--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/CTT/arith.ML Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,497 @@
+(* 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*)
+
+val add_typing = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> a #+ b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (typechk_tac prems) ]);
+
+val add_typingL = prove_goal Arith.thy
+ "[| a=c:N; b=d:N |] ==> a #+ b = c #+ d : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (equal_tac prems) ]);
+
+
+(*computation for add: 0 and successor cases*)
+
+val addC0 = prove_goal Arith.thy
+ "b:N ==> 0 #+ b = b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (rew_tac prems) ]);
+
+val addC_succ = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> succ(a) #+ b = succ(a #+ b) : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (rew_tac prems) ]);
+
+
+(** Multiplication *)
+
+(*typing of mult: short and long versions*)
+
+val mult_typing = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> a #* b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (typechk_tac([add_typing]@prems)) ]);
+
+val mult_typingL = prove_goal Arith.thy
+ "[| a=c:N; b=d:N |] ==> a #* b = c #* d : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (equal_tac (prems@[add_typingL])) ]);
+
+(*computation for mult: 0 and successor cases*)
+
+val multC0 = prove_goal Arith.thy
+ "b:N ==> 0 #* b = 0 : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (rew_tac prems) ]);
+
+val multC_succ = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> succ(a) #* b = b #+ (a #* b) : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (rew_tac prems) ]);
+
+
+(** Difference *)
+
+(*typing of difference*)
+
+val diff_typing = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> a - b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (typechk_tac prems) ]);
+
+val diff_typingL = prove_goal Arith.thy
+ "[| a=c:N; b=d:N |] ==> a - b = c - d : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (equal_tac prems) ]);
+
+
+
+(*computation for difference: 0 and successor cases*)
+
+val diffC0 = prove_goal Arith.thy
+ "a:N ==> a - 0 = a : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (rew_tac prems) ]);
+
+(*Note: rec(a, 0, %z w.z) is pred(a). *)
+
+val diff_0_eq_0 = prove_goal Arith.thy
+ "b:N ==> 0 - b = 0 : N"
+ (fn prems=>
+ [ (NE_tac "b" 1),
+ (rewrite_goals_tac arith_defs),
+ (hyp_rew_tac prems) ]);
+
+
+(*Essential to simplify FIRST!! (Else we get a critical pair)
+ succ(a) - succ(b) rewrites to pred(succ(a) - b) *)
+val diff_succ_succ = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> succ(a) - succ(b) = a - b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (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*)
+val add_assoc = prove_goal 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*)
+val add_commute = prove_goal 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),
+ (resolve_tac [sym_elem] 1),
+ (NE_tac "b" 1),
+ (hyp_arith_rew_tac prems) ]);
+
+
+(****************
+ Multiplication
+ ****************)
+
+(*Commutative law for multiplication
+val mult_commute = prove_goal 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),
+ (resolve_tac [sym_elem] 1),
+ (NE_tac "b" 1),
+ (hyp_arith_rew_tac prems) ]); NEEDS COMMUTATIVE MATCHING
+***************)
+
+(*right annihilation in product*)
+val mult_0_right = prove_goal Arith.thy
+ "a:N ==> a #* 0 = 0 : N"
+ (fn prems=>
+ [ (NE_tac "a" 1),
+ (hyp_arith_rew_tac prems) ]);
+
+(*right successor law for multiplication*)
+val mult_succ_right = prove_goal 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*)
+val mult_commute = prove_goal 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*)
+val add_mult_distrib = prove_goal 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*)
+val mult_assoc = prove_goal 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 *)
+
+val diff_self_eq_0 = prove_goal 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 (resolve_tac [replace_type] 5);
+by (resolve_tac [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);
+val add_diff_inverse_lemma = result();
+
+
+(*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 (resolve_tac [EqE] 1);
+by (resolve_tac [ add_diff_inverse_lemma RS ProdE RS ProdE ] 1);
+by (REPEAT (resolve_tac (prems@[EqI]) 1));
+val add_diff_inverse = result();
+
+
+(********************
+ Absolute difference
+ ********************)
+
+(*typing of absolute difference: short and long versions*)
+
+val absdiff_typing = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> a |-| b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (typechk_tac prems) ]);
+
+val absdiff_typingL = prove_goal Arith.thy
+ "[| a=c:N; b=d:N |] ==> a |-| b = c |-| d : N"
+ (fn prems=>
+ [ (rewrite_goals_tac arith_defs),
+ (equal_tac prems) ]);
+
+val absdiff_self_eq_0 = prove_goal Arith.thy
+ "a:N ==> a |-| a = 0 : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [absdiff_def]),
+ (arith_rew_tac (prems@[diff_self_eq_0])) ]);
+
+val absdiffC0 = prove_goal Arith.thy
+ "a:N ==> 0 |-| a = a : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [absdiff_def]),
+ (hyp_arith_rew_tac prems) ]);
+
+
+val absdiff_succ_succ = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> succ(a) |-| succ(b) = a |-| b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [absdiff_def]),
+ (hyp_arith_rew_tac prems) ]);
+
+(*Note how easy using commutative laws can be? ...not always... *)
+val prems = goal Arith.thy "[| a:N; b:N |] ==> a |-| b = b |-| a : N";
+by (rewrite_goals_tac [absdiff_def]);
+by (resolve_tac [add_commute] 1);
+by (typechk_tac ([diff_typing]@prems));
+val absdiff_commute = result();
+
+(*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 (resolve_tac [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));
+val add_eq0_lemma = result();
+
+(*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 (resolve_tac [EqE] 1);
+by (resolve_tac [add_eq0_lemma RS ProdE] 1);
+by (resolve_tac [EqI] 3);
+by (ALLGOALS (resolve_tac prems));
+val add_eq0 = result();
+
+(*Here is a lemma to infer a-b=0 and b-a=0 from a|-|b=0, below. *)
+val prems = goal Arith.thy
+ "[| 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 (resolve_tac [add_eq0] 2);
+by (resolve_tac [add_eq0] 1);
+by (resolve_tac [add_commute RS trans_elem] 6);
+by (typechk_tac (diff_typing:: map (rewrite_rule [absdiff_def]) prems));
+val absdiff_eq0_lem = result();
+
+(*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 (resolve_tac [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 (resolve_tac [EqE] 3 THEN assume_tac 3);
+by (hyp_arith_rew_tac (prems@[add_0_right]));
+val absdiff_eq0 = result();
+
+(***********************
+ Remainder and Quotient
+ ***********************)
+
+(*typing of remainder: short and long versions*)
+
+val mod_typing = prove_goal Arith.thy
+ "[| a:N; b:N |] ==> a mod b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [mod_def]),
+ (typechk_tac (absdiff_typing::prems)) ]);
+
+val mod_typingL = prove_goal Arith.thy
+ "[| a=c:N; b=d:N |] ==> a mod b = c mod d : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [mod_def]),
+ (equal_tac (prems@[absdiff_typingL])) ]);
+
+
+(*computation for mod : 0 and successor cases*)
+
+val modC0 = prove_goal Arith.thy "b:N ==> 0 mod b = 0 : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [mod_def]),
+ (rew_tac(absdiff_typing::prems)) ]);
+
+val modC_succ = prove_goal Arith.thy
+"[| a:N; b:N |] ==> succ(a) mod b = rec(succ(a mod b) |-| b, 0, %x y.succ(a mod b)) : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [mod_def]),
+ (rew_tac(absdiff_typing::prems)) ]);
+
+
+(*typing of quotient: short and long versions*)
+
+val div_typing = prove_goal Arith.thy "[| a:N; b:N |] ==> a div b : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [div_def]),
+ (typechk_tac ([absdiff_typing,mod_typing]@prems)) ]);
+
+val div_typingL = prove_goal Arith.thy
+ "[| a=c:N; b=d:N |] ==> a div b = c div d : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [div_def]),
+ (equal_tac (prems @ [absdiff_typingL, mod_typingL])) ]);
+
+val div_typing_rls = [mod_typing, div_typing, absdiff_typing];
+
+
+(*computation for quotient: 0 and successor cases*)
+
+val divC0 = prove_goal Arith.thy "b:N ==> 0 div b = 0 : N"
+ (fn prems=>
+ [ (rewrite_goals_tac [div_def]),
+ (rew_tac([mod_typing, absdiff_typing] @ prems)) ]);
+
+val divC_succ =
+prove_goal Arith.thy "[| 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=>
+ [ (rewrite_goals_tac [div_def]),
+ (rew_tac([mod_typing]@prems)) ]);
+
+
+(*Version of above with same condition as the mod one*)
+val divC_succ2 = prove_goal 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*)
+val iszero_decidable = prove_goal 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),
+ (resolve_tac [PlusI_inr] 3),
+ (resolve_tac [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 (resolve_tac [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 (resolve_tac [refl_elem] 3);
+by (hyp_arith_rew_tac (prems @ div_typing_rls));
+val mod_div_equality = result();
+
+writeln"Reached end of file.";