| cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
| cpx_st : ∀G,L,k,l. deg h g k (l+1) → cpx h g G L (⋆k) (⋆(next h k))
| cpx_delta: ∀I,G,L,K,V,V2,W2,i.
| cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
| cpx_st : ∀G,L,k,l. deg h g k (l+1) → cpx h g G L (⋆k) (⋆(next h k))
| cpx_delta: ∀I,G,L,K,V,V2,W2,i.
- â\87©[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
- â\87§[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
+ â¬\87[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
+ â¬\86[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
| cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
| cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
cpx h g G L V1 V2 → cpx h g G L T1 T2 →
cpx h g G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h g G (L.ⓓV) T1 T →
cpx h g G L V1 V2 → cpx h g G L T1 T2 →
cpx h g G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h g G (L.ⓓV) T1 T →
| cpx_eps : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
| cpx_ct : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
| cpx_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
cpx h g G L V1 V2 → cpx h g G L W1 W2 → cpx h g G (L.ⓛW1) T1 T2 →
cpx h g G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
| cpx_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
| cpx_eps : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
| cpx_ct : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
| cpx_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
cpx h g G L V1 V2 → cpx h g G L W1 W2 → cpx h g G (L.ⓛW1) T1 T2 →
cpx h g G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
| cpx_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
-lemma cpx_delift: â\88\80h,g,I,G,K,V,T1,L,d. â\87©[d] L ≡ (K.ⓑ{I}V) →
- â\88\83â\88\83T2,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, g] T2 & â\87§[d, 1] T ≡ T2.
+lemma cpx_delift: â\88\80h,g,I,G,K,V,T1,L,d. â¬\87[d] L ≡ (K.ⓑ{I}V) →
+ â\88\83â\88\83T2,T. â¦\83G, Lâ¦\84 â\8a¢ T1 â\9e¡[h, g] T2 & â¬\86[d, 1] T ≡ T2.
#h #g #I #G #K #V #T1 elim T1 -T1
[ * #i #L #d /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
#h #g #I #G #K #V #T1 elim T1 -T1
[ * #i #L #d /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
∨∨ T2 = ⓪{J}
| ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
∨∨ T2 = ⓪{J}
| ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
- | â\88\83â\88\83I,K,V,V2,i. â\87©[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
- â\87§[O, i+1] V2 ≡ T2 & J = LRef i.
+ | â\88\83â\88\83I,K,V,V2,i. â¬\87[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ â¬\86[O, i+1] V2 ≡ T2 & J = LRef i.
#G #h #g #L #T1 #T2 * -L -T1 -T2
[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
| #G #L #k #l #Hkl #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
#G #h #g #L #T1 #T2 * -L -T1 -T2
[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
| #G #L #k #l #Hkl #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
∨∨ T2 = ⓪{J}
| ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
∨∨ T2 = ⓪{J}
| ∃∃k,l. deg h g k (l+1) & T2 = ⋆(next h k) & J = Sort k
- | â\88\83â\88\83I,K,V,V2,i. â\87©[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
- â\87§[O, i+1] V2 ≡ T2 & J = LRef i.
+ | â\88\83â\88\83I,K,V,V2,i. â¬\87[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ â¬\86[O, i+1] V2 ≡ T2 & J = LRef i.
/2 width=3 by cpx_inv_atom1_aux/ qed-.
lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
/2 width=3 by cpx_inv_atom1_aux/ qed-.
lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
T2 = #i ∨
lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
T2 = #i ∨
- â\88\83â\88\83I,K,V,V2. â\87©[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
- â\87§[O, i+1] V2 ≡ T2.
+ â\88\83â\88\83I,K,V,V2. â¬\87[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+ â¬\86[O, i+1] V2 ≡ T2.
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,J}V2.T2
) ∨
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,J}V2.T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â\87§[0, 1] U2 ≡ T &
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â¬\86[0, 1] U2 ≡ T &
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,I} V2. T2
) ∨
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,I} V2. T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â\87§[0, 1] U2 ≡ T &
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â¬\86[0, 1] U2 ≡ T &
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓓ{a} V2. T2
) ∨
∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓓ{a} V2. T2
) ∨
- â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â\87§[0, 1] U2 ≡ T & a = true.
+ â\88\83â\88\83T. â¦\83G, L.â\93\93V1â¦\84 â\8a¢ T1 â\9e¡[h, g] T & â¬\86[0, 1] U2 ≡ T & a = true.
#h #g #a #G #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
qed-.
#h #g #a #G #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
qed-.
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
- | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â\87§[0,1] V ≡ V2 &
+ | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â¬\86[0,1] V ≡ V2 &
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
- | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â\87§[0,1] V ≡ V2 &
+ | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â¬\86[0,1] V ≡ V2 &
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
| ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
| ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
- | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â\87§[0,1] V ≡ V2 &
+ | â\88\83â\88\83a,V,V2,W1,W2,T1,T2. â¦\83G, Lâ¦\84 â\8a¢ V1 â\9e¡[h, g] V & â¬\86[0,1] V ≡ V2 &
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *