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More examples for first chapter

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Adam Chlipala 2016-01-16 20:32:12 -05:00
parent f8945106da
commit 42d48c6e58
2 changed files with 115 additions and 2 deletions
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111
BasicSyntax.v
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@ -247,4 +247,115 @@ Module ArithWithVariables.
end; equality || linear_arithmetic || ring.
Qed.
Definition isConst (e : arith) : option nat :=
match e with
| Const n => Some n
| _ => None
end.
Fixpoint pushMultiplicationInside' (multiplyBy : nat) (e : arith) : arith :=
match e with
| Const n => Const (multiplyBy * n)
| Var _ => Times (Const multiplyBy) e
| Plus e1 e2 => Plus (pushMultiplicationInside' multiplyBy e1)
(pushMultiplicationInside' multiplyBy e2)
| Times e1 e2 =>
match isConst e1 with
| Some k => pushMultiplicationInside' (k * multiplyBy) e2
| None => Times (pushMultiplicationInside' multiplyBy e1) e2
end
end.
Definition pushMultiplicationInside (e : arith) : arith :=
pushMultiplicationInside' 1 e.
Lemma n_times_0 : forall n, n * 0 = 0.
Proof.
linear_arithmetic.
Qed.
Lemma depth_pushMultiplicationInside'_irrelevance0 : forall e multiplyBy,
depth (pushMultiplicationInside' multiplyBy e)
= depth (pushMultiplicationInside' 0 e).
Proof.
induct e; simplify.
linear_arithmetic.
linear_arithmetic.
rewrite IHe1.
rewrite IHe2.
linear_arithmetic.
cases (isConst e1); simplify.
rewrite IHe2.
rewrite n_times_0.
linear_arithmetic.
rewrite IHe1.
linear_arithmetic.
Qed.
Lemma depth_pushMultiplicationInside'_irrelevance0_snazzy : forall e multiplyBy,
depth (pushMultiplicationInside' multiplyBy e)
= depth (pushMultiplicationInside' 0 e).
Proof.
induct e; simplify;
try match goal with
| [ |- context[match ?E with _ => _ end] ] => cases E; simplify
end; equality.
Qed.
Lemma depth_pushMultiplicationInside'_irrelevance : forall e multiplyBy1 multiplyBy2,
depth (pushMultiplicationInside' multiplyBy1 e)
= depth (pushMultiplicationInside' multiplyBy2 e).
Proof.
intros.
transitivity (depth (pushMultiplicationInside' 0 e)).
apply depth_pushMultiplicationInside'_irrelevance0.
symmetry.
apply depth_pushMultiplicationInside'_irrelevance0.
Qed.
Lemma depth_pushMultiplicationInside' : forall e,
depth (pushMultiplicationInside' 0 e) <= S (depth e).
Proof.
induct e; simplify.
linear_arithmetic.
linear_arithmetic.
linear_arithmetic.
cases (isConst e1); simplify.
rewrite n_times_0.
linear_arithmetic.
linear_arithmetic.
Qed.
Hint Rewrite n_times_0.
Lemma depth_pushMultiplicationInside'_snazzy : forall e,
depth (pushMultiplicationInside' 0 e) <= S (depth e).
Proof.
induct e; simplify;
try match goal with
| [ |- context[match ?E with _ => _ end] ] => cases E; simplify
end; linear_arithmetic.
Qed.
Theorem depth_pushMultiplicationInside : forall e,
depth (pushMultiplicationInside e) <= S (depth e).
Proof.
simplify.
unfold pushMultiplicationInside.
rewrite depth_pushMultiplicationInside'_irrelevance0.
apply depth_pushMultiplicationInside'.
Qed.
End ArithWithVariables.

6
Frap.v
View file

@ -42,9 +42,9 @@ Ltac invert0 e := invert e; fail.
Ltac invert1 e := invert0 e || (invert e; []).
Ltac invert2 e := invert1 e || (invert e; [|]).
Ltac simplify := simpl in *.
Ltac simplify := simpl in *; intros; try autorewrite with core in *.
Ltac linear_arithmetic :=
Ltac linear_arithmetic := intros;
repeat match goal with
| [ |- context[max ?a ?b] ] =>
let Heq := fresh "Heq" in destruct (Max.max_spec a b) as [[? Heq] | [? Heq]];
@ -68,3 +68,5 @@ Ltac cases E :=
| {_} + {_} => destruct E
| _ => let Heq := fresh "Heq" in destruct E eqn:Heq
end.
Global Opaque max min.