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add definition and result about regexp derivatives #73

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Jul 22, 2024
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add definition and result about regexp derivatives #73

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bfe0e6d
add definition and result about regexp derivatives, rename cat-based …
palmskog Jul 22, 2024
6b1a61b
remove nix-action-master CI for now
palmskog Jul 22, 2024
eb70236
rename residual_cat to residual_lang
palmskog Jul 22, 2024
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309 changes: 0 additions & 309 deletions .github/workflows/nix-action-master.yml
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16 changes: 8 additions & 8 deletions theories/dfa.v
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Original file line number Diff line number Diff line change
Expand Up @@ -421,18 +421,18 @@ Section NonRegular.

Implicit Types (L : lang char).

Definition residual L x := fun y => L (x ++ y).
Definition residual_lang L x := fun y => L (x ++ y).

Lemma residualP (f : nat -> word char) (L : lang char) :
(forall n1 n2, residual L (f n1) =p residual L (f n2) -> n1 = n2) ->
Lemma residual_langP (f : nat -> word char) (L : lang char) :
(forall n1 n2, residual_lang L (f n1) =p residual_lang L (f n2) -> n1 = n2) ->
~ inhabited (regular L).
Proof.
move => f_spec [[A E]].
pose f' (n : 'I_#|A|.+1) := delta_s A (f n).
suff: injective f' by move/leq_card; rewrite card_ord ltnn.
move => [n1 H1] [n2 H2]. rewrite /f' /delta_s /= => H.
pose g (n : 'I_#|A|.+1) := delta_s A (f n).
suff: injective g by move/leq_card; rewrite card_ord ltnn.
move => [n1 H1] [n2 H2]. rewrite /g /delta_s /= => H.
apply/eqP; change (n1 == n2); apply/eqP. apply: f_spec => w.
by rewrite /residual !E !inE /dfa_accept !delta_cat H.
by rewrite /residual_lang !E !inE /dfa_accept !delta_cat H.
Qed.

Hypothesis (a b : char) (Hab : a != b).
Expand All @@ -457,7 +457,7 @@ Section NonRegular.
Lemma Lab_not_regular : ~ inhabited (regular Lab).
Proof.
pose f n := nseq n a.
apply: (@residualP f) => n1 n2. move/(_ (nseq n2 b)) => H.
apply: (@residual_langP f) => n1 n2. move/(_ (nseq n2 b)) => H.
apply: Lab_eq. apply/H. by exists n2.
Qed.

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