logrelnew #1
1 changed files with 55 additions and 0 deletions
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@ -379,3 +379,58 @@ Proof.
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have : DJoin.R A0 B by eauto using DJoin.transitive.
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eauto.
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Qed.
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Lemma InterpUniv_Functional n i (A : PTm n) PA PB :
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⟦ A ⟧ i ↘ PA ->
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⟦ A ⟧ i ↘ PB ->
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PA = PB.
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Proof. hauto use:InterpUniv_Join, DJoin.refl. Qed.
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Lemma InterpUniv_Join' n i j (A B : PTm n) PA PB :
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⟦ A ⟧ i ↘ PA ->
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⟦ B ⟧ j ↘ PB ->
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DJoin.R A B ->
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PA = PB.
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Proof.
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have [? ?] : i <= max i j /\ j <= max i j by lia.
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move => hPA hPB.
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have : ⟦ A ⟧ (max i j) ↘ PA by eauto using InterpUniv_cumulative.
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have : ⟦ B ⟧ (max i j) ↘ PB by eauto using InterpUniv_cumulative.
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eauto using InterpUniv_Join.
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Qed.
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Lemma InterpUniv_Functional' n i j A PA PB :
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⟦ A : PTm n ⟧ i ↘ PA ->
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⟦ A ⟧ j ↘ PB ->
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PA = PB.
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Proof.
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hauto l:on use:InterpUniv_Join', DJoin.refl.
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Qed.
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Lemma InterpUniv_Bind_inv_nopf n i p A B P (h : ⟦PBind p A B ⟧ i ↘ P) :
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exists (PA : PTm n -> Prop),
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⟦ A ⟧ i ↘ PA /\
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(forall a, PA a -> exists PB, ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB) /\
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P = BindSpace p PA (fun a PB => ⟦ subst_PTm (scons a VarPTm) B ⟧ i ↘ PB).
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Proof.
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move /InterpUniv_Bind_inv : h.
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move => [PA][PF][hPA][hPF][hPF']?. subst.
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exists PA. repeat split => //.
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- sfirstorder.
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- extensionality b.
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case : p => /=.
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+ extensionality a.
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extensionality PB.
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extensionality ha.
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apply propositional_extensionality.
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split.
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* move => h hPB. apply h.
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have {}/hPF := ha.
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move => [PB0 hPB0].
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have {}/hPF' := hPB0 => ?.
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have : PB = PB0 by hauto l:on use:InterpUniv_Functional.
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congruence.
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* sfirstorder.
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+ rewrite /SumSpace. apply propositional_extensionality.
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split; hauto q:on use:InterpUniv_Functional.
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Qed.
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