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Add ne ered

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Yiyun Liu 2025-01-28 15:13:41 -05:00
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theories/fp_red.v
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@ -105,6 +105,58 @@ Module ERed.
End ERed.
Module NeERed.
Inductive R_nonelim {n} : PTm n -> PTm n -> Prop :=
(****************** Eta ***********************)
| AppEta A a0 a1 :
R_nonelim a0 a1 ->
R_nonelim (PAbs A (PApp (ren_PTm shift a0) (VarPTm var_zero))) a1
| PairEta a0 a1 :
R_nonelim a0 a1 ->
R_nonelim (PPair (PProj PL a0) (PProj PR a0)) a1
(*************** Congruence ********************)
| AbsCong A a0 a1 :
R_nonelim a0 a1 ->
R_nonelim (PAbs A a0) (PAbs A a1)
| AppCong a0 a1 b0 b1 :
R_elim a0 a1 ->
R_nonelim b0 b1 ->
R_nonelim (PApp a0 b0) (PApp a1 b1)
| PairCong a0 a1 b0 b1 :
R_nonelim a0 a1 ->
R_nonelim b0 b1 ->
R_nonelim (PPair a0 b0) (PPair a1 b1)
| ProjCong p a0 a1 :
R_elim a0 a1 ->
R_nonelim (PProj p a0) (PProj p a1)
| VarTm i :
R_nonelim (VarPTm i) (VarPTm i)
with R_elim {n} : PTm n -> PTm n -> Prop :=
| NAbsCong A a0 a1 :
R_nonelim a0 a1 ->
R_elim (PAbs A a0) (PAbs A a1)
| NAppCong a0 a1 b0 b1 :
R_elim a0 a1 ->
R_nonelim b0 b1 ->
R_elim (PApp a0 b0) (PApp a1 b1)
| NPairCong a0 a1 b0 b1 :
R_nonelim a0 a1 ->
R_nonelim b0 b1 ->
R_elim (PPair a0 b0) (PPair a1 b1)
| NProjCong p a0 a1 :
R_elim a0 a1 ->
R_elim (PProj p a0) (PProj p a1)
| NVarTm i :
R_elim (VarPTm i) (VarPTm i).
Scheme ered_elim_ind := Induction for R_elim Sort Prop
with ered_nonelim_ind := Induction for R_nonelim Sort Prop.
Combined Scheme ered_mutual from ered_elim_ind, ered_nonelim_ind.
End NeERed.
Inductive SNe {n} : PTm n -> Prop :=
| N_Var i :
SNe (VarPTm i)