44 lines
1.2 KiB
Text
44 lines
1.2 KiB
Text
|
import data.list
|
||
|
|
|
||
|
|
inductive typ : Type :=
|
||
|
|
| nat : typ
|
||
|
|
| arr : typ → typ → typ
|
||
|
|
|
||
|
|
inductive const : Type :=
|
||
|
|
| z | s
|
||
|
|
|
||
|
|
inductive exp : Type :=
|
||
|
|
| var : nat → exp
|
||
|
|
| cnst : const → exp
|
||
|
|
| lam : nat → typ → exp → exp
|
||
|
|
| ap : exp → exp → exp
|
||
|
|
|
||
|
|
open exp
|
||
|
|
|
||
|
|
inductive is_val : exp → Prop :=
|
||
|
|
| vcnst : Π c, is_val (cnst c)
|
||
|
|
| vlam : Π x t e, is_val (lam x t e)
|
||
|
|
| vcnst_ap : Π {e} c, is_val e → is_val (ap (cnst c) e)
|
||
|
|
|
||
|
|
inductive step : exp → exp → Prop :=
|
||
|
|
infix `➤`:50 := step
|
||
|
|
| stepl : Π {e1 e1'} e2, e1 ➤ e1' → ap e1 e2 ➤ ap e1' e2
|
||
|
|
| stepr : Π {e1 e2 e2'}, is_val e1 → e2 ➤ e2' → ap e1 e2 ➤ ap e1 e2'
|
||
|
|
| subst : Π {x e1 e1' e2} t, is_val e2 → ap (lam x t e1) e2 ➤ e1'
|
||
|
|
|
||
|
|
infix `➤`:50 := step
|
||
|
|
|
||
|
|
open is_val
|
||
|
|
open step
|
||
|
|
|
||
|
|
theorem nostep : ∀ {e} e', is_val e → e ➤ e' → false
|
||
|
|
| nostep e' (vcnst c) Hsteps := by cases Hsteps
|
||
|
|
| nostep e' (vlam x t e) Hsteps := by cases Hsteps
|
||
|
|
| nostep (ap e' e) (@vcnst_ap e c Hval) (stepl e Hbad) :=
|
||
|
|
have Hvalc : is_val (cnst c), from vcnst c,
|
||
|
|
have IH : not (cnst c ➤ e'), from nostep e' Hvalc,
|
||
|
|
absurd Hbad IH
|
||
|
|
| nostep (ap (cnst c) e') (@vcnst_ap e c Hvale) (stepr Hvalc Hbad) :=
|
||
|
|
have IH : not (e ➤ e'), from nostep e' Hvale,
|
||
|
|
absurd Hbad IH
|