+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-notation "hvbox( T1 ➡ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PRed $T1 $T2 }.
-
-include "basic_2/substitution/tps.ma".
-
-(* CONTEXT-FREE PARALLEL REDUCTION ON TERMS *********************************)
-
-(* Basic_1: includes: pr0_delta1 *)
-inductive tpr: relation term ≝
-| tpr_atom : ∀I. tpr (⓪{I}) (⓪{I})
-| tpr_flat : ∀I,V1,V2,T1,T2. tpr V1 V2 → tpr T1 T2 →
- tpr (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-| tpr_beta : ∀a,V1,V2,W,T1,T2.
- tpr V1 V2 → tpr T1 T2 → tpr (ⓐV1. ⓛ{a}W. T1) (ⓓ{a}V2. T2)
-| tpr_delta: ∀a,I,V1,V2,T1,T,T2.
- tpr V1 V2 → tpr T1 T → ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 →
- tpr (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| tpr_theta: ∀a,V,V1,V2,W1,W2,T1,T2.
- tpr V1 V2 → ⇧[0,1] V2 ≡ V → tpr W1 W2 → tpr T1 T2 →
- tpr (ⓐV1. ⓓ{a}W1. T1) (ⓓ{a}W2. ⓐV. T2)
-| tpr_zeta : ∀V,T1,T,T2. tpr T1 T → ⇧[0, 1] T2 ≡ T → tpr (+ⓓV. T1) T2
-| tpr_tau : ∀V,T1,T2. tpr T1 T2 → tpr (ⓝV. T1) T2
-.
-
-interpretation
- "context-free parallel reduction (term)"
- 'PRed T1 T2 = (tpr T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma tpr_bind: ∀a,I,V1,V2,T1,T2. V1 ➡ V2 → T1 ➡ T2 → ⓑ{a,I} V1. T1 ➡ ⓑ{a,I} V2. T2.
-/2 width=3/ qed.
-
-(* Basic_1: was by definition: pr0_refl *)
-lemma tpr_refl: reflexive … tpr.
-#T elim T -T //
-#I elim I -I /2 width=1/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tpr_inv_atom1_aux: ∀U1,U2. U1 ➡ U2 → ∀I. U1 = ⓪{I} → U2 = ⓪{I}.
-#U1 #U2 * -U1 -U2
-[ //
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #k #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #k #H destruct
-| #V #T1 #T #T2 #_ #_ #k #H destruct
-| #V #T1 #T2 #_ #k #H destruct
-]
-qed.
-
-(* Basic_1: was: pr0_gen_sort pr0_gen_lref *)
-lemma tpr_inv_atom1: ∀I,U2. ⓪{I} ➡ U2 → U2 = ⓪{I}.
-/2 width=3/ qed-.
-
-fact tpr_inv_bind1_aux: ∀U1,U2. U1 ➡ U2 → ∀a,I,V1,T1. U1 = ⓑ{a,I} V1. T1 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true & I = Abbr.
-#U1 #U2 * -U1 -U2
-[ #J #a #I #V #T #H destruct
-| #I1 #V1 #V2 #T1 #T2 #_ #_ #a #I #V #T #H destruct
-| #b #V1 #V2 #W #T1 #T2 #_ #_ #a #I #V #T #H destruct
-| #b #I1 #V1 #V2 #T1 #T #T2 #HV12 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=7/
-| #b #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #a #I0 #V0 #T0 #H destruct
-| #V #T1 #T #T2 #HT1 #HT2 #a #I0 #V0 #T0 #H destruct /3 width=3/
-| #V #T1 #T2 #_ #a #I0 #V0 #T0 #H destruct
-]
-qed.
-
-lemma tpr_inv_bind1: ∀V1,T1,U2,a,I. ⓑ{a,I} V1. T1 ➡ U2 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓑ{I} V2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓑ{a,I} V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0,1] U2 ≡ T & a = true & I = Abbr.
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_abbr *)
-lemma tpr_inv_abbr1: ∀a,V1,T1,U2. ⓓ{a}V1. T1 ➡ U2 →
- (∃∃V2,T,T2. V1 ➡ V2 & T1 ➡ T &
- ⋆. ⓓV2 ⊢ T ▶ [0, 1] T2 &
- U2 = ⓓ{a}V2. T2
- ) ∨
- ∃∃T. T1 ➡ T & ⇧[0, 1] U2 ≡ T & a = true.
-#a #V1 #T1 #U2 #H
-elim (tpr_inv_bind1 … H) -H * /3 width=7/
-qed-.
-
-fact tpr_inv_flat1_aux: ∀U1,U2. U1 ➡ U2 → ∀I,V1,U0. U1 = ⓕ{I} V1. U0 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-#U1 #U2 * -U1 -U2
-[ #I #J #V #T #H destruct
-| #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=5/
-| #a #V1 #V2 #W #T1 #T2 #HV12 #HT12 #J #V #T #H destruct /3 width=9/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #J #V0 #T0 #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HV2 #HW12 #HT12 #J #V0 #T0 #H destruct /3 width=13/
-| #V #T1 #T #T2 #_ #_ #J #V0 #T0 #H destruct
-| #V #T1 #T2 #HT12 #J #V0 #T0 #H destruct /3 width=1/
-]
-qed.
-
-lemma tpr_inv_flat1: ∀V1,U0,U2,I. ⓕ{I} V1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓕ{I} V2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2 & I = Appl
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2 &
- I = Appl
- | (U0 ➡ U2 ∧ I = Cast).
-/2 width=3/ qed-.
-
-(* Basic_1: was pr0_gen_appl *)
-lemma tpr_inv_appl1: ∀V1,U0,U2. ⓐV1. U0 ➡ U2 →
- ∨∨ ∃∃V2,T2. V1 ➡ V2 & U0 ➡ T2 &
- U2 = ⓐV2. T2
- | ∃∃a,V2,W,T1,T2. V1 ➡ V2 & T1 ➡ T2 &
- U0 = ⓛ{a}W. T1 &
- U2 = ⓓ{a}V2. T2
- | ∃∃a,V2,V,W1,W2,T1,T2. V1 ➡ V2 & W1 ➡ W2 & T1 ➡ T2 &
- ⇧[0,1] V2 ≡ V &
- U0 = ⓓ{a}W1. T1 &
- U2 = ⓓ{a}W2. ⓐV. T2.
-#V1 #U0 #U2 #H
-elim (tpr_inv_flat1 … H) -H *
-/3 width=5/ /3 width=9/ /3 width=13/
-#_ #H destruct
-qed-.
-
-(* Note: the main property of simple terms *)
-lemma tpr_inv_appl1_simple: ∀V1,T1,U. ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
- ∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 &
- U = ⓐV2. T2.
-#V1 #T1 #U #H #HT1
-elim (tpr_inv_appl1 … H) -H *
-[ /2 width=5/
-| #a #V2 #W #W1 #W2 #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-| #a #V2 #V #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
- elim (simple_inv_bind … HT1)
-]
-qed-.
-
-(* Basic_1: was: pr0_gen_cast *)
-lemma tpr_inv_cast1: ∀V1,T1,U2. ⓝV1. T1 ➡ U2 →
- (∃∃V2,T2. V1 ➡ V2 & T1 ➡ T2 & U2 = ⓝV2. T2)
- ∨ T1 ➡ U2.
-#V1 #T1 #U2 #H
-elim (tpr_inv_flat1 … H) -H * /3 width=5/ #a #V2 #W #W1 #W2
-[ #_ #_ #_ #_ #H destruct
-| #T2 #U1 #_ #_ #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-fact tpr_inv_lref2_aux: ∀T1,T2. T1 ➡ T2 → ∀i. T2 = #i →
- ∨∨ T1 = #i
- | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
-#T1 #T2 * -T1 -T2
-[ #I #i #H destruct /2 width=1/
-| #I #V1 #V2 #T1 #T2 #_ #_ #i #H destruct
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #i #H destruct
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #_ #i #H destruct
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #i #H destruct
-| #V #T1 #T #T2 #HT1 #HT2 #i #H destruct
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2 // #H destruct /3 width=4/
-| #V #T1 #T2 #HT12 #i #H destruct /3 width=4/
-]
-qed.
-
-lemma tpr_inv_lref2: ∀T1,i. T1 ➡ #i →
- ∨∨ T1 = #i
- | ∃∃V,T. T ➡ #(i+1) & T1 = +ⓓV. T
- | ∃∃V,T. T ➡ #i & T1 = ⓝV. T.
-/2 width=3/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tpr_fwd_bind1_minus: ∀I,V1,T1,T. -ⓑ{I}V1.T1 ➡ T → ∀b.
- ∃∃V2,T2. ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
- T = -ⓑ{I}V2.T2.
-#I #V1 #T1 #T #H #b elim (tpr_inv_bind1 … H) -H *
-[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct /3 width=4/
-| #T2 #_ #_ #H destruct
-]
-qed-.
-
-lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
- ∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
-#L1 @(lenv_ind_dx … L1) -L1 normalize
-[ #T1 #T #HT1
- @(ex2_2_intro … (⋆)) // (**) (* explicit constructor *)
-| #I #L1 #V1 #IH #T1 #X
- >shift_append_assoc normalize #H
- elim (tpr_inv_bind1 … H) -H *
- [ #V0 #T0 #X0 #_ #HT10 #H0 #H destruct
- elim (IH … HT10) -IH -T1 #L #T #HL1 #H destruct
- elim (tps_fwd_shift1 … H0) -T #L2 #T2 #HL2 #H destruct
- >append_length >HL1 >HL2 -L1 -L
- @(ex2_2_intro … (⋆.ⓑ{I}V0@@L2) T2) [ >append_length ] // /2 width=3/ (**) (* explicit constructor *)
- | #T #_ #_ #H destruct
- ]
-]
-qed-.
-
-(* Basic_1: removed theorems 3:
- pr0_subst0_back pr0_subst0_fwd pr0_subst0
-*)
-(* Basic_1: removed local theorems: 1: pr0_delta_tau *)
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