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Prove sn_app #1

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Prove sn_app #1

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18 changes: 15 additions & 3 deletions Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/StrongNorm.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -32,9 +32,21 @@ lemma sn_app (t s : Term Var) :
SN s →
(∀ {t' s' : Term Var}, t ↠βᶠ t'.abs → s ↠βᶠ s' → SN (t' ^ s')) →
SN (t.app s) := by
sorry


intro h_sn_t h_sn_s hβ
induction h_sn_t generalizing s with | sn t ht ih_t =>
induction h_sn_s with | sn s hs ih_s =>
constructor
intro u hstep
cases hstep with
| beta _ _ => apply hβ <;> rfl
| appL _ h_s_red =>
apply ih_s _ h_s_red
intro t' s'' hstep1 hstep2
exact hβ hstep1 (Relation.ReflTransGen.head h_s_red hstep2)
| appR _ h_t_red =>
apply ih_t _ h_t_red _ (SN.sn s hs)
intro t' s' hstep1 hstep2
exact hβ (Relation.ReflTransGen.head h_t_red hstep1) hstep2

lemma sn_app_left (M N : Term Var) : Term.LC N → SN (M.app N) → SN M := by
intro lc_N h_sn
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