660 lines
21 KiB
Coq
660 lines
21 KiB
Coq
From Equations Require Import Equations.
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Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax common.
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Require Import Logic.PropExtensionality (propositional_extensionality).
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Require Import ssreflect ssrbool.
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Import Logic (inspect).
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From Ltac2 Require Import Ltac2.
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Import Ltac2.Constr.
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Set Default Proof Mode "Classic".
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Require Import ssreflect ssrbool.
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From Hammer Require Import Tactics.
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Definition tm_nonconf (a b : PTm) : bool :=
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match a, b with
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| PAbs _, _ => (~~ ishf b) || isabs b
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| _, PAbs _ => ~~ ishf a
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| VarPTm _, VarPTm _ => true
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| PPair _ _, _ => (~~ ishf b) || ispair b
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| _, PPair _ _ => ~~ ishf a
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| PZero, PZero => true
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| PSuc _, PSuc _ => true
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| PApp _ _, PApp _ _ => (~~ ishf a) && (~~ ishf b)
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| PProj _ _, PProj _ _ => (~~ ishf a) && (~~ ishf b)
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| PInd _ _ _ _, PInd _ _ _ _ => (~~ ishf a) && (~~ ishf b)
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| PNat, PNat => true
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| PUniv _, PUniv _ => true
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| PBind _ _ _, PBind _ _ _ => true
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| _,_=> false
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end.
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Definition tm_conf (a b : PTm) := ~~ tm_nonconf a b.
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Inductive eq_view : PTm -> PTm -> Type :=
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| V_AbsAbs a b :
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eq_view (PAbs a) (PAbs b)
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| V_AbsNeu a b :
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~~ ishf b ->
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eq_view (PAbs a) b
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| V_NeuAbs a b :
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~~ ishf a ->
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eq_view a (PAbs b)
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| V_VarVar i j :
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eq_view (VarPTm i) (VarPTm j)
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| V_PairPair a0 b0 a1 b1 :
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eq_view (PPair a0 b0) (PPair a1 b1)
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| V_PairNeu a0 b0 u :
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~~ ishf u ->
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eq_view (PPair a0 b0) u
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| V_NeuPair u a1 b1 :
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~~ ishf u ->
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eq_view u (PPair a1 b1)
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| V_ZeroZero :
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eq_view PZero PZero
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| V_SucSuc a b :
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eq_view (PSuc a) (PSuc b)
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| V_AppApp u0 b0 u1 b1 :
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eq_view (PApp u0 b0) (PApp u1 b1)
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| V_ProjProj p0 u0 p1 u1 :
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eq_view (PProj p0 u0) (PProj p1 u1)
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| V_IndInd P0 u0 b0 c0 P1 u1 b1 c1 :
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eq_view (PInd P0 u0 b0 c0) (PInd P1 u1 b1 c1)
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| V_NatNat :
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eq_view PNat PNat
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| V_BindBind p0 A0 B0 p1 A1 B1 :
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eq_view (PBind p0 A0 B0) (PBind p1 A1 B1)
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| V_UnivUniv i j :
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eq_view (PUniv i) (PUniv j)
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| V_Others a b :
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tm_conf a b ->
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eq_view a b.
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Equations tm_to_eq_view (a b : PTm) : eq_view a b :=
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tm_to_eq_view (PAbs a) (PAbs b) := V_AbsAbs a b;
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tm_to_eq_view (PAbs a) (PApp b0 b1) := V_AbsNeu a (PApp b0 b1) _;
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tm_to_eq_view (PAbs a) (VarPTm i) := V_AbsNeu a (VarPTm i) _;
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tm_to_eq_view (PAbs a) (PProj p b) := V_AbsNeu a (PProj p b) _;
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tm_to_eq_view (PAbs a) (PInd P u b c) := V_AbsNeu a (PInd P u b c) _;
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tm_to_eq_view (VarPTm i) (PAbs a) := V_NeuAbs (VarPTm i) a _;
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tm_to_eq_view (PApp b0 b1) (PAbs b) := V_NeuAbs (PApp b0 b1) b _;
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tm_to_eq_view (PProj p b) (PAbs a) := V_NeuAbs (PProj p b) a _;
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tm_to_eq_view (PInd P u b c) (PAbs a) := V_NeuAbs (PInd P u b c) a _;
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tm_to_eq_view (VarPTm i) (VarPTm j) := V_VarVar i j;
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tm_to_eq_view (PPair a0 b0) (PPair a1 b1) := V_PairPair a0 b0 a1 b1;
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(* tm_to_eq_view (PPair a0 b0) u := V_PairNeu a0 b0 u _; *)
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tm_to_eq_view (PPair a0 b0) (VarPTm i) := V_PairNeu a0 b0 (VarPTm i) _;
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tm_to_eq_view (PPair a0 b0) (PApp c0 c1) := V_PairNeu a0 b0 (PApp c0 c1) _;
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tm_to_eq_view (PPair a0 b0) (PProj p c) := V_PairNeu a0 b0 (PProj p c) _;
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tm_to_eq_view (PPair a0 b0) (PInd P t0 t1 t2) := V_PairNeu a0 b0 (PInd P t0 t1 t2) _;
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(* tm_to_eq_view u (PPair a1 b1) := V_NeuPair u a1 b1 _; *)
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tm_to_eq_view (VarPTm i) (PPair a1 b1) := V_NeuPair (VarPTm i) a1 b1 _;
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tm_to_eq_view (PApp t0 t1) (PPair a1 b1) := V_NeuPair (PApp t0 t1) a1 b1 _;
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tm_to_eq_view (PProj t0 t1) (PPair a1 b1) := V_NeuPair (PProj t0 t1) a1 b1 _;
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tm_to_eq_view (PInd t0 t1 t2 t3) (PPair a1 b1) := V_NeuPair (PInd t0 t1 t2 t3) a1 b1 _;
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tm_to_eq_view PZero PZero := V_ZeroZero;
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tm_to_eq_view (PSuc a) (PSuc b) := V_SucSuc a b;
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tm_to_eq_view (PApp a0 b0) (PApp a1 b1) := V_AppApp a0 b0 a1 b1;
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tm_to_eq_view (PProj p0 b0) (PProj p1 b1) := V_ProjProj p0 b0 p1 b1;
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tm_to_eq_view (PInd P0 u0 b0 c0) (PInd P1 u1 b1 c1) := V_IndInd P0 u0 b0 c0 P1 u1 b1 c1;
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tm_to_eq_view PNat PNat := V_NatNat;
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tm_to_eq_view (PUniv i) (PUniv j) := V_UnivUniv i j;
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tm_to_eq_view (PBind p0 A0 B0) (PBind p1 A1 B1) := V_BindBind p0 A0 B0 p1 A1 B1;
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tm_to_eq_view a b := V_Others a b _.
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Inductive algo_dom : PTm -> PTm -> Prop :=
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| A_AbsAbs a b :
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algo_dom_r a b ->
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(* --------------------- *)
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algo_dom (PAbs a) (PAbs b)
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| A_AbsNeu a u :
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ishne u ->
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algo_dom_r a (PApp (ren_PTm shift u) (VarPTm var_zero)) ->
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(* --------------------- *)
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algo_dom (PAbs a) u
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| A_NeuAbs a u :
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ishne u ->
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algo_dom_r (PApp (ren_PTm shift u) (VarPTm var_zero)) a ->
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(* --------------------- *)
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algo_dom u (PAbs a)
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| A_PairPair a0 a1 b0 b1 :
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algo_dom_r a0 a1 ->
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algo_dom_r b0 b1 ->
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(* ---------------------------- *)
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algo_dom (PPair a0 b0) (PPair a1 b1)
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| A_PairNeu a0 a1 u :
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ishne u ->
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algo_dom_r a0 (PProj PL u) ->
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algo_dom_r a1 (PProj PR u) ->
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(* ----------------------- *)
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algo_dom (PPair a0 a1) u
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| A_NeuPair a0 a1 u :
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ishne u ->
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algo_dom_r (PProj PL u) a0 ->
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algo_dom_r (PProj PR u) a1 ->
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(* ----------------------- *)
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algo_dom u (PPair a0 a1)
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| A_ZeroZero :
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algo_dom PZero PZero
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| A_SucSuc a0 a1 :
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algo_dom_r a0 a1 ->
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algo_dom (PSuc a0) (PSuc a1)
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| A_UnivCong i j :
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(* -------------------------- *)
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algo_dom (PUniv i) (PUniv j)
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| A_BindCong p0 p1 A0 A1 B0 B1 :
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algo_dom_r A0 A1 ->
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algo_dom_r B0 B1 ->
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(* ---------------------------- *)
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algo_dom (PBind p0 A0 B0) (PBind p1 A1 B1)
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| A_NatCong :
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algo_dom PNat PNat
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| A_VarCong i j :
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(* -------------------------- *)
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algo_dom (VarPTm i) (VarPTm j)
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| A_ProjCong p0 p1 u0 u1 :
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ishne u0 ->
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ishne u1 ->
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algo_dom u0 u1 ->
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(* --------------------- *)
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algo_dom (PProj p0 u0) (PProj p1 u1)
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| A_AppCong u0 u1 a0 a1 :
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ishne u0 ->
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ishne u1 ->
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algo_dom u0 u1 ->
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algo_dom_r a0 a1 ->
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(* ------------------------- *)
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algo_dom (PApp u0 a0) (PApp u1 a1)
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| A_IndCong P0 P1 u0 u1 b0 b1 c0 c1 :
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ishne u0 ->
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ishne u1 ->
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algo_dom_r P0 P1 ->
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algo_dom u0 u1 ->
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algo_dom_r b0 b1 ->
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algo_dom_r c0 c1 ->
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algo_dom (PInd P0 u0 b0 c0) (PInd P1 u1 b1 c1)
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| A_Conf a b :
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HRed.nf a ->
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HRed.nf b ->
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tm_conf a b ->
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algo_dom a b
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with algo_dom_r : PTm -> PTm -> Prop :=
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| A_NfNf a b :
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algo_dom a b ->
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algo_dom_r a b
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| A_HRedL a a' b :
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HRed.R a a' ->
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algo_dom_r a' b ->
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(* ----------------------- *)
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algo_dom_r a b
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| A_HRedR a b b' :
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HRed.nf a ->
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HRed.R b b' ->
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algo_dom_r a b' ->
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(* ----------------------- *)
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algo_dom_r a b.
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Scheme algo_ind := Induction for algo_dom Sort Prop
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with algor_ind := Induction for algo_dom_r Sort Prop.
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Combined Scheme algo_dom_mutual from algo_ind, algor_ind.
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(* Inductive salgo_dom : PTm -> PTm -> Prop := *)
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(* | S_UnivCong i j : *)
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(* (* -------------------------- *) *)
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(* salgo_dom (PUniv i) (PUniv j) *)
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(* | S_PiCong A0 A1 B0 B1 : *)
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(* salgo_dom_r A1 A0 -> *)
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(* salgo_dom_r B0 B1 -> *)
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(* (* ---------------------------- *) *)
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(* salgo_dom (PBind PPi A0 B0) (PBind PPi A1 B1) *)
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(* | S_SigCong A0 A1 B0 B1 : *)
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(* salgo_dom_r A0 A1 -> *)
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(* salgo_dom_r B0 B1 -> *)
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(* (* ---------------------------- *) *)
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(* salgo_dom (PBind PSig A0 B0) (PBind PSig A1 B1) *)
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(* | S_NatCong : *)
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(* salgo_dom PNat PNat *)
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(* | S_NeuNeu a b : *)
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(* ishne a -> *)
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(* ishne b -> *)
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(* algo_dom a b -> *)
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(* (* ------------------- *) *)
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(* salgo_dom *)
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(* | S_Conf a b : *)
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(* HRed.nf a -> *)
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(* HRed.nf b -> *)
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(* tm_conf a b -> *)
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(* salgo_dom a b *)
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(* with salgo_dom_r : PTm -> PTm -> Prop := *)
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(* | S_NfNf a b : *)
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(* salgo_dom a b -> *)
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(* salgo_dom_r a b *)
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(* | S_HRedL a a' b : *)
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(* HRed.R a a' -> *)
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(* salgo_dom_r a' b -> *)
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(* (* ----------------------- *) *)
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(* salgo_dom_r a b *)
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(* | S_HRedR a b b' : *)
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(* HRed.nf a -> *)
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(* HRed.R b b' -> *)
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(* salgo_dom_r a b' -> *)
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(* (* ----------------------- *) *)
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(* salgo_dom_r a b. *)
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(* Scheme salgo_ind := Induction for salgo_dom Sort Prop *)
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(* with algor_ind := Induction for salgo_dom_r Sort Prop. *)
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Lemma hf_no_hred (a b : PTm) :
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ishf a ->
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HRed.R a b ->
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False.
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Proof. hauto l:on inv:HRed.R. Qed.
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Lemma hne_no_hred (a b : PTm) :
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ishne a ->
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HRed.R a b ->
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False.
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Proof. elim : a b => //=; hauto l:on inv:HRed.R. Qed.
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Lemma algo_dom_no_hred (a b : PTm) :
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algo_dom a b ->
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HRed.nf a /\ HRed.nf b.
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Proof.
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induction 1 =>//=; try hauto inv:HRed.R use:hne_no_hred, hf_no_hred lq:on unfold:HRed.nf.
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Qed.
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Derive Signature for algo_dom algo_dom_r.
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Fixpoint hred (a : PTm) : option (PTm) :=
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match a with
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| VarPTm i => None
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| PAbs a => None
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| PApp (PAbs a) b => Some (subst_PTm (scons b VarPTm) a)
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| PApp a b =>
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match hred a with
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| Some a => Some (PApp a b)
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| None => None
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end
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| PPair a b => None
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| PProj p (PPair a b) => if p is PL then Some a else Some b
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| PProj p a =>
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match hred a with
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| Some a => Some (PProj p a)
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| None => None
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end
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| PUniv i => None
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| PBind p A B => None
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| PNat => None
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| PZero => None
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| PSuc a => None
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| PInd P PZero b c => Some b
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| PInd P (PSuc a) b c =>
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Some (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
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| PInd P a b c =>
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match hred a with
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| Some a => Some (PInd P a b c)
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| None => None
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end
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end.
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Lemma hred_complete (a b : PTm) :
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HRed.R a b -> hred a = Some b.
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Proof.
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induction 1; hauto lq:on rew:off inv:HRed.R b:on.
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Qed.
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Lemma hred_sound (a b : PTm):
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hred a = Some b -> HRed.R a b.
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Proof.
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elim : a b; hauto q:on dep:on ctrs:HRed.R.
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Qed.
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Lemma hred_deter (a b0 b1 : PTm) :
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HRed.R a b0 -> HRed.R a b1 -> b0 = b1.
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Proof.
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move /hred_complete => + /hred_complete. congruence.
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Qed.
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Definition fancy_hred (a : PTm) : HRed.nf a + {b | HRed.R a b}.
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destruct (hred a) eqn:eq.
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right. exists p. by apply hred_sound in eq.
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left. move => b /hred_complete. congruence.
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Defined.
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Ltac2 destruct_algo () :=
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lazy_match! goal with
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| [h : algo_dom ?a ?b |- _ ] =>
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if is_var a then destruct $a; ltac1:(done) else
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(if is_var b then destruct $b; ltac1:(done) else ())
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end.
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Ltac check_equal_triv :=
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intros;subst;
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lazymatch goal with
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(* | [h : algo_dom (VarPTm _) (PAbs _) |- _] => idtac *)
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| [h : algo_dom _ _ |- _] => try (inversion h; subst => //=; ltac2:(Control.enter destruct_algo))
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| _ => idtac
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end.
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Ltac solve_check_equal :=
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try solve [intros *;
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match goal with
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| [|- _ = _] => sauto
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| _ => idtac
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end].
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#[derive(equations=no)]Equations check_equal (a b : PTm) (h : algo_dom a b) :
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bool by struct h :=
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check_equal a b h with tm_to_eq_view a b :=
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check_equal _ _ h (V_VarVar i j) := nat_eqdec i j;
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check_equal _ _ h (V_AbsAbs a b) := check_equal_r a b ltac:(check_equal_triv);
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check_equal _ _ h (V_AbsNeu a b h') := check_equal_r a (PApp (ren_PTm shift b) (VarPTm var_zero)) ltac:(check_equal_triv);
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check_equal _ _ h (V_NeuAbs a b _) := check_equal_r (PApp (ren_PTm shift a) (VarPTm var_zero)) b ltac:(check_equal_triv);
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check_equal _ _ h (V_PairPair a0 b0 a1 b1) :=
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check_equal_r a0 a1 ltac:(check_equal_triv) && check_equal_r b0 b1 ltac:(check_equal_triv);
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check_equal _ _ h (V_PairNeu a0 b0 u _) :=
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check_equal_r a0 (PProj PL u) ltac:(check_equal_triv) && check_equal_r b0 (PProj PR u) ltac:(check_equal_triv);
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check_equal _ _ h (V_NeuPair u a1 b1 _) :=
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check_equal_r (PProj PL u) a1 ltac:(check_equal_triv) && check_equal_r (PProj PR u) b1 ltac:(check_equal_triv);
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check_equal _ _ h V_NatNat := true;
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check_equal _ _ h V_ZeroZero := true;
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check_equal _ _ h (V_SucSuc a b) := check_equal_r a b ltac:(check_equal_triv);
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check_equal _ _ h (V_ProjProj p0 a p1 b) :=
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PTag_eqdec p0 p1 && check_equal a b ltac:(check_equal_triv);
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check_equal _ _ h (V_AppApp a0 b0 a1 b1) :=
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check_equal a0 a1 ltac:(check_equal_triv) && check_equal_r b0 b1 ltac:(check_equal_triv);
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check_equal _ _ h (V_IndInd P0 u0 b0 c0 P1 u1 b1 c1) :=
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check_equal_r P0 P1 ltac:(check_equal_triv) &&
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check_equal u0 u1 ltac:(check_equal_triv) &&
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check_equal_r b0 b1 ltac:(check_equal_triv) &&
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check_equal_r c0 c1 ltac:(check_equal_triv);
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check_equal _ _ h (V_UnivUniv i j) := nat_eqdec i j;
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check_equal _ _ h (V_BindBind p0 A0 B0 p1 A1 B1) :=
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BTag_eqdec p0 p1 && check_equal_r A0 A1 ltac:(check_equal_triv) && check_equal_r B0 B1 ltac:(check_equal_triv);
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check_equal _ _ _ _ := false
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(* check_equal a b h := false; *)
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with check_equal_r (a b : PTm) (h0 : algo_dom_r a b) :
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bool by struct h0 :=
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check_equal_r a b h with (fancy_hred a) :=
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check_equal_r a b h (inr a') := check_equal_r (proj1_sig a') b _;
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check_equal_r a b h (inl h') with (fancy_hred b) :=
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| inr b' := check_equal_r a (proj1_sig b') _;
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| inl h'' := check_equal a b _.
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Next Obligation.
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intros.
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inversion h; subst => //=.
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exfalso. hauto l:on use:hred_complete unfold:HRed.nf.
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exfalso. hauto l:on use:hred_complete unfold:HRed.nf.
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Defined.
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Next Obligation.
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intros.
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destruct h.
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exfalso. sfirstorder use: algo_dom_no_hred.
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exfalso. sfirstorder.
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assert ( b' = b'0)by eauto using hred_deter. subst.
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apply h.
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Defined.
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Next Obligation.
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simpl. intros.
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inversion h; subst =>//=.
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exfalso. sfirstorder use:algo_dom_no_hred.
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move {h}.
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assert (a' = a'0) by eauto using hred_deter, hred_sound. by subst.
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exfalso. sfirstorder use:hne_no_hred, hf_no_hred.
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Defined.
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Lemma check_equal_abs_abs a b h : check_equal (PAbs a) (PAbs b) (A_AbsAbs a b h) = check_equal_r a b h.
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Proof. reflexivity. Qed.
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Lemma check_equal_abs_neu a u neu h : check_equal (PAbs a) u (A_AbsNeu a u neu h) = check_equal_r a (PApp (ren_PTm shift u) (VarPTm var_zero)) h.
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Proof. case : u neu h => //=. Qed.
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Lemma check_equal_neu_abs a u neu h : check_equal u (PAbs a) (A_NeuAbs a u neu h) = check_equal_r (PApp (ren_PTm shift u) (VarPTm var_zero)) a h.
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Proof. case : u neu h => //=. Qed.
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Lemma check_equal_pair_pair a0 b0 a1 b1 a h :
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check_equal (PPair a0 b0) (PPair a1 b1) (A_PairPair a0 a1 b0 b1 a h) =
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check_equal_r a0 a1 a && check_equal_r b0 b1 h.
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Proof. reflexivity. Qed.
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Lemma check_equal_pair_neu a0 a1 u neu h h'
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: check_equal (PPair a0 a1) u (A_PairNeu a0 a1 u neu h h') = check_equal_r a0 (PProj PL u) h && check_equal_r a1 (PProj PR u) h'.
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Proof.
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case : u neu h h' => //=.
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Qed.
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Lemma check_equal_neu_pair a0 a1 u neu h h'
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: check_equal u (PPair a0 a1) (A_NeuPair a0 a1 u neu h h') = check_equal_r (PProj PL u) a0 h && check_equal_r (PProj PR u) a1 h'.
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Proof.
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case : u neu h h' => //=.
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Qed.
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Lemma check_equal_bind_bind p0 A0 B0 p1 A1 B1 h0 h1 :
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check_equal (PBind p0 A0 B0) (PBind p1 A1 B1) (A_BindCong p0 p1 A0 A1 B0 B1 h0 h1) =
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BTag_eqdec p0 p1 && check_equal_r A0 A1 h0 && check_equal_r B0 B1 h1.
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Proof. reflexivity. Qed.
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Lemma check_equal_proj_proj p0 u0 p1 u1 neu0 neu1 h :
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check_equal (PProj p0 u0) (PProj p1 u1) (A_ProjCong p0 p1 u0 u1 neu0 neu1 h) =
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PTag_eqdec p0 p1 && check_equal u0 u1 h.
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Proof. reflexivity. Qed.
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Lemma check_equal_app_app u0 a0 u1 a1 hu0 hu1 hdom hdom' :
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check_equal (PApp u0 a0) (PApp u1 a1) (A_AppCong u0 u1 a0 a1 hu0 hu1 hdom hdom') =
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check_equal u0 u1 hdom && check_equal_r a0 a1 hdom'.
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Proof. reflexivity. Qed.
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Lemma check_equal_ind_ind P0 u0 b0 c0 P1 u1 b1 c1 neu0 neu1 domP domu domb domc :
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check_equal (PInd P0 u0 b0 c0) (PInd P1 u1 b1 c1)
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(A_IndCong P0 P1 u0 u1 b0 b1 c0 c1 neu0 neu1 domP domu domb domc) =
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check_equal_r P0 P1 domP && check_equal u0 u1 domu && check_equal_r b0 b1 domb && check_equal_r c0 c1 domc.
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Proof. reflexivity. Qed.
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Lemma hred_none a : HRed.nf a -> hred a = None.
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Proof.
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destruct (hred a) eqn:eq;
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sfirstorder use:hred_complete, hred_sound.
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Qed.
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Lemma check_equal_nfnf a b dom : check_equal_r a b (A_NfNf a b dom) = check_equal a b dom.
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Proof.
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have [h0 h1] : HRed.nf a /\ HRed.nf b by hauto l:on use:algo_dom_no_hred.
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have [h3 h4] : hred a = None /\ hred b = None by sfirstorder use:hf_no_hred, hne_no_hred, hred_none.
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simpl.
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rewrite /check_equal_r_functional.
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destruct (fancy_hred a).
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simpl.
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destruct (fancy_hred b).
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reflexivity.
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exfalso. hauto l:on use:hred_complete.
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exfalso. hauto l:on use:hred_complete.
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Qed.
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Lemma check_equal_hredl a b a' ha doma :
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check_equal_r a b (A_HRedL a a' b ha doma) = check_equal_r a' b doma.
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Proof.
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simpl.
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rewrite /check_equal_r_functional.
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destruct (fancy_hred a).
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- hauto q:on unfold:HRed.nf.
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- destruct s as [x ?].
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rewrite /check_equal_r_functional.
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have ? : x = a' by eauto using hred_deter. subst.
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simpl.
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f_equal.
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apply PropExtensionality.proof_irrelevance.
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Qed.
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Lemma check_equal_hredr a b b' hu r a0 :
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check_equal_r a b (A_HRedR a b b' hu r a0) =
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check_equal_r a b' a0.
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Proof.
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simpl. rewrite /check_equal_r_functional.
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destruct (fancy_hred a).
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- simpl.
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destruct (fancy_hred b) as [|[b'' hb']].
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+ hauto lq:on unfold:HRed.nf.
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+ simpl.
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have ? : (b'' = b') by eauto using hred_deter. subst.
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f_equal.
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apply PropExtensionality.proof_irrelevance.
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- exfalso.
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sfirstorder use:hne_no_hred, hf_no_hred.
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Qed.
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Lemma check_equal_univ i j :
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check_equal (PUniv i) (PUniv j) (A_UnivCong i j) = nat_eqdec i j.
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Proof. reflexivity. Qed.
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Lemma check_equal_conf a b nfa nfb nfab :
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check_equal a b (A_Conf a b nfa nfb nfab) = false.
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Proof. destruct a; destruct b => //=. Qed.
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#[export]Hint Rewrite check_equal_abs_abs check_equal_abs_neu check_equal_neu_abs check_equal_pair_pair check_equal_pair_neu check_equal_neu_pair check_equal_bind_bind check_equal_hredl check_equal_hredr check_equal_nfnf check_equal_conf : ce_prop.
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Obligation Tactic := try solve [check_equal_triv | sfirstorder].
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Program Fixpoint check_sub (q : bool) (a b : PTm) (h : algo_dom a b) {struct h} :=
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match a, b with
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| PBind PPi A0 B0, PBind PPi A1 B1 =>
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check_sub_r (negb q) A0 A1 _ && check_sub_r q B0 B1 _
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| PBind PSig A0 B0, PBind PSig A1 B1 =>
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check_sub_r q A0 A1 _ && check_sub_r q B0 B1 _
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| PUniv i, PUniv j =>
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if q then PeanoNat.Nat.leb i j else PeanoNat.Nat.leb j i
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| PNat, PNat => true
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| _ ,_ => ishne a && ishne b && check_equal a b h
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end
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with check_sub_r (q : bool) (a b : PTm) (h : algo_dom_r a b) {struct h} :=
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match fancy_hred a with
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| inr a' => check_sub_r q (proj1_sig a') b _
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| inl ha' => match fancy_hred b with
|
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| inr b' => check_sub_r q a (proj1_sig b') _
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| inl hb' => check_sub q a b _
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end
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|
end.
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Next Obligation.
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|
simpl. intros. clear Heq_anonymous. destruct a' as [a' ha']. simpl.
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|
inversion h; subst => //=.
|
|
exfalso. sfirstorder use:algo_dom_no_hred.
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|
assert (a' = a'0) by eauto using hred_deter. by subst.
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|
exfalso. sfirstorder.
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|
Defined.
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|
|
|
Next Obligation.
|
|
simpl. intros. clear Heq_anonymous Heq_anonymous0.
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|
destruct b' as [b' hb']. simpl.
|
|
inversion h; subst.
|
|
- exfalso.
|
|
sfirstorder use:algo_dom_no_hred.
|
|
- exfalso.
|
|
sfirstorder.
|
|
- assert (b' = b'0) by eauto using hred_deter. by subst.
|
|
Defined.
|
|
|
|
(* Need to avoid ssreflect tactics since they generate terms that make the termination checker upset *)
|
|
Next Obligation.
|
|
move => _ /= a b hdom ha _ hb _.
|
|
inversion hdom; subst.
|
|
- assumption.
|
|
- exfalso; sfirstorder.
|
|
- exfalso; sfirstorder.
|
|
Defined.
|
|
|
|
Lemma check_sub_pi_pi q A0 B0 A1 B1 h0 h1 :
|
|
check_sub q (PBind PPi A0 B0) (PBind PPi A1 B1) (A_BindCong PPi PPi A0 A1 B0 B1 h0 h1) =
|
|
check_sub_r (~~ q) A0 A1 h0 && check_sub_r q B0 B1 h1.
|
|
Proof. reflexivity. Qed.
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|
|
|
Lemma check_sub_sig_sig q A0 B0 A1 B1 h0 h1 :
|
|
check_sub q (PBind PSig A0 B0) (PBind PSig A1 B1) (A_BindCong PSig PSig A0 A1 B0 B1 h0 h1) =
|
|
check_sub_r q A0 A1 h0 && check_sub_r q B0 B1 h1.
|
|
reflexivity. Qed.
|
|
|
|
Lemma check_sub_univ_univ i j :
|
|
check_sub true (PUniv i) (PUniv j) (A_UnivCong i j) = PeanoNat.Nat.leb i j.
|
|
Proof. reflexivity. Qed.
|
|
|
|
Lemma check_sub_univ_univ' i j :
|
|
check_sub false (PUniv i) (PUniv j) (A_UnivCong i j) = PeanoNat.Nat.leb j i.
|
|
reflexivity. Qed.
|
|
|
|
Lemma check_sub_nfnf q a b dom : check_sub_r q a b (A_NfNf a b dom) = check_sub q a b dom.
|
|
Proof.
|
|
have [h0 h1] : HRed.nf a /\ HRed.nf b by hauto l:on use:algo_dom_no_hred.
|
|
have [h3 h4] : hred a = None /\ hred b = None by sfirstorder use:hf_no_hred, hne_no_hred, hred_none.
|
|
simpl.
|
|
destruct (fancy_hred b)=>//=.
|
|
destruct (fancy_hred a) =>//=.
|
|
destruct s as [a' ha']. simpl.
|
|
hauto l:on use:hred_complete.
|
|
destruct s as [b' hb']. simpl.
|
|
hauto l:on use:hred_complete.
|
|
Qed.
|
|
|
|
Lemma check_sub_hredl q a b a' ha doma :
|
|
check_sub_r q a b (A_HRedL a a' b ha doma) = check_sub_r q a' b doma.
|
|
Proof.
|
|
simpl.
|
|
destruct (fancy_hred a).
|
|
- hauto q:on unfold:HRed.nf.
|
|
- destruct s as [x ?].
|
|
have ? : x = a' by eauto using hred_deter. subst.
|
|
simpl.
|
|
f_equal.
|
|
apply PropExtensionality.proof_irrelevance.
|
|
Qed.
|
|
|
|
Lemma check_sub_hredr q a b b' hu r a0 :
|
|
check_sub_r q a b (A_HRedR a b b' hu r a0) =
|
|
check_sub_r q a b' a0.
|
|
Proof.
|
|
simpl.
|
|
destruct (fancy_hred a).
|
|
- simpl.
|
|
destruct (fancy_hred b) as [|[b'' hb']].
|
|
+ hauto lq:on unfold:HRed.nf.
|
|
+ simpl.
|
|
have ? : (b'' = b') by eauto using hred_deter. subst.
|
|
f_equal.
|
|
apply PropExtensionality.proof_irrelevance.
|
|
- exfalso.
|
|
sfirstorder use:hne_no_hred, hf_no_hred.
|
|
Qed.
|
|
|
|
#[export]Hint Rewrite check_sub_hredl check_sub_hredr check_sub_nfnf check_sub_univ_univ' check_sub_univ_univ check_sub_pi_pi check_sub_sig_sig : ce_prop.
|