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Prove some simple soundness cases of subtyping

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Yiyun Liu 2025-02-17 21:43:21 -05:00
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14
theories/fp_red.v
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@ -2620,6 +2620,10 @@ Module Sub1.
R a b -> R (subst_PTm ρ a) (subst_PTm ρ b).
Proof. move => h. move : m ρ. elim : n a b /h; hauto lq:on ctrs:R. Qed.
Lemma hne_refl n (a b : PTm n) :
ishne a \/ ishne b -> R a b -> a = b.
Proof. hauto q:on inv:R. Qed.
End Sub1.
Module Sub.
@ -2628,6 +2632,16 @@ Module Sub.
Lemma refl n (a : PTm n) : R a a.
Proof. sfirstorder use:@rtc_refl unfold:R. Qed.
Lemma ToJoin n (a b : PTm n) :
ishne a \/ ishne b ->
R a b ->
DJoin.R a b.
Proof.
move => h [c][d][h0][h1]h2.
have : ishne c \/ ishne d by hauto q:on use:REReds.hne_preservation.
hauto lq:on rew:off use:Sub1.hne_refl.
Qed.
Lemma transitive n (a b c : PTm n) : SN b -> R a b -> R b c -> R a c.
Proof.
rewrite /R.