diff --git a/FirstClassFunctions.v b/FirstClassFunctions.v index 76c9378..138d0c1 100644 --- a/FirstClassFunctions.v +++ b/FirstClassFunctions.v @@ -732,8 +732,6 @@ Fixpoint allSublists {A} (ls : list A) : list (list A) := Compute allSublists [1; 2; 3]. -Definition sum ls := fold_left plus ls 0. - (* This is the main function we want to define. It looks for a sublist whose * sum matches some target. *) Fixpoint sublistSummingTo (ns : list nat) (target : nat) : option (list nat) := @@ -1075,16 +1073,16 @@ Fixpoint flattenKD {A} (fuel : nat) (t : tree A) (acc : list A) (* A somewhat peculiar notion of size for trees. Why that 2 instead of 1? * Because it lets the proof below work out! *) -Fixpoint size {A} (t : tree A) : nat := +Fixpoint tree_size {A} (t : tree A) : nat := match t with | Leaf => 0 - | Node l _ r => 2 + size l + size r + | Node l _ r => 2 + tree_size l + tree_size r end. Fixpoint continuation_size {A} (k : flatten_continuation A) : nat := match k with | KDone => 0 - | KMore l d k' => 1 + size l + continuation_size k' + | KMore l d k' => 1 + tree_size l + continuation_size k' end. (* A continuation encodes a flattening call, waiting to be run. @@ -1100,7 +1098,7 @@ Fixpoint flatten_cont {A} (k : flatten_continuation A) : list A := * *strong induction* via the parameter [fuel], which bounds the actual fuel * amount [fuel']. *) Lemma flattenKD_ok' : forall {A} fuel fuel' (t : tree A) acc k, - size t + continuation_size k < fuel' < fuel + tree_size t + continuation_size k < fuel' < fuel -> flattenKD fuel' t acc k = flatten_cont k ++ flatten t ++ acc. Proof. @@ -1126,10 +1124,10 @@ Qed. (* A nice, simple final theorem can be stated, when we initialize fuel in the * right way. *) Theorem flattenKD_ok : forall {A} (t : tree A), - flattenKD (size t + 1) t [] KDone = flatten t. + flattenKD (tree_size t + 1) t [] KDone = flatten t. Proof. simplify. - rewrite flattenKD_ok' with (fuel := size t + 2). + rewrite flattenKD_ok' with (fuel := tree_size t + 2). simplify. apply app_nil_r. simplify. @@ -1191,7 +1189,7 @@ Proof. Qed. Theorem flattenS_ok : forall {A} (t : tree A), - flattenS (size t + 1) t [] [] = flatten t. + flattenS (tree_size t + 1) t [] [] = flatten t. Proof. simplify. rewrite flattenS_flattenKD.