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Add no confusion lemmas

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Yiyun Liu 2025-02-23 14:58:26 -05:00
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theories/fp_red.v
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@ -2189,6 +2189,10 @@ Module RERed.
R a b -> isuniv a -> isuniv b. R a b -> isuniv a -> isuniv b.
Proof. hauto q:on inv:R. Qed. Proof. hauto q:on inv:R. Qed.
Lemma nat_preservation n (a b : PTm n) :
R a b -> isnat a -> isnat b.
Proof. hauto q:on inv:R. Qed.
Lemma sne_preservation n (a b : PTm n) : Lemma sne_preservation n (a b : PTm n) :
R a b -> SNe a -> SNe b. R a b -> SNe a -> SNe b.
Proof. Proof.
@ -2284,6 +2288,10 @@ Module REReds.
rtc RERed.R a b -> isuniv a -> isuniv b. rtc RERed.R a b -> isuniv a -> isuniv b.
Proof. induction 1; qauto l:on ctrs:rtc use:RERed.univ_preservation. Qed. Proof. induction 1; qauto l:on ctrs:rtc use:RERed.univ_preservation. Qed.
Lemma nat_preservation n (a b : PTm n) :
rtc RERed.R a b -> isnat a -> isnat b.
Proof. induction 1; qauto l:on ctrs:rtc use:RERed.nat_preservation. Qed.
Lemma sne_preservation n (a b : PTm n) : Lemma sne_preservation n (a b : PTm n) :
rtc RERed.R a b -> SNe a -> SNe b. rtc RERed.R a b -> SNe a -> SNe b.
Proof. induction 1; qauto l:on ctrs:rtc use:RERed.sne_preservation. Qed. Proof. induction 1; qauto l:on ctrs:rtc use:RERed.sne_preservation. Qed.
@ -2969,6 +2977,14 @@ Module DJoin.
sfirstorder use:@rtc_refl unfold:R. sfirstorder use:@rtc_refl unfold:R.
Qed. Qed.
Lemma sne_nat_noconf n (a b : PTm n) :
R a b -> SNe a -> isnat b -> False.
Proof.
move => [c [? ?]] *.
have : SNe c /\ isnat c by sfirstorder use:REReds.sne_preservation, REReds.nat_preservation.
qauto l:on inv:SNe.
Qed.
Lemma sne_bind_noconf n (a b : PTm n) : Lemma sne_bind_noconf n (a b : PTm n) :
R a b -> SNe a -> isbind b -> False. R a b -> SNe a -> isbind b -> False.
Proof. Proof.
@ -3016,6 +3032,14 @@ Module DJoin.
case : c => //=. case : c => //=.
Qed. Qed.
Lemma hne_nat_noconf n (a b : PTm n) :
R a b -> ishne a -> isnat b -> False.
Proof.
move => [c [h0 h1]] h2 h3.
have : ishne c /\ isnat c by sfirstorder use:REReds.hne_preservation, REReds.nat_preservation.
clear. case : c => //=; itauto.
Qed.
Lemma bind_inj n p0 p1 (A0 A1 : PTm n) B0 B1 : Lemma bind_inj n p0 p1 (A0 A1 : PTm n) B0 B1 :
DJoin.R (PBind p0 A0 B0) (PBind p1 A1 B1) -> DJoin.R (PBind p0 A0 B0) (PBind p1 A1 B1) ->
p0 = p1 /\ DJoin.R A0 A1 /\ DJoin.R B0 B1. p0 = p1 /\ DJoin.R A0 A1 /\ DJoin.R B0 B1.
@ -3404,6 +3428,22 @@ Module Sub.
@R n (PUniv i) (PUniv j). @R n (PUniv i) (PUniv j).
Proof. hauto lq:on ctrs:Sub1.R, rtc. Qed. Proof. hauto lq:on ctrs:Sub1.R, rtc. Qed.
Lemma sne_nat_noconf n (a b : PTm n) :
R a b -> SNe a -> isnat b -> False.
Proof.
move => [c [d [h0 [h1 h2]]]] *.
have : SNe c /\ isnat d by sfirstorder use:REReds.sne_preservation, REReds.nat_preservation, Sub1.sne_preservation.
move : h2. clear. hauto q:on inv:Sub1.R, SNe.
Qed.
Lemma nat_sne_noconf n (a b : PTm n) :
R a b -> isnat a -> SNe b -> False.
Proof.
move => [c [d [h0 [h1 h2]]]] *.
have : SNe d /\ isnat c by sfirstorder use:REReds.sne_preservation, REReds.nat_preservation.
move : h2. clear. hauto q:on inv:Sub1.R, SNe.
Qed.
Lemma sne_bind_noconf n (a b : PTm n) : Lemma sne_bind_noconf n (a b : PTm n) :
R a b -> SNe a -> isbind b -> False. R a b -> SNe a -> isbind b -> False.
Proof. Proof.
@ -3452,6 +3492,48 @@ Module Sub.
clear. case : c => //=. inversion 2. clear. case : c => //=. inversion 2.
Qed. Qed.
Lemma univ_nat_noconf n (a b : PTm n) :
R a b -> isuniv b -> isnat a -> False.
Proof.
move => [c[d] [? []]] h0 h1 h2 h3.
have : isuniv d by eauto using REReds.univ_preservation.
have : isnat c by sfirstorder use:REReds.nat_preservation.
inversion h1; subst => //=.
clear. case : d => //=.
Qed.
Lemma nat_univ_noconf n (a b : PTm n) :
R a b -> isnat b -> isuniv a -> False.
Proof.
move => [c[d] [? []]] h0 h1 h2 h3.
have : isuniv c by eauto using REReds.univ_preservation.
have : isnat d by sfirstorder use:REReds.nat_preservation.
inversion h1; subst => //=.
clear. case : d => //=.
Qed.
Lemma bind_nat_noconf n (a b : PTm n) :
R a b -> isbind b -> isnat a -> False.
Proof.
move => [c[d] [? []]] h0 h1 h2 h3.
have : isbind d by eauto using REReds.bind_preservation.
have : isnat c by sfirstorder use:REReds.nat_preservation.
move : h1. clear.
inversion 1; subst => //=.
case : d h1 => //=.
Qed.
Lemma nat_bind_noconf n (a b : PTm n) :
R a b -> isnat b -> isbind a -> False.
Proof.
move => [c[d] [? []]] h0 h1 h2 h3.
have : isbind c by eauto using REReds.bind_preservation.
have : isnat d by sfirstorder use:REReds.nat_preservation.
move : h1. clear.
inversion 1; subst => //=.
case : d h1 => //=.
Qed.
Lemma bind_univ_noconf n (a b : PTm n) : Lemma bind_univ_noconf n (a b : PTm n) :
R a b -> isbind a -> isuniv b -> False. R a b -> isbind a -> isuniv b -> False.
Proof. Proof.