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Nicer proof of Permutation_app

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Adam Chlipala 2021-02-28 11:02:46 -05:00
parent cf7d27417d
commit 7a906b1a63
1 changed files with 36 additions and 33 deletions
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69
RuleInduction.v
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@ -166,7 +166,36 @@ Proof.
equality.
Qed.
Lemma Permutation_app' : forall A (ls ls1 ls2 : list A),
Lemma Permutation_refl : forall A (ls : list A),
Permutation ls ls.
Proof.
induct ls.
apply perm_nil.
apply perm_skip.
assumption.
Qed.
Lemma Permutation_app1 : forall A (ls1 ls2 ls : list A),
Permutation ls1 ls2
-> Permutation (ls1 ++ ls) (ls2 ++ ls).
Proof.
induct 1; simplify.
apply Permutation_refl.
apply perm_skip.
apply IHPermutation.
apply perm_swap.
apply perm_trans with (l' ++ ls).
apply IHPermutation1.
apply IHPermutation2.
Qed.
Lemma Permutation_app2 : forall A (ls ls1 ls2 : list A),
Permutation ls1 ls2
-> Permutation (ls ++ ls1) (ls ++ ls2).
Proof.
@ -179,41 +208,15 @@ Proof.
assumption.
Qed.
Lemma Permutation_refl : forall A (ls : list A),
Permutation ls ls.
Proof.
induct ls.
apply perm_nil.
apply perm_skip.
assumption.
Qed.
Theorem Permutation_app : forall A (ls1 ls1' : list A),
Theorem Permutation_app : forall A (ls1 ls1' ls2 ls2' : list A),
Permutation ls1 ls1'
-> forall ls2 ls2', Permutation ls2 ls2'
-> Permutation ls2 ls2'
-> Permutation (ls1 ++ ls2) (ls1' ++ ls2').
Proof.
induct 1; simplify.
simplify.
apply perm_trans with (ls1' ++ ls2).
apply Permutation_app1.
assumption.
apply perm_skip.
apply IHPermutation.
apply Permutation_app2.
assumption.
apply perm_trans with (x :: y :: l ++ ls2).
apply perm_swap.
apply perm_skip.
apply perm_skip.
apply Permutation_app'.
assumption.
apply perm_trans with (l' ++ ls2').
apply IHPermutation1.
assumption.
apply IHPermutation2.
apply Permutation_refl.
Qed.