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+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic        
+    ||A||  Library of Mathematics, developed at the Computer Science     
+    ||T||  Department of the University of Bologna, Italy.                     
+    ||I||                                                                 
+    ||T||  
+    ||A||  This file is distributed under the terms of the 
+    \   /  GNU General Public License Version 2        
+     \ /      
+      V_______________________________________________________________ *)
+
+include "lambdaN/reduction.ma".
+include "lambdaN/inversion.ma". 
+
+(*
+inductive T : Type[0] ≝
+  | Sort: nat → T
+  | Rel: nat → T 
+  | App: T → T → T 
+  | Lambda: T → T → T (* type, body *)
+  | Prod: T → T → T (* type, body *)
+  | D: T →T
+. 
+
+inductive red : T →T → Prop ≝
+  | rbeta: ∀P,M,N. red (App (Lambda P M) N) (M[0 ≝ N])
+  | rappl: ∀M,M1,N. red M M1 → red (App M N) (App M1 N)
+  | rappr: ∀M,N,N1. red N N1 → red (App M N) (App M N1)
+  | rlaml: ∀M,M1,N. red M M1 → red (Lambda M N) (Lambda M1 N)
+  | rlamr: ∀M,N,N1. red N N1 → red(Lambda M N) (Lambda M N1)
+  | rprodl: ∀M,M1,N. red M M1 → red (Prod M N) (Prod M1 N)
+  | rprodr: ∀M,N,N1. red N N1 → red (Prod M N) (Prod M N1)
+  | d: ∀M,M1. red M M1 → red (D M) (D M1). *)
+  
+lemma lift_D: ∀P,M,N. lift P 0 1 = D M N → 
+  ∃M1,N1. M = lift M1 0 1 ∧ N = lift N1 0 1 ∧ P = D M1 N1.   
+#P (cases P) normalize
+  [#i #M #N #H destruct
+  |#i #M #N #H destruct
+  |#A #B #M #N #H destruct
+  |#A #B #M #N #H destruct
+  |#A #B #M #N #H destruct
+  |#A #B #M #N #H destruct @(ex_intro … A) @(ex_intro … B) /3/
+  ]
+qed. 
+
+theorem type_of_type: ∀P,G,A,B. G ⊢_{P} A : B → (∀i. B ≠ Sort i) →
+  ∃i. G ⊢_{P} B : Sort i.
+#Pts #G #A #B #t (elim t)
+  [#i #j #Aij #j @False_ind /2/ 
+  |#G1 #A #i #t1 #_ #P @(ex_intro … i) @(weak … t1 t1)
+  |#G1 #A #B #C #i #t1 #t2 #H1 #H2 #H3 (cases (H1 ?) )
+    [#i #Bi @(ex_intro … i) @(weak … Bi t2)
+    |#i @(not_to_not … (H3 i)) //
+    ]
+  |#G1 #A #B #i #j #k #h #t1 #t2 #_ #_ #H3 @False_ind /2/
+  |#G1 #A #B #C #D #t1 #t2 #H1 #H2 #H3 cases (H1 ?);
+    [#i #t3 cases (prod_inv … t3 … (refl …))
+     #s1 * #s2 * #s3 * * #Ci #H4 #H5 @(ex_intro … s2)
+     @(tj_subst_0 … t2 … H5)
+    |#i % #H destruct
+    ]  
+  |#G1 #A #B #C #i #t1 #t2 #H1 #H2 #H3 /2/
+  |#G1 #A #B #C #i #ch #t1 #t2 #H1 #H2 #H3 /2/
+  |#G1 #A #B #i #t1 #t2 #Hind1 #Hind2 #H /2/
+  ]
+qed.
+
+lemma prod_sort : ∀Pts,G,M,P,Q. G ⊢_{Pts} M :Prod P Q →
+ ∃i. P::G ⊢_{Pts} Q : Sort i.
+#Pts #G #M #P #Q #t cases(type_of_type …t ?);
+  [#s #t2 cases(prod_inv … t2 …(refl …)) #s1 * #s2 * #s3 * * 
+   #_ #_ #H @(ex_intro … s2) //
+  | #i % #H destruct
+  ]
+qed.
+    
+axiom red_lift: ∀M,N. red (lift M 0 1) N → 
+  ∃P. N = lift P 0 1 ∧ red M P. 
+
+theorem tj_d : ∀P,G,M,N,N1. G ⊢_{P} D M N1: N → G ⊢_{P} M : N.
+#Pts #G (cut (∀M,N. G ⊢_{Pts} M : N → ∀P,Q. M = D P Q → G ⊢_{Pts} P : N)) [2: /2/] 
+#M #N #t (elim t)
+  [#i #j #Aij #P #Q #H destruct 
+  |#G1 #A #i #t1 #_ #P #Q #H destruct
+  |#G1 #A #B #C #i #t1 #t2 #H1 #H2 #P #Q #H3 
+   cases (lift_D … H3) #P1 * #Q1 * * #eqP #eqQ #eqA 
+   >eqP @(weak … i) /2/
+  |#G1 #A #B #i #j #k #h #t1 #t2 #_ #_ #P #Q #H destruct
+  |#G1 #A #B #C #D #t1 #t2 #_ #_ #P #Q #H destruct
+  |#G1 #A #B #C #D #t1 #t2 #_ #_ #P #Q #H destruct
+  |#G1 #A #B #C #i #ch #t1 #t2 #H #_ #P #Q #H
+   @(conv… ch …t2) /2/
+  |#G1 #A #B #i #t1 #t2 #Hind1 #Hind2 #P #Q #H destruct //
+  ]
+qed.
+
+definition red0 ≝ λM,N. M = N ∨ red M N.
+
+axiom red_to_conv : ∀P,M,N. red M N → Co P M N.
+axiom red0_to_conv : ∀P,M,N. red0 M N → Co P M N.
+axiom conv_prod: ∀P,A,B,M,N. Co P A B → Co P M N → 
+  Co P (Prod A M) (Prod B N). 
+axiom conv_subst_1: ∀Pts,M,P,Q. red P Q → Co Pts (M[0≝Q]) (M[0≝P]).
+
+inductive redG : list T → list T → Prop ≝
+ | rnil : redG (nil T) (nil T)
+ | rstep : ∀A,B,G1,G2. red0 A B → redG G1 G2 → 
+     redG (A::G1) (B::G2).
+
+lemma redG_inv: ∀A,G,G1. redG (A::G) G1 → 
+  ∃B. ∃G2. red0 A B ∧ redG G G2 ∧ G1 = B::G2.
+#A #G #G1 #rg (inversion rg) 
+  [#H destruct
+  |#A1 #B1 #G2 #G3 #r1 #r2 #_ #H destruct
+   #H1 @(ex_intro … B1) @(ex_intro … G3) % // % //
+  ]
+qed.
+
+lemma redG_nil: ∀G. redG (nil T) G → G = nil T.
+#G #rg (inversion rg) // 
+#A #B #G1 #G2 #r1 #r2 #_ #H destruct
+qed.
+
+axiom conv_prod_split: ∀P,A,A1,B,B1. 
+ Co P(Prod A B) (Prod A1 B1) → Co P A A1 ∧ Co P B B1.
+
+axiom red0_prod : ∀M,N,P. red0 (Prod M N) P → 
+  (∃Q. P = Prod Q N ∧ red0 M Q) ∨
+  (∃Q. P = Prod M Q ∧ red0 N Q).
+
+theorem subject_reduction: ∀P,G,M,N. G ⊢_{P} M:N → 
+ ∀M1. red0 M M1 → ∀G1. redG G G1 → G1 ⊢_{P} M1:N.
+#Pts #G #M #N #d (elim d)
+  [#i #j #Aij #M1 * 
+    [#eqM1 <eqM1 #G1 #H >(redG_nil …H) /2/
+    |#H @False_ind /2/
+    ]
+  |#G1 #A #i #t1 #Hind #M1 * 
+    [#eqM1 <eqM1 #G2 #H cases (redG_inv … H)
+     #P * #G3 * * #r1 #r2 #eqG2 >eqG2
+     @(conv ??? (lift P O 1) ? i);
+      [@conv_lift @sym_conv @red0_to_conv //
+      |@(start … i) @Hind //
+      |@(weak … (Sort i) ? i); [@Hind /2/ | @Hind //]
+      ]
+    |#H @False_ind /2/
+    ]  
+  |#G1 #A #B #C #i #t1 #t2 #H1 #H2 #M1 
+   #H cases H;
+    [#eqM1 <eqM1 #G2 #rg (cases (redG_inv … rg)) 
+     #Q * #G3 * * #r2 #rg1 #eqG2 >eqG2 @(weak … i);
+      [@H1 /2/ | @H2 //] 
+    |#H (elim (red_lift … H)) #P * #eqM1 >eqM1 #redAP
+      #G2 #rg (cases (redG_inv … rg)) #Q * #G3 * * #r2 
+      #rg1 #eqG2 >eqG2 @(weak … i); 
+      [@H1 /2/ | @H2 //]
+    ]
+  |#G #A #B #i #j #k #Rjk #t1 #t2 #Hind1 #Hind2 #M1 #redP
+   (cases (red0_prod … redP))
+    [* #M2 * #eqM1 #redA >eqM1 #G1 #rg @(prod … Rjk);
+      [@Hind1 // | @Hind2 /2/]
+    |* #M2 * #eqM1 #redA >eqM1 #G1 #rg @(prod … Rjk); 
+      [@Hind1 /2/ | @Hind2 /3/] 
+    ]
+  |#G #A #B #C #P #t1 #t2 #Hind1 #Hind2 #M1 #red0a
+   (cases red0a)
+    [#eqM1 <eqM1 #G1 #rg @(app … B);
+      [@Hind1 /2/ | @Hind2 /2/ ]
+    |#reda (cases (red_app …reda))
+      [*
+        [* 
+          #M2 * #N1 * #eqA #eqM1 >eqM1 #G1 #rg
+           cut (G1 ⊢_{Pts} A: Prod B C); [@Hind1 /2/] #H1
+           (cases (abs_inv … H1 … eqA)) #i * #N2 * * 
+           #cProd #t3 #t4 
+           (cut (Co Pts B M2 ∧ Co Pts C N2) ) [/2/] * #convB #convC
+           (cases (prod_inv … t3 … (refl …) )) #i * #j * #k * *
+           #cik #t5 #t6 (cut (G1 ⊢_{Pts} P:B)) 
+            [@Hind2 /2/
+            |#Hcut cut (G1 ⊢_{Pts} N1[0:=P] : N2 [0:=P]);
+              [@(tj_subst_0 … M2) // @(conv … convB Hcut t5)
+              |#Hcut1 cases (prod_sort … H1) #s #Csort
+               @(conv … s  … Hcut1); 
+                [@conv_subst /2/ | @(tj_subst_0 … Csort) //]
+              ]
+            ]
+        |* #M2 * #eqM1 >eqM1 #H #G1 #rg @(app … B);
+          [@Hind1 /2/ | @Hind2 /2/] 
+        ]      
+      |* #M2 * #eqM1 >eqM1 #H #G1 #rg 
+       cut (G1 ⊢_{Pts} A:Prod B C); [@Hind1 /2/] #t3
+       cases (prod_sort …t3) #i #Csort @(conv ??? C[O≝ M2] … i);
+        [@conv_subst_1 //
+        |@(app … B)  // @Hind2 /2/
+        |@(tj_subst_0 … Csort) @Hind2 /2/
+        ]
+      ]
+    ]
+  |#G #A #B #C #i #t1 #t2 #Hind1 #Hind2 #M2 #red0l #G1 #rg
+   cut (A::G1⊢_{Pts} C:B); [@Hind1 /3/] #t3
+   cut (G1 ⊢_{Pts} Prod A B : Sort i); [@Hind2 /2/] #t4
+   cases red0l;
+    [#eqM2 <eqM2 @(abs … t3 t4)
+    |#redl (cases (red_lambda … redl))  
+      [* #M3 * #eqM2 #redA >eqM2 
+         @(conv ??? (Prod M3 B) … t4);
+          [@conv_prod /2/
+          |@(abs … i); [@Hind1 /3/ |@Hind2 /3/]
+          ]
+        |* #M3 * #eqM3 #redC >eqM3
+         @(abs … t4) @Hind1 /3/
+      ]
+    ]
+  |#G #A #B #C #i #convBC #t1 #t2 #Hind1 #Hind2 #M1 #redA
+   #G1 #rg @(conv … i … convBC); [@Hind1 // |@Hind2 /2/]
+  |#G #A #B #i #t1 #t2 #Hind1 #Hind2 #M1 #red0d #G1 #rg
+   cases red0d; 
+    [#eqM1 <eqM1 @(dummy … i); [@Hind1 /2/ |@Hind2 /2/] 
+    |#redd (cases (red_d … redd))
+      [* #A1 * #eqM1 #redA >eqM1 
+       @(dummy … i);[@Hind1 /2/ |@Hind2 /2/]
+      |* #B1 * #eqM1 #redB >eqM1 
+       cut (G1 ⊢_{Pts} B: Sort i); [@Hind2 /2/] #sB
+       cut (G1 ⊢_{Pts} B1: Sort i); [@Hind2 /2/] #sB1
+       @(conv … B1 … i) /2/ @(dummy … i) // @(conv … B … i) /2/ 
+       @Hind1 /2/
+      ]
+    ]
+  ]
+qed.
+