X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fsyntax%2Flveq.ma;h=43d4e636f0266eccf465a92a29d150c639ff7edd;hp=e127fa23654f19aab07b5687cb781ae25b8b4fe8;hb=156d974ad89aa04a086fdf9d332c8b04adf279fd;hpb=8fe4dc148d50a0352313633bea61441bc817afbf diff --git a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma index e127fa236..43d4e636f 100644 --- a/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma +++ b/matita/matita/contribs/lambdadelta/static_2/syntax/lveq.ma @@ -12,21 +12,23 @@ (* *) (**************************************************************************) +include "ground/xoa/ex_3_1.ma". include "ground/xoa/ex_3_4.ma". include "ground/xoa/ex_4_1.ma". +include "ground/arith/nat_succ.ma". include "static_2/notation/relations/voidstareq_4.ma". include "static_2/syntax/lenv.ma". (* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************) inductive lveq: bi_relation nat lenv ≝ -| lveq_atom : lveq 0 (⋆) 0 (⋆) -| lveq_bind : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 → - lveq 0 (K1.ⓘ[I1]) 0 (K2.ⓘ[I2]) -| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 0 K2 → - lveq (↑n1) (K1.ⓧ) 0 K2 -| lveq_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 → - lveq 0 K1 (↑n2) (K2.ⓧ) +| lveq_atom : lveq 𝟎 (⋆) 𝟎 (⋆) +| lveq_bind : ∀I1,I2,K1,K2. lveq 𝟎 K1 𝟎 K2 → + lveq 𝟎 (K1.ⓘ[I1]) 𝟎 (K2.ⓘ[I2]) +| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 𝟎 K2 → + lveq (↑n1) (K1.ⓧ) 𝟎 K2 +| lveq_void_dx: ∀K1,K2,n2. lveq 𝟎 K1 n2 K2 → + lveq 𝟎 K1 (↑n2) (K2.ⓧ) . interpretation "equivalence up to exclusion binders (local environment)" @@ -34,7 +36,7 @@ interpretation "equivalence up to exclusion binders (local environment)" (* Basic properties *********************************************************) -lemma lveq_refl: ∀L. L ≋ⓧ*[0,0] L. +lemma lveq_refl: ∀L. L ≋ⓧ*[𝟎,𝟎] L. #L elim L -L /2 width=1 by lveq_atom, lveq_bind/ qed. @@ -46,37 +48,41 @@ qed-. (* Basic inversion lemmas ***************************************************) fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → - 0 = n1 → 0 = n2 → + (𝟎 = n1) → 𝟎 = n2 → ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2 - | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2. + | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[𝟎,𝟎] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2. #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2 [1: /3 width=1 by or_introl, conj/ |2: /3 width=7 by ex3_4_intro, or_intror/ -|*: #K1 #K2 #n #_ #H1 #H2 destruct +|*: #K1 #K2 #n #_ [ #H #_ | #_ #H ] + elim (eq_inv_zero_nsucc … H) ] qed-. -lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0,0] L2 → +lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[𝟎,𝟎] L2 → ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2 - | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0,0] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2. + | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[𝟎,𝟎] K2 & K1.ⓘ[I1] = L1 & K2.ⓘ[I2] = L2. /2 width=5 by lveq_inv_zero_aux/ qed-. fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → ∀m1. ↑m1 = n1 → - ∃∃K1. K1 ≋ⓧ*[m1,0] L2 & K1.ⓧ = L1 & 0 = n2. + ∃∃K1. K1 ≋ⓧ*[m1,𝟎] L2 & K1.ⓧ = L1 & 𝟎 = n2. #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2 -[1: #m #H destruct -|2: #I1 #I2 #K1 #K2 #_ #m #H destruct -|*: #K1 #K2 #n #HK #m #H destruct /2 width=3 by ex3_intro/ +[1: #m #H elim (eq_inv_nsucc_zero … H) +|2: #I1 #I2 #K1 #K2 #_ #m #H elim (eq_inv_nsucc_zero … H) +|*: #K1 #K2 #n #HK #m #H + [ >(eq_inv_nsucc_bi … H) -m /2 width=3 by ex3_intro/ + | elim (eq_inv_nsucc_zero … H) + ] ] qed-. lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1,n2] K2 → - ∃∃K1. K1 ≋ⓧ*[n1,0] K2 & K1.ⓧ = L1 & 0 = n2. + ∃∃K1. K1 ≋ⓧ*[n1,𝟎] K2 & K1.ⓧ = L1 & 𝟎 = n2. /2 width=3 by lveq_inv_succ_sn_aux/ qed-. lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1,↑n2] L2 → - ∃∃K2. K1 ≋ⓧ*[0,n2] K2 & K2.ⓧ = L2 & 0 = n1. + ∃∃K2. K1 ≋ⓧ*[𝟎,n2] K2 & K2.ⓧ = L2 & 𝟎 = n1. #K1 #L2 #n1 #n2 #H lapply (lveq_sym … H) -H #H elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/ @@ -85,9 +91,10 @@ qed-. fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → ∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥. #L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2 -[1: #m1 #m2 #H1 #H2 destruct -|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct -|*: #K1 #K2 #n #_ #m1 #m2 #H1 #H2 destruct +[1: #m1 #m2 #H #_ elim (eq_inv_nsucc_zero … H) +|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H #_ elim (eq_inv_nsucc_zero … H) +|*: #K1 #K2 #n #_ #m1 #m2 [ #_ #H | #H #_ ] + elim (eq_inv_nsucc_zero … H) ] qed-. @@ -96,13 +103,15 @@ lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1,↑n2] L2 → ⊥. (* Advanced inversion lemmas ************************************************) -lemma lveq_inv_bind_O: ∀I1,I2,K1,K2. K1.ⓘ[I1] ≋ⓧ*[0,0] K2.ⓘ[I2] → K1 ≋ⓧ*[0,0] K2. +lemma lveq_inv_bind_O: ∀I1,I2,K1,K2. K1.ⓘ[I1] ≋ⓧ*[𝟎,𝟎] K2.ⓘ[I2] → K1 ≋ⓧ*[𝟎,𝟎] K2. #I1 #I2 #K1 #K2 #H elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct // qed-. -lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 = n2. -* [2: #n1 ] * [2,4: #n2 ] #H +lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 𝟎 = n1 & 𝟎 = n2. +#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ] +#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ] +#H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct @@ -111,8 +120,11 @@ lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1,n2] ⋆ → ∧∧ 0 = n1 & 0 qed-. lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ[I1] ≋ⓧ*[n1,n2] ⋆ → - ∃∃m1. K1 ≋ⓧ*[m1,0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2. -#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H + ∃∃m1. K1 ≋ⓧ*[m1,𝟎] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 𝟎 = n2. +#I1 #K1 +#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ] +#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ] +#H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct | elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=3 by ex4_intro/ @@ -124,7 +136,7 @@ lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ[I1] ≋ⓧ*[n1,n2] ⋆ → qed-. lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1,n2] K2.ⓘ[I2] → - ∃∃m2. ⋆ ≋ⓧ*[0,m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2. + ∃∃m2. ⋆ ≋ⓧ*[𝟎,m2] K2 & BUnit Void = I2 & 𝟎 = n1 & ↑m2 = n2. #I2 #K2 #n1 #n2 #H lapply (lveq_sym … H) -H #H elim (lveq_inv_bind_atom … H) -H @@ -132,8 +144,11 @@ elim (lveq_inv_bind_atom … H) -H qed-. lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 → - ∧∧ K1 ≋ⓧ*[0,0] K2 & 0 = n1 & 0 = n2. -#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H + ∧∧ K1 ≋ⓧ*[𝟎,𝟎] K2 & 𝟎 = n1 & 𝟎 = n2. +#I1 #I2 #K1 #K2 #V1 #V2 +#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ] +#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ] +#H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct @@ -145,13 +160,13 @@ lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2 qed-. lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1,n2] L2 → - ∧∧ L1 ≋ ⓧ*[n1,0] L2 & 0 = n2. + ∧∧ L1 ≋ ⓧ*[n1,𝟎] L2 & 𝟎 = n2. #L1 #L2 #n1 #n2 #H elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/ qed-. lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,↑n2] L2.ⓧ → - ∧∧ L1 ≋ ⓧ*[0,n2] L2 & 0 = n1. + ∧∧ L1 ≋ ⓧ*[𝟎,n2] L2 & 𝟎 = n1. #L1 #L2 #n1 #n2 #H lapply (lveq_sym … H) -H #H elim (lveq_inv_void_succ_sn … H) -H @@ -161,18 +176,25 @@ qed-. (* Advanced forward lemmas **************************************************) lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1,n2] L2 → - ∨∨ 0 = n1 | 0 = n2. -#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H + ∨∨ 𝟎 = n1 | 𝟎 = n2. +#L1 #L2 +#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ] +#n2 @(nat_ind_succ … n2) -n2 [2,4: #n2 #_ ] +#H [ elim (lveq_inv_succ … H) ] /2 width=1 by or_introl, or_intror/ qed-. -lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] L2 → 0 = n1. -#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H +lemma lveq_fwd_pair_sn: + ∀I1,K1,L2,V1,n1,n2. K1.ⓑ[I1]V1 ≋ⓧ*[n1,n2] L2 → 𝟎 = n1. +#I1 #K1 #L2 #V1 +#n1 @(nat_ind_succ … n1) -n1 [2: #n1 #_ ] // +#n2 @(nat_ind_succ … n2) -n2 [2: #n2 #_ ] #H [ elim (lveq_inv_succ … H) | elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct ] qed-. -lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 → 0 = n2. +lemma lveq_fwd_pair_dx: + ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1,n2] K2.ⓑ[I2]V2 → 𝟎 = n2. /3 width=6 by lveq_fwd_pair_sn, lveq_sym/ qed-.