8 (*#* #caption "main properties of the relation $\\PrT{}{}{}$" *)
13 Section pr3_context. (****************************************************)
15 Theorem pr3_pr0_pr2_t: (u1,u2:?) (pr0 u1 u2) ->
16 (c:?; t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
17 (pr3 (CTail c k u1) t1 t2).
18 Intros; Inversion H0; Clear H0; XAuto.
20 (* case 1 : pr2_delta i = 0 *)
21 DropGenBase; Inversion H0; Clear H0 H4 H5 H6 c k t.
22 Rewrite H7 in H; Clear H7 u2.
24 (* case 2 : pr2_delta i > 0 *)
25 NewInduction k; DropGenBase; XEAuto.
28 Theorem pr3_pr2_pr2_t: (c:?; u1,u2:?) (pr2 c u1 u2) ->
29 (t1,t2:?; k:?) (pr2 (CTail c k u2) t1 t2) ->
30 (pr3 (CTail c k u1) t1 t2).
31 Intros until 1; Inversion H; Clear H; Intros.
32 (* case 1 : pr2_free *)
33 EApply pr3_pr0_pr2_t; [ Apply H0 | XAuto ].
34 (* case 2 : pr2_delta *)
35 Inversion H; [ XAuto | NewInduction i0 ].
36 (* case 2.1 : i0 = 0 *)
37 DropGenBase; Inversion H4; Clear H3 H4 H7 t t4.
38 Rewrite <- H9; Rewrite H10 in H; Rewrite <- H11 in H6; Clear H9 H10 H11 d0 k u0.
39 Subst0Subst0; Arith9'In H4 i; Clear H2 H H6 u2.
40 Pr0Subst0; Apply pr3_sing with t2:=x0; XEAuto.
41 (* case 2.2 : i0 > 0 *)
42 Clear IHi0; NewInduction k; DropGenBase; XEAuto.
45 Theorem pr3_pr2_pr3_t: (c:?; u2,t1,t2:?; k:?)
46 (pr3 (CTail c k u2) t1 t2) ->
47 (u1:?) (pr2 c u1 u2) ->
48 (pr3 (CTail c k u1) t1 t2).
49 Intros until 1; XElim H; Intros.
50 (* case 1 : pr3_refl *)
52 (* case 2 : pr3_sing *)
54 EApply pr3_pr2_pr2_t; [ Apply H2 | Apply H ].
58 (*#* #caption "reduction inside context items" *)
59 (*#* #cap #cap t1, t2 #alpha c in E, u1 in V1, u2 in V2, k in z *)
61 Theorem pr3_pr3_pr3_t: (c:?; u1,u2:?) (pr3 c u1 u2) ->
62 (t1,t2:?; k:?) (pr3 (CTail c k u2) t1 t2) ->
63 (pr3 (CTail c k u1) t1 t2).
64 Intros until 1; XElim H; Intros.
65 (* case 1 : pr3_refl *)
67 (* case 2 : pr3_sing *)
68 EApply pr3_pr2_pr3_t; [ Apply H1; XAuto | XAuto ].
73 Tactic Definition Pr3Context :=
75 | [ H1: (pr0 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
76 LApply (pr3_pr0_pr2_t ?2 ?3); [ Intros H_x | XAuto ];
77 LApply (H_x ?1 ?5 ?6 ?4); [ Clear H_x H2; Intros | XAuto ]
78 | [ H1: (pr0 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
79 LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
80 LApply (H2 ?2); [ Clear H2; Intros | XAuto ]
81 | [ H1: (pr2 ?1 ?2 ?3); H2: (pr2 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
82 LApply (pr3_pr2_pr2_t ?1 ?2 ?3); [ Intros H_x | XAuto ];
83 LApply (H_x ?5 ?6 ?4); [ Clear H_x H2; Intros | XAuto ]
84 | [ H1: (pr2 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
85 LApply (pr3_pr2_pr3_t ?1 ?3 ?5 ?6 ?4); [ Clear H2; Intros H2 | XAuto ];
86 LApply (H2 ?2); [ Clear H2; Intros | XAuto ]
87 | [ H1: (pr3 ?1 ?2 ?3); H2: (pr3 (CTail ?1 ?4 ?3) ?5 ?6) |- ? ] ->
88 LApply (pr3_pr3_pr3_t ?1 ?2 ?3); [ Intros H_x | XAuto ];
89 LApply (H_x ?5 ?6 ?4); [ Clear H_x H2; Intros | XAuto ].
91 Section pr3_lift. (*******************************************************)
93 (*#* #caption "conguence with lift" *)
94 (*#* #cap #cap c, t1, t2 #alpha e in D, d in i *)
96 Theorem pr3_lift: (c,e:?; h,d:?) (drop h d c e) ->
97 (t1,t2:?) (pr3 e t1 t2) ->
98 (pr3 c (lift h d t1) (lift h d t2)).
99 Intros until 2; XElim H0; Intros; XEAuto.
104 Hints Resolve pr3_lift : ltlc.
106 Section pr3_cpr0. (*******************************************************)
108 Theorem pr3_cpr0_t: (c1,c2:?) (cpr0 c2 c1) -> (t1,t2:?) (pr3 c1 t1 t2) ->
110 Intros until 1; XElim H; Intros.
111 (* case 1 : cpr0_refl *)
113 (* case 2 : cpr0_comp *)
114 Pr3Context; Clear H1.
116 (* case 2.1 : pr3_refl *)
118 (* case 2.2 : pr3_sing *)
119 EApply pr3_t; [ Idtac | XEAuto ]. Clear H2 H3 c1 c2 t1 t2 t4 u2.
121 (* case 2.2.1 : pr2_free *)
123 (* case 2.2.1 : pr2_delta *)
124 Cpr0Drop; Pr0Subst0; Apply pr3_sing with t2:=x; XEAuto.