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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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15 include "basic_2/unwind/sstas_sstas.ma".
16 include "basic_2/computation/ygt.ma".
17 include "basic_2/equivalence/cpcs_ltpr.ma".
18 include "basic_2/dynamic/snv_ltpss_dx.ma".
19 include "basic_2/dynamic/snv_sstas.ma".
21 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
23 (* Inductive premises for the preservation results **************************)
25 definition IH_snv_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
26 λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
27 ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
29 definition IH_ssta_ltpr_tpr: ∀h:sh. sd h → relation2 lenv term ≝
30 λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
31 ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
32 ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 →
33 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ⬌* U2.
35 definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
36 λh,g,L1,T1. ⦃h, L1⦄ ⊩ T1 :[g] →
37 ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l + 1] U1 → ⦃h, L1⦄ ⊩ U1 :[g].
39 fact snv_ltpr_cpr_aux: ∀h,g,L1,T1. IH_snv_ltpr_tpr h g L1 T1 →
41 ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 → ⦃h, L2⦄ ⊩ T2 :[g].
42 #h #g #L1 #T1 #IH #HT1 #L2 #HL12 #T2 * #T #HT1T #HTT2
43 lapply (IH … HL12 … HT1T) -HL12 // -T1 #HT0
44 lapply (snv_tpss_conf … HT0 … HTT2) -T //
47 fact ssta_ltpr_cpr_aux: ∀h,g,L1,T1. IH_ssta_ltpr_tpr h g L1 T1 →
49 ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
50 ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡ T2 →
51 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ⬌* U2.
52 #h #g #L1 #T1 #IH #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 * #T #HT1T #HTT2
53 elim (IH … HTU1 … HL12 … HT1T) // -L1 -T1 #U #HTU #HU1
54 elim (ssta_tpss_conf … HTU … HTT2) -T #U2 #HTU2 #HU2
55 lapply (cpcs_cpr_strap1 … HU1 U2 ?) /2 width=3/
58 fact snv_ltpr_cprs_aux: ∀h,g,L0,T0.
59 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
60 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
61 ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → ⦃h, L2⦄ ⊩ T2 :[g].
62 #h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #L2 #HL12 #T2 #H
63 @(cprs_ind … H) -T2 [ /2 width=6 by snv_ltpr_cpr_aux/ ] -HT1
64 /5 width=6 by snv_ltpr_cpr_aux, ygt_yprs_trans, ltpr_cprs_yprs/
67 fact ssta_ltpr_cprs_aux: ∀h,g,L0,T0.
68 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
69 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
70 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
71 ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g, l] U1 →
72 ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
73 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g, l] U2 & L2 ⊢ U1 ⬌* U2.
74 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #L2 #HL12 #T2 #H
75 @(cprs_ind … H) -T2 [ /2 width=7 by ssta_ltpr_cpr_aux/ ]
76 #T #T2 #HT1T #HTT2 * #U #HTU #HU1
77 elim (ssta_ltpr_cpr_aux … HTU … HTT2) //
78 [2: /3 width=9 by snv_ltpr_cprs_aux/
79 |3: /5 width=6 by ygt_yprs_trans, ltpr_cprs_yprs/
80 ] -L0 -L1 -T0 -T1 -T #U2 #HTU2 #HU2
81 lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
84 fact ssta_ltpr_cpcs_aux: ∀h,g,L0,T0.
85 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
86 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
87 ∀L1,L2,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L2, T2⦄ →
88 ⦃h, L1⦄ ⊩ T1 :[g] → ⦃h, L2⦄ ⊩ T2 :[g] →
89 ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g, l1] U1 →
90 ∀U2,l2. ⦃h, L2⦄ ⊢ T2 •[g, l2] U2 →
91 L1 ➡ L2 → L2 ⊢ T1 ⬌* T2 →
92 l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
93 #h #g #L0 #T0 #IH2 #IH1 #L1 #L2 #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #HL12 #H
94 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
95 elim (ssta_ltpr_cprs_aux … HLT01 HT1 … HTU1 … H1) -T1 /2 width=1/ #W1 #H1 #HUW1
96 elim (ssta_ltpr_cprs_aux … HLT02 HT2 … HTU2 … H2) -T2 /2 width=1/ #W2 #H2 #HUW2 -L1 -L0 -T0
97 elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
98 lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
101 fact snv_sstas_aux: ∀h,g,L0,T0.
102 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
103 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
104 ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ⦃h, L1⦄ ⊩ U1 :[g].
105 #h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #U1 #H
106 @(sstas_ind … H) -U1 // -HT1 /4 width=5 by ygt_yprs_trans, sstas_yprs/
109 fact sstas_ltpr_cprs_aux: ∀h,g,L0,T0.
110 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
111 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
112 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
113 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
114 ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 →
115 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ⬌* U2.
116 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #L2 #HL12 #T2 #HT12 #U1 #H
117 @(sstas_ind … H) -U1 [ /3 width=3/ ]
118 #U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
119 lapply (snv_ltpr_cprs_aux … IH2 … HT1 … HT12) // #HT2
120 elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
121 elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12 //
122 [2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, ltpr_cprs_yprs/
123 |3: /3 width=7 by snv_sstas_aux, ygt_yprs_trans, cprs_yprs/
124 |4: /4 width=5 by ygt_yprs_trans, ltpr_cprs_yprs, sstas_yprs/
125 |5: /3 width=4 by ygt_yprs_trans, cprs_yprs, sstas_yprs/
126 ] -L0 -T0 -T1 -HT2 #H #HW12 destruct /3 width=4/
129 fact dxprs_ltpr_cprs_aux: ∀h,g,L0,T0.
130 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
131 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
132 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
133 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
134 ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
135 ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
136 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
137 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #L2 #HL12 #T2 #HT12
138 elim (sstas_ltpr_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
139 lapply (ltpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
140 lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
141 elim (cpcs_inv_cprs … H) -H /3 width=3/
144 fact ssta_dxprs_aux: ∀h,g,L0,T0.
145 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
146 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
147 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊩ T1 :[g] →
148 ∀l,U1. ⦃h, L1⦄ ⊢ T1 •[g, l+1] U1 → ∀T2. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T2 →
149 ∃∃U,U2. ⦃h, L1⦄ ⊢ U1 •*[g] U & ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U ⬌* U2.
150 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
151 elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
152 [ elim (ssta_ltpr_cprs_aux … IH2 IH1 … HTU1 L1 … HTT2) // -L0 -T0 -T /3 width=5/
153 | @(ex3_2_intro …T2 HU1T) // /2 width=1/