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Revising for Wednesday's lecture

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Adam Chlipala 2022-03-06 14:47:05 -05:00
parent eb53c0d714
commit 955fd59327
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38
ProofByReflection.v
View file

@ -50,7 +50,7 @@ Unset Printing All.
* theorem. This Ltac procedure always works (at least on machines with * theorem. This Ltac procedure always works (at least on machines with
* infinite resources), but it has a serious drawback, which we see when we * infinite resources), but it has a serious drawback, which we see when we
* print the proof it generates that 256 is even. The final proof term has * print the proof it generates that 256 is even. The final proof term has
* length super-linear in the input value, which we reveal with * length superlinear in the input value, which we reveal with
* [Set Printing All], to disable all syntactic niceties and show every node of * [Set Printing All], to disable all syntactic niceties and show every node of
* the internal proof AST. The problem is that each [Even_SS] application needs * the internal proof AST. The problem is that each [Even_SS] application needs
* a choice of [n], and we wind up giving every even number from 0 to 254 in * a choice of [n], and we wind up giving every even number from 0 to 254 in
@ -59,7 +59,7 @@ Unset Printing All.
* It is also unfortunate not to have static-typing guarantees that our tactic * It is also unfortunate not to have static-typing guarantees that our tactic
* always behaves appropriately. Other invocations of similar tactics might * always behaves appropriately. Other invocations of similar tactics might
* fail with dynamic type errors, and we would not know about the bugs behind * fail with dynamic type errors, and we would not know about the bugs behind
* these errors until we happened to attempt to prove complex enough goals. * these errors until we happened to attempt to prove complex-enough goals.
* *
* The techniques of proof by reflection address both complaints. We will be * The techniques of proof by reflection address both complaints. We will be
* able to write proofs like in the example above with constant size overhead * able to write proofs like in the example above with constant size overhead
@ -290,9 +290,9 @@ Section monoid.
Fixpoint mdenote (me : mexp) : A := Fixpoint mdenote (me : mexp) : A :=
match me with match me with
| Ident => e | Ident => e
| Var v => v | Var v => v
| Op me1 me2 => mdenote me1 + mdenote me2 | Op me1 me2 => mdenote me1 + mdenote me2
end. end.
(* We will normalize expressions by flattening them into lists, via (* We will normalize expressions by flattening them into lists, via
@ -301,17 +301,17 @@ Section monoid.
Fixpoint mldenote (ls : list A) : A := Fixpoint mldenote (ls : list A) : A :=
match ls with match ls with
| nil => e | nil => e
| x :: ls' => x + mldenote ls' | x :: ls' => x + mldenote ls'
end. end.
(* The flattening function itself is easy to implement. *) (* The flattening function itself is easy to implement. *)
Fixpoint flatten (me : mexp) : list A := Fixpoint flatten (me : mexp) : list A :=
match me with match me with
| Ident => [] | Ident => []
| Var x => [x] | Var x => [x]
| Op me1 me2 => flatten me1 ++ flatten me2 | Op me1 me2 => flatten me1 ++ flatten me2
end. end.
(* This function has a straightforward correctness proof in terms of our (* This function has a straightforward correctness proof in terms of our
@ -344,12 +344,12 @@ Section monoid.
Ltac reify me := Ltac reify me :=
match me with match me with
| e => Ident | e => Ident
| ?me1 + ?me2 => | ?me1 + ?me2 =>
let r1 := reify me1 in let r1 := reify me1 in
let r2 := reify me2 in let r2 := reify me2 in
constr:(Op r1 r2) constr:(Op r1 r2)
| _ => constr:(Var me) | _ => constr:(Var me)
end. end.
(* The final [monoid] tactic works on goals that equate two monoid terms. We (* The final [monoid] tactic works on goals that equate two monoid terms. We
@ -358,11 +358,11 @@ Section monoid.
Ltac monoid := Ltac monoid :=
match goal with match goal with
| [ |- ?me1 = ?me2 ] => | [ |- ?me1 = ?me2 ] =>
let r1 := reify me1 in let r1 := reify me1 in
let r2 := reify me2 in let r2 := reify me2 in
change (mdenote r1 = mdenote r2); change (mdenote r1 = mdenote r2);
apply monoid_reflect; simplify apply monoid_reflect; simplify
end. end.
(* We can make short work of theorems like this one: *) (* We can make short work of theorems like this one: *)
@ -416,7 +416,7 @@ Proof.
apply multiStepClosure_ok. apply multiStepClosure_ok.
simplify. simplify.
(* Here we'll see that the Frap libary uses slightly different, optimized (* Here we'll see that the Frap library uses slightly different, optimized
* versions of the model-checking relations. For instance, [multiStepClosure] * versions of the model-checking relations. For instance, [multiStepClosure]
* takes an extra set argument, the _worklist_ recording newly discovered * takes an extra set argument, the _worklist_ recording newly discovered
* states. There is no point in following edges out of states that were * states. There is no point in following edges out of states that were
@ -604,8 +604,8 @@ Section my_tauto.
Fixpoint allTrue (s : set propvar) : Prop := Fixpoint allTrue (s : set propvar) : Prop :=
match s with match s with
| nil => True | nil => True
| v :: s' => atomics v /\ allTrue s' | v :: s' => atomics v /\ allTrue s'
end. end.
Theorem allTrue_add : forall v s, Theorem allTrue_add : forall v s,

2
ProofByReflection_template.v
View file

@ -171,7 +171,7 @@ Proof.
apply multiStepClosure_ok. apply multiStepClosure_ok.
simplify. simplify.
(* Here we'll see that the Frap libary uses slightly different, optimized (* Here we'll see that the Frap library uses slightly different, optimized
* versions of the model-checking relations. For instance, [multiStepClosure] * versions of the model-checking relations. For instance, [multiStepClosure]
* takes an extra set argument, the _worklist_ recording newly discovered * takes an extra set argument, the _worklist_ recording newly discovered
* states. There is no point in following edges out of states that were * states. There is no point in following edges out of states that were