From c8f9324f016be3f7545815269bc416bafea6caed Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Tue, 9 Aug 2011 18:46:44 +0000 Subject: [PATCH] confluence of parallel substitution (tps) started ... --- matita/matita/lib/lambda-delta/ground.ma | 9 +++ matita/matita/lib/lambda-delta/names.txt | 3 + .../lib/lambda-delta/substitution/drop.ma | 22 ++++++- .../lambda-delta/substitution/drop_drop.ma | 52 +++++++++++----- .../lambda-delta/substitution/lift_lift.ma | 30 ++++----- .../lib/lambda-delta/substitution/tps.ma | 23 +++++++ .../lib/lambda-delta/substitution/tps_lift.ma | 1 - .../lib/lambda-delta/substitution/tps_tps.ma | 62 +++++++++++++++++-- matita/matita/lib/lambda-delta/syntax/item.ma | 1 - 9 files changed, 164 insertions(+), 39 deletions(-) diff --git a/matita/matita/lib/lambda-delta/ground.ma b/matita/matita/lib/lambda-delta/ground.ma index 92a3f8167..e09e7d47a 100644 --- a/matita/matita/lib/lambda-delta/ground.ma +++ b/matita/matita/lib/lambda-delta/ground.ma @@ -10,6 +10,7 @@ V_______________________________________________________________ *) include "arithmetics/nat.ma". +include "lambda-delta/xoa_props.ma". (* ARITHMETICAL PROPERTIES **************************************************) @@ -42,6 +43,14 @@ lemma lt_or_ge: ∀m,n. m < n ∨ n ≤ m. #m #n elim (decidable_lt m n) /3/ qed. +lemma lt_or_eq_or_gt: ∀m,n. ∨∨ m < n | n = m | n < m. +#m elim m -m +[ * /2/ +| #m #IHm * [ /2/ ] + #n elim (IHm n) -IHm #H + [ @or3_intro0 | @or3_intro1 destruct | @or3_intro2 ] /2/ (**) (* /3/ is slow *) + qed. + lemma le_to_lt_or_eq: ∀m,n. m ≤ n → m < n ∨ m = n. #m #n * -n /3/ qed. diff --git a/matita/matita/lib/lambda-delta/names.txt b/matita/matita/lib/lambda-delta/names.txt index 61ef3e314..2a030fb4f 100644 --- a/matita/matita/lib/lambda-delta/names.txt +++ b/matita/matita/lib/lambda-delta/names.txt @@ -1,8 +1,11 @@ NAMING CONVENTIONS FOR METAVARIABLES +H : reserved: transient premise +IH : reserved: inductive premise I,J : item K,L : local environment T,U,V,W: term +X,Y,Z : reserved: transient objet denoted by a capital letter d : relocation depth e : relocation height diff --git a/matita/matita/lib/lambda-delta/substitution/drop.ma b/matita/matita/lib/lambda-delta/substitution/drop.ma index f5911775b..29b57405f 100644 --- a/matita/matita/lib/lambda-delta/substitution/drop.ma +++ b/matita/matita/lib/lambda-delta/substitution/drop.ma @@ -80,6 +80,26 @@ elim (drop_inv_O1 … H) -H * // #H destruct -e; elim (lt_refl_false … He) qed. +lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d → + ∀I,K1,V1. L1 = K1. 𝕓{I} V1 → + ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 & + ↑[d - 1, e] V2 ≡ V1 & + L2 = K2. 𝕓{I} V2. +#d #e #L1 #L2 * -d e L1 L2 +[ #d #e #_ #I #K #V #H destruct +| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H) +| #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) +| #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z + /2 width=5/ +] +qed. + +lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d → + ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 & + ↑[d - 1, e] V2 ≡ V1 & + L2 = K2. 𝕓{I} V2. +/2/ qed. + lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d → ∀I,K2,V2. L2 = K2. 𝕓{I} V2 → ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & @@ -89,7 +109,7 @@ lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d → [ #d #e #_ #I #K #V #H destruct | #L1 #L2 #I #V #_ #H elim (lt_refl_false … H) | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H) -| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z; +| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z /2 width=5/ ] qed. diff --git a/matita/matita/lib/lambda-delta/substitution/drop_drop.ma b/matita/matita/lib/lambda-delta/substitution/drop_drop.ma index 135db64ca..812fb7e06 100644 --- a/matita/matita/lib/lambda-delta/substitution/drop_drop.ma +++ b/matita/matita/lib/lambda-delta/substitution/drop_drop.ma @@ -12,15 +12,33 @@ (* *) (**************************************************************************) +include "lambda-delta/substitution/lift_lift.ma". include "lambda-delta/substitution/drop.ma". (* DROPPING *****************************************************************) (* Main properties **********************************************************) -lemma drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → - ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 → - ↓[0, e2 - e1] L1 ≡ L2. +theorem drop_mono: ∀d,e,L,L1. ↓[d, e] L ≡ L1 → + ∀L2. ↓[d, e] L ≡ L2 → L1 = L2. +#d #e #L #L1 #H elim H -H d e L L1 +[ #d #e #L2 #H + >(drop_inv_sort1 … H) -H L2 // +| #K1 #K2 #I #V #HK12 #_ #L2 #HL12 + <(drop_inv_refl … HK12) -HK12 K2 + <(drop_inv_refl … HL12) -HL12 L2 // +| #L #K #I #V #e #_ #IHLK #L2 #H + lapply (drop_inv_drop1 … H ?) -H /2/ +| #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H + elim (drop_inv_skip1 … H ?) -H // (lift_inj … HVT1 … HVT2) -HVT1 HVT2 + >(IHLK1 … HLK2) -IHLK1 HLK2 // +] +qed. + +theorem drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → + ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 + e1 ≤ e2 → + ↓[0, e2 - e1] L1 ≡ L2. #d1 #e1 #L #L1 #H elim H -H d1 e1 L L1 [ #d #e #e2 #L2 #H >(drop_inv_sort1 … H) -H L2 // @@ -32,16 +50,16 @@ lemma drop_conf_ge: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → | #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2 lapply (transitive_le 1 … Hdee2) // #He2 lapply (drop_inv_drop1 … H ?) -H // -He2 #HL2 - lapply (transitive_le (1+e) … Hdee2) // #Hee2 + lapply (transitive_le (1 + e) … Hdee2) // #Hee2 @drop_drop_lt >minus_minus_comm /3/ (**) (* explicit constructor *) ] qed. -lemma drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → - ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 → - e2 < d1 → let d ≝ d1 - e2 - 1 in - ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 & - ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2. +theorem drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → + ∀e2,K2,I,V2. ↓[0, e2] L ≡ K2. 𝕓{I} V2 → + e2 < d1 → let d ≝ d1 - e2 - 1 in + ∃∃K1,V1. ↓[0, e2] L1 ≡ K1. 𝕓{I} V1 & + ↓[d, e1] K2 ≡ K1 & ↑[d, e1] V1 ≡ V2. #d1 #e1 #L #L1 #H elim H -H d1 e1 L L1 [ #d #e #e2 #K2 #I #V2 #H lapply (drop_inv_sort1 … H) -H #H destruct @@ -58,9 +76,9 @@ lemma drop_conf_lt: ∀d1,e1,L,L1. ↓[d1, e1] L ≡ L1 → ] qed. -lemma drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → - ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 → - ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2. +theorem drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → + ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 → + ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2. #d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L [ #d #e #e2 #L2 #H >(drop_inv_sort1 … H) -H L2 /2/ @@ -81,8 +99,8 @@ lemma drop_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → ] qed. -lemma drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → - ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2. +theorem drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → + ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2. #d1 #e1 #L1 #L #H elim H -H d1 e1 L1 L [ #d #e #e2 #L2 #H >(drop_inv_sort1 … H) -H L2 // @@ -97,9 +115,9 @@ lemma drop_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L → ] qed. -lemma drop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L. - ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 → - ↓[0, e2 + e1] L1 ≡ L2. +theorem drop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L. + ↓[d1, e1] L1 ≡ L → ↓[0, e2] L ≡ L2 → d1 ≤ e2 → + ↓[0, e2 + e1] L1 ≡ L2. #e1 #e1 #e2 >commutative_plus /2 width=5/ qed. diff --git a/matita/matita/lib/lambda-delta/substitution/lift_lift.ma b/matita/matita/lib/lambda-delta/substitution/lift_lift.ma index 4281046cb..205eab2bf 100644 --- a/matita/matita/lib/lambda-delta/substitution/lift_lift.ma +++ b/matita/matita/lib/lambda-delta/substitution/lift_lift.ma @@ -18,7 +18,7 @@ include "lambda-delta/substitution/lift.ma". (* Main properies ***********************************************************) -lemma lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2. +theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2. #d #e #T1 #U #H elim H -H d e T1 U [ #k #d #e #X #HX lapply (lift_inv_sort2 … HX) -HX // @@ -33,10 +33,10 @@ lemma lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → ] qed. -lemma lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T → - d1 ≤ d2 → - ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1. +theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T → + d1 ≤ d2 → + ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1. #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T [ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12 lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/ @@ -63,7 +63,7 @@ lemma lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → ] qed. -lemma lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2. +theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2. #d #e #T #U1 #H elim H -H d e T U1 [ #k #d #e #X #HX lapply (lift_inv_sort1 … HX) -HX // @@ -78,9 +78,9 @@ lemma lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 ] qed. -lemma lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → - d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2. +theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → + d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2. #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T [ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_ >(lift_inv_sort1 … HT2) -HT2 // @@ -103,9 +103,9 @@ lemma lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → ] qed. -lemma lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 → - ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2. +theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 → + ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2. #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T [ #k #d1 #e1 #d2 #e2 #X #HX #_ >(lift_inv_sort1 … HX) -HX /2/ @@ -127,9 +127,9 @@ lemma lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → ] qed. -lemma lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → - ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → - ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2. +theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T → + ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 → + ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2. #d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T [ #k #d1 #e1 #d2 #e2 #X #HX #_ >(lift_inv_sort1 … HX) -HX /2/ diff --git a/matita/matita/lib/lambda-delta/substitution/tps.ma b/matita/matita/lib/lambda-delta/substitution/tps.ma index 9fe0c29e7..56d8f3211 100644 --- a/matita/matita/lib/lambda-delta/substitution/tps.ma +++ b/matita/matita/lib/lambda-delta/substitution/tps.ma @@ -91,6 +91,29 @@ qed. (* Basic inversion lemmas ***************************************************) +lemma tps_inv_lref1_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → ∀i. T1 = #i → + T2 = #i ∨ + ∃∃K,V1,V2,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V1 & + K ⊢ V1 [O, d + e - i - 1] ≫ V2 & + ↑[O, i + 1] V2 ≡ T2. +#L #T1 #T2 #d #e * -L T1 T2 d e +[ #L #k #d #e #i #H destruct +| /2/ +| #L #K #V1 #V2 #T2 #i #d #e #Hdi #Hide #HLK #HV12 #HVT2 #j #H destruct -i /3 width=9/ +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct +| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct +] +qed. + +lemma tps_inv_lref1: ∀L,T2,i,d,e. L ⊢ #i [d, e] ≫ T2 → + T2 = #i ∨ + ∃∃K,V1,V2,i. d ≤ i & i < d + e & + ↓[O, i] L ≡ K. 𝕓{Abbr} V1 & + K ⊢ V1 [O, d + e - i - 1] ≫ V2 & + ↑[O, i + 1] V2 ≡ T2. +/2/ qed. + lemma tps_inv_bind1_aux: ∀d,e,L,U1,U2. L ⊢ U1 [d, e] ≫ U2 → ∀I,V1,T1. U1 = 𝕓{I} V1. T1 → ∃∃V2,T2. L ⊢ V1 [d, e] ≫ V2 & diff --git a/matita/matita/lib/lambda-delta/substitution/tps_lift.ma b/matita/matita/lib/lambda-delta/substitution/tps_lift.ma index 70dc0903d..1f8d7a88c 100644 --- a/matita/matita/lib/lambda-delta/substitution/tps_lift.ma +++ b/matita/matita/lib/lambda-delta/substitution/tps_lift.ma @@ -9,7 +9,6 @@ \ / V_______________________________________________________________ *) -include "lambda-delta/substitution/lift_lift.ma". include "lambda-delta/substitution/drop_drop.ma". include "lambda-delta/substitution/tps.ma". diff --git a/matita/matita/lib/lambda-delta/substitution/tps_tps.ma b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma index 55e8be812..8d801d50e 100644 --- a/matita/matita/lib/lambda-delta/substitution/tps_tps.ma +++ b/matita/matita/lib/lambda-delta/substitution/tps_tps.ma @@ -11,12 +11,15 @@ include "lambda-delta/substitution/tps_split.ma". +lemma arith_i2: ∀a,c1,c2. c1 + c2 ≤ a → c1 + c2 + (a - c1 - c2) = a. +/2/ qed. + (* PARTIAL SUBSTITUTION ON TERMS ********************************************) (* Main properties **********************************************************) - -lemma tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 → - L ⊢ T1 [d, e] ≫ T2. +(* +theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 → + L ⊢ T1 [d, e] ≫ T2. #L #T1 #T #d #e #H elim H -L T1 T d e [ // | // @@ -31,9 +34,60 @@ lemma tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/ ] qed. +*) -axiom tps_conf: ∀L,T0,d,e,T1. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 → +axiom tps_conf_subst_subst_lt: ∀L,K1,V1,W1,T1,i1,d,e,T2,K2,V2,W2,i2. + ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 → + K1 ⊢ V1 [O, d + e - i1 - 1] ≫ W1 → K2 ⊢ V2 [O, d + e - i2 - 1] ≫ W2 → + ↑[O, i1 + 1] W1 ≡ T1 → ↑[O, i2 + 1] W2 ≡ T2 → + d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 → + ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T. +(* +#L #K1 #V1 #W1 #T1 #i1 #d #e #T2 #K2 #V2 #W2 #i2 +#HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12 +*) + +theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 → ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T. +#L #T0 #T1 #d #e #H elim H -H L T0 T1 d e +[ /2/ +| /2/ +| #L #K1 #V1 #W1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVW1 #HWT1 #IHVW1 #T2 #H + elim (tps_inv_lref1 … H) -H + [ -IHVW1 #HX destruct -T2 + @ex2_1_intro [2: // | skip ] /2 width=6/ (**) (* /3 width=9/ is slow *) + | * #K2 #V2 #W2 #i2 #Hdi2 #Hi2de #HLK2 #HVW2 #HWT2 + elim (lt_or_eq_or_gt i1 i2) #Hi12 + [ @tps_conf_subst_subst_lt /width=19/ + | -HVW1; destruct -i2; + lapply (transitive_le ? ? (i1 + 1) Hdi2 ?) -Hdi2 // #Hdi2 + lapply (drop_mono … HLK1 … HLK2) -HLK1 Hdi1 Hi1de #H destruct -V1 K1; + elim (IHVW1 … HVW2) -IHVW1 HVW2 #W #HW1 #HW2 + lapply (drop_fwd_drop2 … HLK2) -HLK2 #HLK2 + elim (lift_total W 0 (i1 + 1)) #T #HWT + lapply (tps_lift_ge … HW1 … HLK2 HWT1 HWT ?) -HW1 HWT1 // + lapply (tps_lift_ge … HW2 … HLK2 HWT2 HWT ?) -HW2 HWT2 HLK2 HWT // normalize #HT2 #HT1 + lapply (tps_weak … HT1 d e ? ?) -HT1 [ >arith_i2 // | // ] + lapply (tps_weak … HT2 d e ? ?) -HT2 [ >arith_i2 // | // ] + /2/ + | @ex2_1_comm @tps_conf_subst_subst_lt /width=19/ + ] + ] +| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX + elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2 + elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2 + @ex2_1_intro + [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ] + |1: skip + |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ] + ] // (**) (* /5/ is too slow *) +| #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX + elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; + elim (IHV01 … HV02) -IHV01 HV02; + elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/ +] +qed. (* Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) -> diff --git a/matita/matita/lib/lambda-delta/syntax/item.ma b/matita/matita/lib/lambda-delta/syntax/item.ma index d813a2dc8..ea7a45362 100644 --- a/matita/matita/lib/lambda-delta/syntax/item.ma +++ b/matita/matita/lib/lambda-delta/syntax/item.ma @@ -16,7 +16,6 @@ *) include "lambda-delta/ground.ma". -include "lambda-delta/xoa_props.ma". include "lambda-delta/notation.ma". (* BINARY ITEMS *************************************************************) -- 2.39.2