--- a/src/HOL/Numeral_Simprocs.thy Sat Jul 18 21:44:18 2015 +0200
+++ b/src/HOL/Numeral_Simprocs.thy Sat Jul 18 22:58:50 2015 +0200
@@ -1,6 +1,6 @@
(* Author: Various *)
-section {* Combination and Cancellation Simprocs for Numeral Expressions *}
+section \<open>Combination and Cancellation Simprocs for Numeral Expressions\<close>
theory Numeral_Simprocs
imports Divides
@@ -23,12 +23,12 @@
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} *}
+text \<open>For @{text combine_numerals}\<close>
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} *}
+text \<open>For @{text cancel_numerals}\<close>
lemma nat_diff_add_eq1:
"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
@@ -62,7 +62,7 @@
"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} *}
+text \<open>For @{text cancel_numeral_factors}\<close>
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
by auto
@@ -83,7 +83,7 @@
lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
by(auto)
-text {* For @{text cancel_factor} *}
+text \<open>For @{text cancel_factor}\<close>
lemmas nat_mult_le_cancel_disj = mult_le_cancel1
@@ -107,17 +107,17 @@
simproc_setup semiring_assoc_fold
("(a::'a::comm_semiring_1_cancel) * b") =
- {* fn phi => Numeral_Simprocs.assoc_fold *}
+ \<open>fn phi => Numeral_Simprocs.assoc_fold\<close>
(* 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 *}
+ \<open>fn phi => Numeral_Simprocs.combine_numerals\<close>
simproc_setup field_combine_numerals
("(i::'a::{field,ring_char_0}) + j"
|"(i::'a::{field,ring_char_0}) - j") =
- {* fn phi => Numeral_Simprocs.field_combine_numerals *}
+ \<open>fn phi => Numeral_Simprocs.field_combine_numerals\<close>
simproc_setup inteq_cancel_numerals
("(l::'a::comm_ring_1) + m = n"
@@ -128,7 +128,7 @@
|"(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 *}
+ \<open>fn phi => Numeral_Simprocs.eq_cancel_numerals\<close>
simproc_setup intless_cancel_numerals
("(l::'a::linordered_idom) + m < n"
@@ -139,7 +139,7 @@
|"(l::'a::linordered_idom) < m * n"
|"- (l::'a::linordered_idom) < m"
|"(l::'a::linordered_idom) < - m") =
- {* fn phi => Numeral_Simprocs.less_cancel_numerals *}
+ \<open>fn phi => Numeral_Simprocs.less_cancel_numerals\<close>
simproc_setup intle_cancel_numerals
("(l::'a::linordered_idom) + m \<le> n"
@@ -150,140 +150,140 @@
|"(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 *}
+ \<open>fn phi => Numeral_Simprocs.le_cancel_numerals\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.eq_cancel_numeral_factor\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.less_cancel_numeral_factor\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.le_cancel_numeral_factor\<close>
(* 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 *}
+ \<open>fn phi => Numeral_Simprocs.div_cancel_numeral_factor\<close>
simproc_setup divide_cancel_numeral_factor
("((l::'a::{field,ring_char_0}) * m) / n"
|"(l::'a::{field,ring_char_0}) / (m * n)"
|"((numeral v)::'a::{field,ring_char_0}) / (numeral w)") =
- {* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *}
+ \<open>fn phi => Numeral_Simprocs.divide_cancel_numeral_factor\<close>
simproc_setup ring_eq_cancel_factor
("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =
- {* fn phi => Numeral_Simprocs.eq_cancel_factor *}
+ \<open>fn phi => Numeral_Simprocs.eq_cancel_factor\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.le_cancel_factor\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.less_cancel_factor\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.div_cancel_factor\<close>
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 *}
+ \<open>fn phi => Numeral_Simprocs.mod_cancel_factor\<close>
simproc_setup dvd_cancel_factor
("((l::'a::idom) * m) dvd n"
|"(l::'a::idom) dvd (m * n)") =
- {* fn phi => Numeral_Simprocs.dvd_cancel_factor *}
+ \<open>fn phi => Numeral_Simprocs.dvd_cancel_factor\<close>
simproc_setup divide_cancel_factor
("((l::'a::field) * m) / n"
|"(l::'a::field) / (m * n)") =
- {* fn phi => Numeral_Simprocs.divide_cancel_factor *}
+ \<open>fn phi => Numeral_Simprocs.divide_cancel_factor\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.combine_numerals\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numerals\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numerals\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals\<close>
simproc_setup nat_eq_cancel_numeral_factor
("(l::nat) * m = n" | "(l::nat) = m * n") =
- {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor\<close>
simproc_setup nat_less_cancel_numeral_factor
("(l::nat) * m < n" | "(l::nat) < m * n") =
- {* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor\<close>
simproc_setup nat_le_cancel_numeral_factor
("(l::nat) * m <= n" | "(l::nat) <= m * n") =
- {* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor\<close>
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 *}
+ \<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor\<close>
simproc_setup nat_eq_cancel_factor
("(l::nat) * m = n" | "(l::nat) = m * n") =
- {* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_factor\<close>
simproc_setup nat_less_cancel_factor
("(l::nat) * m < n" | "(l::nat) < m * n") =
- {* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.less_cancel_factor\<close>
simproc_setup nat_le_cancel_factor
("(l::nat) * m <= n" | "(l::nat) <= m * n") =
- {* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.le_cancel_factor\<close>
simproc_setup nat_div_cancel_factor
("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
- {* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.div_cancel_factor\<close>
simproc_setup nat_dvd_cancel_factor
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
- {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *}
+ \<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor\<close>
-declaration {*
+declaration \<open>
K (Lin_Arith.add_simprocs
[@{simproc semiring_assoc_fold},
@{simproc int_combine_numerals},
@@ -297,6 +297,6 @@
@{simproc natless_cancel_numerals},
@{simproc natle_cancel_numerals},
@{simproc natdiff_cancel_numerals}])
-*}
+\<close>
end