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Add some more injection lemmas for neutrals

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Yiyun Liu 2025-02-14 20:41:56 -05:00
parent 186f2138e6
commit 8d765c495d
2 changed files with 137 additions and 1 deletions
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53
theories/algorithmic.v
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@ -567,6 +567,53 @@ Proof.
- hauto lq:on ctrs:nsteps inv:LoRed.R.
Qed.
Lemma lored_hne_preservation n (a b : PTm n) :
LoRed.R a b -> ishne a -> ishne b.
Proof. induction 1; sfirstorder. Qed.
Lemma lored_nsteps_app_inv k n (a0 b0 C : PTm n) :
nsteps LoRed.R k (PApp a0 b0) C ->
ishne a0 ->
exists i j a1 b1,
i <= k /\ j <= k /\
C = PApp a1 b1 /\
nsteps LoRed.R i a0 a1 /\
nsteps LoRed.R j b0 b1.
Proof.
move E : (PApp a0 b0) => u hu. move : a0 b0 E.
elim : k u C / hu.
- sauto lq:on.
- move => k a0 a1 a2 ha01 ha12 ih a3 b0 ?. subst.
inversion ha01; subst => //=.
spec_refl.
move => h.
have : ishne a4 by sfirstorder use:lored_hne_preservation.
move : ih => /[apply]. move => [i][j][a1][b1][?][?][?][h0]h1.
subst. exists (S i),j,a1,b1. sauto lq:on solve+:lia.
spec_refl. move : ih => /[apply].
move => [i][j][a1][b1][?][?][?][h0]h1. subst.
exists i, (S j), a1, b1. sauto lq:on solve+:lia.
Qed.
Lemma algo_metric_app n k (a0 b0 a1 b1 : PTm n) :
algo_metric k (PApp a0 b0) (PApp a1 b1) ->
ishne a0 ->
ishne a1 ->
exists j, j < k /\ algo_metric j a0 a1 /\ algo_metric j b0 b1.
Proof.
move => [i][j][va][vb][v][h0][h1][h2][h3][h4][h5]h6.
move => hne0 hne1.
move : lored_nsteps_app_inv h0 (hne0);repeat move/[apply].
move => [i0][i1][a2][b2][?][?][?][ha02]hb02. subst.
move : lored_nsteps_app_inv h1 (hne1);repeat move/[apply].
move => [j0][j1][a3][b3][?][?][?][ha13]hb13. subst.
simpl in *. exists (k - 1).
split. lia.
split.
+ rewrite /algo_metric.
have : exists a4 b4, rtc ERed.R a2 a4 /\ ERed.R
exists i0,i1,a2,b2.
Lemma algo_metric_join n k (a b : PTm n) :
algo_metric k a b ->
DJoin.R a b.
@ -663,6 +710,12 @@ Proof.
* move => b1 a1 b0 a0 halg hne1 hne0 Γ A B wtA wtB.
move /App_Inv : wtA => [A0][B0][hb0][ha0]hS0.
move /App_Inv : wtB => [A1][B1][hb1][ha1]hS1.
have : DJoin.R (PApp b0 a0) (PApp b1 a1)
by hauto l:on use:algo_metric_join.
move : DJoin.hne_app_inj (hne0) (hne1). repeat move/[apply].
move => [hjb hja].
split. apply CE_AppCong => //=.
eapply ih; eauto.
admit.
* admit.
* sfirstorder use:T_Bot_Imp.