# Theory Equality

```(*  Title:      CTT/ex/Equality.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

section "Equality reasoning by rewriting"

theory Equality
imports "../CTT"
begin

lemma split_eq: "p : Sum(A,B) ⟹ split(p,pair) = p : Sum(A,B)"
apply (rule EqE)
apply (rule elim_rls, assumption)
apply rew
done

lemma when_eq: "⟦A type; B type; p : A+B⟧ ⟹ when(p,inl,inr) = p : A + B"
apply (rule EqE)
apply (rule elim_rls, assumption)
apply rew
done

(*in the "rec" formulation of addition, 0+n=n *)
lemma "p:N ⟹ rec(p,0, λy z. succ(y)) = p : N"
apply (rule EqE)
apply (rule elim_rls, assumption)
apply rew
done

(*the harder version, n+0=n: recursive, uses induction hypothesis*)
lemma "p:N ⟹ rec(p,0, λy z. succ(z)) = p : N"
apply (rule EqE)
apply (rule elim_rls, assumption)
apply hyp_rew
done

lemma "⟦a:N; b:N; c:N⟧
⟹ rec(rec(a, b, λx y. succ(y)), c, λx y. succ(y)) =
rec(a, rec(b, c, λx y. succ(y)), λx y. succ(y)) : N"
apply (NE a)
apply hyp_rew
done

(*Martin-Löf (1984) page 62: pairing is surjective*)
lemma "p : Sum(A,B) ⟹ <split(p,λx y. x), split(p,λx y. y)> = p : Sum(A,B)"
apply (rule EqE)
apply (rule elim_rls, assumption)
apply (tactic ‹DEPTH_SOLVE_1 (rew_tac \<^context> [])›) (*!!!!!!!*)
done

lemma "⟦a : A; b : B⟧ ⟹ (❙λu. split(u, λv w.<w,v>)) ` <a,b> = <b,a> : ∑x:B. A"
apply rew
done

(*a contrived, complicated simplication, requires sum-elimination also*)
lemma "(❙λf. ❙λx. f`(f`x)) ` (❙λu. split(u, λv w.<w,v>)) =
❙λx. x  :  ∏x:(∑y:N. N). (∑y:N. N)"
apply (rule reduction_rls)
apply (rule_tac [3] intrL_rls)
apply (rule_tac [4] EqE)
apply (erule_tac [4] SumE)
(*order of unifiers is essential here*)
apply rew
done

end
```