Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Prove that 0 != 1 with the W-encoded natural number types #711

Merged
merged 1 commit into from
Mar 1, 2025
Merged

Prove that 0 != 1 with the W-encoded natural number types #711

Changes from all commits
Commits
Show all changes
1 commit
Select commit
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
46 changes: 31 additions & 15 deletions proofs/type_theory/w_types.v
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -22,17 +22,14 @@ Inductive W [A] (B : A -> Type) :=

Arguments sup [_ _] _ _.

Check W_ind.
Check W_rect.

(*
```
W_ind :
forall (A : Type) (B : A -> Type) (P : W B -> Prop),
(
forall (a : A) (f : B a -> W B),
(forall b : B a, P (f b)) ->
P (sup a f)
) ->
W_rect :
forall (A : Type) (B : A -> Type) (P : W B -> Type),
(forall (a : A) (f : B a -> W B), (forall b : B a, P (f b)) ->
P (sup a f)) ->
forall w : W B, P w
```
*)
Expand Down Expand Up @@ -130,6 +127,16 @@ Proof.
reflexivity.
Qed.

Goal pre_zero <> pre_succ pre_zero.
Proof.
intro.
pose (f := pre_recursor Type unit (fun _ => Empty_set)).
pose (x := tt : f pre_zero).
rewrite H in x.
cbn in x.
destruct x.
Qed.

(*
There are two situations above where function extensionality was required:

Expand All @@ -142,9 +149,9 @@ Qed.
η-conversion on Π types, which we already have in Coq).

This encoding of natural numbers suffers from the general issue that the `f`
argument of the `sup` constructor interacts poorly with judgmental equality
by virtue of being a function. For example, here's another definition of zero
that is extensionally but not judgmentally equal to `zero`:
argument of the `sup` constructor isn't sufficiently constrained. For
example, here's another definition of zero that is extensionally but not
judgmentally equal to `zero`:
*)

Definition other_zero : PreNat := sup true (fun _ : Empty_set => pre_zero).
Expand Down Expand Up @@ -245,18 +252,17 @@ Definition eliminator
| eq_refl => p_zero
end
| false => fun f h c =>
let eq := c.(y).(x) in
match eq
match c.(y).(x) as e
in _ = z
return
forall p : Canonical (z tt),
P {| x := z tt; y := p |} ->
P {|
x := sup false z;
y := {| x := c.(x); y := {| x := eq; y := p |} |}
y := {| x := c.(x); y := {| x := e; y := p |} |}
|}
with
| eq_refl => fun p ih => p_succ {| x := c.(x); y := p |} ih
| eq_refl => fun p i => p_succ {| x := c.(x); y := p |} i
end
c.(y).(y) (h tt c.(y).(y))
end
Expand Down Expand Up @@ -285,3 +291,13 @@ Proof.
rewrite H.
reflexivity.
Qed.

Goal zero <> succ zero.
Proof.
intro.
pose (f := recursor Type unit (fun _ => Empty_set)).
pose (x := tt : f zero).
rewrite H in x.
cbn in x.
destruct x.
Qed.