matita/matita/contribs/lambdadelta/basic_2/etc/frsup/ssta_frsups.etc

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  14
  15 include "basic_2/unfold/frsups.ma".
  16 include "basic_2/static/ssta.ma".
  17
  18 (* STRATIFIED STATIC TYPE ASSIGNMENT ON TERMS *******************************)
  19
  20 (* Advanced inversion lemmas ************************************************)
  21
  22 lemma ssta_inv_frsupp: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁+ ⦃L, T⦄ → ⊥.
  23 #h #g #L #T #U #l #H elim H -L -T -U -l
  24 [ #L #k #l #_ #H
  25   elim (frsupp_inv_atom1_frsups … H)
  26 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H
  27   elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
  28   elim (lift_inv_lref2_be … H ? ?) -H //
  29 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H
  30   elim (lift_frsupp_trans … (⋆) … H … HWU) -U #X #H
  31   elim (lift_inv_lref2_be … H ? ?) -H //
  32 | #a #I #L #V #T #U #l #_ #IHTU #H
  33   elim (frsupp_inv_bind1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
  34   lapply (frsups_fwd_fw … H) -H normalize
  35   <associative_plus <associative_plus #H
  36   elim (le_plus_xySz_x_false … H)
  37 | #L #V #T #U #l #_ #IHTU #H
  38   elim (frsupp_inv_flat1_frsups … H) -H #H [2: /4 width=4/ ] -IHTU
  39   lapply (frsups_fwd_fw … H) -H normalize
  40   <associative_plus <associative_plus #H
  41   elim (le_plus_xySz_x_false … H)
  42 | /3 width=4/
  43 ]
  44 qed-.
  45
  46 fact ssta_inv_refl_aux: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → T = U → ⊥.
  47 #h #g #L #T #U #l #H elim H -L -T -U -l
  48 [ #L #k #l #_ #H
  49   lapply (next_lt h k) destruct -H -e0 (**) (* destruct: these premises are not erased *)
  50   <e1 -e1 #H elim (lt_refl_false … H)
  51 | #L #K #V #W #U #i #l #_ #_ #HWU #_ #H destruct
  52   elim (lift_inv_lref2_be … HWU ? ?) -HWU //
  53 | #L #K #W #V #U #i #l #_ #_ #HWU #_ #H destruct
  54   elim (lift_inv_lref2_be … HWU ? ?) -HWU //
  55 | #a #I #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
  56 | #L #V #T #U #l #_ #IHTU #H destruct /2 width=1/
  57 | #L #W #T #U #l #HTU #_ #H destruct
  58   elim (ssta_inv_frsupp … HTU ?) -HTU /2 width=1/
  59 ]
  60 qed-.
  61
  62 lemma ssta_inv_refl: ∀h,g,T,L,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, T⦄ → ⊥.
  63 /2 width=8 by ssta_inv_refl_aux/ qed-.
  64
  65 lemma ssta_inv_frsups: ∀h,g,L,T,U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l, U⦄ → ⦃L, U⦄ ⧁* ⦃L, T⦄ → ⊥.
  66 #h #g #L #T #U #L #HTU #H elim (frsups_inv_all … H) -H
  67 [ * #_ #H destruct /2 width=6 by ssta_inv_refl/
  68 | /2 width=8 by ssta_inv_frsupp/
  69 ]
  70 qed-.