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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/unfold/sstas_sstas.ma".
16 include "basic_2/equivalence/cpcs_cpcs.ma".
17 include "basic_2/dynamic/snv_sstas.ma".
18 include "basic_2/dynamic/ygt.ma".
20 (* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
22 (* Inductive premises for the preservation results **************************)
24 definition IH_snv_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
25 λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
26 ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
28 definition IH_ssta_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
29 λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
30 ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
31 ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 →
32 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
34 definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
35 λh,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] →
36 ∀U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → ⦃h, L⦄ ⊢ U ¡[g].
38 definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
39 λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[g] →
40 ∀L1. h ⊢ L1 ¡⊑[g] L2 → ⦃h, L1⦄ ⊢ T ¡[g].
42 (* Properties for the preservation results **********************************)
44 fact snv_cprs_lpr_aux: ∀h,g,L0,T0.
45 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
46 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
47 ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ T2 ¡[g].
48 #h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #T2 #H
49 elim H -T2 [ /2 width=6/ ] -HT1
50 /4 width=6 by ygt_yprs_trans, cprs_yprs/
53 fact ssta_cprs_lpr_aux: ∀h,g,L0,T0.
54 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
55 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
56 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
57 ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
58 ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
59 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
60 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
61 elim H -T2 [ /2 width=7/ ]
62 #T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
63 elim (IHT1 L1) // -IHT1 #U #HTU #HU1
64 elim (IH1 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
65 [2: /3 width=9 by snv_cprs_lpr_aux/
66 |3: /5 width=6 by ygt_yprs_trans, cprs_yprs/
67 ] -L0 -T0 -T1 -T #U2 #HTU2 #HU2
68 lapply (lpr_cpcs_conf … HL12 … HU1) -L1 #HU1
69 lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
72 fact ssta_cpcs_lpr_aux: ∀h,g,L0,T0.
73 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
74 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
75 ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T2⦄ →
76 ⦃h, L1⦄ ⊢ T1 ¡[g] → ⦃h, L1⦄ ⊢ T2 ¡[g] →
77 ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
78 ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[g] ⦃l2, U2⦄ →
79 L1 ⊢ T1 ⬌* T2 → ∀L2. L1 ⊢ ➡ L2 →
80 l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
81 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #T2 #H01 #H02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H #L2 #HL12
82 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
83 elim (ssta_cprs_lpr_aux … H01 HT1 … HTU1 … H1 … HL12) -T1 /2 width=1/ #W1 #H1 #HUW1
84 elim (ssta_cprs_lpr_aux … H02 HT2 … HTU2 … H2 … HL12) -T2 /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
85 elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
86 lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
89 fact snv_sstas_aux: ∀h,g,L0,T0.
90 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
91 ∀L,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T⦄ → ⦃h, L⦄ ⊢ T ¡[g] →
92 ∀U. ⦃h, L⦄ ⊢ T •*[g] U → ⦃h, L⦄ ⊢ U ¡[g].
93 #h #g #L0 #T0 #IH #L #T #H01 #HT #U #H
94 @(sstas_ind … H) -U // -HT /4 width=5 by ygt_yprs_trans, sstas_yprs/
97 fact snv_sstas_lpr_aux: ∀h,g,L0,T0.
98 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
99 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
100 ∀L1,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[g] →
101 ∀U. ⦃h, L1⦄ ⊢ T •*[g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[g].
102 /4 width=7 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
105 fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
106 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
107 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
108 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
109 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
110 ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
111 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ⬌* U2.
112 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #H
113 @(sstas_ind … H) -U1 [ /3 width=5 by lpr_cprs_conf, ex2_intro/ ]
114 #U1 #W1 #l1 #HTU1 #HUW1 #IHTU1 #T2 #HT12 #L2 #HL12
115 elim (IHTU1 … HT12 … HL12) -IHTU1 #U2 #HTU2 #HU12
116 lapply (snv_cprs_lpr_aux … IH2 … HT1 … HT12 … HL12) // #HT2
117 elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
118 elim (IH1 … HUW1 U1 … HL12) -HUW1 //
119 [2: /3 width=7 by snv_sstas_aux/
120 |3: /3 width=4 by ygt_yprs_trans, sstas_yprs/
122 elim (ssta_cpcs_lpr_aux … IH2 IH1 … HU1W … HUW2 … HU12 L2) // -IH1 -HU1W -HU12
123 [2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, cprs_lpr_yprs/
124 |3: /3 width=10 by snv_sstas_lpr_aux/
125 |4,5: /4 width=5 by ygt_yprs_trans, cprs_lpr_yprs, sstas_yprs/
126 ] -L0 -T0 -L1 -T1 -HT2 #H #HW12 destruct
127 lapply (cpcs_trans … HW1 … HW12) -W /3 width=4/
130 fact cpds_cprs_lpr_aux: ∀h,g,L0,T0.
131 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
132 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
133 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
134 ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
135 ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
136 ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
137 ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
138 #h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
139 elim (sstas_cprs_lpr_aux … IH3 IH2 IH1 … H01 … HTW1 … HT12 … HL12) // -L0 -T0 -T1 #W2 #HTW2 #HW12
140 lapply (lpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
141 lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
142 elim (cpcs_inv_cprs … H) -H /3 width=3/
145 fact ssta_cpds_aux: ∀h,g,L0,T0.
146 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
147 (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
148 ∀L,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → ⦃h, L⦄ ⊢ T1 ¡[g] →
149 ∀l,U1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L⦄ ⊢ T1 •*➡*[g] T2 →
150 ∃∃U,U2. ⦃h, L⦄ ⊢ U1 •*[g] U & ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U ⬌* U2.
151 #h #g #L0 #T0 #IH2 #IH1 #L #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
152 elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
153 [ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -L0 -T0 -T /3 width=5/
154 | @(ex3_2_intro …T2 HU1T) // /2 width=1/