matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpm.ma

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  14
  15 include "basic_2/notation/relations/pred_6.ma".
  16 include "basic_2/notation/relations/pred_5.ma".
  17 include "basic_2/rt_transition/cpg.ma".
  18
  19 (* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
  20
  21 (* Basic_2A1: includes: cpr *)
  22 definition cpm (n) (h): relation4 genv lenv term term ≝
  23                         λG,L,T1,T2. ∃∃c. 𝐑𝐓⦃n, c⦄ & ⦃G, L⦄ ⊢ T1 ⬈[c, h] T2.
  24
  25 interpretation
  26    "semi-counted context-sensitive parallel rt-transition (term)"
  27    'PRed n h G L T1 T2 = (cpm n h G L T1 T2).
  28
  29 interpretation
  30    "context-sensitive parallel r-transition (term)"
  31    'PRed h G L T1 T2 = (cpm O h G L T1 T2).
  32
  33 (* Basic properties *********************************************************)
  34
  35 lemma cpm_ess: ∀h,G,L,s. ⦃G, L⦄ ⊢ ⋆s ➡[1, h] ⋆(next h s).
  36 /2 width=3 by cpg_ess, ex2_intro/ qed.
  37
  38 lemma cpm_delta: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
  39                  ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓓV1⦄ ⊢ #0 ➡[n, h] W2.
  40 #n #h #G #K #V1 #V2 #W2 *
  41 /3 width=5 by cpg_delta, ex2_intro/
  42 qed.
  43
  44 lemma cpm_ell: ∀n,h,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 →
  45                ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓛV1⦄ ⊢ #0 ➡[⫯n, h] W2.
  46 #n #h #G #K #V1 #V2 #W2 *
  47 /3 width=5 by cpg_ell, ex2_intro, isrt_succ/
  48 qed.
  49
  50 lemma cpm_lref: ∀n,h,I,G,K,V,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T →
  51                 ⬆*[1] T ≡ U → ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ➡[n, h] U.
  52 #n #h #I #G #K #V #T #U #i *
  53 /3 width=5 by cpg_lref, ex2_intro/
  54 qed.
  55
  56 (* Basic_2A1: includes: cpr_bind *)
  57 lemma cpm_bind: ∀n,h,p,I,G,L,V1,V2,T1,T2.
  58                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
  59                 ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2.
  60 #n #h #p #I #G #L #V1 #V2 #T1 #T2 * #riV #rhV #HV12 *
  61 /5 width=5 by cpg_bind, isrt_plus_O1, isr_shift, ex2_intro/
  62 qed.
  63
  64 (* Note: cpr_flat: does not hold in basic_1 *)
  65 (* Basic_1: includes: pr2_thin_dx *)
  66 (* Basic_2A1: includes: cpr_flat *)
  67 lemma cpm_flat: ∀n,h,I,G,L,V1,V2,T1,T2.
  68                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
  69                 ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡[n, h] ⓕ{I}V2.T2.
  70 #n #h #I #G #L #V1 #V2 #T1 #T2 * #riV #rhV #HV12 *
  71 /5 width=5 by isrt_plus_O1, isr_shift, cpg_flat, ex2_intro/
  72 qed.
  73
  74 (* Basic_2A1: includes: cpr_zeta *)
  75 lemma cpm_zeta: ∀n,h,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[n, h] T →
  76                 ⬆*[1] T2 ≡ T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡[n, h] T2.
  77 #n #h #G #L #V #T1 #T #T2 *
  78 /3 width=5 by cpg_zeta, isrt_plus_O2, ex2_intro/
  79 qed.
  80
  81 (* Basic_2A1: includes: cpr_eps *)
  82 lemma cpm_eps: ∀n,h,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → ⦃G, L⦄ ⊢ ⓝV.T1 ➡[n, h] T2.
  83 #n #h #G #L #V #T1 #T2 *
  84 /3 width=3 by cpg_eps, isrt_plus_O2, ex2_intro/
  85 qed.
  86
  87 lemma cpm_ee: ∀n,h,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ ⓝV1.T ➡[⫯n, h] V2.
  88 #n #h #G #L #V1 #V2 #T *
  89 /3 width=3 by cpg_ee, isrt_succ, ex2_intro/
  90 qed.
  91
  92 (* Basic_2A1: includes: cpr_beta *)
  93 lemma cpm_beta: ∀n,h,p,G,L,V1,V2,W1,W2,T1,T2.
  94                 ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
  95                 ⦃G, L⦄ ⊢ ⓐV1.ⓛ{p}W1.T1 ➡[n, h] ⓓ{p}ⓝW2.V2.T2.
  96 #n #h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV12 * #riW #rhW #HW12 *
  97 /6 width=7 by cpg_beta, isrt_plus_O2, isrt_plus, isr_shift, ex2_intro/
  98 qed.
  99
 100 (* Basic_2A1: includes: cpr_theta *)
 101 lemma cpm_theta: ∀n,h,p,G,L,V1,V,V2,W1,W2,T1,T2.
 102                  ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆*[1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 →
 103                  ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
 104                  ⦃G, L⦄ ⊢ ⓐV1.ⓓ{p}W1.T1 ➡[n, h] ⓓ{p}W2.ⓐV2.T2.
 105 #n #h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #riV #rhV #HV1 #HV2 * #riW #rhW #HW12 *
 106 /6 width=9 by cpg_theta, isrt_plus_O2, isrt_plus, isr_shift, ex2_intro/
 107 qed.
 108
 109 (* Basic properties on r-transition *****************************************)
 110
 111 (* Basic_1: includes by definition: pr0_refl *)
 112 (* Basic_2A1: includes: cpr_atom *)
 113 lemma cpr_refl: ∀h,G,L. reflexive … (cpm 0 h G L).
 114 /2 width=3 by ex2_intro/ qed.
 115
 116 (* Basic_1: was: pr2_head_1 *)
 117 lemma cpr_pair_sn: ∀h,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 →
 118                    ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h] ②{I}V2.T.
 119 #h #I #G #L #V1 #V2 *
 120 /3 width=3 by cpg_pair_sn, isr_shift, ex2_intro/
 121 qed.
 122
 123 (* Basic inversion lemmas ***************************************************)
 124
 125 lemma cpm_inv_atom1: ∀n,h,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[n, h] T2 →
 126                      ∨∨ T2 = ⓪{J} ∧ n = 0
 127                       | ∃∃s. T2 = ⋆(next h s) & J = Sort s & n = 1
 128                       | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≡ T2 &
 129                                    L = K.ⓓV1 & J = LRef 0
 130                       | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≡ T2 &
 131                                      L = K.ⓛV1 & J = LRef 0 & n = ⫯m
 132                       | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≡ T2 &
 133                                      L = K.ⓑ{I}V & J = LRef (⫯i).
 134 #n #h #J #G #L #T2 * #c #Hc #H elim (cpg_inv_atom1 … H) -H *
 135 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or5_intro0, conj/
 136 | #s #H1 #H2 #H3 destruct /4 width=3 by isrt_inv_01, or5_intro1, ex3_intro/
 137 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
 138   /4 width=6 by or5_intro2, ex4_3_intro, ex2_intro/
 139 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 #H3 destruct
 140   elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
 141   /4 width=9 by or5_intro3, ex5_4_intro, ex2_intro/
 142 | #I #K #V1 #V2 #i #HV2 #HVT2 #H1 #H2 destruct
 143   /4 width=9 by or5_intro4, ex4_5_intro, ex2_intro/
 144 ]
 145 qed-.
 146
 147 lemma cpm_inv_sort1: ∀n,h,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[n,h] T2 →
 148                      (T2 = ⋆s ∧ n = 0) ∨
 149                      (T2 = ⋆(next h s) ∧ n = 1).
 150 #n #h #G #L #T2 #s * #c #Hc #H elim (cpg_inv_sort1 … H) -H *
 151 #H1 #H2 destruct
 152 /4 width=1 by isrt_inv_01, isrt_inv_00, or_introl, or_intror, conj/
 153 qed-.
 154
 155 lemma cpm_inv_zero1: ∀n,h,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[n, h] T2 →
 156                      ∨∨ (T2 = #0 ∧ n = 0)
 157                       | ∃∃K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 & ⬆*[1] V2 ≡ T2 &
 158                                    L = K.ⓓV1
 159                       | ∃∃m,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[m, h] V2 & ⬆*[1] V2 ≡ T2 &
 160                                      L = K.ⓛV1 & n = ⫯m.
 161 #n #h #G #L #T2 * #c #Hc #H elim (cpg_inv_zero1 … H) -H *
 162 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or3_intro0, conj/
 163 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
 164   /4 width=8 by or3_intro1, ex3_3_intro, ex2_intro/
 165 | #cV #K #V1 #V2 #HV12 #HVT2 #H1 #H2 destruct
 166   elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
 167   /4 width=8 by or3_intro2, ex4_4_intro, ex2_intro/
 168 ]
 169 qed-.
 170
 171 lemma cpm_inv_lref1: ∀n,h,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[n, h] T2 →
 172                      (T2 = #(⫯i) ∧ n = 0) ∨
 173                      ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[n, h] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
 174 #n #h #G #L #T2 #i * #c #Hc #H elim (cpg_inv_lref1 … H) -H *
 175 [ #H1 #H2 destruct /4 width=1 by isrt_inv_00, or_introl, conj/
 176 | #I #K #V1 #V2 #HV2 #HVT2 #H1 destruct
 177  /4 width=7 by ex3_4_intro, ex2_intro, or_intror/
 178 ]
 179 qed-.
 180
 181 lemma cpm_inv_gref1: ∀n,h,G,L,T2,l. ⦃G, L⦄ ⊢ §l ➡[n, h] T2 → T2 = §l ∧ n = 0.
 182 #n #h #G #L #T2 #l * #c #Hc #H elim (cpg_inv_gref1 … H) -H
 183 #H1 #H2 destruct /3 width=1 by isrt_inv_00, conj/
 184 qed-.
 185
 186 (* Basic_2A1: includes: cpr_inv_bind1 *)
 187 lemma cpm_inv_bind1: ∀n,h,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] U2 → (
 188                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 &
 189                               U2 = ⓑ{p,I}V2.T2
 190                      ) ∨
 191                      ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T & ⬆*[1] U2 ≡ T &
 192                           p = true & I = Abbr.
 193 #n #h #p #I #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_bind1 … H) -H *
 194 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 195   elim (isrt_inv_plus … Hc) -Hc #nV #nT #HcV #HcT #H destruct
 196   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 197   /4 width=5 by ex3_2_intro, ex2_intro, or_introl/
 198 | #cT #T2 #HT12 #HUT2 #H1 #H2 #H3 destruct
 199   /5 width=3 by isrt_inv_plus_O_dx, ex4_intro, ex2_intro, or_intror/
 200 ]
 201 qed-.
 202
 203 (* Basic_1: includes: pr0_gen_abbr pr2_gen_abbr *)
 204 (* Basic_2A1: includes: cpr_inv_abbr1 *)
 205 lemma cpm_inv_abbr1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[n, h] U2 → (
 206                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T2 &
 207                               U2 = ⓓ{p}V2.T2
 208                      ) ∨
 209                      ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[n, h] T & ⬆*[1] U2 ≡ T & p = true.
 210 #n #h #p #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_abbr1 … H) -H *
 211 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 212   elim (isrt_inv_plus … Hc) -Hc #nV #nT #HcV #HcT #H destruct
 213   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 214   /4 width=5 by ex3_2_intro, ex2_intro, or_introl/
 215 | #cT #T2 #HT12 #HUT2 #H1 #H2 destruct
 216   /5 width=3 by isrt_inv_plus_O_dx, ex3_intro, ex2_intro, or_intror/
 217 ]
 218 qed-.
 219
 220 (* Basic_1: includes: pr0_gen_abst pr2_gen_abst *)
 221 (* Basic_2A1: includes: cpr_inv_abst1 *)
 222 lemma cpm_inv_abst1: ∀n,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[n, h] U2 →
 223                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[n, h] T2 &
 224                               U2 = ⓛ{p}V2.T2.
 225 #n #h #p #G #L #V1 #T1 #U2 * #c #Hc #H elim (cpg_inv_abst1 … H) -H
 226 #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 227 elim (isrt_inv_plus … Hc) -Hc #nV #nT #HcV #HcT #H destruct
 228 elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 229 /3 width=5 by ex3_2_intro, ex2_intro/
 230 qed-.
 231
 232 lemma cpm_inv_flat1: ∀n,h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[n, h] U2 →
 233                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
 234                                  U2 = ⓕ{I}V2.T2
 235                       | (⦃G, L⦄ ⊢ U1 ➡[n, h] U2 ∧ I = Cast)
 236                       | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & I = Cast & n = ⫯m
 237                       | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
 238                                             ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 &
 239                                             U1 = ⓛ{p}W1.T1 &
 240                                             U2 = ⓓ{p}ⓝW2.V2.T2 & I = Appl
 241                       | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≡ V2 &
 242                                               ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 &
 243                                               U1 = ⓓ{p}W1.T1 &
 244                                               U2 = ⓓ{p}W2.ⓐV2.T2 & I = Appl.
 245 #n #h #I #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_flat1 … H) -H *
 246 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 247   elim (isrt_inv_plus … Hc) -Hc #nV #nT #HcV #HcT #H destruct
 248   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 249   /4 width=5 by or5_intro0, ex3_2_intro, ex2_intro/
 250 | #cU #U12 #H1 #H2 destruct
 251   /5 width=3 by isrt_inv_plus_O_dx, or5_intro1, conj, ex2_intro/
 252 | #cU #H12 #H1 #H2 destruct
 253   elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
 254   /4 width=3 by or5_intro2, ex3_intro, ex2_intro/
 255 | #cV #cW #cT #p #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #H1 #H2 #H3 #H4 destruct
 256   lapply (isrt_inv_plus_O_dx … Hc ?) -Hc // #Hc
 257   elim (isrt_inv_plus … Hc) -Hc #n0 #nT #Hc #HcT #H destruct
 258   elim (isrt_inv_plus … Hc) -Hc #nV #nW #HcV #HcW #H destruct
 259   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 260   elim (isrt_inv_shift … HcW) -HcW #HcW #H destruct
 261   /4 width=11 by or5_intro3, ex6_6_intro, ex2_intro/
 262 | #cV #cW #cT #p #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #H1 #H2 #H3 #H4 destruct
 263   lapply (isrt_inv_plus_O_dx … Hc ?) -Hc // #Hc
 264   elim (isrt_inv_plus … Hc) -Hc #n0 #nT #Hc #HcT #H destruct
 265   elim (isrt_inv_plus … Hc) -Hc #nV #nW #HcV #HcW #H destruct
 266   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 267   elim (isrt_inv_shift … HcW) -HcW #HcW #H destruct
 268   /4 width=13 by or5_intro4, ex7_7_intro, ex2_intro/
 269 ]
 270 qed-.
 271
 272 (* Basic_1: includes: pr0_gen_appl pr2_gen_appl *)
 273 (* Basic_2A1: includes: cpr_inv_appl1 *)
 274 lemma cpm_inv_appl1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[n, h] U2 →
 275                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n, h] T2 &
 276                                  U2 = ⓐV2.T2
 277                       | ∃∃p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ W1 ➡[h] W2 &
 278                                             ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 &
 279                                             U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2
 280                       | ∃∃p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V & ⬆*[1] V ≡ V2 &
 281                                               ⦃G, L⦄ ⊢ W1 ➡[h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 &
 282                                               U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2.
 283 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_appl1 … H) -H *
 284 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 285   elim (isrt_inv_plus … Hc) -Hc #nV #nT #HcV #HcT #H destruct
 286   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 287   /4 width=5 by or3_intro0, ex3_2_intro, ex2_intro/
 288 | #cV #cW #cT #p #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #H1 #H2 #H3 destruct
 289   lapply (isrt_inv_plus_O_dx … Hc ?) -Hc // #Hc
 290   elim (isrt_inv_plus … Hc) -Hc #n0 #nT #Hc #HcT #H destruct
 291   elim (isrt_inv_plus … Hc) -Hc #nV #nW #HcV #HcW #H destruct
 292   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 293   elim (isrt_inv_shift … HcW) -HcW #HcW #H destruct
 294   /4 width=11 by or3_intro1, ex5_6_intro, ex2_intro/
 295 | #cV #cW #cT #p #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #H1 #H2 #H3 destruct
 296   lapply (isrt_inv_plus_O_dx … Hc ?) -Hc // #Hc
 297   elim (isrt_inv_plus … Hc) -Hc #n0 #nT #Hc #HcT #H destruct
 298   elim (isrt_inv_plus … Hc) -Hc #nV #nW #HcV #HcW #H destruct
 299   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 300   elim (isrt_inv_shift … HcW) -HcW #HcW #H destruct
 301   /4 width=13 by or3_intro2, ex6_7_intro, ex2_intro/
 302 ]
 303 qed-.
 304
 305 lemma cpm_inv_cast1: ∀n,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[n, h] U2 →
 306                      ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 & ⦃G, L⦄ ⊢ U1 ➡[n,h] T2 &
 307                                  U2 = ⓝV2.T2
 308                       | ⦃G, L⦄ ⊢ U1 ➡[n, h] U2
 309                       | ∃∃m. ⦃G, L⦄ ⊢ V1 ➡[m, h] U2 & n = ⫯m.
 310 #n #h #G #L #V1 #U1 #U2 * #c #Hc #H elim (cpg_inv_cast1 … H) -H *
 311 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct
 312   elim (isrt_inv_plus … Hc) -Hc #nV #nT #HcV #HcT #H destruct
 313   elim (isrt_inv_shift … HcV) -HcV #HcV #H destruct
 314   /4 width=5 by or3_intro0, ex3_2_intro, ex2_intro/
 315 | #cU #U12 #H destruct
 316   /4 width=3 by isrt_inv_plus_O_dx, or3_intro1, ex2_intro/
 317 | #cU #H12 #H destruct
 318   elim (isrt_inv_plus_SO_dx … Hc) -Hc // #m #Hc #H destruct
 319   /4 width=3 by or3_intro2, ex2_intro/
 320 ]
 321 qed-.
 322
 323 (* Basic forward lemmas *****************************************************)
 324
 325 (* Basic_2A1: includes: cpr_fwd_bind1_minus *)
 326 lemma cpm_fwd_bind1_minus: ∀n,h,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[n, h] T → ∀p.
 327                            ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[n, h] ⓑ{p,I}V2.T2 &
 328                                     T = -ⓑ{I}V2.T2.
 329 #n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
 330 /3 width=4 by ex2_2_intro, ex2_intro/
 331 qed-.