matita/matita/contribs/lambdadelta/basic_2A/multiple/llor_alt.ma

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  14
  15 include "basic_2A/multiple/frees_append.ma".
  16 include "basic_2A/multiple/llor.ma".
  17
  18 (* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
  19
  20 (* Alternative definition ***************************************************)
  21
  22 lemma llor_tail_frees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L → l ≤ yinj (|L1|) →
  23                        ∀I1,W1. ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ →
  24                        ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I2}W2.L.
  25 #L1 #L2 #L #U #l * #HL12 #HL1 #IH #Hl #I1 #W1 #HU #I2 #W2
  26 @and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
  27 #J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
  28 lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H
  29 lapply (lt_plus_SO_to_le … H) -H #H
  30 elim (le_to_or_lt_eq … H) -H #H
  31 [ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1
  32   elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2
  33   elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY
  34   lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct
  35   lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct
  36   lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct
  37   elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY *
  38   [ /3 width=1 by and3_intro, or3_intro0/
  39   | /6 width=2 by frees_inv_append, lt_to_le, or3_intro1, and3_intro/
  40   | /5 width=1 by frees_append, lt_to_le, or3_intro2, and4_intro/
  41   ]
  42 | -IH -HLK1 destruct
  43   lapply (drop_O1_inv_append1_le … HLK2 … (⋆) ?) // -HLK2 normalize #H destruct
  44   lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct
  45   /3 width=1 by or3_intro2, and4_intro/
  46 ]
  47 qed.
  48
  49 lemma llor_tail_cofrees: ∀L1,L2,L,U,l. L1 ⋓[U, l] L2 ≡ L →
  50                          ∀I1,W1. (ⓑ{I1}W1.L1 ⊢ |L1| ϵ 𝐅*[l]⦃U⦄ → ⊥) →
  51                          ∀I2,W2. ⓑ{I1}W1.L1 ⋓[U, l] ⓑ{I2}W2.L2 ≡ ⓑ{I1}W1.L.
  52 #L1 #L2 #L #U #l * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2
  53 @and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
  54 #J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
  55 lapply (drop_fwd_length_lt2 … HLK1) >ltail_length #H
  56 lapply (lt_plus_SO_to_le … H) -H #H
  57 elim (le_to_or_lt_eq … H) -H #H
  58 [ elim (drop_O1_lt (Ⓕ) … H) #Z1 #Y1 #X1 #HLY1
  59   elim (drop_O1_lt (Ⓕ) L2 i) // #Z2 #Y2 #X2 #HLY2
  60   elim (drop_O1_lt (Ⓕ) L i) // #Z #Y #X #HLY
  61   lapply (drop_O1_inv_append1_le … HLK1 … HLY1) /2 width=1 by lt_to_le/ -HLK1 normalize #H destruct
  62   lapply (drop_O1_inv_append1_le … HLK2 … HLY2) /2 width=1 by lt_to_le/ -HLK2 normalize #H destruct
  63   lapply (drop_O1_inv_append1_le … HLK … HLY) /2 width=1 by lt_to_le/ -HLK normalize #H destruct
  64   elim (IH … HLY1 HLY2 HLY) -IH -HLY1 -HLY2 -HLY *
  65   [ /3 width=1 by and3_intro, or3_intro0/
  66   | /6 width=2 by frees_inv_append, lt_to_le, or3_intro1, and3_intro/
  67   | /5 width=1 by frees_append, lt_to_le, or3_intro2, and4_intro/
  68   ]
  69 | -IH -HLK2 destruct
  70   lapply (drop_O1_inv_append1_le … HLK1 … (⋆) ?) // -HLK1 normalize #H destruct
  71   lapply (drop_O1_inv_append1_le … HLK … (⋆) ?) // -HLK normalize #H destruct
  72   /4 width=1 by or3_intro1, and3_intro/
  73 ]
  74 qed.