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Polymorphism: syntax trees
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188
Polymorphism.v
188
Polymorphism.v
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@ -414,3 +414,191 @@ Proof.
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simplify.
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equality.
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Qed.
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(** * Syntax trees *)
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(* Trees are particularly important to us in studying program proof, since it is
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* natural to represent programs as *syntax trees*. Here's a quick example, for
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* a tiny imperative language. *)
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Inductive expression : Set :=
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| Const (n : nat)
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| Var (x : var)
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| Plus (e1 e2 : expression)
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| Minus (e1 e2 : expression)
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| Times (e1 e2 : expression)
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| GreaterThan (e1 e2 : expression)
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| Not (e : expression).
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Inductive statement : Set :=
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| Assign (x : var) (e : expression)
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| Sequence (s1 s2 : statement)
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| IfThenElse (e : expression) (s1 s2 : statement)
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| WhileLoop (e : expression) (s : statement).
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(* First, here's a quick sample of nifty notations to write
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* almost-natural-looking embedded programs in Coq. *)
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Coercion Const : nat >-> expression.
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Coercion Var : string >-> expression.
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Infix "+" := Plus : embedded_scope.
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Infix "-" := Minus : embedded_scope.
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Infix "*" := Times : embedded_scope.
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Infix ">" := GreaterThan : embedded_scope.
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Infix "<-" := Assign (at level 75) : embedded_scope.
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Infix ";" := Sequence (at level 76) : embedded_scope.
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Notation "'If' e {{ s1 }} 'else' {{ s2 }}" := (IfThenElse e s1 s2) (at level 75) : embedded_scope.
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Notation "'While' e {{ s }}" := (WhileLoop e s) (at level 75) : embedded_scope.
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Delimit Scope embedded_scope with embedded.
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Example factorial :=
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("answer" <- 1;
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While ("input" > 0) {{
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"answer" <- "answer" * "input";
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"input" <- "input" - 1
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}})%embedded.
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(* A variety of compiler-style operations can be coded on top of this type.
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* Here's one to count total variable occurrences. *)
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Fixpoint varsInExpression (e : expression) : nat :=
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match e with
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| Const _ => 0
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| Var _ => 1
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| Plus e1 e2
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| Minus e1 e2
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| Times e1 e2
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| GreaterThan e1 e2 => varsInExpression e1 + varsInExpression e2
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| Not e1 => varsInExpression e1
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end.
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Fixpoint varsInStatement (s : statement) : nat :=
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match s with
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| Assign _ e => 1 + varsInExpression e
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| Sequence s1 s2 => varsInStatement s1 + varsInStatement s2
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| IfThenElse e s1 s2 => varsInExpression e + varsInStatement s1 + varsInStatement s2
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| WhileLoop e s1 => varsInExpression e + varsInStatement s1
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end.
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(* We will need to wait for a few more lectures' worth of conceptual progress
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* before we can prove that transformations on programs preserve meaning, but we
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* do already have enough tools that prove that transformations preserve more
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* basic properties, like number of variables. Here's one such transformation,
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* which flips "then" and "else" cases while also negating "if" conditions. *)
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Fixpoint flipper (s : statement) : statement :=
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match s with
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| Assign _ _ => s
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| Sequence s1 s2 => Sequence (flipper s1) (flipper s2)
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| IfThenElse e s1 s2 => IfThenElse (Not e) (flipper s2) (flipper s1)
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| WhileLoop e s1 => WhileLoop e (flipper s1)
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end.
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Theorem varsIn_flipper : forall s,
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varsInStatement (flipper s) = varsInStatement s.
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Proof.
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induct s; simplify.
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equality.
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equality.
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linear_arithmetic.
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equality.
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Qed.
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(* Just for the sheer madcap fun of it, let's write some translations of
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* programs into our lists from before, with variables as data values. *)
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Fixpoint listifyExpression (e : expression) : list var :=
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match e with
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| Const _ => []
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| Var x => [x]
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| Plus e1 e2
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| Minus e1 e2
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| Times e1 e2
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| GreaterThan e1 e2 => listifyExpression e1 ++ listifyExpression e2
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| Not e1 => listifyExpression e1
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end.
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Fixpoint listifyStatement (s : statement) : list var :=
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match s with
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| Assign x e => x :: listifyExpression e
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| Sequence s1 s2 => listifyStatement s1 ++ listifyStatement s2
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| IfThenElse e s1 s2 => listifyExpression e ++ listifyStatement s1 ++ listifyStatement s2
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| WhileLoop e s1 => listifyExpression e ++ listifyStatement s1
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end.
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Compute listifyStatement factorial.
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(* At this point, I can't resist switching to a more automated proof style,
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* though still a pretty tame one. *)
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Hint Rewrite length_app.
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Lemma length_listifyExpression : forall e,
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length (listifyExpression e) = varsInExpression e.
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Proof.
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induct e; simplify; linear_arithmetic.
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Qed.
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Hint Rewrite length_listifyExpression.
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Theorem length_listifyStatement : forall s,
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length (listifyStatement s) = varsInStatement s.
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Proof.
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induct s; simplify; linear_arithmetic.
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Qed.
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(* Other transformations are also possible, like the Swedish-Chef optimization,
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* that turns every variable into "bork". It saves many bits when most variable
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* names are longer than 4 characters. *)
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Fixpoint swedishExpression (e : expression) : expression :=
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match e with
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| Const _ => e
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| Var _ => Var "bork"
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| Plus e1 e2 => Plus (swedishExpression e1) (swedishExpression e2)
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| Minus e1 e2 => Minus (swedishExpression e1) (swedishExpression e2)
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| Times e1 e2 => Times (swedishExpression e1) (swedishExpression e2)
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| GreaterThan e1 e2 => GreaterThan (swedishExpression e1) (swedishExpression e2)
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| Not e1 => Not (swedishExpression e1)
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end.
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Fixpoint swedishStatement (s : statement) : statement :=
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match s with
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| Assign _ e => Assign "bork" (swedishExpression e)
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| Sequence s1 s2 => Sequence (swedishStatement s1) (swedishStatement s2)
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| IfThenElse e s1 s2 => IfThenElse (swedishExpression e) (swedishStatement s1) (swedishStatement s2)
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| WhileLoop e s1 => WhileLoop (swedishExpression e) (swedishStatement s1)
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end.
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Compute swedishStatement factorial.
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Fixpoint swedishList (ls : list var) : list var :=
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match ls with
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| [] => []
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| _ :: ls => "bork" :: swedishList ls
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end.
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Lemma swedishList_app : forall ls1 ls2,
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swedishList (ls1 ++ ls2) = swedishList ls1 ++ swedishList ls2.
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Proof.
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induct ls1; simplify; equality.
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Qed.
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Hint Rewrite swedishList_app.
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Lemma listifyExpression_swedishExpression : forall e,
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listifyExpression (swedishExpression e) = swedishList (listifyExpression e).
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Proof.
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induct e; simplify; equality.
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Qed.
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Hint Rewrite listifyExpression_swedishExpression.
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Lemma listifyStatement_swedishStatement : forall s,
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listifyStatement (swedishStatement s) = swedishList (listifyStatement s).
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Proof.
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induct s; simplify; equality.
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Qed.
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@ -10,6 +10,7 @@ AbstractInterpret.v
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Frap.v
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BasicSyntax_template.v
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BasicSyntax.v
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Polymorphism.v
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Interpreters_template.v
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Interpreters.v
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TransitionSystems_template.v
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