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(*       ___                                                              *)
(*      ||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                  *)
(*                                                                        *)
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include "basic_2/reduction/lpx_lpx.ma".
include "basic_2/reduction/fpn.ma".

(* REDUCTION FOR "BIG TREE" NORMAL FORMS ************************************)

(* Advanced properties ******************************************************)

lemma fpn_fqu_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
                     ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
                     ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqu_trans … H12 … HKL1 HV1) -L1
/3 width=5 by fpn_intro, ex2_3_intro/
qed-.

lemma fpn_fquq_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
                      ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃⸮ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fquq_trans … H12 … HKL1 HV1) -L1
/3 width=5 by fpn_intro, ex2_3_intro/
qed-.

lemma fpn_fqup_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
                      ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃+ ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqup_trans … H12 … HKL1 HV1) -L1
/3 width=5 by fpn_intro, ex2_3_intro/
qed-.

lemma fpn_fqus_trans: ∀h,g,F1,G1,K1,L1,V1,T1. ⦃F1, K1, V1⦄ ⊢ ⋕➡[h, g] ⦃G1, L1, T1⦄ →
                      ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
                      ∃∃F2,K2,V2. ⦃F1, K1, V1⦄ ⊃* ⦃F2, K2, V2⦄ & ⦃F2, K2, V2⦄ ⊢ ⋕➡[h, g] ⦃G2, L2, T2⦄.
#h #g #F1 #G1 #K1 #L1 #V1 #T1 * -G1 -L1 -T1
#L1 #HKL1 #HV1 #G2 #L2 #T2 #H12 elim (lpx_lleq_fqus_trans … H12 … HKL1 HV1) -L1
/3 width=5 by fpn_intro, ex2_3_intro/
qed-.