qed-.
(* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
-lemma cpx_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
+lemma cpx_refl: ∀h,g,T,L. ⦃G, L⦄ ⊢ T ➡[h, g] T.
#h #g #T elim T -T // * /2 width=1/
qed.
-lemma cpr_cpx: ∀h,g,L,T1,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+lemma cpr_cpx: ∀h,g,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
#h #g #L #T1 #T2 #H elim H -L -T1 -T2 // /2 width=1/ /2 width=3/ /2 width=7/
qed.
-fact ssta_cpx_aux: ∀h,g,L,T1,T2,l0. ⦃h, L⦄ ⊢ T1 •[g] ⦃l0, T2⦄ →
- ∀l. l0 = l+1 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+fact ssta_cpx_aux: ∀h,g,L,T1,T2,l0. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l0, T2⦄ →
+ ∀l. l0 = l+1 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
#h #g #L #T1 #T2 #l0 #H elim H -L -T1 -T2 -l0 /2 width=2/ /2 width=7/ /3 width=2/ /3 width=7/
qed-.
-lemma ssta_cpx: ∀h,g,L,T1,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
+lemma ssta_cpx: ∀h,g,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 •[h, g] ⦃l+1, T2⦄ → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
/2 width=4 by ssta_cpx_aux/ qed.
-lemma cpx_pair_sn: ∀h,g,I,L,V1,V2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 →
- ∀T. ⦃h, L⦄ ⊢ ②{I}V1.T ➡[g] ②{I}V2.T.
+lemma cpx_pair_sn: ∀h,g,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
+ ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, g] ②{I}V2.T.
#h #g * /2 width=1/ qed.
lemma cpx_delift: ∀h,g,I,K,V,T1,L,d. ⇩[0, d] L ≡ (K.ⓑ{I}V) →
- ∃∃T2,T. ⦃h, L⦄ ⊢ T1 ➡[g] T2 & ⇧[d, 1] T ≡ T2.
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⇧[d, 1] T ≡ T2.
#h #g #I #K #V #T1 elim T1 -T1
[ * #i #L #d #HLK /2 width=4/
elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4/ ]
(* Basic inversion lemmas ***************************************************)
-fact cpx_inv_atom1_aux: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 ➡[g] T2 → ∀J. T1 = ⓪{J} →
+fact cpx_inv_atom1_aux: ∀h,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
- | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
+ | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[h, g] V2 &
⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
#h #g #L #T1 #T2 * -L -T1 -T2
[ #I #L #J #H destruct /2 width=1/
]
qed-.
-lemma cpx_inv_atom1: ∀h,g,J,L,T2. ⦃h, L⦄ ⊢ ⓪{J} ➡[g] T2 →
+lemma cpx_inv_atom1: ∀h,g,J,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
- | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
+ | ∃∃I,K,V,V2,i. ⇩[O, i] L ≡ K.ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[h, g] V2 &
⇧[O, i + 1] V2 ≡ T2 & J = LRef i.
/2 width=3 by cpx_inv_atom1_aux/ qed-.
-lemma cpx_inv_sort1: ∀h,g,L,T2,k. ⦃h, L⦄ ⊢ ⋆k ➡[g] T2 → T2 = ⋆k ∨
+lemma cpx_inv_sort1: ∀h,g,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
∃∃l. deg h g k (l+1) & T2 = ⋆(next h k).
#h #g #L #T2 #k #H
elim (cpx_inv_atom1 … H) -H /2 width=1/ *
]
qed-.
-lemma cpx_inv_lref1: ∀h,g,L,T2,i. ⦃h, L⦄ ⊢ #i ➡[g] T2 →
+lemma cpx_inv_lref1: ∀h,g,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
T2 = #i ∨
- ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[g] V2 &
+ ∃∃I,K,V,V2. ⇩[O, i] L ≡ K. ⓑ{I}V & ⦃h, K⦄ ⊢ V ➡[h, g] V2 &
⇧[O, i + 1] V2 ≡ T2.
#h #g #L #T2 #i #H
elim (cpx_inv_atom1 … H) -H /2 width=1/ *
]
qed-.
-lemma cpx_inv_gref1: ∀h,g,L,T2,p. ⦃h, L⦄ ⊢ §p ➡[g] T2 → T2 = §p.
+lemma cpx_inv_gref1: ∀h,g,L,T2,p. ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
#h #g #L #T2 #p #H
elim (cpx_inv_atom1 … H) -H // *
[ #k #l #_ #_ #H destruct
]
qed-.
-fact cpx_inv_bind1_aux: ∀h,g,L,U1,U2. ⦃h, L⦄ ⊢ U1 ➡[g] U2 →
+fact cpx_inv_bind1_aux: ∀h,g,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 → (
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓑ{J}V1⦄ ⊢ T1 ➡[g] T2 &
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,J}V2.T2
) ∨
- ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T &
+ ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T &
a = true & J = Abbr.
#h #g #L #U1 #U2 * -L -U1 -U2
[ #I #L #b #J #W #U1 #H destruct
]
qed-.
-lemma cpx_inv_bind1: ∀h,g,a,I,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[g] U2 → (
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_bind1: ∀h,g,a,I,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, g] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓑ{a,I} V2. T2
) ∨
- ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T &
+ ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T &
a = true & I = Abbr.
/2 width=3 by cpx_inv_bind1_aux/ qed-.
-lemma cpx_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓓ{a}V1.T1 ➡[g] U2 → (
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_abbr1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, g] U2 → (
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓓ{a} V2. T2
) ∨
- ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[g] T & ⇧[0, 1] U2 ≡ T & a = true.
+ ∃∃T. ⦃h, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⇧[0, 1] U2 ≡ T & a = true.
#h #g #a #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H * /3 width=3/ /3 width=5/
qed-.
-lemma cpx_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃h, L⦄ ⊢ ⓛ{a}V1.T1 ➡[g] U2 →
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_abst1: ∀h,g,a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃h, L.ⓛV1⦄ ⊢ T1 ➡[h, g] T2 &
U2 = ⓛ{a} V2. T2.
#h #g #a #L #V1 #T1 #U2 #H
elim (cpx_inv_bind1 … H) -H *
]
qed-.
-fact cpx_inv_flat1_aux: ∀h,g,L,U,U2. ⦃h, L⦄ ⊢ U ➡[g] U2 →
+fact cpx_inv_flat1_aux: ∀h,g,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
∀J,V1,U1. U = ⓕ{J}V1.U1 →
- ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
U2 = ⓕ{J}V2.T2
- | (⦃h, L⦄ ⊢ U1 ➡[g] U2 ∧ J = Cast)
- | (⦃h, L⦄ ⊢ V1 ➡[g] U2 ∧ J = Cast)
- | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
- ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
+ | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ J = Cast)
+ | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ J = Cast)
+ | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+ ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
- | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
- ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
+ | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⇧[0,1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
#h #g #L #U #U2 * -L -U -U2
]
qed-.
-lemma cpx_inv_flat1: ∀h,g,I,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓕ{I}V1.U1 ➡[g] U2 →
- ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+lemma cpx_inv_flat1: ∀h,g,I,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, g] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
U2 = ⓕ{I} V2. T2
- | (⦃h, L⦄ ⊢ U1 ➡[g] U2 ∧ I = Cast)
- | (⦃h, L⦄ ⊢ V1 ➡[g] U2 ∧ I = Cast)
- | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
- ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
+ | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ I = Cast)
+ | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ I = Cast)
+ | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+ ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 &
U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
- | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
- ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
+ | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⇧[0,1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 &
U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
/2 width=3 by cpx_inv_flat1_aux/ qed-.
-lemma cpx_inv_appl1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓐ V1.U1 ➡[g] U2 →
- ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+lemma cpx_inv_appl1: ∀h,g,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
U2 = ⓐ V2. T2
- | ∃∃a,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ W1 ➡[g] W2 &
- ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[g] T2 &
+ | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
+ ⦃h, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
- | ∃∃a,V,V2,W1,W2,T1,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V & ⇧[0,1] V ≡ V2 &
- ⦃h, L⦄ ⊢ W1 ➡[g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[g] T2 &
+ | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⇧[0,1] V ≡ V2 &
+ ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃h, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
#h #g #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
[ /3 width=5/
qed-.
(* Note: the main property of simple terms *)
-lemma cpx_inv_appl1_simple: ∀h,g,L,V1,T1,U. ⦃h, L⦄ ⊢ ⓐV1.T1 ➡[g] U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ T1 ➡[g] T2 &
+lemma cpx_inv_appl1_simple: ∀h,g,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, g] U → 𝐒⦃T1⦄ →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 &
U = ⓐV2.T2.
#h #g #L #V1 #T1 #U #H #HT1
elim (cpx_inv_appl1 … H) -H *
]
qed-.
-lemma cpx_inv_cast1: ∀h,g,L,V1,U1,U2. ⦃h, L⦄ ⊢ ⓝV1.U1 ➡[g] U2 →
- ∨∨ ∃∃V2,T2. ⦃h, L⦄ ⊢ V1 ➡[g] V2 & ⦃h, L⦄ ⊢ U1 ➡[g] T2 &
+lemma cpx_inv_cast1: ∀h,g,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, g] U2 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
U2 = ⓝ V2. T2
- | ⦃h, L⦄ ⊢ U1 ➡[g] U2
- | ⦃h, L⦄ ⊢ V1 ➡[g] U2.
+ | ⦃G, L⦄ ⊢ U1 ➡[h, g] U2
+ | ⦃G, L⦄ ⊢ V1 ➡[h, g] U2.
#h #g #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
[ /3 width=5/
|2,3: /2 width=1/
(* Basic forward lemmas *****************************************************)
-lemma cpx_fwd_bind1_minus: ∀h,g,I,L,V1,T1,T. ⦃h, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[g] T → ∀b.
- ∃∃V2,T2. ⦃h, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[g] ⓑ{b,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,g,I,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, g] T → ∀b.
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, g] ⓑ{b,I}V2.T2 &
T = -ⓑ{I}V2.T2.
#h #g #I #L #V1 #T1 #T #H #b
elim (cpx_inv_bind1 … H) -H *
]
qed-.
-lemma cpx_fwd_shift1: ∀h,g,L1,L,T1,T. ⦃h, L⦄ ⊢ L1 @@ T1 ➡[g] T →
+lemma cpx_fwd_shift1: ∀h,g,L1,L,T1,T. ⦃G, L⦄ ⊢ L1 @@ T1 ➡[h, g] T →
∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
#h #g #L1 @(lenv_ind_dx … L1) -L1 normalize
[ #L #T1 #T #HT1
projects / helm.git / blobdiff