From 670ad7822d59e598a38d9037d482d3de188b170c Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Thu, 6 Apr 2017 10:51:28 +0000 Subject: [PATCH] advances on lfsx ... --- .../basic_2/etc/lleq/lleq_drop.etc | 3 - .../basic_2/etc/llpx_sn/llpx_sn_drop.etc | 42 ------------- .../lambdadelta/basic_2/etc/lsx/lsx.etc | 25 ++++++++ .../lambdadelta/basic_2/relocation/lexs.ma | 23 +++++++ .../basic_2/rt_computation/lfsx_lfsx.ma | 9 ++- .../basic_2/rt_computation/lsx_lpx.ma | 63 ------------------- .../lambdadelta/basic_2/static/lfdeq.ma | 3 +- .../lambdadelta/basic_2/static/lfdeq_lfdeq.ma | 5 ++ .../lambdadelta/basic_2/static/lfxs.ma | 5 +- .../lambdadelta/basic_2/static/lfxs_lfxs.ma | 13 ++++ .../lambdadelta/basic_2/syntax/lenv.ma | 5 +- 11 files changed, 83 insertions(+), 113 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/basic_2/etc/lsx/lsx.etc delete mode 100644 matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx_lpx.ma diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc index 87ceaf096..10e8a919a 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lleq/lleq_drop.etc @@ -23,9 +23,6 @@ lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ≡[T, 0] L2.ⓑ{I}V → ∀J,W. L1 ≡[W, 0] L2 → L1.ⓑ{J}W ≡[T, 0] L2.ⓑ{J}W. /2 width=7 by llpx_sn_bind_repl_O/ qed-. -lemma lleq_dec: ∀T,L1,L2,l. Decidable (L1 ≡[T, l] L2). -/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-. - lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R → ∀L1,L2,T,l. L1 ≡[T, l] L2 → ∀L. llpx_sn R l T L2 L → llpx_sn R l T L1 L. diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_drop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_drop.etc index 9bb605e39..34980b9f6 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_drop.etc +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/llpx_sn/llpx_sn_drop.etc @@ -125,48 +125,6 @@ lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2 ∀J,W1,W2. llpx_sn R 0 W1 L1 L2 → R L1 W1 W2 → llpx_sn R 0 T (L1.ⓑ{J}W1) (L2.ⓑ{J}W2). /3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-. -lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → - ∀T,L1,L2,l. Decidable (llpx_sn R l T L1 L2). -#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T -#x #IH #L1 * * -[ #s #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/ -| #i #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) - [ #HL12 #l elim (ylt_split i l) /3 width=1 by llpx_sn_skip, or_introl/ - #Hli elim (lt_or_ge i (|L1|)) #HiL1 - elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/ - elim (drop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2 - elim (drop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1 - elim (eq_bind2_dec I2 I1) - [ #H2 elim (HR K1 V1 V2) -HR - [ #H3 elim (IH K1 V1 … K2 0) destruct - /3 width=9 by llpx_sn_lref, drop_fwd_rfw, or_introl/ - ] - ] - -IH #H3 @or_intror - #H elim (llpx_sn_fwd_lref … H) -H [1,3,4,6,7,9: * ] - [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/ - |7,8,9: /2 width=4 by ylt_yle_false/ - ] - #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12 - lapply (drop_mono … HLY1 … HLK1) -HLY1 -HLK1 - lapply (drop_mono … HLY2 … HLK2) -HLY2 -HLK2 - #H #H0 destruct /2 width=1 by/ - ] -| #p #Hx #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/ -| #a #I #V #T #Hx #L2 #l destruct - elim (IH L1 V … L2 l) /2 width=1 by/ - elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V) (⫯l)) -IH /3 width=1 by or_introl, llpx_sn_bind/ - #H1 #H2 @or_intror - #H elim (llpx_sn_inv_bind … H) -H /2 width=1 by/ -| #I #V #T #Hx #L2 #l destruct - elim (IH L1 V … L2 l) /2 width=1 by/ - elim (IH L1 T … L2 l) -IH /3 width=1 by or_introl, llpx_sn_flat/ - #H1 #H2 @or_intror - #H elim (llpx_sn_inv_flat … H) -H /2 width=1 by/ -] --x /4 width=4 by llpx_sn_fwd_length, or_intror/ -qed-. - (* Inversion lemmas on negated lazy pointwise extension *********************) lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsx/lsx.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsx/lsx.etc new file mode 100644 index 000000000..b5407abdf --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsx/lsx.etc @@ -0,0 +1,25 @@ +lemma lsx_lreq_conf: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 → + ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, o, T, l] L2. +#h #o #G #L1 #T #l #H @(lsx_ind … H) -L1 +#L1 #_ #IHL1 #L2 #HL12 @lsx_intro +#L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq … HL12 … HL23) -HL12 -HL23 +#L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/ +qed-. + +(* these two are better expressed with the binder \chi *) + +lemma lsx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓑ{a,I}V.T, l] L → + G ⊢ ⬈*[h, o, T, ⫯l] L.ⓑ{I}V. +#h #o #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L +#L1 #_ #IHL1 @lsx_intro +#Y #H #HT elim (lpx_inv_pair1 … H) -H +#L2 #V2 #HL12 #_ #H destruct +@(lsx_lreq_conf … (L2.ⓑ{I}V1)) /2 width=1 by lreq_succ/ +@IHL1 // #H @HT -IHL1 -HL12 -HT +@(lleq_lreq_trans … (L2.ⓑ{I}V1)) +/2 width=2 by lleq_fwd_bind_dx, lreq_succ/ +qed-. + +lemma lsx_inv_bind: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬈*[h, o, ⓑ{a, I}V.T, l] L → + G ⊢ ⬈*[h, o, V, l] L ∧ G ⊢ ⬈*[h, o, T, ⫯l] L.ⓑ{I}V. +/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma index b002e073a..e4e501f09 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lexs.ma @@ -255,3 +255,26 @@ lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. refle elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/ ] qed-. + +lemma lexs_dec: ∀RN,RP. + (∀L,T1,T2. Decidable (RN L T1 T2)) → + (∀L,T1,T2. Decidable (RP L T1 T2)) → + ∀L1,L2,f. Decidable (L1 ⦻*[RN, RP, f] L2). +#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #V1 #IH * ] +[ /2 width=1 by lexs_atom, or_introl/ +| #L2 #I2 #V2 #f @or_intror #H + lapply (lexs_inv_atom1 … H) -H #H destruct +| #f @or_intror #H + lapply (lexs_inv_atom2 … H) -H #H destruct +| #L2 #I2 #V2 #f elim (eq_bind2_dec I1 I2) #H destruct + [2: @or_intror #H elim (lexs_fwd_pair … H) -H /2 width=1 by/ ] + elim (IH L2 (⫱f)) -IH #HL12 + [2: @or_intror #H elim (lexs_fwd_pair … H) -H /2 width=1 by/ ] + elim (pn_split f) * #g #H destruct + [ elim (HRP L1 V1 V2) | elim (HRN L1 V1 V2) ] -HRP -HRN #HV12 + [1,3: /3 width=1 by lexs_push, lexs_next, or_introl/ ] + @or_intror #H + [ elim (lexs_inv_push … H) | elim (lexs_inv_next … H) ] -H + /2 width=1 by/ +] +qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfsx_lfsx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfsx_lfsx.ma index 7ca826d11..7e1d5cffb 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfsx_lfsx.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lfsx_lfsx.ma @@ -20,6 +20,7 @@ include "basic_2/rt_computation/lfsx.ma". (* Advanced properties ******************************************************) +(* Basic_2A1: was just: lsx_lleq_trans *) lemma lfsx_lfdeq_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ → ∀L2. L1 ≡[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄. #h #o #G #L1 #T #H @(lfsx_ind … H) -L1 @@ -28,6 +29,13 @@ lemma lfsx_lfdeq_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ → /4 width=5 by lfdeq_repl/ qed-. +(* Basic_2A1: was: lsx_lpx_trans *) +lemma lfsx_lfpx_trans: ∀h,o,G,L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ → + ∀L2. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄. +#h #o #G #L1 #T #H @(lfsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 +elim (lfdeq_dec h o L1 L2 T) /3 width=4 by lfsx_lfdeq_trans, lfxs_refl/ +qed-. + (* Advanced forward lemmas **************************************************) (* Basic_2A1: includes: lsx_fwd_bind_sn lsx_fwd_flat_sn *) @@ -40,7 +48,6 @@ lemma lfsx_fwd_pair_sn: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ②{I}V.T] 𝐒⦃L /6 width=4 by lfsx_lfdeq_trans, lfdeq_trans, lfdeq_fwd_pair_sn/ qed-. - (* Basic_2A1: was: lsx_fwd_flat_dx *) lemma lfsx_fwd_flat_dx: ∀h,o,I,G,L,V,T. G ⊢ ⬈*[h, o, ⓕ{I}V.T] 𝐒⦃L⦄ → G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄. diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx_lpx.ma deleted file mode 100644 index 81dc6e73d..000000000 --- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lsx_lpx.ma +++ /dev/null @@ -1,63 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||M|| *) -(* ||A|| A project by Andrea Asperti *) -(* ||T|| *) -(* ||I|| Developers: *) -(* ||T|| The HELM team. *) -(* ||A|| http://helm.cs.unibo.it *) -(* \ / *) -(* \ / This file is distributed under the terms of the *) -(* v GNU General Public License Version 2 *) -(* *) -(**************************************************************************) - -include "basic_2/multiple/lleq_lleq.ma". -include "basic_2/reduction/lpx_lleq.ma". -include "basic_2/computation/lsx.ma". - -(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************) - -(* Advanced properties ******************************************************) - -lemma lsx_lleq_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊*[h, o, T, l] L1 → - ∀L2. L1 ≡[T, l] L2 → G ⊢ ⬊*[h, o, T, l] L2. -#h #o #T #G #L1 #l #H @(lsx_ind … H) -L1 -#L1 #_ #IHL1 #L2 #HL12 @lsx_intro -#K2 #HLK2 #HnLK2 elim (lleq_lpx_trans … HLK2 … HL12) -HLK2 -/5 width=4 by lleq_canc_sn, lleq_trans/ -qed-. - -lemma lsx_lpx_trans: ∀h,o,T,G,L1,l. G ⊢ ⬊*[h, o, T, l] L1 → - ∀L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → G ⊢ ⬊*[h, o, T, l] L2. -#h #o #T #G #L1 #l #H @(lsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12 -elim (lleq_dec T L1 L2 l) /3 width=4 by lsx_lleq_trans/ -qed-. - -lemma lsx_lreq_conf: ∀h,o,G,L1,T,l. G ⊢ ⬊*[h, o, T, l] L1 → - ∀L2. L1 ⩬[l, ∞] L2 → G ⊢ ⬊*[h, o, T, l] L2. -#h #o #G #L1 #T #l #H @(lsx_ind … H) -L1 -#L1 #_ #IHL1 #L2 #HL12 @lsx_intro -#L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq … HL12 … HL23) -HL12 -HL23 -#L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/ -qed-. - -(* Advanced forward lemmas **************************************************) - -lemma lsx_fwd_bind_dx: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a,I}V.T, l] L → - G ⊢ ⬊*[h, o, T, ⫯l] L.ⓑ{I}V. -#h #o #a #I #G #L #V1 #T #l #H @(lsx_ind … H) -L -#L1 #_ #IHL1 @lsx_intro -#Y #H #HT elim (lpx_inv_pair1 … H) -H -#L2 #V2 #HL12 #_ #H destruct -@(lsx_lreq_conf … (L2.ⓑ{I}V1)) /2 width=1 by lreq_succ/ -@IHL1 // #H @HT -IHL1 -HL12 -HT -@(lleq_lreq_trans … (L2.ⓑ{I}V1)) -/2 width=2 by lleq_fwd_bind_dx, lreq_succ/ -qed-. - -(* Advanced inversion lemmas ************************************************) - -lemma lsx_inv_bind: ∀h,o,a,I,G,L,V,T,l. G ⊢ ⬊*[h, o, ⓑ{a, I}V.T, l] L → - G ⊢ ⬊*[h, o, V, l] L ∧ G ⊢ ⬊*[h, o, T, ⫯l] L.ⓑ{I}V. -/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma index aa6484ace..f6bc891d7 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq.ma @@ -176,11 +176,12 @@ lemma lfdeq_fwd_dx: ∀h,o,I,L1,K2,V2. ∀T:term. L1 ≡[h, o, T] K2.ⓑ{I}V2 ∃∃K1,V1. L1 = K1.ⓑ{I}V1. /2 width=5 by lfxs_fwd_dx/ qed-. -(* Basic_2A1: removed theorems 30: +(* Basic_2A1: removed theorems 31: lleq_ind lleq_inv_bind lleq_inv_flat lleq_fwd_length lleq_fwd_lref lleq_fwd_drop_sn lleq_fwd_drop_dx lleq_fwd_bind_sn lleq_fwd_bind_dx lleq_fwd_flat_sn lleq_fwd_flat_dx lleq_sort lleq_skip lleq_lref lleq_free lleq_gref lleq_bind lleq_flat lleq_refl lleq_Y lleq_sym lleq_ge_up lleq_ge lleq_bind_O llpx_sn_lrefl lleq_trans lleq_canc_sn lleq_canc_dx lleq_nlleq_trans nlleq_lleq_div + lleq_dec *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma index c42edd196..9f92ecd27 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma @@ -18,6 +18,11 @@ include "basic_2/static/lfdeq.ma". (* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******) +(* Advanced properties ******************************************************) + +lemma lfdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≡[h, o, T] L2). +/3 width=1 by lfxs_dec, tdeq_dec/ qed-. + (* Main properties **********************************************************) theorem lfdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T. diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma index 1d7781d7e..d7dae247e 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs.ma @@ -268,12 +268,13 @@ lemma lfxs_fwd_dx: ∀R,I,L1,K2,T,V2. L1 ⦻*[R, T] K2.ⓑ{I}V2 → /2 width=3 by ex1_2_intro/ qed-. -(* Basic_2A1: removed theorems 24: +(* Basic_2A1: removed theorems 25: llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref llpx_sn_bind llpx_sn_flat llpx_sn_inv_bind llpx_sn_inv_flat llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co - llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx + llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_dx + llpx_sn_dec *) diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma index 3b94bdb61..54fdb9e43 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma @@ -21,6 +21,19 @@ include "basic_2/static/lfxs.ma". (* Advanced properties ******************************************************) +lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 → + ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⦻*[R, cfull, f] L2. +#R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/ +qed-. + +lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) → + ∀L1,L2,T. Decidable (L1 ⦻*[R, T] L2). +#R #HR #L1 #L2 #T +elim (frees_total L1 T) #f #Hf +elim (lexs_dec R cfull HR … L1 L2 f) +/4 width=3 by lfxs_inv_frees, cfull_dec, ex2_intro, or_intror, or_introl/ +qed-. + lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) → lexs_frees_confluent … R1 cfull → ∀L1,L2,V. L1 ⦻*[R1, V] L2 → ∀I,T. diff --git a/matita/matita/contribs/lambdadelta/basic_2/syntax/lenv.ma b/matita/matita/contribs/lambdadelta/basic_2/syntax/lenv.ma index cfefa7fd8..05b8fde62 100644 --- a/matita/matita/contribs/lambdadelta/basic_2/syntax/lenv.ma +++ b/matita/matita/contribs/lambdadelta/basic_2/syntax/lenv.ma @@ -56,6 +56,9 @@ lemma eq_lenv_dec: ∀L1,L2:lenv. Decidable (L1 = L2). ] qed-. +lemma cfull_dec: ∀L,T1,T2. Decidable (cfull L T1 T2). +/2 width=1 by or_introl/ qed-. + (* Basic inversion lemmas ***************************************************) fact destruct_lpair_lpair_aux: ∀I1,I2,L1,L2,V1,V2. L1.ⓑ{I1}V1 = L2.ⓑ{I2}V2 → @@ -71,5 +74,5 @@ lemma discr_lpair_x_xy: ∀I,V,L. L = L.ⓑ{I}V → ⊥. ] qed-. -lemma discr_lpair_xy_x: ∀I,V,L. L.ⓑ{I}V = L→ ⊥. +lemma discr_lpair_xy_x: ∀I,V,L. L.ⓑ{I}V = L → ⊥. /2 width=4 by discr_lpair_x_xy/ qed-. -- 2.39.2