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Nicer proof of Permutation_app
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1 changed files with 36 additions and 33 deletions
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@ -166,7 +166,36 @@ Proof.
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equality.
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Qed.
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Lemma Permutation_app' : forall A (ls ls1 ls2 : list A),
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Lemma Permutation_refl : forall A (ls : list A),
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Permutation ls ls.
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Proof.
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induct ls.
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apply perm_nil.
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apply perm_skip.
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assumption.
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Qed.
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Lemma Permutation_app1 : forall A (ls1 ls2 ls : list A),
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Permutation ls1 ls2
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-> Permutation (ls1 ++ ls) (ls2 ++ ls).
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Proof.
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induct 1; simplify.
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apply Permutation_refl.
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apply perm_skip.
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apply IHPermutation.
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apply perm_swap.
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apply perm_trans with (l' ++ ls).
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apply IHPermutation1.
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apply IHPermutation2.
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Qed.
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Lemma Permutation_app2 : forall A (ls ls1 ls2 : list A),
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Permutation ls1 ls2
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-> Permutation (ls ++ ls1) (ls ++ ls2).
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Proof.
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@ -179,41 +208,15 @@ Proof.
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assumption.
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Qed.
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Lemma Permutation_refl : forall A (ls : list A),
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Permutation ls ls.
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Proof.
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induct ls.
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apply perm_nil.
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apply perm_skip.
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assumption.
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Qed.
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Theorem Permutation_app : forall A (ls1 ls1' : list A),
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Theorem Permutation_app : forall A (ls1 ls1' ls2 ls2' : list A),
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Permutation ls1 ls1'
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-> forall ls2 ls2', Permutation ls2 ls2'
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-> Permutation ls2 ls2'
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-> Permutation (ls1 ++ ls2) (ls1' ++ ls2').
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Proof.
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induct 1; simplify.
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simplify.
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apply perm_trans with (ls1' ++ ls2).
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apply Permutation_app1.
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assumption.
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apply perm_skip.
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apply IHPermutation.
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apply Permutation_app2.
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assumption.
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apply perm_trans with (x :: y :: l ++ ls2).
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apply perm_swap.
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apply perm_skip.
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apply perm_skip.
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apply Permutation_app'.
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assumption.
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apply perm_trans with (l' ++ ls2').
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apply IHPermutation1.
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assumption.
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apply IHPermutation2.
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apply Permutation_refl.
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Qed.
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