author | paulson |
Wed, 15 Jul 1998 10:15:13 +0200 | |
changeset 5143 | b94cd208f073 |
parent 5078 | 7b5ea59c0275 |
child 5183 | 89f162de39cf |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/Arith.ML |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1998 University of Cambridge |
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|
6 |
Proofs about elementary arithmetic: addition, multiplication, etc. |
|
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Some from the Hoare example from Norbert Galm |
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*) |
9 |
||
10 |
(*** Basic rewrite rules for the arithmetic operators ***) |
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|
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(** Difference **) |
14 |
||
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qed_goal "diff_0_eq_0" thy |
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"0 - n = 0" |
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(fn _ => [induct_tac "n" 1, ALLGOALS Asm_simp_tac]); |
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|
19 |
(*Must simplify BEFORE the induction!! (Else we get a critical pair) |
|
20 |
Suc(m) - Suc(n) rewrites to pred(Suc(m) - n) *) |
|
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qed_goal "diff_Suc_Suc" thy |
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"Suc(m) - Suc(n) = m - n" |
23 |
(fn _ => |
|
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[Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]); |
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|
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Addsimps [diff_0_eq_0, diff_Suc_Suc]; |
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(* Could be (and is, below) generalized in various ways; |
29 |
However, none of the generalizations are currently in the simpset, |
|
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and I dread to think what happens if I put them in *) |
|
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Goal "0 < n ==> Suc(n-1) = n"; |
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by (asm_simp_tac (simpset() addsplits [split_nat_case]) 1); |
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qed "Suc_pred"; |
34 |
Addsimps [Suc_pred]; |
|
35 |
||
36 |
Delsimps [diff_Suc]; |
|
37 |
||
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|
39 |
(**** Inductive properties of the operators ****) |
|
40 |
||
41 |
(*** Addition ***) |
|
42 |
||
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qed_goal "add_0_right" thy "m + 0 = m" |
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(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
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|
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qed_goal "add_Suc_right" thy "m + Suc(n) = Suc(m+n)" |
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(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
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Addsimps [add_0_right,add_Suc_right]; |
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|
51 |
(*Associative law for addition*) |
|
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qed_goal "add_assoc" thy "(m + n) + k = m + ((n + k)::nat)" |
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(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
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|
55 |
(*Commutative law for addition*) |
|
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qed_goal "add_commute" thy "m + n = n + (m::nat)" |
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(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
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|
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qed_goal "add_left_commute" thy "x+(y+z)=y+((x+z)::nat)" |
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(fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1, |
61 |
rtac (add_commute RS arg_cong) 1]); |
|
62 |
||
63 |
(*Addition is an AC-operator*) |
|
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val add_ac = [add_assoc, add_commute, add_left_commute]; |
|
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||
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Goal "!!k::nat. (k + m = k + n) = (m=n)"; |
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by (induct_tac "k" 1); |
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by (Simp_tac 1); |
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69 |
by (Asm_simp_tac 1); |
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qed "add_left_cancel"; |
71 |
||
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Goal "!!k::nat. (m + k = n + k) = (m=n)"; |
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by (induct_tac "k" 1); |
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by (Simp_tac 1); |
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by (Asm_simp_tac 1); |
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qed "add_right_cancel"; |
77 |
||
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Goal "!!k::nat. (k + m <= k + n) = (m<=n)"; |
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by (induct_tac "k" 1); |
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by (Simp_tac 1); |
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81 |
by (Asm_simp_tac 1); |
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qed "add_left_cancel_le"; |
83 |
||
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Goal "!!k::nat. (k + m < k + n) = (m<n)"; |
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by (induct_tac "k" 1); |
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by (Simp_tac 1); |
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by (Asm_simp_tac 1); |
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qed "add_left_cancel_less"; |
89 |
||
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|
90 |
Addsimps [add_left_cancel, add_right_cancel, |
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add_left_cancel_le, add_left_cancel_less]; |
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|
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(** Reasoning about m+0=0, etc. **) |
94 |
||
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Goal "(m+n = 0) = (m=0 & n=0)"; |
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by (induct_tac "m" 1); |
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|
97 |
by (ALLGOALS Asm_simp_tac); |
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qed "add_is_0"; |
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AddIffs [add_is_0]; |
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|
100 |
|
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Goal "(0<m+n) = (0<m | 0<n)"; |
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by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1); |
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qed "add_gr_0"; |
104 |
AddIffs [add_gr_0]; |
|
105 |
||
106 |
(* FIXME: really needed?? *) |
|
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Goal "((m+n)-1 = 0) = (m=0 & n-1 = 0 | m-1 = 0 & n=0)"; |
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by (exhaust_tac "m" 1); |
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by (ALLGOALS (fast_tac (claset() addss (simpset())))); |
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qed "pred_add_is_0"; |
111 |
Addsimps [pred_add_is_0]; |
|
112 |
||
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(* Could be generalized, eg to "!!n. k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *) |
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Goal "0<n ==> m + (n-1) = (m+n)-1"; |
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by (exhaust_tac "m" 1); |
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc] |
|
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addsplits [split_nat_case]))); |
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qed "add_pred"; |
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Addsimps [add_pred]; |
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|
120 |
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Goal "!!m::nat. m + n = m ==> n = 0"; |
122 |
by (dtac (add_0_right RS ssubst) 1); |
|
123 |
by (asm_full_simp_tac (simpset() addsimps [add_assoc] |
|
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delsimps [add_0_right]) 1); |
|
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qed "add_eq_self_zero"; |
|
126 |
||
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|
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(**** Additional theorems about "less than" ****) |
129 |
||
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(*Deleted less_natE; instead use less_eq_Suc_add RS exE*) |
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Goal "m<n --> (? k. n=Suc(m+k))"; |
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by (induct_tac "n" 1); |
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by (ALLGOALS (simp_tac (simpset() addsimps [less_Suc_eq]))); |
134 |
by (blast_tac (claset() addSEs [less_SucE] |
|
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addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1); |
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qed_spec_mp "less_eq_Suc_add"; |
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|
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Goal "n <= ((m + n)::nat)"; |
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by (induct_tac "m" 1); |
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by (ALLGOALS Simp_tac); |
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by (etac le_trans 1); |
142 |
by (rtac (lessI RS less_imp_le) 1); |
|
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qed "le_add2"; |
|
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||
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Goal "n <= ((n + m)::nat)"; |
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by (simp_tac (simpset() addsimps add_ac) 1); |
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by (rtac le_add2 1); |
148 |
qed "le_add1"; |
|
149 |
||
150 |
bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans))); |
|
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bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans))); |
|
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||
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(*"i <= j ==> i <= j+m"*) |
|
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bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans)); |
|
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||
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(*"i <= j ==> i <= m+j"*) |
|
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bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans)); |
|
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||
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(*"i < j ==> i < j+m"*) |
|
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bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans)); |
|
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||
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(*"i < j ==> i < m+j"*) |
|
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bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans)); |
|
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||
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165 |
Goal "i+j < (k::nat) ==> i<k"; |
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by (etac rev_mp 1); |
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by (induct_tac "j" 1); |
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by (ALLGOALS Asm_simp_tac); |
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by (blast_tac (claset() addDs [Suc_lessD]) 1); |
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qed "add_lessD1"; |
171 |
||
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Goal "!!i::nat. ~ (i+j < i)"; |
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by (rtac notI 1); |
174 |
by (etac (add_lessD1 RS less_irrefl) 1); |
|
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qed "not_add_less1"; |
176 |
||
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Goal "!!i::nat. ~ (j+i < i)"; |
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by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1); |
3234 | 179 |
qed "not_add_less2"; |
180 |
AddIffs [not_add_less1, not_add_less2]; |
|
181 |
||
5069 | 182 |
Goal "!!k::nat. m <= n ==> m <= n+k"; |
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by (etac le_trans 1); |
184 |
by (rtac le_add1 1); |
|
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qed "le_imp_add_le"; |
186 |
||
5069 | 187 |
Goal "!!k::nat. m < n ==> m < n+k"; |
1552 | 188 |
by (etac less_le_trans 1); |
189 |
by (rtac le_add1 1); |
|
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qed "less_imp_add_less"; |
191 |
||
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Goal "m+k<=n --> m<=(n::nat)"; |
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by (induct_tac "k" 1); |
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194 |
by (ALLGOALS Asm_simp_tac); |
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by (blast_tac (claset() addDs [Suc_leD]) 1); |
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196 |
qed_spec_mp "add_leD1"; |
923 | 197 |
|
5069 | 198 |
Goal "!!n::nat. m+k<=n ==> k<=n"; |
4089 | 199 |
by (full_simp_tac (simpset() addsimps [add_commute]) 1); |
2498 | 200 |
by (etac add_leD1 1); |
201 |
qed_spec_mp "add_leD2"; |
|
202 |
||
5069 | 203 |
Goal "!!n::nat. m+k<=n ==> m<=n & k<=n"; |
4089 | 204 |
by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1); |
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bind_thm ("add_leE", result() RS conjE); |
206 |
||
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Goal "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n"; |
4089 | 208 |
by (safe_tac (claset() addSDs [less_eq_Suc_add])); |
923 | 209 |
by (asm_full_simp_tac |
4089 | 210 |
(simpset() delsimps [add_Suc_right] |
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211 |
addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1); |
1552 | 212 |
by (etac subst 1); |
4089 | 213 |
by (simp_tac (simpset() addsimps [less_add_Suc1]) 1); |
923 | 214 |
qed "less_add_eq_less"; |
215 |
||
216 |
||
1713 | 217 |
(*** Monotonicity of Addition ***) |
923 | 218 |
|
219 |
(*strict, in 1st argument*) |
|
5069 | 220 |
Goal "!!i j k::nat. i < j ==> i + k < j + k"; |
3339 | 221 |
by (induct_tac "k" 1); |
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222 |
by (ALLGOALS Asm_simp_tac); |
923 | 223 |
qed "add_less_mono1"; |
224 |
||
225 |
(*strict, in both arguments*) |
|
5069 | 226 |
Goal "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l"; |
923 | 227 |
by (rtac (add_less_mono1 RS less_trans) 1); |
1198 | 228 |
by (REPEAT (assume_tac 1)); |
3339 | 229 |
by (induct_tac "j" 1); |
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230 |
by (ALLGOALS Asm_simp_tac); |
923 | 231 |
qed "add_less_mono"; |
232 |
||
233 |
(*A [clumsy] way of lifting < monotonicity to <= monotonicity *) |
|
4732 | 234 |
val [lt_mono,le] = goal thy |
1465 | 235 |
"[| !!i j::nat. i<j ==> f(i) < f(j); \ |
236 |
\ i <= j \ |
|
923 | 237 |
\ |] ==> f(i) <= (f(j)::nat)"; |
238 |
by (cut_facts_tac [le] 1); |
|
4089 | 239 |
by (asm_full_simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1); |
240 |
by (blast_tac (claset() addSIs [lt_mono]) 1); |
|
923 | 241 |
qed "less_mono_imp_le_mono"; |
242 |
||
243 |
(*non-strict, in 1st argument*) |
|
5069 | 244 |
Goal "!!i j k::nat. i<=j ==> i + k <= j + k"; |
3842 | 245 |
by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1); |
1552 | 246 |
by (etac add_less_mono1 1); |
923 | 247 |
by (assume_tac 1); |
248 |
qed "add_le_mono1"; |
|
249 |
||
250 |
(*non-strict, in both arguments*) |
|
5069 | 251 |
Goal "!!k l::nat. [|i<=j; k<=l |] ==> i + k <= j + l"; |
923 | 252 |
by (etac (add_le_mono1 RS le_trans) 1); |
4089 | 253 |
by (simp_tac (simpset() addsimps [add_commute]) 1); |
923 | 254 |
(*j moves to the end because it is free while k, l are bound*) |
1552 | 255 |
by (etac add_le_mono1 1); |
923 | 256 |
qed "add_le_mono"; |
1713 | 257 |
|
3234 | 258 |
|
259 |
(*** Multiplication ***) |
|
260 |
||
261 |
(*right annihilation in product*) |
|
4732 | 262 |
qed_goal "mult_0_right" thy "m * 0 = 0" |
3339 | 263 |
(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
3234 | 264 |
|
3293 | 265 |
(*right successor law for multiplication*) |
4732 | 266 |
qed_goal "mult_Suc_right" thy "m * Suc(n) = m + (m * n)" |
3339 | 267 |
(fn _ => [induct_tac "m" 1, |
4089 | 268 |
ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); |
3234 | 269 |
|
3293 | 270 |
Addsimps [mult_0_right, mult_Suc_right]; |
3234 | 271 |
|
5069 | 272 |
Goal "1 * n = n"; |
3234 | 273 |
by (Asm_simp_tac 1); |
274 |
qed "mult_1"; |
|
275 |
||
5069 | 276 |
Goal "n * 1 = n"; |
3234 | 277 |
by (Asm_simp_tac 1); |
278 |
qed "mult_1_right"; |
|
279 |
||
280 |
(*Commutative law for multiplication*) |
|
4732 | 281 |
qed_goal "mult_commute" thy "m * n = n * (m::nat)" |
3339 | 282 |
(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
3234 | 283 |
|
284 |
(*addition distributes over multiplication*) |
|
4732 | 285 |
qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)" |
3339 | 286 |
(fn _ => [induct_tac "m" 1, |
4089 | 287 |
ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); |
3234 | 288 |
|
4732 | 289 |
qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)" |
3339 | 290 |
(fn _ => [induct_tac "m" 1, |
4089 | 291 |
ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]); |
3234 | 292 |
|
293 |
(*Associative law for multiplication*) |
|
4732 | 294 |
qed_goal "mult_assoc" thy "(m * n) * k = m * ((n * k)::nat)" |
3339 | 295 |
(fn _ => [induct_tac "m" 1, |
4089 | 296 |
ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib]))]); |
3234 | 297 |
|
4732 | 298 |
qed_goal "mult_left_commute" thy "x*(y*z) = y*((x*z)::nat)" |
3234 | 299 |
(fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1, |
300 |
rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]); |
|
301 |
||
302 |
val mult_ac = [mult_assoc,mult_commute,mult_left_commute]; |
|
303 |
||
5069 | 304 |
Goal "(m*n = 0) = (m=0 | n=0)"; |
3339 | 305 |
by (induct_tac "m" 1); |
306 |
by (induct_tac "n" 2); |
|
3293 | 307 |
by (ALLGOALS Asm_simp_tac); |
308 |
qed "mult_is_0"; |
|
309 |
Addsimps [mult_is_0]; |
|
310 |
||
5069 | 311 |
Goal "!!m::nat. m <= m*m"; |
4158 | 312 |
by (induct_tac "m" 1); |
313 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym]))); |
|
314 |
by (etac (le_add2 RSN (2,le_trans)) 1); |
|
315 |
qed "le_square"; |
|
316 |
||
3234 | 317 |
|
318 |
(*** Difference ***) |
|
319 |
||
320 |
||
4732 | 321 |
qed_goal "diff_self_eq_0" thy "m - m = 0" |
3339 | 322 |
(fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]); |
3234 | 323 |
Addsimps [diff_self_eq_0]; |
324 |
||
325 |
(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *) |
|
5069 | 326 |
Goal "~ m<n --> n+(m-n) = (m::nat)"; |
3234 | 327 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
3352 | 328 |
by (ALLGOALS Asm_simp_tac); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
329 |
qed_spec_mp "add_diff_inverse"; |
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
330 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
331 |
Goal "n<=m ==> n+(m-n) = (m::nat)"; |
4089 | 332 |
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
333 |
qed "le_add_diff_inverse"; |
3234 | 334 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
335 |
Goal "n<=m ==> (m-n)+n = (m::nat)"; |
4089 | 336 |
by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
337 |
qed "le_add_diff_inverse2"; |
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
338 |
|
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
339 |
Addsimps [le_add_diff_inverse, le_add_diff_inverse2]; |
3234 | 340 |
|
341 |
||
342 |
(*** More results about difference ***) |
|
343 |
||
4732 | 344 |
val [prem] = goal thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)"; |
3352 | 345 |
by (rtac (prem RS rev_mp) 1); |
346 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
347 |
by (ALLGOALS Asm_simp_tac); |
|
348 |
qed "Suc_diff_n"; |
|
349 |
||
5069 | 350 |
Goal "m - n < Suc(m)"; |
3234 | 351 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
352 |
by (etac less_SucE 3); |
|
4089 | 353 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq]))); |
3234 | 354 |
qed "diff_less_Suc"; |
355 |
||
5069 | 356 |
Goal "!!m::nat. m - n <= m"; |
3234 | 357 |
by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1); |
358 |
by (ALLGOALS Asm_simp_tac); |
|
359 |
qed "diff_le_self"; |
|
3903
1b29151a1009
New simprule diff_le_self, requiring a new proof of diff_diff_cancel
paulson
parents:
3896
diff
changeset
|
360 |
Addsimps [diff_le_self]; |
3234 | 361 |
|
4732 | 362 |
(* j<k ==> j-n < k *) |
363 |
bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans); |
|
364 |
||
5069 | 365 |
Goal "!!i::nat. i-j-k = i - (j+k)"; |
3352 | 366 |
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); |
367 |
by (ALLGOALS Asm_simp_tac); |
|
368 |
qed "diff_diff_left"; |
|
369 |
||
5069 | 370 |
Goal "(Suc m - n) - Suc k = m - n - k"; |
4423 | 371 |
by (simp_tac (simpset() addsimps [diff_diff_left]) 1); |
4736 | 372 |
qed "Suc_diff_diff"; |
373 |
Addsimps [Suc_diff_diff]; |
|
4360 | 374 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
375 |
Goal "0<n ==> n - Suc i < n"; |
4732 | 376 |
by (res_inst_tac [("n","n")] natE 1); |
377 |
by Safe_tac; |
|
378 |
by (asm_simp_tac (simpset() addsimps [le_eq_less_Suc RS sym]) 1); |
|
379 |
qed "diff_Suc_less"; |
|
380 |
Addsimps [diff_Suc_less]; |
|
381 |
||
5069 | 382 |
Goal "!!n::nat. m - n <= Suc m - n"; |
4732 | 383 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
384 |
by (ALLGOALS Asm_simp_tac); |
|
385 |
qed "diff_le_Suc_diff"; |
|
386 |
||
3396 | 387 |
(*This and the next few suggested by Florian Kammueller*) |
5069 | 388 |
Goal "!!i::nat. i-j-k = i-k-j"; |
4089 | 389 |
by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1); |
3352 | 390 |
qed "diff_commute"; |
391 |
||
5069 | 392 |
Goal "!!i j k:: nat. k<=j --> j<=i --> i - (j - k) = i - j + k"; |
3352 | 393 |
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1); |
394 |
by (ALLGOALS Asm_simp_tac); |
|
395 |
by (asm_simp_tac |
|
4089 | 396 |
(simpset() addsimps [Suc_diff_n, le_imp_less_Suc, le_Suc_eq]) 1); |
3352 | 397 |
qed_spec_mp "diff_diff_right"; |
398 |
||
5069 | 399 |
Goal "!!i j k:: nat. k<=j --> (i + j) - k = i + (j - k)"; |
3352 | 400 |
by (res_inst_tac [("m","j"),("n","k")] diff_induct 1); |
401 |
by (ALLGOALS Asm_simp_tac); |
|
402 |
qed_spec_mp "diff_add_assoc"; |
|
403 |
||
5069 | 404 |
Goal "!!i j k:: nat. k<=j --> (j + i) - k = i + (j - k)"; |
4732 | 405 |
by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1); |
406 |
qed_spec_mp "diff_add_assoc2"; |
|
407 |
||
5069 | 408 |
Goal "!!n::nat. (n+m) - n = m"; |
3339 | 409 |
by (induct_tac "n" 1); |
3234 | 410 |
by (ALLGOALS Asm_simp_tac); |
411 |
qed "diff_add_inverse"; |
|
412 |
Addsimps [diff_add_inverse]; |
|
413 |
||
5069 | 414 |
Goal "!!n::nat.(m+n) - n = m"; |
4089 | 415 |
by (simp_tac (simpset() addsimps [diff_add_assoc]) 1); |
3234 | 416 |
qed "diff_add_inverse2"; |
417 |
Addsimps [diff_add_inverse2]; |
|
418 |
||
5069 | 419 |
Goal "!!i j k::nat. i<=j ==> (j-i=k) = (j=k+i)"; |
3724 | 420 |
by Safe_tac; |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
421 |
by (ALLGOALS Asm_simp_tac); |
3366 | 422 |
qed "le_imp_diff_is_add"; |
423 |
||
4732 | 424 |
val [prem] = goal thy "m < Suc(n) ==> m-n = 0"; |
3234 | 425 |
by (rtac (prem RS rev_mp) 1); |
426 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
4089 | 427 |
by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1); |
3352 | 428 |
by (ALLGOALS Asm_simp_tac); |
3234 | 429 |
qed "less_imp_diff_is_0"; |
430 |
||
4732 | 431 |
val prems = goal thy "m-n = 0 --> n-m = 0 --> m=n"; |
3234 | 432 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
433 |
by (REPEAT(Simp_tac 1 THEN TRY(atac 1))); |
|
434 |
qed_spec_mp "diffs0_imp_equal"; |
|
435 |
||
4732 | 436 |
val [prem] = goal thy "m<n ==> 0<n-m"; |
3234 | 437 |
by (rtac (prem RS rev_mp) 1); |
438 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
3352 | 439 |
by (ALLGOALS Asm_simp_tac); |
3234 | 440 |
qed "less_imp_diff_positive"; |
441 |
||
5078 | 442 |
Goal "!! (i::nat). i < j ==> ? k. 0<k & i+k = j"; |
443 |
by (res_inst_tac [("x","j - i")] exI 1); |
|
444 |
by (fast_tac (claset() addDs [less_trans, less_irrefl] |
|
445 |
addIs [less_imp_diff_positive, add_diff_inverse]) 1); |
|
446 |
qed "less_imp_add_positive"; |
|
447 |
||
5069 | 448 |
Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))"; |
4686 | 449 |
by (simp_tac (simpset() addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]) 1); |
3234 | 450 |
qed "if_Suc_diff_n"; |
451 |
||
5069 | 452 |
Goal "Suc(m)-n <= Suc(m-n)"; |
4686 | 453 |
by (simp_tac (simpset() addsimps [if_Suc_diff_n]) 1); |
4672
9d55bc687e1e
New theorem diff_Suc_le_Suc_diff; tidied another proof
paulson
parents:
4423
diff
changeset
|
454 |
qed "diff_Suc_le_Suc_diff"; |
9d55bc687e1e
New theorem diff_Suc_le_Suc_diff; tidied another proof
paulson
parents:
4423
diff
changeset
|
455 |
|
5069 | 456 |
Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)"; |
3234 | 457 |
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1); |
3718 | 458 |
by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac)); |
3234 | 459 |
qed "zero_induct_lemma"; |
460 |
||
4732 | 461 |
val prems = goal thy "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)"; |
3234 | 462 |
by (rtac (diff_self_eq_0 RS subst) 1); |
463 |
by (rtac (zero_induct_lemma RS mp RS mp) 1); |
|
464 |
by (REPEAT (ares_tac ([impI,allI]@prems) 1)); |
|
465 |
qed "zero_induct"; |
|
466 |
||
5069 | 467 |
Goal "!!k::nat. (k+m) - (k+n) = m - n"; |
3339 | 468 |
by (induct_tac "k" 1); |
3234 | 469 |
by (ALLGOALS Asm_simp_tac); |
470 |
qed "diff_cancel"; |
|
471 |
Addsimps [diff_cancel]; |
|
472 |
||
5069 | 473 |
Goal "!!m::nat. (m+k) - (n+k) = m - n"; |
3234 | 474 |
val add_commute_k = read_instantiate [("n","k")] add_commute; |
4089 | 475 |
by (asm_simp_tac (simpset() addsimps ([add_commute_k])) 1); |
3234 | 476 |
qed "diff_cancel2"; |
477 |
Addsimps [diff_cancel2]; |
|
478 |
||
479 |
(*From Clemens Ballarin*) |
|
5069 | 480 |
Goal "!!n::nat. [| k<=n; n<=m |] ==> (m-k) - (n-k) = m-n"; |
3234 | 481 |
by (subgoal_tac "k<=n --> n<=m --> (m-k) - (n-k) = m-n" 1); |
482 |
by (Asm_full_simp_tac 1); |
|
3339 | 483 |
by (induct_tac "k" 1); |
3234 | 484 |
by (Simp_tac 1); |
485 |
(* Induction step *) |
|
486 |
by (subgoal_tac "Suc na <= m --> n <= m --> Suc na <= n --> \ |
|
487 |
\ Suc (m - Suc na) - Suc (n - Suc na) = m-n" 1); |
|
488 |
by (Asm_full_simp_tac 1); |
|
4089 | 489 |
by (blast_tac (claset() addIs [le_trans]) 1); |
490 |
by (auto_tac (claset() addIs [Suc_leD], simpset() delsimps [diff_Suc_Suc])); |
|
491 |
by (asm_full_simp_tac (simpset() delsimps [Suc_less_eq] |
|
3234 | 492 |
addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1); |
493 |
qed "diff_right_cancel"; |
|
494 |
||
5069 | 495 |
Goal "!!n::nat. n - (n+m) = 0"; |
3339 | 496 |
by (induct_tac "n" 1); |
3234 | 497 |
by (ALLGOALS Asm_simp_tac); |
498 |
qed "diff_add_0"; |
|
499 |
Addsimps [diff_add_0]; |
|
500 |
||
501 |
(** Difference distributes over multiplication **) |
|
502 |
||
5069 | 503 |
Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)"; |
3234 | 504 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
505 |
by (ALLGOALS Asm_simp_tac); |
|
506 |
qed "diff_mult_distrib" ; |
|
507 |
||
5069 | 508 |
Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)"; |
3234 | 509 |
val mult_commute_k = read_instantiate [("m","k")] mult_commute; |
4089 | 510 |
by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1); |
3234 | 511 |
qed "diff_mult_distrib2" ; |
512 |
(*NOT added as rewrites, since sometimes they are used from right-to-left*) |
|
513 |
||
514 |
||
1713 | 515 |
(*** Monotonicity of Multiplication ***) |
516 |
||
5069 | 517 |
Goal "!!i::nat. i<=j ==> i*k<=j*k"; |
3339 | 518 |
by (induct_tac "k" 1); |
4089 | 519 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono]))); |
1713 | 520 |
qed "mult_le_mono1"; |
521 |
||
522 |
(*<=monotonicity, BOTH arguments*) |
|
5069 | 523 |
Goal "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l"; |
2007 | 524 |
by (etac (mult_le_mono1 RS le_trans) 1); |
1713 | 525 |
by (rtac le_trans 1); |
2007 | 526 |
by (stac mult_commute 2); |
527 |
by (etac mult_le_mono1 2); |
|
4089 | 528 |
by (simp_tac (simpset() addsimps [mult_commute]) 1); |
1713 | 529 |
qed "mult_le_mono"; |
530 |
||
531 |
(*strict, in 1st argument; proof is by induction on k>0*) |
|
5069 | 532 |
Goal "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j"; |
5078 | 533 |
by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1); |
1713 | 534 |
by (Asm_simp_tac 1); |
3339 | 535 |
by (induct_tac "x" 1); |
4089 | 536 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono]))); |
1713 | 537 |
qed "mult_less_mono2"; |
538 |
||
5069 | 539 |
Goal "!!i::nat. [| i<j; 0<k |] ==> i*k < j*k"; |
3457 | 540 |
by (dtac mult_less_mono2 1); |
4089 | 541 |
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute]))); |
3234 | 542 |
qed "mult_less_mono1"; |
543 |
||
5069 | 544 |
Goal "(0 < m*n) = (0<m & 0<n)"; |
3339 | 545 |
by (induct_tac "m" 1); |
546 |
by (induct_tac "n" 2); |
|
1713 | 547 |
by (ALLGOALS Asm_simp_tac); |
548 |
qed "zero_less_mult_iff"; |
|
4356 | 549 |
Addsimps [zero_less_mult_iff]; |
1713 | 550 |
|
5069 | 551 |
Goal "(m*n = 1) = (m=1 & n=1)"; |
3339 | 552 |
by (induct_tac "m" 1); |
1795 | 553 |
by (Simp_tac 1); |
3339 | 554 |
by (induct_tac "n" 1); |
1795 | 555 |
by (Simp_tac 1); |
4089 | 556 |
by (fast_tac (claset() addss simpset()) 1); |
1795 | 557 |
qed "mult_eq_1_iff"; |
4356 | 558 |
Addsimps [mult_eq_1_iff]; |
1795 | 559 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
560 |
Goal "0<k ==> (m*k < n*k) = (m<n)"; |
4089 | 561 |
by (safe_tac (claset() addSIs [mult_less_mono1])); |
3234 | 562 |
by (cut_facts_tac [less_linear] 1); |
4389 | 563 |
by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1); |
3234 | 564 |
qed "mult_less_cancel2"; |
565 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
566 |
Goal "0<k ==> (k*m < k*n) = (m<n)"; |
3457 | 567 |
by (dtac mult_less_cancel2 1); |
4089 | 568 |
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1); |
3234 | 569 |
qed "mult_less_cancel1"; |
570 |
Addsimps [mult_less_cancel1, mult_less_cancel2]; |
|
571 |
||
5069 | 572 |
Goal "(Suc k * m < Suc k * n) = (m < n)"; |
4423 | 573 |
by (rtac mult_less_cancel1 1); |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
574 |
by (Simp_tac 1); |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
575 |
qed "Suc_mult_less_cancel1"; |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
576 |
|
5069 | 577 |
Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)"; |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
578 |
by (simp_tac (simpset_of HOL.thy) 1); |
4423 | 579 |
by (rtac Suc_mult_less_cancel1 1); |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
580 |
qed "Suc_mult_le_cancel1"; |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
581 |
|
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
582 |
Goal "0<k ==> (m*k = n*k) = (m=n)"; |
3234 | 583 |
by (cut_facts_tac [less_linear] 1); |
3724 | 584 |
by Safe_tac; |
3457 | 585 |
by (assume_tac 2); |
3234 | 586 |
by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac)); |
587 |
by (ALLGOALS Asm_full_simp_tac); |
|
588 |
qed "mult_cancel2"; |
|
589 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
590 |
Goal "0<k ==> (k*m = k*n) = (m=n)"; |
3457 | 591 |
by (dtac mult_cancel2 1); |
4089 | 592 |
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1); |
3234 | 593 |
qed "mult_cancel1"; |
594 |
Addsimps [mult_cancel1, mult_cancel2]; |
|
595 |
||
5069 | 596 |
Goal "(Suc k * m = Suc k * n) = (m = n)"; |
4423 | 597 |
by (rtac mult_cancel1 1); |
4297
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
598 |
by (Simp_tac 1); |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
599 |
qed "Suc_mult_cancel1"; |
5defc2105cc8
added Suc_mult_less_cancel1, Suc_mult_le_cancel1, Suc_mult_cancel1;
wenzelm
parents:
4158
diff
changeset
|
600 |
|
3234 | 601 |
|
1795 | 602 |
(** Lemma for gcd **) |
603 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
604 |
Goal "m = m*n ==> n=1 | m=0"; |
1795 | 605 |
by (dtac sym 1); |
606 |
by (rtac disjCI 1); |
|
607 |
by (rtac nat_less_cases 1 THEN assume_tac 2); |
|
4089 | 608 |
by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1); |
4356 | 609 |
by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1); |
1795 | 610 |
qed "mult_eq_self_implies_10"; |
611 |
||
612 |
||
4736 | 613 |
(*** Subtraction laws -- mostly from Clemens Ballarin ***) |
3234 | 614 |
|
5069 | 615 |
Goal "!! a b c::nat. [| a < b; c <= a |] ==> a-c < b-c"; |
3234 | 616 |
by (subgoal_tac "c+(a-c) < c+(b-c)" 1); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
617 |
by (Full_simp_tac 1); |
3234 | 618 |
by (subgoal_tac "c <= b" 1); |
4089 | 619 |
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2); |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
620 |
by (Asm_simp_tac 1); |
3234 | 621 |
qed "diff_less_mono"; |
622 |
||
5069 | 623 |
Goal "!! a b c::nat. a+b < c ==> a < c-b"; |
3457 | 624 |
by (dtac diff_less_mono 1); |
625 |
by (rtac le_add2 1); |
|
3234 | 626 |
by (Asm_full_simp_tac 1); |
627 |
qed "add_less_imp_less_diff"; |
|
628 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
629 |
Goal "n <= m ==> Suc m - n = Suc (m - n)"; |
4672
9d55bc687e1e
New theorem diff_Suc_le_Suc_diff; tidied another proof
paulson
parents:
4423
diff
changeset
|
630 |
by (asm_full_simp_tac (simpset() addsimps [Suc_diff_n, le_eq_less_Suc]) 1); |
3234 | 631 |
qed "Suc_diff_le"; |
632 |
||
5143
b94cd208f073
Removal of leading "\!\!..." from most Goal commands
paulson
parents:
5078
diff
changeset
|
633 |
Goal "Suc i <= n ==> Suc (n - Suc i) = n - i"; |
3234 | 634 |
by (asm_full_simp_tac |
4089 | 635 |
(simpset() addsimps [Suc_diff_n RS sym, le_eq_less_Suc]) 1); |
3234 | 636 |
qed "Suc_diff_Suc"; |
637 |
||
5069 | 638 |
Goal "!! i::nat. i <= n ==> n - (n - i) = i"; |
3903
1b29151a1009
New simprule diff_le_self, requiring a new proof of diff_diff_cancel
paulson
parents:
3896
diff
changeset
|
639 |
by (etac rev_mp 1); |
1b29151a1009
New simprule diff_le_self, requiring a new proof of diff_diff_cancel
paulson
parents:
3896
diff
changeset
|
640 |
by (res_inst_tac [("m","n"),("n","i")] diff_induct 1); |
4089 | 641 |
by (ALLGOALS (asm_simp_tac (simpset() addsimps [Suc_diff_le]))); |
3234 | 642 |
qed "diff_diff_cancel"; |
3381
2bac33ec2b0d
New theorems le_add_diff_inverse, le_add_diff_inverse2
paulson
parents:
3366
diff
changeset
|
643 |
Addsimps [diff_diff_cancel]; |
3234 | 644 |
|
5069 | 645 |
Goal "!!k::nat. k <= n ==> m <= n + m - k"; |
3457 | 646 |
by (etac rev_mp 1); |
3234 | 647 |
by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1); |
648 |
by (Simp_tac 1); |
|
4089 | 649 |
by (simp_tac (simpset() addsimps [less_add_Suc2, less_imp_le]) 1); |
3234 | 650 |
by (Simp_tac 1); |
651 |
qed "le_add_diff"; |
|
652 |
||
5069 | 653 |
Goal "!!i::nat. 0<k ==> j<i --> j+k-i < k"; |
4736 | 654 |
by (res_inst_tac [("m","j"),("n","i")] diff_induct 1); |
655 |
by (ALLGOALS Asm_simp_tac); |
|
656 |
qed_spec_mp "add_diff_less"; |
|
657 |
||
3234 | 658 |
|
4732 | 659 |
|
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
660 |
(** (Anti)Monotonicity of subtraction -- by Stefan Merz **) |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
661 |
|
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
662 |
(* Monotonicity of subtraction in first argument *) |
5069 | 663 |
Goal "!!n::nat. m<=n --> (m-l) <= (n-l)"; |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
664 |
by (induct_tac "n" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
665 |
by (Simp_tac 1); |
4089 | 666 |
by (simp_tac (simpset() addsimps [le_Suc_eq]) 1); |
4732 | 667 |
by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
668 |
qed_spec_mp "diff_le_mono"; |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
669 |
|
5069 | 670 |
Goal "!!n::nat. m<=n ==> (l-n) <= (l-m)"; |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
671 |
by (induct_tac "l" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
672 |
by (Simp_tac 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
673 |
by (case_tac "n <= l" 1); |
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
674 |
by (subgoal_tac "m <= l" 1); |
4089 | 675 |
by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1); |
676 |
by (fast_tac (claset() addEs [le_trans]) 1); |
|
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
677 |
by (dtac not_leE 1); |
4089 | 678 |
by (asm_simp_tac (simpset() addsimps [if_Suc_diff_n]) 1); |
3484
1e93eb09ebb9
Added the following lemmas tp Divides and a few others to Arith and NatDef:
nipkow
parents:
3457
diff
changeset
|
679 |
qed_spec_mp "diff_le_mono2"; |