366 lines
11 KiB
Coq
366 lines
11 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 ssreflect ssrbool.
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Import Logic (inspect).
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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 _, _ => ishne b || isabs b
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| _, PAbs _ => ishne a
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| VarPTm _, VarPTm _ => true
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| PPair _ _, _ => ishne b || ispair b
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| _, PPair _ _ => ishne a
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| PZero, PZero => true
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| PSuc _, PSuc _ => true
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| PApp _ _, PApp _ _ => ishne a && ishne b
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| PProj _ _, PProj _ _ => ishne a && ishne b
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| PInd _ _ _ _, PInd _ _ _ _ => ishne a && ishne 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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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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ishne a \/ ishf 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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Lemma algo_dom_hf_hne (a b : PTm) :
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algo_dom a b ->
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(ishf a \/ ishne a) /\ (ishf b \/ ishne b).
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Proof.
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induction 1 =>//=; hauto lq:on.
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Qed.
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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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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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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; by subst)
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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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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. apply algo_dom_hf_hne in H0.
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destruct H0 as [h0 h1].
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sfirstorder use:hf_no_hred, hne_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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move {h}. hauto lq:on use:algo_dom_hf_hne, hf_no_hred, hne_no_hred, hred_sound.
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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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Next Obligation.
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qauto inv:algo_dom, algo_dom_r.
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Defined.
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