(* *)
(**************************************************************************)
-include "ground_2/steps/rtc_shift.ma".
include "ground_2/steps/rtc_plus.ma".
include "basic_2/notation/relations/predty_6.ma".
include "basic_2/grammar/lenv.ma".
| cpg_delta: ∀c,G,L,V1,V2,W2. cpg h c G L V1 V2 →
⬆*[1] V2 ≡ W2 → cpg h c G (L.ⓓV1) (#0) W2
| cpg_ell : ∀c,G,L,V1,V2,W2. cpg h c G L V1 V2 →
- ⬆*[1] V2 ≡ W2 → cpg h ((↓c)+𝟘𝟙) G (L.ⓛV1) (#0) W2
+ ⬆*[1] V2 ≡ W2 → cpg h (c+𝟘𝟙) G (L.ⓛV1) (#0) W2
| cpg_lref : ∀c,I,G,L,V,T,U,i. cpg h c G L (#i) T →
⬆*[1] T ≡ U → cpg h c G (L.ⓑ{I}V) (#⫯i) U
| cpg_bind : ∀cV,cT,p,I,G,L,V1,V2,T1,T2.
cpg h cV G L V1 V2 → cpg h cT G L T1 T2 →
cpg h ((↓cV)+cT) G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
| cpg_zeta : ∀c,G,L,V,T1,T,T2. cpg h c G (L.ⓓV) T1 T →
- ⬆*[1] T2 ≡ T → cpg h ((↓c)+𝟙𝟘) G L (+ⓓV.T1) T2
-| cpg_eps : ∀c,G,L,V,T1,T2. cpg h c G L T1 T2 → cpg h ((↓c)+𝟙𝟘) G L (ⓝV.T1) T2
-| cpg_ee : ∀c,G,L,V1,V2,T. cpg h c G L V1 V2 → cpg h ((↓c)+𝟘𝟙) G L (ⓝV1.T) V2
+ ⬆*[1] T2 ≡ T → cpg h (c+𝟙𝟘) G L (+ⓓV.T1) T2
+| cpg_eps : ∀c,G,L,V,T1,T2. cpg h c G L T1 T2 → cpg h (c+𝟙𝟘) G L (ⓝV.T1) T2
+| cpg_ee : ∀c,G,L,V1,V2,T. cpg h c G L V1 V2 → cpg h (c+𝟘𝟙) G L (ⓝV1.T) V2
| cpg_beta : ∀cV,cW,cT,p,G,L,V1,V2,W1,W2,T1,T2.
cpg h cV G L V1 V2 → cpg h cW G L W1 W2 → cpg h cT G (L.ⓛW1) T1 T2 →
- cpg h ((↓cV)+(↓cW)+(↓cT)+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+ cpg h ((↓cV)+(↓cW)+cT+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
| cpg_theta: ∀cV,cW,cT,p,G,L,V1,V,V2,W1,W2,T1,T2.
cpg h cV G L V1 V → ⬆*[1] V ≡ V2 → cpg h cW G L W1 W2 →
cpg h cT G (L.ⓓW1) T1 T2 →
- cpg h ((↓cV)+(↓cW)+(↓cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ cpg h ((↓cV)+(↓cW)+cT+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
.
interpretation
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
- L = K.ⓛV1 & J = LRef 0 & c = (↓cV)+𝟘𝟙
+ L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
| ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[c, h] T & ⬆*[1] T ≡ T2 &
L = K.ⓑ{I}V & J = LRef (⫯i).
#c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓓV1 & J = LRef 0 & c = cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
- L = K.ⓛV1 & J = LRef 0 & c = (↓cV)+𝟘𝟙
+ L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
| ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈[c, h] T & ⬆*[1] T ≡ T2 &
L = K.ⓑ{I}V & J = LRef (⫯i).
/2 width=3 by cpg_inv_atom1_aux/ qed-.
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
L = K.ⓓV1 & c = cV
| ∃∃cV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈[cV, h] V2 & ⬆*[1] V2 ≡ T2 &
- L = K.ⓛV1 & c = (↓cV)+𝟘𝟙.
+ L = K.ⓛV1 & c = cV+𝟘𝟙.
#c #h #G #L #T2 #H
elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
[ #s #H destruct
U2 = ⓑ{p,J}V2.T2 & c = (↓cV)+cT
) ∨
∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ U1 ⬈[cT, h] T & ⬆*[1] U2 ≡ T &
- p = true & J = Abbr & c = (↓cT)+𝟙𝟘.
+ p = true & J = Abbr & c = cT+𝟙𝟘.
#c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #q #J #W #U1 #H destruct
| #G #L #s #q #J #W #U1 #H destruct
U2 = ⓑ{p,I}V2.T2 & c = (↓cV)+cT
) ∨
∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[cT, h] T & ⬆*[1] U2 ≡ T &
- p = true & I = Abbr & c = (↓cT)+𝟙𝟘.
+ p = true & I = Abbr & c = cT+𝟙𝟘.
/2 width=3 by cpg_inv_bind1_aux/ qed-.
lemma cpg_inv_abbr1: ∀c,h,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ⬈[c, h] U2 → (
U2 = ⓓ{p}V2.T2 & c = (↓cV)+cT
) ∨
∃∃cT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ⬈[cT, h] T & ⬆*[1] U2 ≡ T &
- p = true & c = (↓cT)+𝟙𝟘.
+ p = true & c = cT+𝟙𝟘.
#c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
/3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/
qed-.
∀J,V1,U1. U = ⓕ{J}V1.U1 →
∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
U2 = ⓕ{J}V2.T2 & c = (↓cV)+cT
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & J = Cast & c = (↓cT)+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & J = Cast & c = (↓cV)+𝟘𝟙
+ | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & J = Cast & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & J = Cast & c = cV+𝟘𝟙
| ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[cT, h] T2 &
- J = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘
+ J = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+cT+𝟙𝟘
| ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[cT, h] T2 &
- J = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘.
+ J = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+cT+𝟙𝟘.
#c #h #G #L #U #U2 * -c -G -L -U -U2
[ #I #G #L #J #W #U1 #H destruct
| #G #L #s #J #W #U1 #H destruct
lemma cpg_inv_flat1: ∀c,h,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ⬈[c, h] U2 →
∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
U2 = ⓕ{I}V2.T2 & c = (↓cV)+cT
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & I = Cast & c = (↓cT)+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & I = Cast & c = (↓cV)+𝟘𝟙
+ | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & I = Cast & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & I = Cast & c = cV+𝟘𝟙
| ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[cT, h] T2 &
- I = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘
+ I = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+cT+𝟙𝟘
| ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[cT, h] T2 &
- I = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘.
+ I = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+cT+𝟙𝟘.
/2 width=3 by cpg_inv_flat1_aux/ qed-.
lemma cpg_inv_appl1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ⬈[c, h] U2 →
∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
U2 = ⓐV2.T2 & c = (↓cV)+cT
| ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬈[cT, h] T2 &
- U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘
+ U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & c = (↓cV)+(↓cW)+cT+𝟙𝟘
| ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ⬈[cW, h] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ⬈[cT, h] T2 &
- U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+(↓cT)+𝟙𝟘.
+ U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = (↓cV)+(↓cW)+cT+𝟙𝟘.
#c #h #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H *
[ /3 width=8 by or3_intro0, ex4_4_intro/
|2,3: #c #_ #H destruct
lemma cpg_inv_cast1: ∀c,h,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ⬈[c, h] U2 →
∨∨ ∃∃cV,cT,V2,T2. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] V2 & ⦃G, L⦄ ⊢ U1 ⬈[cT, h] T2 &
U2 = ⓝV2.T2 & c = (↓cV)+cT
- | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & c = (↓cT)+𝟙𝟘
- | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & c = (↓cV)+𝟘𝟙.
+ | ∃∃cT. ⦃G, L⦄ ⊢ U1 ⬈[cT, h] U2 & c = cT+𝟙𝟘
+ | ∃∃cV. ⦃G, L⦄ ⊢ V1 ⬈[cV, h] U2 & c = cV+𝟘𝟙.
#c #h #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H *
[ /3 width=8 by or3_intro0, ex4_4_intro/
|2,3: /3 width=3 by or3_intro1, or3_intro2, ex2_intro/