doc-src/Ref/simplifier-eg.txt

final Springer version

Pretty.setmargin 70;

> goal Nat.thy "m+0 = m";
Level 0
m + 0 = m
 1. m + 0 = m
> by (res_inst_tac [("n","m")] induct 1);
Level 1
m + 0 = m
 1. 0 + 0 = 0
 2. !!x. x + 0 = x ==> Suc(x) + 0 = Suc(x)
> by (simp_tac add_ss 1);
Level 2
m + 0 = m
 1. !!x. x + 0 = x ==> Suc(x) + 0 = Suc(x)
> by (asm_simp_tac add_ss 1);
Level 3
m + 0 = m
No subgoals!


> goal Nat.thy "m+Suc(n) = Suc(m+n)";
Level 0
m + Suc(n) = Suc(m + n)
 1. m + Suc(n) = Suc(m + n)
val it = [] : thm list   
> by (res_inst_tac [("n","m")] induct 1);
Level 1
m + Suc(n) = Suc(m + n)
 1. 0 + Suc(n) = Suc(0 + n)
 2. !!x. x + Suc(n) = Suc(x + n) ==> Suc(x) + Suc(n) = Suc(Suc(x) + n)
val it = () : unit   
> by (simp_tac add_ss 1);
Level 2
m + Suc(n) = Suc(m + n)
 1. !!x. x + Suc(n) = Suc(x + n) ==> Suc(x) + Suc(n) = Suc(Suc(x) + n)
val it = () : unit   
> trace_simp := true;
val it = () : unit   
> by (asm_simp_tac add_ss 1);
Rewriting:
Suc(x) + Suc(n) == Suc(x + Suc(n))
Rewriting:
x + Suc(n) == Suc(x + n)
Rewriting:
Suc(x) + n == Suc(x + n)
Rewriting:
Suc(Suc(x + n)) = Suc(Suc(x + n)) == True
Level 3
m + Suc(n) = Suc(m + n)
No subgoals!
val it = () : unit   



> val prems = goal Nat.thy "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)";
Level 0
f(i + j) = i + f(j)
 1. f(i + j) = i + f(j)
> prths prems;
f(Suc(?n)) = Suc(f(?n))  [!!n. f(Suc(n)) = Suc(f(n))]

> by (res_inst_tac [("n","i")] induct 1);
Level 1
f(i + j) = i + f(j)
 1. f(0 + j) = 0 + f(j)
 2. !!x. f(x + j) = x + f(j) ==> f(Suc(x) + j) = Suc(x) + f(j)
> by (simp_tac f_ss 1);
Level 2
f(i + j) = i + f(j)
 1. !!x. f(x + j) = x + f(j) ==> f(Suc(x) + j) = Suc(x) + f(j)
> by (asm_simp_tac (f_ss addrews prems) 1);
Level 3
f(i + j) = i + f(j)
No subgoals!



> goal NatSum.thy "Suc(Suc(0))*sum(%i.i,Suc(n)) = n*Suc(n)";
Level 0
Suc(Suc(0)) * sum(%i. i, Suc(n)) = n * Suc(n)
 1. Suc(Suc(0)) * sum(%i. i, Suc(n)) = n * Suc(n)

> by (simp_tac natsum_ss 1);
Level 1
Suc(Suc(0)) * sum(%i. i, Suc(n)) = n * Suc(n)
 1. n + (n + (sum(%i. i, n) + sum(%i. i, n))) = n + n * n

> by (nat_ind_tac "n" 1);
Level 2
Suc(Suc(0)) * sum(%i. i, Suc(n)) = n * Suc(n)
 1. 0 + (0 + (sum(%i. i, 0) + sum(%i. i, 0))) = 0 + 0 * 0
 2. !!n1. n1 + (n1 + (sum(%i. i, n1) + sum(%i. i, n1))) =
          n1 + n1 * n1 ==>
          Suc(n1) +
          (Suc(n1) + (sum(%i. i, Suc(n1)) + sum(%i. i, Suc(n1)))) =
          Suc(n1) + Suc(n1) * Suc(n1)

> by (simp_tac natsum_ss 1);
Level 3
Suc(Suc(0)) * sum(%i. i, Suc(n)) = n * Suc(n)
 1. !!n1. n1 + (n1 + (sum(%i. i, n1) + sum(%i. i, n1))) =
          n1 + n1 * n1 ==>
          Suc(n1) +
          (Suc(n1) + (sum(%i. i, Suc(n1)) + sum(%i. i, Suc(n1)))) =
          Suc(n1) + Suc(n1) * Suc(n1)

> by (asm_simp_tac natsum_ss 1);
Level 4
Suc(Suc(0)) * sum(%i. i, Suc(n)) = n * Suc(n)
No subgoals!