the theory of extended multiple substitution for therms is complete

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / substitution / gr2_minus.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/gr2_minus.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/gr2_minus.ma
index 3a98ab7282a45a9e433a93c11af530aad498189b..3c32514d2ef4d505b7cc7e088ec3191989269307 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/gr2_minus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/gr2_minus.ma
@@ -12,12 +12,13 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/relations/rminus_3.ma".
 include "basic_2/substitution/gr2.ma".
 
 (* GENERIC RELOCATION WITH PAIRS ********************************************)
 
 inductive minuss: nat → relation (list2 nat nat) ≝
-| minuss_nil: ∀i. minuss i ⟠ ⟠
+| minuss_nil: ∀i. minuss i (⟠) (⟠)
 | minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 →
               minuss i ({d, e} @ des1) ({d - i, e} @ des2)
 | minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
@@ -35,10 +36,10 @@ fact minuss_inv_nil1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 → des1 = ⟠ →
 | #des1 #des2 #d #e #i #_ #_ #H destruct
 | #des1 #des2 #d #e #i #_ #_ #H destruct
 ]
-qed.
+qed-.
 
 lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠.
-/2 width=4/ qed-.
+/2 width=4 by minuss_inv_nil1_aux/ qed-.
 
 fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
                            ∀d,e,des. des1 = {d, e} @ des →
@@ -47,16 +48,16 @@ fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
                                    des2 = {d - i, e} @ des0.
 #des1 #des2 #i * -des1 -des2 -i
 [ #i #d #e #des #H destruct
-| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3/
-| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1/
+| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3 by ex3_intro, or_intror/
+| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
 ]
-qed.
+qed-.
 
 lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
                         d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨
                         ∃∃des. i < d & des1 ▭ i ≡ des &
                                des2 = {d - i, e} @ des.
-/2 width=3/ qed-.
+/2 width=3 by minuss_inv_cons1_aux/ qed-.
 
 lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
                            d ≤ i → des1 ▭ e + i ≡ des2.
@@ -69,8 +70,7 @@ qed-.
 lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
                            i < d →
                            ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} @ des.
-#des1 #des2 #d #e #i #H
-elim (minuss_inv_cons1 … H) -H * /2 width=3/ #Hdi #_ #Hid
-lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi
-elim (lt_refl_false … Hi)
+#des1 #des2 #d #e #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
+#Hdi #_ #Hid lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi
+#Hi elim (lt_refl_false … Hi)
 qed-.