# File Arith.ML

    (*  Title:      CTT/Arith.ML
        ID:         $Id: Arith.ML,v 1.8 2005/09/16 21:01:29 wenzelm Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1991  University of Cambridge

    Proofs about elementary arithmetic: addition, multiplication, etc.
    Tests definitions and simplifier.
    *)

    val arith_defs = [add_def, diff_def, absdiff_def, mult_def, mod_def, div_def];

    (** Addition *)

    (*typing of add: short and long versions*)

    Goalw arith_defs "[| a:N;  b:N |] ==> a #+ b : N";
    by (typechk_tac []) ;
    qed "add_typing";

    Goalw arith_defs "[| a=c:N;  b=d:N |] ==> a #+ b = c #+ d : N";
    by (equal_tac []) ;
    qed "add_typingL";

    (*computation for add: 0 and successor cases*)

    Goalw arith_defs "b:N ==> 0 #+ b = b : N";
    by (rew_tac []) ;
    qed "addC0";

    Goalw arith_defs "[| a:N;  b:N |] ==> succ(a) #+ b = succ(a #+ b) : N";
    by (rew_tac []) ;
    qed "addC_succ";

    (** Multiplication *)

    (*typing of mult: short and long versions*)

    Goalw arith_defs "[| a:N;  b:N |] ==> a #* b : N";
    by (typechk_tac [add_typing]) ;
    qed "mult_typing";

    Goalw arith_defs "[| a=c:N;  b=d:N |] ==> a #* b = c #* d : N";
    by (equal_tac [add_typingL]) ;
    qed "mult_typingL";

    (*computation for mult: 0 and successor cases*)

    Goalw arith_defs "b:N ==> 0 #* b = 0 : N";
    by (rew_tac []) ;
    qed "multC0";

    Goalw arith_defs "[| a:N;  b:N |] ==> succ(a) #* b = b #+ (a #* b) : N";
    by (rew_tac []) ;
    qed "multC_succ";

    (** Difference *)

    (*typing of difference*)

    Goalw arith_defs "[| a:N;  b:N |] ==> a - b : N";
    by (typechk_tac []) ;
    qed "diff_typing";

    Goalw arith_defs "[| a=c:N;  b=d:N |] ==> a - b = c - d : N";
    by (equal_tac []) ;
    qed "diff_typingL";

    (*computation for difference: 0 and successor cases*)

    Goalw arith_defs "a:N ==> a - 0 = a : N";
    by (rew_tac []) ;
    qed "diffC0";

    (*Note: rec(a, 0, %z w.z) is pred(a). *)

    Goalw arith_defs "b:N ==> 0 - b = 0 : N";
    by (NE_tac "b" 1);
    by (hyp_rew_tac []) ;
    qed "diff_0_eq_0";

    (*Essential to simplify FIRST!!  (Else we get a critical pair)
      succ(a) - succ(b) rewrites to   pred(succ(a) - b)  *)
    Goalw arith_defs "[| a:N;  b:N |] ==> succ(a) - succ(b) = a - b : N";
    by (hyp_rew_tac []);
    by (NE_tac "b" 1);
    by (hyp_rew_tac []) ;
    qed "diff_succ_succ";

    (*** 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*)
    Goal "[| a:N;  b:N;  c:N |] ==> (a #+ b) #+ c = a #+ (b #+ c) : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac []) ;
    qed "add_assoc";

    (*Commutative law for addition.  Can be proved using three inductions.
      Must simplify after first induction!  Orientation of rewrites is delicate*)
    Goal "[| a:N;  b:N |] ==> a #+ b = b #+ a : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac []);
    by (NE_tac "b" 2);
    by (rtac sym_elem 1);
    by (NE_tac "b" 1);
    by (hyp_arith_rew_tac []) ;
    qed "add_commute";

    (****************
      Multiplication
     ****************)

    (*Commutative law for multiplication
    Goal "[| a:N;  b:N |] ==> a #* b = b #* a : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac []);
    by (NE_tac "b" 2);
    by (rtac sym_elem 1);
    by (NE_tac "b" 1);
    by (hyp_arith_rew_tac []) ;
    qed "mult_commute";   NEEDS COMMUTATIVE MATCHING
    ***************)

    (*right annihilation in product*)
    Goal "a:N ==> a #* 0 = 0 : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac []) ;
    qed "mult_0_right";

    (*right successor law for multiplication*)
    Goal "[| a:N;  b:N |] ==> a #* succ(b) = a #+ (a #* b) : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac [add_assoc RS sym_elem]);
    by (REPEAT (assume_tac 1
         ORELSE resolve_tac ([add_commute,mult_typingL,add_typingL]@ intrL_rls@
                             [refl_elem])   1)) ;
    qed "mult_succ_right";

    (*Commutative law for multiplication*)
    Goal "[| a:N;  b:N |] ==> a #* b = b #* a : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac [mult_0_right, mult_succ_right]) ;
    qed "mult_commute";

    (*addition distributes over multiplication*)
    Goal "[| a:N;  b:N;  c:N |] ==> (a #+ b) #* c = (a #* c) #+ (b #* c) : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac [add_assoc RS sym_elem]) ;
    qed "add_mult_distrib";

    (*Associative law for multiplication*)
    Goal "[| a:N;  b:N;  c:N |] ==> (a #* b) #* c = a #* (b #* c) : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac [add_mult_distrib]) ;
    qed "mult_assoc";

    (************
      Difference
     ************

    Difference on natural numbers, without negative numbers
      a - b = 0  iff  a<=b    a - b = succ(c) iff a>b   *)

    Goal "a:N ==> a - a = 0 : N";
    by (NE_tac "a" 1);
    by (hyp_arith_rew_tac []) ;
    qed "diff_self_eq_0";

    (*  [| 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.*)
    Goal "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 []);
    (*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 [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. *)
    Goal "[| 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 (ares_tac [EqI] 1));
    qed "add_diff_inverse";

    (********************
      Absolute difference
     ********************)

    (*typing of absolute difference: short and long versions*)

    Goalw arith_defs "[| a:N;  b:N |] ==> a |-| b : N";
    by (typechk_tac []) ;
    qed "absdiff_typing";

    Goalw arith_defs "[| a=c:N;  b=d:N |] ==> a |-| b = c |-| d : N";
    by (equal_tac []) ;
    qed "absdiff_typingL";

    Goalw [absdiff_def] "a:N ==> a |-| a = 0 : N";
    by (arith_rew_tac [diff_self_eq_0]) ;
    qed "absdiff_self_eq_0";

    Goalw [absdiff_def] "a:N ==> 0 |-| a = a : N";
    by (hyp_arith_rew_tac []);
    qed "absdiffC0";

    Goalw [absdiff_def] "[| a:N;  b:N |] ==> succ(a) |-| succ(b)  =  a |-| b : N";
    by (hyp_arith_rew_tac []) ;
    qed "absdiff_succ_succ";

    (*Note how easy using commutative laws can be?  ...not always... *)
    Goalw [absdiff_def] "[| a:N;  b:N |] ==> a |-| b = b |-| a : N";
    by (rtac add_commute 1);
    by (typechk_tac [diff_typing]);
    qed "absdiff_commute";

    (*If a+b=0 then a=0.   Surprisingly tedious*)
    Goal "[| 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 []);
    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]);
    qed "add_eq0_lemma";

    (*Version of above with the premise  a+b=0.
      Again, resolution instantiates variables in ProdE *)
    Goal "[| 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 (typechk_tac []) ;
    qed "add_eq0";

    (*Here is a lemma to infer a-b=0 and b-a=0 from a|-|b=0, below. *)
    Goalw [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]);
    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*)
    Goal "[| 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 assume_tac);
    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 [add_0_right]);
    qed "absdiff_eq0";

    (***********************
      Remainder and Quotient
     ***********************)

    (*typing of remainder: short and long versions*)

    Goalw [mod_def] "[| a:N;  b:N |] ==> a mod b : N";
    by (typechk_tac [absdiff_typing]) ;
    qed "mod_typing";

    Goalw [mod_def] "[| a=c:N;  b=d:N |] ==> a mod b = c mod d : N";
    by (equal_tac [absdiff_typingL]) ;
    qed "mod_typingL";

    (*computation for  mod : 0 and successor cases*)

    Goalw [mod_def]   "b:N ==> 0 mod b = 0 : N";
    by (rew_tac [absdiff_typing]) ;
    qed "modC0";

    Goalw [mod_def]
    "[| a:N; b:N |] ==> succ(a) mod b = rec(succ(a mod b) |-| b, 0, %x y. succ(a mod b)) : N";
    by (rew_tac [absdiff_typing]) ;
    qed "modC_succ";

    (*typing of quotient: short and long versions*)

    Goalw [div_def]   "[| a:N;  b:N |] ==> a div b : N";
    by (typechk_tac [absdiff_typing,mod_typing]) ;
    qed "div_typing";

    Goalw [div_def] "[| a=c:N;  b=d:N |] ==> a div b = c div d : N";
    by (equal_tac [absdiff_typingL, mod_typingL]);
    qed "div_typingL";

    val div_typing_rls = [mod_typing, div_typing, absdiff_typing];

    (*computation for quotient: 0 and successor cases*)

    Goalw [div_def]   "b:N ==> 0 div b = 0 : N";
    by (rew_tac [mod_typing, absdiff_typing]) ;
    qed "divC0";

    Goalw [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";
    by (rew_tac [mod_typing]) ;
    qed "divC_succ";

    (*Version of above with same condition as the  mod  one*)
    Goal "[| a:N;  b:N |] ==> \
    \    succ(a) div b =rec(succ(a mod b) |-| b, succ(a div b), %x y. a div b) : N";
    by (resolve_tac [ divC_succ RS trans_elem ] 1);
    by (rew_tac(div_typing_rls @ [modC_succ]));
    by (NE_tac "succ(a mod b)|-|b" 1);
    by (rew_tac [mod_typing, div_typing, absdiff_typing]);
    qed "divC_succ2";

    (*for case analysis on whether a number is 0 or a successor*)
    Goal "a:N ==> rec(a, inl(eq), %ka kb. inr(<ka, eq>)) : \
    \                     Eq(N,a,0) + (SUM x:N. Eq(N,a, succ(x)))";
    by (NE_tac "a" 1);
    by (rtac PlusI_inr 3);
    by (rtac PlusI_inl 2);
    by eqintr_tac;
    by (equal_tac []) ;
    qed "iszero_decidable";

    (*Main Result.  Holds when b is 0 since   a mod 0 = a     and    a div 0 = 0  *)
    Goal "[| 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@[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 (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 (div_typing_rls));
    qed "mod_div_equality";