author | hoelzl |
Mon, 03 Dec 2012 18:19:08 +0100 | |
changeset 50328 | 25b1e8686ce0 |
parent 48891 | c0eafbd55de3 |
child 54249 | ce00f2fef556 |
permissions | -rw-r--r-- |
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(* Author: Various *) |
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header {* Combination and Cancellation Simprocs for Numeral Expressions *} |
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theory Numeral_Simprocs |
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imports Divides |
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begin |
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ML_file "~~/src/Provers/Arith/assoc_fold.ML" |
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ML_file "~~/src/Provers/Arith/cancel_numerals.ML" |
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ML_file "~~/src/Provers/Arith/combine_numerals.ML" |
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ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML" |
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ML_file "~~/src/Provers/Arith/extract_common_term.ML" |
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lemmas semiring_norm = |
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Let_def arith_simps nat_arith rel_simps |
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if_False if_True |
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add_0 add_Suc add_numeral_left |
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add_neg_numeral_left mult_numeral_left |
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numeral_1_eq_1 [symmetric] Suc_eq_plus1 |
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eq_numeral_iff_iszero not_iszero_Numeral1 |
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declare split_div [of _ _ "numeral k", arith_split] for k |
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declare split_mod [of _ _ "numeral k", arith_split] for k |
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text {* For @{text combine_numerals} *} |
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lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)" |
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by (simp add: add_mult_distrib) |
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text {* For @{text cancel_numerals} *} |
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lemma nat_diff_add_eq1: |
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"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)" |
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by (simp split add: nat_diff_split add: add_mult_distrib) |
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lemma nat_diff_add_eq2: |
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"i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))" |
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by (simp split add: nat_diff_split add: add_mult_distrib) |
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lemma nat_eq_add_iff1: |
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"j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)" |
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by (auto split add: nat_diff_split simp add: add_mult_distrib) |
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lemma nat_eq_add_iff2: |
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"i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)" |
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by (auto split add: nat_diff_split simp add: add_mult_distrib) |
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lemma nat_less_add_iff1: |
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"j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)" |
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by (auto split add: nat_diff_split simp add: add_mult_distrib) |
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lemma nat_less_add_iff2: |
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"i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)" |
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by (auto split add: nat_diff_split simp add: add_mult_distrib) |
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lemma nat_le_add_iff1: |
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"j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)" |
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by (auto split add: nat_diff_split simp add: add_mult_distrib) |
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lemma nat_le_add_iff2: |
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"i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)" |
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by (auto split add: nat_diff_split simp add: add_mult_distrib) |
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text {* For @{text cancel_numeral_factors} *} |
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lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)" |
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by auto |
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lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)" |
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by auto |
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lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)" |
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by auto |
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lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)" |
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by auto |
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lemma nat_mult_dvd_cancel_disj[simp]: |
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"(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))" |
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by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1) |
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lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)" |
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by(auto) |
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text {* For @{text cancel_factor} *} |
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lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)" |
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by auto |
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lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)" |
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by auto |
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lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)" |
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by auto |
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lemma nat_mult_div_cancel_disj[simp]: |
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"(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)" |
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by (simp add: nat_mult_div_cancel1) |
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ML_file "Tools/numeral_simprocs.ML" |
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simproc_setup semiring_assoc_fold |
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("(a::'a::comm_semiring_1_cancel) * b") = |
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{* fn phi => Numeral_Simprocs.assoc_fold *} |
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(* TODO: see whether the type class can be generalized further *) |
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simproc_setup int_combine_numerals |
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("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") = |
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{* fn phi => Numeral_Simprocs.combine_numerals *} |
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simproc_setup field_combine_numerals |
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("(i::'a::{field_inverse_zero,ring_char_0}) + j" |
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|"(i::'a::{field_inverse_zero,ring_char_0}) - j") = |
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{* fn phi => Numeral_Simprocs.field_combine_numerals *} |
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simproc_setup inteq_cancel_numerals |
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("(l::'a::comm_ring_1) + m = n" |
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|"(l::'a::comm_ring_1) = m + n" |
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|"(l::'a::comm_ring_1) - m = n" |
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|"(l::'a::comm_ring_1) = m - n" |
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|"(l::'a::comm_ring_1) * m = n" |
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|"(l::'a::comm_ring_1) = m * n" |
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|"- (l::'a::comm_ring_1) = m" |
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|"(l::'a::comm_ring_1) = - m") = |
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{* fn phi => Numeral_Simprocs.eq_cancel_numerals *} |
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simproc_setup intless_cancel_numerals |
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("(l::'a::linordered_idom) + m < n" |
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|"(l::'a::linordered_idom) < m + n" |
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|"(l::'a::linordered_idom) - m < n" |
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|"(l::'a::linordered_idom) < m - n" |
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|"(l::'a::linordered_idom) * m < n" |
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|"(l::'a::linordered_idom) < m * n" |
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|"- (l::'a::linordered_idom) < m" |
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|"(l::'a::linordered_idom) < - m") = |
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{* fn phi => Numeral_Simprocs.less_cancel_numerals *} |
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simproc_setup intle_cancel_numerals |
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("(l::'a::linordered_idom) + m \<le> n" |
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|"(l::'a::linordered_idom) \<le> m + n" |
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|"(l::'a::linordered_idom) - m \<le> n" |
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|"(l::'a::linordered_idom) \<le> m - n" |
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|"(l::'a::linordered_idom) * m \<le> n" |
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|"(l::'a::linordered_idom) \<le> m * n" |
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|"- (l::'a::linordered_idom) \<le> m" |
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|"(l::'a::linordered_idom) \<le> - m") = |
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{* fn phi => Numeral_Simprocs.le_cancel_numerals *} |
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simproc_setup ring_eq_cancel_numeral_factor |
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("(l::'a::{idom,ring_char_0}) * m = n" |
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|"(l::'a::{idom,ring_char_0}) = m * n") = |
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{* fn phi => Numeral_Simprocs.eq_cancel_numeral_factor *} |
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simproc_setup ring_less_cancel_numeral_factor |
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("(l::'a::linordered_idom) * m < n" |
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|"(l::'a::linordered_idom) < m * n") = |
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{* fn phi => Numeral_Simprocs.less_cancel_numeral_factor *} |
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simproc_setup ring_le_cancel_numeral_factor |
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("(l::'a::linordered_idom) * m <= n" |
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|"(l::'a::linordered_idom) <= m * n") = |
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{* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *} |
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(* TODO: remove comm_ring_1 constraint if possible *) |
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simproc_setup int_div_cancel_numeral_factors |
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("((l::'a::{semiring_div,comm_ring_1,ring_char_0}) * m) div n" |
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|"(l::'a::{semiring_div,comm_ring_1,ring_char_0}) div (m * n)") = |
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{* fn phi => Numeral_Simprocs.div_cancel_numeral_factor *} |
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simproc_setup divide_cancel_numeral_factor |
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("((l::'a::{field_inverse_zero,ring_char_0}) * m) / n" |
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|"(l::'a::{field_inverse_zero,ring_char_0}) / (m * n)" |
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|"((numeral v)::'a::{field_inverse_zero,ring_char_0}) / (numeral w)") = |
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{* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *} |
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176 |
|
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177 |
simproc_setup ring_eq_cancel_factor |
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|
178 |
("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") = |
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179 |
{* fn phi => Numeral_Simprocs.eq_cancel_factor *} |
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180 |
|
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181 |
simproc_setup linordered_ring_le_cancel_factor |
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182 |
("(l::'a::linordered_idom) * m <= n" |
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183 |
|"(l::'a::linordered_idom) <= m * n") = |
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184 |
{* fn phi => Numeral_Simprocs.le_cancel_factor *} |
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185 |
|
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|
186 |
simproc_setup linordered_ring_less_cancel_factor |
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187 |
("(l::'a::linordered_idom) * m < n" |
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188 |
|"(l::'a::linordered_idom) < m * n") = |
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189 |
{* fn phi => Numeral_Simprocs.less_cancel_factor *} |
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190 |
|
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|
191 |
simproc_setup int_div_cancel_factor |
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|
192 |
("((l::'a::semiring_div) * m) div n" |
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193 |
|"(l::'a::semiring_div) div (m * n)") = |
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194 |
{* fn phi => Numeral_Simprocs.div_cancel_factor *} |
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|
195 |
|
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196 |
simproc_setup int_mod_cancel_factor |
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|
197 |
("((l::'a::semiring_div) * m) mod n" |
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|
198 |
|"(l::'a::semiring_div) mod (m * n)") = |
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199 |
{* fn phi => Numeral_Simprocs.mod_cancel_factor *} |
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200 |
|
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|
201 |
simproc_setup dvd_cancel_factor |
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parents:
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changeset
|
202 |
("((l::'a::idom) * m) dvd n" |
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changeset
|
203 |
|"(l::'a::idom) dvd (m * n)") = |
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|
204 |
{* fn phi => Numeral_Simprocs.dvd_cancel_factor *} |
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|
205 |
|
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|
206 |
simproc_setup divide_cancel_factor |
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use simproc_setup for cancellation simprocs, to get proper name bindings
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parents:
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|
207 |
("((l::'a::field_inverse_zero) * m) / n" |
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use simproc_setup for cancellation simprocs, to get proper name bindings
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parents:
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|
208 |
|"(l::'a::field_inverse_zero) / (m * n)") = |
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|
209 |
{* fn phi => Numeral_Simprocs.divide_cancel_factor *} |
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|
210 |
|
48891 | 211 |
ML_file "Tools/nat_numeral_simprocs.ML" |
33366 | 212 |
|
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213 |
simproc_setup nat_combine_numerals |
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|
214 |
("(i::nat) + j" | "Suc (i + j)") = |
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215 |
{* fn phi => Nat_Numeral_Simprocs.combine_numerals *} |
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|
216 |
|
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|
217 |
simproc_setup nateq_cancel_numerals |
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|
218 |
("(l::nat) + m = n" | "(l::nat) = m + n" | |
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219 |
"(l::nat) * m = n" | "(l::nat) = m * n" | |
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220 |
"Suc m = n" | "m = Suc n") = |
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221 |
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *} |
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222 |
|
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|
223 |
simproc_setup natless_cancel_numerals |
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|
224 |
("(l::nat) + m < n" | "(l::nat) < m + n" | |
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|
225 |
"(l::nat) * m < n" | "(l::nat) < m * n" | |
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|
226 |
"Suc m < n" | "m < Suc n") = |
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227 |
{* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *} |
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228 |
|
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|
229 |
simproc_setup natle_cancel_numerals |
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|
230 |
("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | |
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|
231 |
"(l::nat) * m \<le> n" | "(l::nat) \<le> m * n" | |
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|
232 |
"Suc m \<le> n" | "m \<le> Suc n") = |
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233 |
{* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *} |
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|
234 |
|
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|
235 |
simproc_setup natdiff_cancel_numerals |
62bc9474d04b
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|
236 |
("((l::nat) + m) - n" | "(l::nat) - (m + n)" | |
62bc9474d04b
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237 |
"(l::nat) * m - n" | "(l::nat) - m * n" | |
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|
238 |
"Suc m - n" | "m - Suc n") = |
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|
239 |
{* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *} |
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|
240 |
|
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|
241 |
simproc_setup nat_eq_cancel_numeral_factor |
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|
242 |
("(l::nat) * m = n" | "(l::nat) = m * n") = |
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|
243 |
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *} |
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|
244 |
|
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|
245 |
simproc_setup nat_less_cancel_numeral_factor |
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|
246 |
("(l::nat) * m < n" | "(l::nat) < m * n") = |
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|
247 |
{* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *} |
9a588a835c1e
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|
248 |
|
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|
249 |
simproc_setup nat_le_cancel_numeral_factor |
9a588a835c1e
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changeset
|
250 |
("(l::nat) * m <= n" | "(l::nat) <= m * n") = |
9a588a835c1e
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|
251 |
{* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *} |
9a588a835c1e
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|
252 |
|
9a588a835c1e
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|
253 |
simproc_setup nat_div_cancel_numeral_factor |
9a588a835c1e
use simproc_setup for the remaining nat_numeral simprocs
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changeset
|
254 |
("((l::nat) * m) div n" | "(l::nat) div (m * n)") = |
9a588a835c1e
use simproc_setup for the remaining nat_numeral simprocs
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|
255 |
{* fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor *} |
9a588a835c1e
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|
256 |
|
9a588a835c1e
use simproc_setup for the remaining nat_numeral simprocs
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changeset
|
257 |
simproc_setup nat_dvd_cancel_numeral_factor |
9a588a835c1e
use simproc_setup for the remaining nat_numeral simprocs
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changeset
|
258 |
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") = |
9a588a835c1e
use simproc_setup for the remaining nat_numeral simprocs
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|
259 |
{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *} |
9a588a835c1e
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|
260 |
|
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|
261 |
simproc_setup nat_eq_cancel_factor |
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|
262 |
("(l::nat) * m = n" | "(l::nat) = m * n") = |
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|
263 |
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *} |
aba629d6cee5
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|
264 |
|
aba629d6cee5
use simproc_setup for more nat_numeral simprocs; add simproc tests
huffman
parents:
45436
diff
changeset
|
265 |
simproc_setup nat_less_cancel_factor |
aba629d6cee5
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huffman
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|
266 |
("(l::nat) * m < n" | "(l::nat) < m * n") = |
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parents:
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|
267 |
{* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *} |
aba629d6cee5
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|
268 |
|
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|
269 |
simproc_setup nat_le_cancel_factor |
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|
270 |
("(l::nat) * m <= n" | "(l::nat) <= m * n") = |
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|
271 |
{* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *} |
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|
272 |
|
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|
273 |
simproc_setup nat_div_cancel_factor |
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|
274 |
("((l::nat) * m) div n" | "(l::nat) div (m * n)") = |
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|
275 |
{* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *} |
45462
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|
276 |
|
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|
277 |
simproc_setup nat_dvd_cancel_factor |
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|
278 |
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") = |
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|
279 |
{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *} |
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|
280 |
|
47108
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merged fork with new numeral representation (see NEWS)
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|
281 |
(* FIXME: duplicate rule warnings for: |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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|
282 |
ring_distribs |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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parents:
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changeset
|
283 |
numeral_plus_numeral numeral_times_numeral |
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
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|
284 |
numeral_eq_iff numeral_less_iff numeral_le_iff |
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|
285 |
numeral_neq_zero zero_neq_numeral zero_less_numeral |
2a1953f0d20d
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|
286 |
if_True if_False *) |
33366 | 287 |
declaration {* |
47108
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K (Lin_Arith.add_simps ([@{thm Suc_numeral}, @{thm int_numeral}]) |
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#> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1}, |
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@{thm nat_0}, @{thm nat_1}, |
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@{thm numeral_plus_numeral}, @{thm diff_nat_numeral}, @{thm numeral_times_numeral}, |
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@{thm numeral_eq_iff}, @{thm numeral_less_iff}, @{thm numeral_le_iff}, |
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@{thm le_Suc_numeral}, @{thm le_numeral_Suc}, |
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@{thm less_Suc_numeral}, @{thm less_numeral_Suc}, |
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@{thm Suc_eq_numeral}, @{thm eq_numeral_Suc}, |
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@{thm mult_Suc}, @{thm mult_Suc_right}, |
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@{thm add_Suc}, @{thm add_Suc_right}, |
|
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@{thm numeral_neq_zero}, @{thm zero_neq_numeral}, @{thm zero_less_numeral}, |
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@{thm of_int_numeral}, @{thm of_nat_numeral}, @{thm nat_numeral}, |
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@{thm if_True}, @{thm if_False}]) |
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#> Lin_Arith.add_simprocs |
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[@{simproc semiring_assoc_fold}, |
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@{simproc int_combine_numerals}, |
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@{simproc inteq_cancel_numerals}, |
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@{simproc intless_cancel_numerals}, |
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@{simproc intle_cancel_numerals}] |
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#> Lin_Arith.add_simprocs |
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[@{simproc nat_combine_numerals}, |
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@{simproc nateq_cancel_numerals}, |
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@{simproc natless_cancel_numerals}, |
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@{simproc natle_cancel_numerals}, |
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@{simproc natdiff_cancel_numerals}]) |
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*} |
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||
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end |