--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "lambdaN/terms.ma".
+
+(* arguments: k is the depth (starts from 0), p is the height (starts from 0) *)
+let rec lift t k p ≝
+ match t with
+ [ Sort n ⇒ Sort n
+ | Rel n ⇒ if_then_else T (leb k n) (Rel (n+p)) (Rel n)
+ | App m n ⇒ App (lift m k p) (lift n k p)
+ | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
+ | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
+ | D n m ⇒ D (lift n k p) (lift m k p)
+ ].
+
+(*
+ndefinition lift ≝ λt.λp.lift_aux t 0 p.
+
+notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift O $M}.
+notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}.
+*)
+(* interpretation "Lift" 'Lift n M = (lift M n). *)
+interpretation "Lift" 'Lift n k M = (lift M k n).
+
+let rec subst t k a ≝
+ match t with
+ [ Sort n ⇒ Sort n
+ | Rel n ⇒ if_then_else T (leb k n)
+ (if_then_else T (eqb k n) (lift a 0 n) (Rel (n-1))) (Rel n)
+ | App m n ⇒ App (subst m k a) (subst n k a)
+ | Lambda m n ⇒ Lambda (subst m k a) (subst n (k+1) a)
+ | Prod m n ⇒ Prod (subst m k a) (subst n (k+1) a)
+ | D n m ⇒ D (subst n k a) (subst m k a)
+ ].
+
+(* meglio non definire
+ndefinition subst ≝ λa.λt.subst_aux t 0 a.
+notation "M [ N ]" non associative with precedence 90 for @{'Subst $N $M}.
+*)
+
+(* interpretation "Subst" 'Subst N M = (subst N M). *)
+interpretation "Subst" 'Subst1 M k N = (subst M k N).
+
+(*** properties of lift and subst ***)
+
+lemma lift_0: ∀t:T.∀k. lift t k 0 = t.
+#t (elim t) normalize // #n #k cases (leb k n) normalize //
+qed.
+
+(* nlemma lift_0: ∀t:T. lift t 0 = t.
+#t; nelim t; nnormalize; //; nqed. *)
+
+lemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
+// qed.
+
+lemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
+// qed.
+
+lemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
+#i (change with (lift (Rel i) 0 1 = Rel (1 + i))) //
+qed.
+
+lemma lift_rel_lt : ∀n,k,i. i < k → lift (Rel i) k n = Rel i.
+#n #k #i #ltik change with
+(if_then_else ? (leb k i) (Rel (i+n)) (Rel i) = Rel i)
+>(lt_to_leb_false … ltik) //
+qed.
+
+lemma lift_rel_ge : ∀n,k,i. k ≤ i → lift (Rel i) k n = Rel (i+n).
+#n #k #i #leki change with
+(if_then_else ? (leb k i) (Rel (i+n)) (Rel i) = Rel (i+n))
+>le_to_leb_true //
+qed.
+
+lemma lift_lift: ∀t.∀m,j.j ≤ m → ∀n,k.
+ lift (lift t k m) (j+k) n = lift t k (m+n).
+#t #i #j #h (elim t) normalize // #n #h #k
+@(leb_elim k n) #Hnk normalize
+ [>(le_to_leb_true (j+k) (n+i) ?)
+ normalize // >(commutative_plus j k) @le_plus //
+ |>(lt_to_leb_false (j+k) n ?) normalize //
+ @(transitive_le ? k) // @not_le_to_lt //
+ ]
+qed.
+
+lemma lift_lift_up: ∀n,m,t,k,i.
+ lift (lift t i m) (m+k+i) n = lift (lift t (k+i) n) i m.
+#n #m #N (elim N)
+ [1,3,4,5,6: normalize //
+ |#p #k #i @(leb_elim i p);
+ [#leip >lift_rel_ge // @(leb_elim (k+i) p);
+ [#lekip >lift_rel_ge;
+ [>lift_rel_ge // >lift_rel_ge // @(transitive_le … leip) //
+ |>associative_plus >commutative_plus @monotonic_le_plus_l //
+ ]
+ |#lefalse (cut (p < k+i)) [@not_le_to_lt //] #ltpki
+ >lift_rel_lt; [|>associative_plus >commutative_plus @monotonic_lt_plus_r //]
+ >lift_rel_lt // >lift_rel_ge //
+ ]
+ |#lefalse (cut (p < i)) [@not_le_to_lt //] #ltpi
+ >lift_rel_lt // >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
+ >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
+ >lift_rel_lt //
+ ]
+ ]
+qed.
+
+lemma lift_lift1: ∀t.∀i,j,k.
+ lift(lift t k j) k i = lift t k (j+i).
+/2/ qed.
+
+lemma lift_lift2: ∀t.∀i,j,k.
+ lift (lift t k j) (j+k) i = lift t k (j+i).
+/2/ qed.
+
+(*
+nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
+nnormalize; //; nqed. *)
+
+lemma subst_lift_k: ∀A,B.∀k. (lift B k 1)[k ≝ A] = B.
+#A #B (elim B) normalize /2/ #n #k
+@(leb_elim k n) normalize #Hnk
+ [cut (k ≤ n+1) [@transitive_le //] #H
+ >(le_to_leb_true … H) normalize
+ >(not_eq_to_eqb_false k (n+1)) normalize /2/
+ |>(lt_to_leb_false … (not_le_to_lt … Hnk)) normalize //
+ ]
+qed.
+
+(*
+nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
+nnormalize; //; nqed. *)
+
+lemma subst_sort: ∀A.∀n,k.(Sort n) [k ≝ A] = Sort n.
+// qed.
+
+lemma subst_rel: ∀A.(Rel 0) [0 ≝ A] = A.
+normalize // qed.
+
+lemma subst_rel1: ∀A.∀k,i. i < k →
+ (Rel i) [k ≝ A] = Rel i.
+#A #k #i normalize #ltik >(lt_to_leb_false … ltik) //
+qed.
+
+lemma subst_rel2: ∀A.∀k.
+ (Rel k) [k ≝ A] = lift A 0 k.
+#A #k normalize >(le_to_leb_true k k) // >(eq_to_eqb_true … (refl …)) //
+qed.
+
+lemma subst_rel3: ∀A.∀k,i. k < i →
+ (Rel i) [k ≝ A] = Rel (i-1).
+#A #k #i normalize #ltik >(le_to_leb_true k i) /2/
+>(not_eq_to_eqb_false k i) // @lt_to_not_eq //
+qed.
+
+lemma lift_subst_ijk: ∀A,B.∀i,j,k.
+ lift (B [j+k := A]) k i = (lift B k i) [j+k+i ≝ A].
+#A #B #i #j (elim B) normalize /2/ #n #k
+@(leb_elim (j+k) n) normalize #Hnjk
+ [@(eqb_elim (j+k) n) normalize #Heqnjk
+ [>(le_to_leb_true k n) //
+ (cut (j+k+i = n+i)) [//] #Heq
+ >Heq >(subst_rel2 A ?) (applyS lift_lift) //
+ |(cut (j + k < n))
+ [@not_eq_to_le_to_lt; /2/] #ltjkn
+ (cut (O < n)) [/2/] #posn (cut (k ≤ n)) [/2/] #lekn
+ >(le_to_leb_true k (n-1)) normalize
+ [>(le_to_leb_true … lekn)
+ >(subst_rel3 A (j+k+i) (n+i)); [/3/ |/2/]
+ |(applyS monotonic_pred) @le_plus_b //
+ ]
+ ]
+ |(elim (leb k n))
+ [>(subst_rel1 A (j+k+i) (n+i)) // @monotonic_lt_plus_l /2/
+ |>(subst_rel1 A (j+k+i) n) // @(lt_to_le_to_lt ? (j+k)) /2/
+ ]
+ ]
+qed.
+
+lemma lift_subst_up: ∀M,N,n,i,j.
+ lift M[i≝N] (i+j) n = (lift M (i+j+1) n)[i≝ (lift N j n)].
+#M (elim M)
+ [//
+ |#p #N #n #i #j (cases (true_or_false (leb p i)))
+ [#lepi (cases (le_to_or_lt_eq … (leb_true_to_le … lepi)))
+ [#ltpi >(subst_rel1 … ltpi)
+ (cut (p < i+j)) [@(lt_to_le_to_lt … ltpi) //] #ltpij
+ >(lift_rel_lt … ltpij); >(lift_rel_lt ?? p ?);
+ [>subst_rel1 // | @(lt_to_le_to_lt … ltpij) //]
+ |#eqpi >eqpi >subst_rel2 >lift_rel_lt;
+ [>subst_rel2 >(plus_n_O (i+j))
+ applyS lift_lift_up
+ |@(le_to_lt_to_lt ? (i+j)) //
+ ]
+ ]
+ |#lefalse (cut (i < p)) [@not_le_to_lt /2/] #ltip
+ (cut (0 < p)) [@(le_to_lt_to_lt … ltip) //] #posp
+ >(subst_rel3 … ltip) (cases (true_or_false (leb (S p) (i+j+1))))
+ [#Htrue (cut (p < i+j+1)) [@(leb_true_to_le … Htrue)] #Hlt
+ >lift_rel_lt;
+ [>lift_rel_lt // >(subst_rel3 … ltip) // | @lt_plus_to_minus //]
+ |#Hfalse >lift_rel_ge;
+ [>lift_rel_ge;
+ [>subst_rel3; [@eq_f /2/ | @(lt_to_le_to_lt … ltip) //]
+ |@not_lt_to_le @(leb_false_to_not_le … Hfalse)
+ ]
+ |@le_plus_to_minus_r @not_lt_to_le
+ @(leb_false_to_not_le … Hfalse)
+ ]
+ ]
+ ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |@HindQ ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
+ <associative_plus @HindQ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
+ <associative_plus @HindQ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |@HindQ ]
+ ]
+qed.
+
+theorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
+ (lift B i (S k)) [j ≝ A] = lift B i k.
+#A #B (elim B) normalize /2/
+ [2,3,4,5: #T #T0 #Hind1 #Hind2 #i #j #k #leij #lejk
+ @eq_f2 /2/ @Hind2 (applyS (monotonic_le_plus_l 1)) //
+ |#n #i #j #k #leij #ltjk @(leb_elim i n) normalize #len
+ [cut (j < n + S k)
+ [<plus_n_Sm @le_S_S @(transitive_le … ltjk) /2/] #H
+ >(le_to_leb_true j (n+S k));
+ [normalize >(not_eq_to_eqb_false j (n+S k)) normalize /2/
+ |/2/
+ ]
+ |>(lt_to_leb_false j n) // @(lt_to_le_to_lt … leij)
+ @not_le_to_lt //
+ ]
+ ]
+qed.
+
+(********************* substitution lemma ***********************)
+
+lemma subst_lemma: ∀A,B,C.∀k,i.
+ (A [i ≝ B]) [k+i ≝ C] =
+ (A [(k+i)+1:= C]) [i ≝ B [k ≝ C]].
+#A #B #C #k (elim A) normalize // (* WOW *)
+#n #i @(leb_elim i n) #Hle
+ [@(eqb_elim i n) #eqni
+ [<eqni >(lt_to_leb_false (k+i+1) i) // >(subst_rel2 …);
+ normalize @sym_eq (applyS (lift_subst_ijk C B i k O))
+ |(cut (i < n))
+ [cases (le_to_or_lt_eq … Hle) // #eqin @False_ind /2/] #ltin
+ (cut (O < n)) [@(le_to_lt_to_lt … ltin) //] #posn
+ normalize @(leb_elim (k+i) (n-1)) #nk
+ [@(eqb_elim (k+i) (n-1)) #H normalize
+ [cut (k+i+1 = n); [/2/] #H1
+ >(le_to_leb_true (k+i+1) n) /2/
+ >(eq_to_eqb_true … H1) normalize
+ (generalize in match ltin)
+ @(lt_O_n_elim … posn) #m #leim >delift // /2/
+ |(cut (k+i < n-1)) [@not_eq_to_le_to_lt; //] #Hlt
+ >(le_to_leb_true (k+i+1) n);
+ [>(not_eq_to_eqb_false (k+i+1) n);
+ [>(subst_rel3 ? i (n-1));
+ // @(le_to_lt_to_lt … Hlt) //
+ |@(not_to_not … H) #Hn /2/
+ ]
+ |@le_minus_to_plus_r //
+ ]
+ ]
+ |>(not_le_to_leb_false (k+i+1) n);
+ [>(subst_rel3 ? i n) normalize //
+ |@(not_to_not … nk) #H @le_plus_to_minus_r //
+ ]
+ ]
+ ]
+ |(cut (n < k+i)) [@(lt_to_le_to_lt ? i) /2/] #ltn (* lento *)
+ (* (cut (n ≤ k+i)) [/2/] #len *)
+ >(subst_rel1 C (k+i) n ltn) >(lt_to_leb_false (k+i+1) n);
+ [>subst_rel1 /2/ | @(transitive_lt …ltn) // ]
+ ]
+qed.
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