2 include "basic_2/rt_computation/cpms_lpr.ma".
4 lemma cpm_lsubr_trans (h) (n) (G) (L1) (T1):
5 ∀T2. ⦃G,L1⦄ ⊢ T1 ➡[↑n,h] T2 → ∀L2. L1 ⫃ L2 →
6 ∃∃T0. ⦃G,L2⦄ ⊢ T1 ➡[↑n,h] T0 & ⦃G,L2⦄ ⊢ T0 ➡*[h] T2.
8 @(insert_eq_0 … (↑m)) #n #H
9 @(cpm_ind … H) -n -G -L1 -T1 -T2
12 | #n #G #K1 #V1 #V2 #W2 #_ #IH #HVW2 #Hm #L2 #H destruct
13 elim (lsubr_inv_bind1 … H) -H *
15 elim (IH … HK) -K1 [| // ] #V0 #HV10 #HV02
16 elim (lifts_total V0 (𝐔❴1❵)) #W0 #HVW0
17 lapply (cpms_lifts_bi … HV02 (Ⓣ) … (K2.ⓓV1) … HVW0 … HVW2) -V2
18 [ /3 width=2 by drops_refl, drops_drop/ ] #HW02
19 /3 width=3 by cpm_delta, ex2_intro/
20 | #K2 #VX #WX #HK #H1 #H2 destruct
21 elim (IH … HK) -K1 [| // ] #X0 #H1 #H2
22 elim (cpm_inv_cast1 … H1) -H1 [ * || * ]
23 [ #W0 #V0 #HW0 #HV0 #H destruct
24 @(ex2_intro … (#0)) [ // |
25 | #I1 #I2 #K2 #VX #HK #H1 #H2 destruct
29 | #n #p #I #G #L1 #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #L2 #HL
30 elim (IHV … HL) -IHV #V0 #HV01 #HV02
31 elim (IHT (L2.ⓑ{I}V1)) [| /2 width=1 by lsubr_bind/ ] -L1 #T0 #HT10 #HT02
32 @(ex2_intro … (ⓑ{p,I}V1.T0)) /3 width=3 by cprs_step_sn, cpms_bind, cpm_bind/ (**) (* full auto a bit slow *)
38 lemma cpr_cpm_trans_swap_lsubr_lpr (h) (G) (L1) (T1):
39 ∀T. ⦃G,L1⦄ ⊢ T1 ➡[h] T → ∀L. L ⫃ L1 →
40 ∀L2. ⦃G,L⦄ ⊢ ➡[h] L2 → ∀n2,T2. ⦃G,L2⦄ ⊢ T ➡[n2,h] T2 →
41 ∃∃T0. ⦃G,L⦄ ⊢ T1 ➡[n2,h] T0 & ⦃G,L⦄ ⊢ T0 ➡*[h] T2.
42 #h #G #L1 #T1 @(fqup_wf_ind_eq (Ⓣ) … G L1 T1) -G -L1 -T1
43 #G0 #L0 #T0 #IH #G #L1 * [| * [| * ]]
45 #I #HG #HL #HT #X #H1 #L2 #HL12 #m2 #X2 #H2 destruct
46 elim (cpr_inv_atom1 … H1) -H1 [|*: * ]
48 elim (cpm_inv_atom1 … H2) -H2 *
49 [ #H1 #H2 destruct /3 width=3 by cpm_cpms, ex2_intro/
50 | #s #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
51 | #K2 #V #V2 #HV2 #HVT2 #H1 #H2 destruct
52 elim (lpr_inv_pair_dx … HL12) -HL12 #K1 #V1 #HK12 #HV1 #H destruct
53 elim (IH … HV1 … HK12 … HV2) -K2 -V -IH
54 [| /2 width=1 by fqu_fqup, fqu_lref_O/ ] #V0 #HV10 #HV02
55 elim (lifts_total V0 (𝐔❴1❵)) #T0 #HVT0
56 lapply (cpms_lifts_bi … HV02 (Ⓣ) … (K1.ⓓV1) … HVT0 … HVT2) -V2
57 [ /3 width=2 by drops_refl, drops_drop/ ] #HT02
58 /3 width=3 by cpm_delta, ex2_intro/
59 | #n2 #K2 #W #W2 #HW2 #HWT2 #H1 #H2 #H3 destruct
60 elim (lpr_inv_pair_dx … HL12) -HL12 #K1 #W1 #HK12 #HW1 #H destruct
61 elim (IH … HW1 … HK12 … HW2) -K2 -W -IH
62 [| /2 width=1 by fqu_fqup, fqu_lref_O/ ] #W0 #HW10 #HW02
63 elim (lifts_total W0 (𝐔❴1❵)) #T0 #HWT0
64 lapply (cpms_lifts_bi … HW02 (Ⓣ) … (K1.ⓛW1) … HWT0 … HWT2) -W2
65 [ /3 width=2 by drops_refl, drops_drop/ ] #HT02
66 /3 width=3 by cpm_ell, ex2_intro/
67 | #I2 #K2 #T2 #i #HT2 #HTU2 #H1 #H2 destruct
68 elim (lpr_inv_bind_dx … HL12) -HL12 #I1 #K1 #HK12 #_ #H destruct
69 elim (IH … (#i) … HK12 … HT2) -I2 -K2 -IH
70 [|*: /2 width=1 by fqu_fqup/ ] #T0 #HT10 #HT02
71 elim (lifts_total T0 (𝐔❴1❵)) #U0 #HTU0
72 lapply (cpms_lifts_bi … HT02 (Ⓣ) … (K1.ⓘ{I1}) … HTU0 … HTU2) -T2
73 [ /3 width=2 by drops_refl, drops_drop/ ] #HU02
74 /3 width=3 by cpm_lref, ex2_intro/
76 | #K1 #V1 #V #HV1 #HVT #H1 #H2 destruct
77 elim (lpr_inv_pair_sn … HL12) -HL12 #K2 #V0 #HK12 #_ #H destruct
78 elim (cpm_inv_lifts_sn … H2 (Ⓣ) … HVT) -X
79 [|*: /3 width=2 by drops_refl, drops_drop/ ] -V0 #V2 #HVT2 #HV2
80 elim (IH … HV1 … HK12 … HV2) -K2 -V -IH
81 [| /2 width=1 by fqu_fqup, fqu_lref_O/ ] #V0 #HV10 #HV02
82 elim (lifts_total V0 (𝐔❴1❵)) #T0 #HVT0
83 lapply (cpms_lifts_bi … HV02 (Ⓣ) … (K1.ⓓV1) … HVT0 … HVT2) -V2
84 [ /3 width=2 by drops_refl, drops_drop/ ] #HT02
85 /3 width=3 by cpm_delta, ex2_intro/
86 | #I1 #K1 #T #i #HT1 #HTU #H1 #H2 destruct
87 elim (lpr_inv_bind_sn … HL12) -HL12 #I2 #K2 #HK12 #_ #H destruct
88 elim (cpm_inv_lifts_sn … H2 (Ⓣ) … HTU) -X
89 [|*: /3 width=2 by drops_refl, drops_drop/ ] -I2 #T2 #HTU2 #HT2
90 elim (IH … HT1 … HK12 … HT2) -K2 -T -IH
91 [| /2 width=1 by fqu_fqup/ ] #T0 #HT10 #HT02
92 elim (lifts_total T0 (𝐔❴1❵)) #U0 #HTU0
93 lapply (cpms_lifts_bi … HT02 (Ⓣ) … (K1.ⓘ{I1}) … HTU0 … HTU2) -T2
94 [ /3 width=2 by drops_refl, drops_drop/ ] #HU02
95 /3 width=3 by cpm_lref, ex2_intro/
99 #p #I #V1 #T1 #HG #HL #HT #X #H1 #L2 #HL12 #n2 #X2 #H2
100 elim (cpm_inv_bind1 … H1) -H1 *
101 [ #V #T #HV1 #HT1 #H destruct
102 elim (cpm_inv_bind1 … H2) -H2 *
103 [ #V2 #T2 #HV2 #HT2 #H destruct
104 elim (IH … HT1 … HT2) -T
105 [|*: /2 width=1 by lpr_pair/ ] #T0 #HT10 #HT02
106 elim (IH … HV1 … HL12 … HV2) -L2 -V -IH
107 [| // ] #V0 #HV10 #HV02
108 /4 width=7 by cpms_bind, cpms_step_sn, cpm_bind, ex2_intro/
109 | #X #HXT #HX2 #H1 #H2 destruct
110 elim (cpm_lifts_sn … HX2 (Ⓣ) … (L2.ⓓV) … HXT) -HX2
111 [| /3 width=2 by drops_refl, drops_drop/ ] #T2 #HXT2 #HT2
112 elim (IH … HT1 … HT2) -HT2 -IH
113 [|*: /2 width=1 by lpr_pair/ ] -L2 #T0 #HT10 #HT02
114 /3 width=6 by cpms_zeta_dx, cpm_bind, ex2_intro/
116 | #X1 #HXT1 #HX1 #H1 #H2 destruct
117 elim (IH … HX1 … HL12 … H2) -L2 -X -IH
118 [| /2 width=1 by fqup_zeta/ ] #X0 #HX10 #HX02
119 /3 width=3 by cpm_zeta, ex2_intro/
122 | #V1 #T1 #HG #HL #HT #X #H1 #L #HL1 #L2 #HL2 #m2 #X2 #H2 destruct
123 elim (cpm_inv_appl1 … H1) -H1 *
125 #V #T #HV1 #HT1 #H destruct
126 elim (cpm_inv_appl1 … H2) -H2 *
127 [ #V2 #T2 #HV2 #HT2 #H destruct
128 elim (IH … HV1 … HL12 … HV2) -V [| // ] #V0 #HV10 #HV02
129 elim (IH … HT1 … HL12 … HT2) -L2 -T -IH [| // ] #T0 #HT10 #HT02
130 /3 width=5 by cpms_appl, cpm_appl, ex2_intro/
131 | #q #V2 #WX #W2 #TX #T2 #HV2 #HW2 #HT2 #H1 #H2 destruct
132 elim (IH … HV1 … HL12 … HV2) -V [| // ] #V0 #HV10 #HV02
133 elim (IH … HT1 … HL12 m2 (ⓛ{q}W2.T2)) -IH -HT1
134 [|*: /2 width=1 by cpm_bind/ ] -L2 -WX -TX #T0 #HT10 #HT02
135 /4 width=9 by cprs_step_dx, cpms_appl, cpm_beta, cpm_appl, ex2_intro/
136 | #q #V2 #U2 #WX #W2 #TX #T2 #HV2 #HVU2 #HW2 #HT2 #H1 #H2 destruct
137 elim (IH … HV1 … HL12 … HV2) -V [| // ] #V0 #HV10 #HV02
138 elim (IH … HT1 … HL12 m2 (ⓓ{q}W2.T2)) -IH -HT1
139 [|*: /2 width=1 by cpm_bind/ ] -L2 -WX -TX #T0 #HT10 #HT02
140 /4 width=11 by cprs_step_dx, cpms_appl, cpm_theta, cpm_appl, ex2_intro/
143 | #p #V #W1 #W #TX1 #T #HV1 #HW1 #HT1 #H1 #H3 destruct
144 elim (cpm_inv_abbr1 … H2) -H2 *
145 [ #X3 #T2 #H2 #HT2 #H destruct
146 elim (cpr_inv_cast1 … H2) -H2 [ * ]
147 [ #W2 #V2 #HW2 #HV2 #H destruct
148 elim (IH … HT1 (L.ⓓⓝW1.V1) … HT2) -T
149 [|*: /4 width=3 by lsubr_beta, lpr_pair, cpm_cast, lsubr_cpm_trans/ ] #T0 #HT10 #HT02
150 elim (IH … HV1 … HL1 … HL2 … HV2) -V [| // ] #V0 #HV10 #HV02
151 elim (IH … HW1 … HL1 … HL2 … HW2) -L2 -W -IH [| // ] #W0 #HW10 #HW02
152 @(ex2_intro … (ⓓ{p}ⓝW1.V1.T0))
155 /2 width=1 by cpm_beta/
156 | /3 width=7 by cprs_step_dx, cpms_appl, cpm_beta/
158 @cprs_step_dx [| @(cpms_appl … HT02 … HV02) | /2 width=1 by cpm_beta/