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Revising for this week's lectures
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@ -141,13 +141,10 @@ Section ilist.
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| Cons _ h _ => h
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end.
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(* Unlike in ML, we cannot use inexhaustive pattern matching, because there is
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* no conception of a <<Match>> exception to be thrown. In fact, recent
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* versions of Coq _do_ allow this, by implicit translation to a [match] that
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* considers all constructors; the error message above was generated by an
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* older Coq version. It is educational to discover for ourselves the
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* encoding that the most recent Coq versions use. We might try using an [in]
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* clause somehow. *)
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(* Actually, these days, Coq is smart enough to make that definition work!
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* However, it will be educational to look at how Coq elaborates this code
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* into its core language, where, unlike in ML, all pattern matching must be
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* _exhaustive_. We might try using an [in] clause somehow. *)
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Fail Fail Definition hd n (ls : ilist (S n)) : A :=
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match ls in (ilist (S n)) with
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@ -173,13 +170,22 @@ Section ilist.
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Check hd'.
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Definition hd n (ls : ilist (S n)) : A := hd' ls.
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End ilist.
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(* We annotate our main [match] with a type that is itself a [match]. We write
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* that the function [hd'] returns [unit] when the list is empty and returns the
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* carried type [A] in all other cases. In the definition of [hd], we just call
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* [hd']. Because the index of [ls] is known to be nonzero, the type checker
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* reduces the [match] in the type of [hd'] to [A]. *)
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(* We annotate our main [match] with a type that is itself a [match]. We
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* write that the function [hd'] returns [unit] when the list is empty and
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* returns the carried type [A] in all other cases. In the definition of [hd],
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* we just call [hd']. Because the index of [ls] is known to be nonzero, the
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* type checker reduces the [match] in the type of [hd'] to [A]. *)
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(* In fact, when we "got lucky" earlier with Coq accepting simpler
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* definitions, under the hood it was desugaring _almost_ to this one. *)
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Definition easy_hd n (ls : ilist (S n)) : A :=
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match ls with
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| Cons _ h _ => h
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end.
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Print easy_hd.
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End ilist.
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(** * The One Rule of Dependent Pattern Matching in Coq *)
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@ -330,7 +336,7 @@ Fixpoint expDenote t (e : exp t) : typeDenote t :=
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match e with
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| NConst n => n
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| Plus e1 e2 => expDenote e1 + expDenote e2
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| Eq e1 e2 => if eq_nat_dec (expDenote e1) (expDenote e2) then true else false
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| Eq e1 e2 => if expDenote e1 ==n expDenote e2 then true else false
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| BConst b => b
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| And e1 e2 => expDenote e1 && expDenote e2
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@ -345,10 +351,10 @@ Fixpoint expDenote t (e : exp t) : typeDenote t :=
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* less complicated than what we would write in ML or Haskell 98, since we do
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* not need to worry about pushing final values in and out of an algebraic
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* datatype. The only unusual thing is the use of an expression of the form
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* [if E then true else false] in the [Eq] case. Remember that [eq_nat_dec] has
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* [if E then true else false] in the [Eq] case. Remember that [==n] has
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* a rich dependent type, rather than a simple Boolean type. Coq's native [if]
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* is overloaded to work on a test of any two-constructor type, so we can use
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* [if] to build a simple Boolean from the [sumbool] that [eq_nat_dec] returns.
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* [if] to build a simple Boolean from the [sumbool] that [==n] returns.
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*
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* We can implement our old favorite, a constant-folding function, and prove it
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* correct. It will be useful to write a function [pairOut] that checks if an
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@ -387,10 +393,7 @@ Definition pairOut t (e : exp t) :=
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(* With [pairOut] available, we can write [cfold] in a straightforward way.
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* There are really no surprises beyond that Coq verifies that this code has
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* such an expressive type, given the small annotation burden. In some places,
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* we see that Coq's [match] annotation inference is too smart for its own
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* good, and we have to turn that inference off with explicit [return]
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* clauses. *)
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* such an expressive type, given the small annotation burden. *)
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Fixpoint cfold t (e : exp t) : exp t :=
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match e with
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@ -398,14 +401,14 @@ Fixpoint cfold t (e : exp t) : exp t :=
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| Plus e1 e2 =>
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let e1' := cfold e1 in
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let e2' := cfold e2 in
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match e1', e2' return exp Nat with
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match e1', e2' with
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| NConst n1, NConst n2 => NConst (n1 + n2)
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| _, _ => Plus e1' e2'
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end
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| Eq e1 e2 =>
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let e1' := cfold e1 in
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let e2' := cfold e2 in
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match e1', e2' return exp Bool with
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match e1', e2' with
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| NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
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| _, _ => Eq e1' e2'
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end
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@ -414,7 +417,7 @@ Fixpoint cfold t (e : exp t) : exp t :=
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| And e1 e2 =>
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let e1' := cfold e1 in
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let e2' := cfold e2 in
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match e1', e2' return exp Bool with
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match e1', e2' with
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| BConst b1, BConst b2 => BConst (b1 && b2)
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| _, _ => And e1' e2'
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end
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@ -488,9 +491,6 @@ Proof.
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end; simplify); try equality.
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Qed.
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(* With this example, we get a first taste of how to build automated proofs that
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* adapt automatically to changes in function definitions. *)
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(** * Interlude: The Convoy Pattern *)
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@ -676,7 +676,7 @@ Section present.
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End present.
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(* Insertion relies on two balancing operations. It will be useful to give types
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* to these operations using a relative of the subset types from last chapter.
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* to these operations using a relative of the subset types from SubsetTypes.
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* While subset types let us pair a value with a proof about that value, here we
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* want to pair a value with another non-proof dependently typed value. The
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* [sigT] type fills this role. *)
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@ -1071,7 +1071,7 @@ Ltac substring :=
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destruct N; simplify
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end; try linear_arithmetic; eauto; try equality.
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Hint Resolve le_n_S : core.
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Local Hint Resolve le_n_S : core.
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Lemma substring_le : forall s n m,
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length (substring n m s) <= m.
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@ -1105,7 +1105,7 @@ Proof.
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induct s1; substring.
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Qed.
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Hint Resolve length_emp append_emp substring_le substring_split length_app1 : core.
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Local Hint Resolve length_emp append_emp substring_le substring_split length_app1 : core.
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Lemma substring_app_fst : forall s2 s1 n,
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length s1 = n
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@ -1151,7 +1151,7 @@ End sumbool_and.
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Infix "&&" := sumbool_and (at level 40, left associativity).
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Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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Local Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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Section split.
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Variables P1 P2 : string -> Prop.
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@ -1253,7 +1253,7 @@ Proof.
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induct s; substring.
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Qed.
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Hint Extern 1 (String _ _ = String _ _) => f_equal : core.
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Local Hint Extern 1 (String _ _ = String _ _) => f_equal : core.
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Lemma substring_stack : forall s n2 m1 m2,
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m1 <= m2
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@ -1507,7 +1507,7 @@ Proof.
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equality.
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Qed.
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Hint Resolve app_cong : core.
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Local Hint Resolve app_cong : core.
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(* With these helper functions completed, the implementation of our [matches]
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* function is refreshingly straightforward. *)
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@ -91,14 +91,14 @@ Fixpoint cfold t (e : exp t) : exp t :=
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| Plus e1 e2 =>
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let e1' := cfold e1 in
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let e2' := cfold e2 in
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match e1', e2' return exp Nat with
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match e1', e2' with
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| NConst n1, NConst n2 => NConst (n1 + n2)
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| _, _ => Plus e1' e2'
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end
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| Eq e1 e2 =>
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let e1' := cfold e1 in
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let e2' := cfold e2 in
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match e1', e2' return exp Bool with
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match e1', e2' with
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| NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
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| _, _ => Eq e1' e2'
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end
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| And e1 e2 =>
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let e1' := cfold e1 in
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let e2' := cfold e2 in
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match e1', e2' return exp Bool with
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match e1', e2' with
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| BConst b1, BConst b2 => BConst (b1 && b2)
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| _, _ => And e1' e2'
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end
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destruct N; simplify
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end; try linear_arithmetic; eauto; try equality.
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Hint Resolve le_n_S : core.
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Local Hint Resolve le_n_S : core.
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Lemma substring_le : forall s n m,
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length (substring n m s) <= m.
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induct s1; substring.
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Qed.
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Hint Resolve length_emp append_emp substring_le substring_split length_app1 : core.
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Local Hint Resolve length_emp append_emp substring_le substring_split length_app1 : core.
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Lemma substring_app_fst : forall s2 s1 n,
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length s1 = n
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induct s1; simplify; subst; simplify; auto.
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Qed.
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Hint Rewrite substring_app_fst substring_app_snd using solve [trivial].
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Local Hint Rewrite substring_app_fst substring_app_snd using solve [trivial].
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(* BOREDOM'S END! *)
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@ -563,7 +563,7 @@ End sumbool_and.
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Infix "&&" := sumbool_and (at level 40, left associativity).
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Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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Local Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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Section split.
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Variables P1 P2 : string -> Prop.
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equality.
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Qed.
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Hint Resolve app_cong : core.
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Local Hint Resolve app_cong : core.
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Definition matches : forall P (r : regexp P) s, {P s} + {~ P s}.
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refine (fix F P (r : regexp P) s : {P s} + {~ P s} :=
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12
HoareLogic.v
12
HoareLogic.v
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@ -61,7 +61,7 @@ Fixpoint eval (e : exp) (h : heap) (v : valuation) : nat :=
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end.
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(* Meaning of Boolean expressions *)
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Fixpoint beval (b : bexp) (h : heap) (v : valuation) : bool :=
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Definition beval (b : bexp) (h : heap) (v : valuation) : bool :=
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match b with
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| Equal e1 e2 => if eval e1 h v ==n eval e2 h v then true else false
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| Less e1 e2 => if eval e2 h v <=? eval e1 h v then false else true
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ht.
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Qed.
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Hint Resolve leq_f : core.
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Hint Extern 1 (@eq nat _ _) => linear_arithmetic : core.
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Hint Extern 1 (_ < _) => linear_arithmetic : core.
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Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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Local Hint Resolve leq_f : core.
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Local Hint Extern 1 (@eq nat _ _) => linear_arithmetic : core.
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Local Hint Extern 1 (_ < _) => linear_arithmetic : core.
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Local Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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(* We also register [linear_arithmetic] as a step to try during proof search. *)
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(* These invariants are fairly hairy, but probably the best way to understand
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a h v
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-> step (h, v, Assert a) (h, v, Skip).
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Hint Constructors step : core.
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Local Hint Constructors step : core.
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Definition trsys_of (st : heap * valuation * cmd) := {|
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Initial := {st};
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ht.
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Qed.
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Hint Resolve leq_f : core.
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Hint Extern 1 (@eq nat _ _) => linear_arithmetic : core.
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Hint Extern 1 (_ < _) => linear_arithmetic : core.
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Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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Local Hint Resolve leq_f : core.
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Local Hint Extern 1 (@eq nat _ _) => linear_arithmetic : core.
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Local Hint Extern 1 (_ < _) => linear_arithmetic : core.
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Local Hint Extern 1 (_ <= _) => linear_arithmetic : core.
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(* We also register [linear_arithmetic] as a step to try during proof search. *)
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Theorem selectionSort_ok :
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a h v
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-> step (h, v, Assert a) (h, v, Skip).
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Hint Constructors step : core.
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Local Hint Constructors step : core.
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Definition trsys_of (st : heap * valuation * cmd) := {|
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Initial := {st};
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\encoding
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A first rule for $\skipe$ is easy: anything that was true before is also true after.
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$$\infer{\hoare{P}{\skipe}{P}}{}$$
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A rule for assignment is slightly more involved: to state what we know is true after, we recall that there existed a prestate satisfying the precondition, which then evolved into the poststate in the expected way.
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Even an infinite-looping program must satisfy its $\mathsf{assert}$ commands, every time it passes one of them.
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For that reason, it's interesting to consider how to show that a command never gets stuck on a false assertion.
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We work up to that result with a few intermediate ones.
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First, we define \emph{stuck} much the same way as in the last two chapters: a state $(h, v, c)$ is stuck if $c$ is not $\skipe$, but there is also nowhere to step to from this state.
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First, we define \emph{stuck} much the same way as in the last three chapters: a state $(h, v, c)$ is stuck if $c$ is not $\skipe$, but there is also nowhere to step to from this state.
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An example of a stuck state would be one beginning with an $\mathsf{assert}$ of an assertion that does not hold on $h$ and $v$.
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In fact, we can prove that any other state is unstuck, though we won't bother here.
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