X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Flift.ma;fp=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Flift.ma;h=7d0f43c8188b42d29f392aa4775808fe440f37a3;hb=d38087520d6ce1d696b28da40f3811291fc8a311;hp=1934c37bcce9305b589f020d1b547211e9e8dcc4;hpb=016603891d57b4c6b41da6a398bf8ae466019f9e;p=helm.git diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma index 1934c37bc..7d0f43c81 100644 --- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma +++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma @@ -34,59 +34,6 @@ inductive lift: nat → nat → relation term ≝ interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2). -(* Basic properties *********************************************************) - -(* Basic_1: was: lift_lref_gt *) -lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i. -#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/ -qed. - -(* Basic_1: was: lift_r *) -lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T. -#T elim T -T -[ * #i // #d elim (lt_or_ge i d) /2/ -| * /2/ -] -qed. - -lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2. -#T1 elim T1 -T1 -[ * #i /2/ #d #e elim (lt_or_ge i d) /3/ -| * #I #V1 #T1 #IHV1 #IHT1 #d #e - elim (IHV1 d e) -IHV1 #V2 #HV12 - [ elim (IHT1 (d+1) e) -IHT1 /3/ - | elim (IHT1 d e) -IHT1 /3/ - ] -] -qed. - -(* Basic_1: was: lift_free (right to left) *) -lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1. - d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 → - ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2. -#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2 -[ /3/ -| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_ - lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/ -| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12 - lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21 - <(arith_d1 i e2 e1) // /3/ -| /3/ -| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 - elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b - elim (IHT (d2+1) … ? ? He12) /3 width = 5/ -| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 - elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b - elim (IHT d2 … ? ? He12) /3 width = 5/ -] -qed. - -(* Basic forward lemmas *****************************************************) - -lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #[T1] = #[T2]. -#d #e #T1 #T2 #H elim H -d e T1 T2; normalize // -qed. - (* Basic inversion lemmas ***************************************************) fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2. @@ -94,7 +41,7 @@ fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2. qed. lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2. -/2/ qed. +/2/ qed-. fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k. #d #e #T1 #T2 * -d e T1 T2 // @@ -105,7 +52,7 @@ fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k qed. lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k. -/2 width=5/ qed. +/2 width=5/ qed-. fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i → (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). @@ -121,19 +68,19 @@ qed. lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)). -/2/ qed. +/2/ qed-. lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i. #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd elim (lt_refl_false … Hdd) -qed. +qed-. lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e). #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * // #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd elim (lt_refl_false … Hdd) -qed. +qed-. fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p. #d #e #T1 #T2 * -d e T1 T2 // @@ -144,7 +91,7 @@ fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → qed. lemma lift_inv_gref1: ∀d,e,T2,p. ↑[d,e] §p ≡ T2 → T2 = §p. -/2 width=5/ qed. +/2 width=5/ qed-. fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀I,V1,U1. T1 = 𝕓{I} V1.U1 → @@ -163,7 +110,7 @@ qed. lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 → ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & T2 = 𝕓{I} V2. U2. -/2/ qed. +/2/ qed-. fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀I,V1,U1. T1 = 𝕗{I} V1.U1 → @@ -182,7 +129,7 @@ qed. lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 → ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & T2 = 𝕗{I} V2. U2. -/2/ qed. +/2/ qed-. fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k. #d #e #T1 #T2 * -d e T1 T2 // @@ -194,7 +141,7 @@ qed. (* Basic_1: was: lift_gen_sort *) lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k. -/2 width=5/ qed. +/2 width=5/ qed-. fact lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i → (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). @@ -211,23 +158,30 @@ qed. (* Basic_1: was: lift_gen_lref *) lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → (i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)). -/2/ qed. +/2/ qed-. (* Basic_1: was: lift_gen_lref_lt *) lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i. #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * // #Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd elim (plus_lt_false … Hdd) -qed. +qed-. (* Basic_1: was: lift_gen_lref_false *) +lemma lift_inv_lref2_be: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → + d ≤ i → i < d + e → False. +#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * +[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ] +lapply (le_to_lt_to_lt … H2 H1) -H2 H1 #H +elim (lt_refl_false … H) +qed-. (* Basic_1: was: lift_gen_lref_ge *) lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e). #d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * // #Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd elim (plus_lt_false … Hdd) -qed. +qed-. fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p. #d #e #T1 #T2 * -d e T1 T2 // @@ -238,7 +192,7 @@ fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → qed. lemma lift_inv_gref2: ∀d,e,T1,p. ↑[d,e] T1 ≡ §p → T1 = §p. -/2 width=5/ qed. +/2 width=5/ qed-. fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀I,V2,U2. T2 = 𝕓{I} V2.U2 → @@ -258,7 +212,7 @@ qed. lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 → ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 & T1 = 𝕓{I} V1. U1. -/2/ qed. +/2/ qed-. fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀I,V2,U2. T2 = 𝕗{I} V2.U2 → @@ -278,7 +232,124 @@ qed. lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 → ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 & T1 = 𝕗{I} V1. U1. -/2/ qed. +/2/ qed-. + +lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ↑[d, e] 𝕔{I} V. T ≡ V → False. +#d #e #J #V elim V -V +[ * #i #T #H + [ lapply (lift_inv_sort2 … H) -H #H destruct + | elim (lift_inv_lref2 … H) -H * #_ #H destruct + | lapply (lift_inv_gref2 … H) -H #H destruct + ] +| * #I #W2 #U2 #IHW2 #_ #T #H + [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct -J T W1 /2/ + | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct -J T W1 /2/ + ] +] +qed-. + +lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ↑[d, e] 𝕔{I} V. T ≡ T → False. +#J #T elim T -T +[ * #i #V #d #e #H + [ lapply (lift_inv_sort2 … H) -H #H destruct + | elim (lift_inv_lref2 … H) -H * #_ #H destruct + | lapply (lift_inv_gref2 … H) -H #H destruct + ] +| * #I #W2 #U2 #_ #IHU2 #V #d #e #H + [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct -J U1 W1 /2/ + | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct -J U1 W1 /2/ + ] +] +qed-. + +(* Basic forward lemmas *****************************************************) + +lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #[T1] = #[T2]. +#d #e #T1 #T2 #H elim H -d e T1 T2; normalize // +qed-. + +(* Basic properties *********************************************************) + +(* Basic_1: was: lift_lref_gt *) +lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i. +#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3 width=2/ +qed. + +(* Basic_1: was: lift_r *) +lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T. +#T elim T -T +[ * #i // #d elim (lt_or_ge i d) /2/ +| * /2/ +] +qed. + +lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2. +#T1 elim T1 -T1 +[ * #i /2/ #d #e elim (lt_or_ge i d) /3/ +| * #I #V1 #T1 #IHV1 #IHT1 #d #e + elim (IHV1 d e) -IHV1 #V2 #HV12 + [ elim (IHT1 (d+1) e) -IHT1 /3/ + | elim (IHT1 d e) -IHT1 /3/ + ] +] +qed. + +(* Basic_1: was: lift_free (right to left) *) +lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → + ∀d2,e1. d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 → + ∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2. +#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2 +[ /3/ +| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_ + lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/ +| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12 + lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21 + <(arith_d1 i e2 e1) // /3/ +| /3/ +| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 + elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b + elim (IHT (d2+1) … ? ? He12) /3 width=5/ +| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12 + elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b + elim (IHT d2 … ? ? He12) /3 width=5/ +] +qed. + +(* Basic_1: was only: dnf_dec2 dnf_dec *) +lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ↑[d,e] T1 ≡ T2). +#T1 elim T1 -T1 +[ * [1,3: /3 width=2/ ] #i #d #e + elim (lt_dec i d) #Hid + [ /4 width=2/ + | lapply (false_lt_to_le … Hid) -Hid #Hid + elim (lt_dec i (d + e)) #Hide + [ @or_intror * #T1 #H + elim (lift_inv_lref2_be … H Hid Hide) + | lapply (false_lt_to_le … Hide) -Hide /4 width=2/ + ] + ] +| * #I #V2 #T2 #IHV2 #IHT2 #d #e + [ elim (IHV2 d e) -IHV2 + [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2 + [ * #T1 #HT12 @or_introl /3/ + | -V1 #HT2 @or_intror * #X #H + elim (lift_inv_bind2 … H) -H /3 width=2/ + ] + | -IHT2 #HV2 @or_intror * #X #H + elim (lift_inv_bind2 … H) -H /3 width=2/ + ] + | elim (IHV2 d e) -IHV2 + [ * #V1 #HV12 elim (IHT2 d e) -IHT2 + [ * #T1 #HT12 /4 width=2/ + | -V1 #HT2 @or_intror * #X #H + elim (lift_inv_flat2 … H) -H /3 width=2/ + ] + | -IHT2 #HV2 @or_intror * #X #H + elim (lift_inv_flat2 … H) -H /3 width=2/ + ] + ] +] +qed. (* Basic_1: removed theorems 7: lift_head lift_gen_head