"(l::'a::linordered_idom) < m * n") =
{* fn phi => Numeral_Simprocs.less_cancel_numeral_factor *}

simproc_setup ring_le_cancel_numeral_factor
("(l::'a::linordered_idom) * m <= n"
|"(l::'a::linordered_idom) <= m * n") =
{* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *}

(* TODO: remove comm_ring_1 constraint if possible *)
simproc_setup int_div_cancel_numeral_factors
("((l::'a::{semiring_div,comm_ring_1,ring_char_0}) * m) div n"
|"(l::'a::{semiring_div,comm_ring_1,ring_char_0}) div (m * n)") =
{* fn phi => Numeral_Simprocs.div_cancel_numeral_factor *}

simproc_setup divide_cancel_numeral_factor
("((l::'a::{field_inverse_zero,ring_char_0}) * m) / n"
|"(l::'a::{field_inverse_zero,ring_char_0}) / (m * n)"
|"((numeral v)::'a::{field_inverse_zero,ring_char_0}) / (numeral w)") =
{* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *}

simproc_setup ring_eq_cancel_factor
("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =
{* fn phi => Numeral_Simprocs.eq_cancel_factor *}

simproc_setup linordered_ring_le_cancel_factor
("(l::'a::linordered_idom) * m <= n"
|"(l::'a::linordered_idom) <= m * n") =
{* fn phi => Numeral_Simprocs.le_cancel_factor *}

simproc_setup linordered_ring_less_cancel_factor
("(l::'a::linordered_idom) * m < n"
|"(l::'a::linordered_idom) < m * n") =
{* fn phi => Numeral_Simprocs.less_cancel_factor *}

simproc_setup int_div_cancel_factor
("((l::'a::semiring_div) * m) div n"
|"(l::'a::semiring_div) div (m * n)") =
{* fn phi => Numeral_Simprocs.div_cancel_factor *}

simproc_setup int_mod_cancel_factor
("((l::'a::semiring_div) * m) mod n"
|"(l::'a::semiring_div) mod (m * n)") =
{* fn phi => Numeral_Simprocs.mod_cancel_factor *}

simproc_setup dvd_cancel_factor
("((l::'a::idom) * m) dvd n"
|"(l::'a::idom) dvd (m * n)") =
{* fn phi => Numeral_Simprocs.dvd_cancel_factor *}

simproc_setup divide_cancel_factor
("((l::'a::field_inverse_zero) * m) / n"
|"(l::'a::field_inverse_zero) / (m * n)") =
{* fn phi => Numeral_Simprocs.divide_cancel_factor *}

ML_file "Tools/nat_numeral_simprocs.ML"

simproc_setup nat_combine_numerals
("(i::nat) + j" | "Suc (i + j)") =
{* fn phi => Nat_Numeral_Simprocs.combine_numerals *}

simproc_setup nateq_cancel_numerals
("(l::nat) + m = n" | "(l::nat) = m + n" |
"(l::nat) * m = n" | "(l::nat) = m * n" |
"Suc m = n" | "m = Suc n") =
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *}

simproc_setup natless_cancel_numerals
("(l::nat) + m < n" | "(l::nat) < m + n" |
"(l::nat) * m < n" | "(l::nat) < m * n" |
"Suc m < n" | "m < Suc n") =
{* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *}

simproc_setup natle_cancel_numerals
("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" |
"(l::nat) * m \<le> n" | "(l::nat) \<le> m * n" |
"Suc m \<le> n" | "m \<le> Suc n") =
{* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *}

simproc_setup natdiff_cancel_numerals
("((l::nat) + m) - n" | "(l::nat) - (m + n)" |
"(l::nat) * m - n" | "(l::nat) - m * n" |
"Suc m - n" | "m - Suc n") =
{* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *}

simproc_setup nat_eq_cancel_numeral_factor
("(l::nat) * m = n" | "(l::nat) = m * n") =
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *}

simproc_setup nat_less_cancel_numeral_factor
("(l::nat) * m < n" | "(l::nat) < m * n") =
{* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *}

simproc_setup nat_le_cancel_numeral_factor
("(l::nat) * m <= n" | "(l::nat) <= m * n") =
{* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *}

simproc_setup nat_div_cancel_numeral_factor
("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
{* fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor *}

simproc_setup nat_dvd_cancel_numeral_factor
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *}

simproc_setup nat_eq_cancel_factor
("(l::nat) * m = n" | "(l::nat) = m * n") =
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *}

simproc_setup nat_less_cancel_factor
("(l::nat) * m < n" | "(l::nat) < m * n") =
{* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *}

simproc_setup nat_le_cancel_factor
("(l::nat) * m <= n" | "(l::nat) <= m * n") =
{* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *}

simproc_setup nat_div_cancel_factor
("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
{* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *}

simproc_setup nat_dvd_cancel_factor
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *}

declaration {*
K (Lin_Arith.add_simprocs
[@{simproc semiring_assoc_fold},
@{simproc int_combine_numerals},
@{simproc inteq_cancel_numerals},
@{simproc intless_cancel_numerals},
@{simproc intle_cancel_numerals}]
#> Lin_Arith.add_simprocs
[@{simproc nat_combine_numerals},
@{simproc nateq_cancel_numerals},
@{simproc natless_cancel_numerals},
@{simproc natle_cancel_numerals},
@{simproc natdiff_cancel_numerals}])
*}

end
```