(* Title: CTT/arith
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Theorems for arith.thy (Arithmetic operators)
Proofs about elementary arithmetic: addition, multiplication, etc.
Tests definitions and simplifier.
*)
open Arith;
val arith_defs = [add_def, diff_def, absdiff_def, mult_def, mod_def, div_def];
(** Addition *)
(*typing of add: short and long versions*)
qed_goalw "add_typing" Arith.thy arith_defs
"[| a:N; b:N |] ==> a #+ b : N"
(fn prems=> [ (typechk_tac prems) ]);
qed_goalw "add_typingL" Arith.thy arith_defs
"[| a=c:N; b=d:N |] ==> a #+ b = c #+ d : N"
(fn prems=> [ (equal_tac prems) ]);
(*computation for add: 0 and successor cases*)
qed_goalw "addC0" Arith.thy arith_defs
"b:N ==> 0 #+ b = b : N"
(fn prems=> [ (rew_tac prems) ]);
qed_goalw "addC_succ" Arith.thy arith_defs
"[| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N"
(fn prems=> [ (rew_tac prems) ]);
(** Multiplication *)
(*typing of mult: short and long versions*)
qed_goalw "mult_typing" Arith.thy arith_defs
"[| a:N; b:N |] ==> a #* b : N"
(fn prems=>
[ (typechk_tac([add_typing]@prems)) ]);
qed_goalw "mult_typingL" Arith.thy arith_defs
"[| a=c:N; b=d:N |] ==> a #* b = c #* d : N"
(fn prems=>
[ (equal_tac (prems@[add_typingL])) ]);
(*computation for mult: 0 and successor cases*)
qed_goalw "multC0" Arith.thy arith_defs
"b:N ==> 0 #* b = 0 : N"
(fn prems=> [ (rew_tac prems) ]);
qed_goalw "multC_succ" Arith.thy arith_defs
"[| a:N; b:N |] ==> succ(a) #* b = b #+ (a #* b) : N"
(fn prems=> [ (rew_tac prems) ]);
(** Difference *)
(*typing of difference*)
qed_goalw "diff_typing" Arith.thy arith_defs
"[| a:N; b:N |] ==> a - b : N"
(fn prems=> [ (typechk_tac prems) ]);
qed_goalw "diff_typingL" Arith.thy arith_defs
"[| a=c:N; b=d:N |] ==> a - b = c - d : N"
(fn prems=> [ (equal_tac prems) ]);
(*computation for difference: 0 and successor cases*)
qed_goalw "diffC0" Arith.thy arith_defs
"a:N ==> a - 0 = a : N"
(fn prems=> [ (rew_tac prems) ]);
(*Note: rec(a, 0, %z w.z) is pred(a). *)
qed_goalw "diff_0_eq_0" Arith.thy arith_defs
"b:N ==> 0 - b = 0 : N"
(fn prems=>
[ (NE_tac "b" 1),
(hyp_rew_tac prems) ]);
(*Essential to simplify FIRST!! (Else we get a critical pair)
succ(a) - succ(b) rewrites to pred(succ(a) - b) *)
qed_goalw "diff_succ_succ" Arith.thy arith_defs
"[| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N"
(fn prems=>
[ (hyp_rew_tac prems),
(NE_tac "b" 1),
(hyp_rew_tac prems) ]);
(*** Simplification *)
val arith_typing_rls =
[add_typing, mult_typing, diff_typing];
val arith_congr_rls =
[add_typingL, mult_typingL, diff_typingL];
val congr_rls = arith_congr_rls@standard_congr_rls;
val arithC_rls =
[addC0, addC_succ,
multC0, multC_succ,
diffC0, diff_0_eq_0, diff_succ_succ];
structure Arith_simp_data: TSIMP_DATA =
struct
val refl = refl_elem
val sym = sym_elem
val trans = trans_elem
val refl_red = refl_red
val trans_red = trans_red
val red_if_equal = red_if_equal
val default_rls = arithC_rls @ comp_rls
val routine_tac = routine_tac (arith_typing_rls @ routine_rls)
end;
structure Arith_simp = TSimpFun (Arith_simp_data);
fun arith_rew_tac prems = make_rew_tac
(Arith_simp.norm_tac(congr_rls, prems));
fun hyp_arith_rew_tac prems = make_rew_tac
(Arith_simp.cond_norm_tac(prove_cond_tac, congr_rls, prems));
(**********
Addition
**********)
(*Associative law for addition*)
qed_goal "add_assoc" Arith.thy
"[| a:N; b:N; c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac prems) ]);
(*Commutative law for addition. Can be proved using three inductions.
Must simplify after first induction! Orientation of rewrites is delicate*)
qed_goal "add_commute" Arith.thy
"[| a:N; b:N |] ==> a #+ b = b #+ a : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac prems),
(NE_tac "b" 2),
(rtac sym_elem 1),
(NE_tac "b" 1),
(hyp_arith_rew_tac prems) ]);
(****************
Multiplication
****************)
(*Commutative law for multiplication
qed_goal "mult_commute" Arith.thy
"[| a:N; b:N |] ==> a #* b = b #* a : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac prems),
(NE_tac "b" 2),
(rtac sym_elem 1),
(NE_tac "b" 1),
(hyp_arith_rew_tac prems) ]); NEEDS COMMUTATIVE MATCHING
***************)
(*right annihilation in product*)
qed_goal "mult_0_right" Arith.thy
"a:N ==> a #* 0 = 0 : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac prems) ]);
(*right successor law for multiplication*)
qed_goal "mult_succ_right" Arith.thy
"[| a:N; b:N |] ==> a #* succ(b) = a #+ (a #* b) : N"
(fn prems=>
[ (NE_tac "a" 1),
(*swap round the associative law of addition*)
(hyp_arith_rew_tac (prems @ [add_assoc RS sym_elem])),
(*leaves a goal involving a commutative law*)
(REPEAT (assume_tac 1 ORELSE
resolve_tac
(prems@[add_commute,mult_typingL,add_typingL]@
intrL_rls@[refl_elem]) 1)) ]);
(*Commutative law for multiplication*)
qed_goal "mult_commute" Arith.thy
"[| a:N; b:N |] ==> a #* b = b #* a : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac (prems @ [mult_0_right, mult_succ_right])) ]);
(*addition distributes over multiplication*)
qed_goal "add_mult_distrib" Arith.thy
"[| a:N; b:N; c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N"
(fn prems=>
[ (NE_tac "a" 1),
(*swap round the associative law of addition*)
(hyp_arith_rew_tac (prems @ [add_assoc RS sym_elem])) ]);
(*Associative law for multiplication*)
qed_goal "mult_assoc" Arith.thy
"[| a:N; b:N; c:N |] ==> (a #* b) #* c = a #* (b #* c) : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac (prems @ [add_mult_distrib])) ]);
(************
Difference
************
Difference on natural numbers, without negative numbers
a - b = 0 iff a<=b a - b = succ(c) iff a>b *)
qed_goal "diff_self_eq_0" Arith.thy
"a:N ==> a - a = 0 : N"
(fn prems=>
[ (NE_tac "a" 1),
(hyp_arith_rew_tac prems) ]);
(* [| c : N; 0 : N; c : N |] ==> c #+ 0 = c : N *)
val add_0_right = addC0 RSN (3, add_commute RS trans_elem);
(*Addition is the inverse of subtraction: if b<=x then b#+(x-b) = x.
An example of induction over a quantified formula (a product).
Uses rewriting with a quantified, implicative inductive hypothesis.*)
val prems =
goal Arith.thy
"b:N ==> ?a : PROD x:N. Eq(N, b-x, 0) --> Eq(N, b #+ (x-b), x)";
by (NE_tac "b" 1);
(*strip one "universal quantifier" but not the "implication"*)
by (resolve_tac intr_rls 3);
(*case analysis on x in
(succ(u) <= x) --> (succ(u)#+(x-succ(u)) = x) *)
by (NE_tac "x" 4 THEN assume_tac 4);
(*Prepare for simplification of types -- the antecedent succ(u)<=x *)
by (rtac replace_type 5);
by (rtac replace_type 4);
by (arith_rew_tac prems);
(*Solves first 0 goal, simplifies others. Two sugbgoals remain.
Both follow by rewriting, (2) using quantified induction hyp*)
by (intr_tac[]); (*strips remaining PRODs*)
by (hyp_arith_rew_tac (prems@[add_0_right]));
by (assume_tac 1);
qed "add_diff_inverse_lemma";
(*Version of above with premise b-a=0 i.e. a >= b.
Using ProdE does not work -- for ?B(?a) is ambiguous.
Instead, add_diff_inverse_lemma states the desired induction scheme;
the use of RS below instantiates Vars in ProdE automatically. *)
val prems =
goal Arith.thy "[| a:N; b:N; b-a = 0 : N |] ==> b #+ (a-b) = a : N";
by (rtac EqE 1);
by (resolve_tac [ add_diff_inverse_lemma RS ProdE RS ProdE ] 1);
by (REPEAT (resolve_tac (prems@[EqI]) 1));
qed "add_diff_inverse";
(********************
Absolute difference
********************)
(*typing of absolute difference: short and long versions*)
qed_goalw "absdiff_typing" Arith.thy arith_defs
"[| a:N; b:N |] ==> a |-| b : N"
(fn prems=> [ (typechk_tac prems) ]);
qed_goalw "absdiff_typingL" Arith.thy arith_defs
"[| a=c:N; b=d:N |] ==> a |-| b = c |-| d : N"
(fn prems=> [ (equal_tac prems) ]);
qed_goalw "absdiff_self_eq_0" Arith.thy [absdiff_def]
"a:N ==> a |-| a = 0 : N"
(fn prems=>
[ (arith_rew_tac (prems@[diff_self_eq_0])) ]);
qed_goalw "absdiffC0" Arith.thy [absdiff_def]
"a:N ==> 0 |-| a = a : N"
(fn prems=>
[ (hyp_arith_rew_tac prems) ]);
qed_goalw "absdiff_succ_succ" Arith.thy [absdiff_def]
"[| a:N; b:N |] ==> succ(a) |-| succ(b) = a |-| b : N"
(fn prems=>
[ (hyp_arith_rew_tac prems) ]);
(*Note how easy using commutative laws can be? ...not always... *)
val prems = goalw Arith.thy [absdiff_def]
"[| a:N; b:N |] ==> a |-| b = b |-| a : N";
by (rtac add_commute 1);
by (typechk_tac ([diff_typing]@prems));
qed "absdiff_commute";
(*If a+b=0 then a=0. Surprisingly tedious*)
val prems =
goal Arith.thy "[| a:N; b:N |] ==> ?c : PROD u: Eq(N,a#+b,0) . Eq(N,a,0)";
by (NE_tac "a" 1);
by (rtac replace_type 3);
by (arith_rew_tac prems);
by (intr_tac[]); (*strips remaining PRODs*)
by (resolve_tac [ zero_ne_succ RS FE ] 2);
by (etac (EqE RS sym_elem) 3);
by (typechk_tac ([add_typing] @prems));
qed "add_eq0_lemma";
(*Version of above with the premise a+b=0.
Again, resolution instantiates variables in ProdE *)
val prems =
goal Arith.thy "[| a:N; b:N; a #+ b = 0 : N |] ==> a = 0 : N";
by (rtac EqE 1);
by (resolve_tac [add_eq0_lemma RS ProdE] 1);
by (rtac EqI 3);
by (ALLGOALS (resolve_tac prems));
qed "add_eq0";
(*Here is a lemma to infer a-b=0 and b-a=0 from a|-|b=0, below. *)
val prems = goalw Arith.thy [absdiff_def]
"[| a:N; b:N; a |-| b = 0 : N |] ==> \
\ ?a : SUM v: Eq(N, a-b, 0) . Eq(N, b-a, 0)";
by (intr_tac[]);
by eqintr_tac;
by (rtac add_eq0 2);
by (rtac add_eq0 1);
by (resolve_tac [add_commute RS trans_elem] 6);
by (typechk_tac (diff_typing::prems));
qed "absdiff_eq0_lem";
(*if a |-| b = 0 then a = b
proof: a-b=0 and b-a=0, so b = a+(b-a) = a+0 = a*)
val prems =
goal Arith.thy "[| a |-| b = 0 : N; a:N; b:N |] ==> a = b : N";
by (rtac EqE 1);
by (resolve_tac [absdiff_eq0_lem RS SumE] 1);
by (TRYALL (resolve_tac prems));
by eqintr_tac;
by (resolve_tac [add_diff_inverse RS sym_elem RS trans_elem] 1);
by (rtac EqE 3 THEN assume_tac 3);
by (hyp_arith_rew_tac (prems@[add_0_right]));
qed "absdiff_eq0";
(***********************
Remainder and Quotient
***********************)
(*typing of remainder: short and long versions*)
qed_goalw "mod_typing" Arith.thy [mod_def]
"[| a:N; b:N |] ==> a mod b : N"
(fn prems=>
[ (typechk_tac (absdiff_typing::prems)) ]);
qed_goalw "mod_typingL" Arith.thy [mod_def]
"[| a=c:N; b=d:N |] ==> a mod b = c mod d : N"
(fn prems=>
[ (equal_tac (prems@[absdiff_typingL])) ]);
(*computation for mod : 0 and successor cases*)
qed_goalw "modC0" Arith.thy [mod_def] "b:N ==> 0 mod b = 0 : N"
(fn prems=>
[ (rew_tac(absdiff_typing::prems)) ]);
qed_goalw "modC_succ" Arith.thy [mod_def]
"[| a:N; b:N |] ==> succ(a) mod b = rec(succ(a mod b) |-| b, 0, %x y.succ(a mod b)) : N"
(fn prems=>
[ (rew_tac(absdiff_typing::prems)) ]);
(*typing of quotient: short and long versions*)
qed_goalw "div_typing" Arith.thy [div_def] "[| a:N; b:N |] ==> a div b : N"
(fn prems=>
[ (typechk_tac ([absdiff_typing,mod_typing]@prems)) ]);
qed_goalw "div_typingL" Arith.thy [div_def]
"[| a=c:N; b=d:N |] ==> a div b = c div d : N"
(fn prems=>
[ (equal_tac (prems @ [absdiff_typingL, mod_typingL])) ]);
val div_typing_rls = [mod_typing, div_typing, absdiff_typing];
(*computation for quotient: 0 and successor cases*)
qed_goalw "divC0" Arith.thy [div_def] "b:N ==> 0 div b = 0 : N"
(fn prems=>
[ (rew_tac([mod_typing, absdiff_typing] @ prems)) ]);
val divC_succ =
prove_goalw Arith.thy [div_def] "[| a:N; b:N |] ==> succ(a) div b = \
\ rec(succ(a) mod b, succ(a div b), %x y. a div b) : N"
(fn prems=>
[ (rew_tac([mod_typing]@prems)) ]);
(*Version of above with same condition as the mod one*)
qed_goal "divC_succ2" Arith.thy
"[| a:N; b:N |] ==> \
\ succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), %x y. a div b) : N"
(fn prems=>
[ (resolve_tac [ divC_succ RS trans_elem ] 1),
(rew_tac(div_typing_rls @ prems @ [modC_succ])),
(NE_tac "succ(a mod b)|-|b" 1),
(rew_tac ([mod_typing, div_typing, absdiff_typing] @prems)) ]);
(*for case analysis on whether a number is 0 or a successor*)
qed_goal "iszero_decidable" Arith.thy
"a:N ==> rec(a, inl(eq), %ka kb.inr(<ka, eq>)) : \
\ Eq(N,a,0) + (SUM x:N. Eq(N,a, succ(x)))"
(fn prems=>
[ (NE_tac "a" 1),
(rtac PlusI_inr 3),
(rtac PlusI_inl 2),
eqintr_tac,
(equal_tac prems) ]);
(*Main Result. Holds when b is 0 since a mod 0 = a and a div 0 = 0 *)
val prems =
goal Arith.thy "[| a:N; b:N |] ==> a mod b #+ (a div b) #* b = a : N";
by (NE_tac "a" 1);
by (arith_rew_tac (div_typing_rls@prems@[modC0,modC_succ,divC0,divC_succ2]));
by (rtac EqE 1);
(*case analysis on succ(u mod b)|-|b *)
by (res_inst_tac [("a1", "succ(u mod b) |-| b")]
(iszero_decidable RS PlusE) 1);
by (etac SumE 3);
by (hyp_arith_rew_tac (prems @ div_typing_rls @
[modC0,modC_succ, divC0, divC_succ2]));
(*Replace one occurence of b by succ(u mod b). Clumsy!*)
by (resolve_tac [ add_typingL RS trans_elem ] 1);
by (eresolve_tac [EqE RS absdiff_eq0 RS sym_elem] 1);
by (rtac refl_elem 3);
by (hyp_arith_rew_tac (prems @ div_typing_rls));
qed "mod_div_equality";
writeln"Reached end of file.";