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Revising before tomorrow's lecture
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2 changed files with 24 additions and 24 deletions
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@ -44,7 +44,7 @@ Qed.
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* automate searching through sequences of that kind, when we prime it with good
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* automate searching through sequences of that kind, when we prime it with good
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* suggestions of single proof steps to try, as with this command: *)
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* suggestions of single proof steps to try, as with this command: *)
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Hint Constructors plusR : core.
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Local Hint Constructors plusR : core.
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(* That is, every constructor of [plusR] should be considered as an atomic proof
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(* That is, every constructor of [plusR] should be considered as an atomic proof
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* step, from which we enumerate step sequences. *)
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* step, from which we enumerate step sequences. *)
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@ -195,17 +195,17 @@ Qed.
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* programs in the same way as we did above for a logic program. Let us prove
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* programs in the same way as we did above for a logic program. Let us prove
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* that the constructors of [plusR] have natural interpretations as lemmas about
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* that the constructors of [plusR] have natural interpretations as lemmas about
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* [plus]. We can find the first such lemma already proved in the standard
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* [plus]. We can find the first such lemma already proved in the standard
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* library, using the [SearchRewrite] command to find a library function proving
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* library, using the [Search] command to find a library function proving
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* an equality whose lefthand or righthand side matches a pattern with
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* an equality whose lefthand or righthand side matches a pattern with
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* wildcards. *)
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* wildcards. *)
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SearchRewrite (O + _).
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Search (O + _).
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(* The command [Hint Immediate] asks [auto] and [eauto] to consider this lemma
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(* The command [Hint Immediate] asks [auto] and [eauto] to consider this lemma
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* as a candidate step for any leaf of a proof tree, meaning that all premises
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* as a candidate step for any leaf of a proof tree, meaning that all premises
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* of the rule need to match hypotheses. *)
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* of the rule need to match hypotheses. *)
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Hint Immediate plus_O_n : core.
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Local Hint Immediate plus_O_n : core.
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(* The counterpart to [PlusS] we will prove ourselves. *)
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(* The counterpart to [PlusS] we will prove ourselves. *)
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@ -219,7 +219,7 @@ Qed.
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(* The command [Hint Resolve] adds a new candidate proof step, to be attempted
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(* The command [Hint Resolve] adds a new candidate proof step, to be attempted
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* at any level of a proof tree, not just at leaves. *)
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* at any level of a proof tree, not just at leaves. *)
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Hint Resolve plusS : core.
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Local Hint Resolve plusS : core.
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(* Now that we have registered the proper hints, we can replicate our previous
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(* Now that we have registered the proper hints, we can replicate our previous
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* examples with the normal, functional addition [plus]. *)
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* examples with the normal, functional addition [plus]. *)
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@ -251,7 +251,7 @@ Proof.
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linear_arithmetic.
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linear_arithmetic.
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Qed.
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Qed.
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Hint Resolve plusO : core.
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Local Hint Resolve plusO : core.
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(* Note that, if we consider the inputs to [plus] as the inputs of a
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(* Note that, if we consider the inputs to [plus] as the inputs of a
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* corresponding logic program, the new rule [plusO] introduces an ambiguity.
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* corresponding logic program, the new rule [plusO] introduces an ambiguity.
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@ -304,7 +304,7 @@ End slow.
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* _hint databases_ to segregate hints into different groups that may be called
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* _hint databases_ to segregate hints into different groups that may be called
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* on as needed. Here we put [eq_trans] into the database [slow]. *)
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* on as needed. Here we put [eq_trans] into the database [slow]. *)
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Hint Resolve eq_trans : slow.
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Local Hint Resolve eq_trans : slow.
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Example from_one_to_zero : exists x, 1 + x = 0.
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Example from_one_to_zero : exists x, 1 + x = 0.
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Proof.
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Proof.
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@ -367,7 +367,7 @@ Proof.
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simplify; equality.
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simplify; equality.
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Qed.
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Qed.
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Hint Resolve length_O length_S : core.
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Local Hint Resolve length_O length_S : core.
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(* Let us apply these hints to prove that a [list nat] of length 2 exists.
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(* Let us apply these hints to prove that a [list nat] of length 2 exists.
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* (Here we register [length_O] with [Hint Resolve] instead of [Hint Immediate]
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* (Here we register [length_O] with [Hint Resolve] instead of [Hint Immediate]
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@ -424,7 +424,7 @@ Proof.
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linear_arithmetic.
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linear_arithmetic.
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Qed.
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Qed.
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Hint Resolve plusO' : core.
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Local Hint Resolve plusO' : core.
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(* Finally, we meet [Hint Extern], the command to register a custom hint. That
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(* Finally, we meet [Hint Extern], the command to register a custom hint. That
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* is, we provide a pattern to match against goals during proof search.
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* is, we provide a pattern to match against goals during proof search.
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@ -434,7 +434,7 @@ Hint Resolve plusO' : core.
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* effect on proof-search time, i.e. when we manage to give lower priorities to
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* effect on proof-search time, i.e. when we manage to give lower priorities to
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* the cheaper rules. *)
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* the cheaper rules. *)
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Hint Extern 1 (sum _ = _) => simplify : core.
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Local Hint Extern 1 (sum _ = _) => simplify : core.
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(* Now we can find a length-2 list whose sum is 0. *)
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(* Now we can find a length-2 list whose sum is 0. *)
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@ -497,7 +497,7 @@ Inductive eval (var : nat) : exp -> nat -> Prop :=
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-> eval var e2 n2
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-> eval var e2 n2
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-> eval var (Plus e1 e2) (n1 + n2).
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-> eval var (Plus e1 e2) (n1 + n2).
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Hint Constructors eval : core.
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Local Hint Constructors eval : core.
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(* We can use [auto] to execute the semantics for specific expressions. *)
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(* We can use [auto] to execute the semantics for specific expressions. *)
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@ -531,7 +531,7 @@ Proof.
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simplify; subst; auto.
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simplify; subst; auto.
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Qed.
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Qed.
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Hint Resolve EvalPlus' : core.
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Local Hint Resolve EvalPlus' : core.
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(* Further, we instruct [eauto] to apply [ring], via [Hint Extern]. We should
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(* Further, we instruct [eauto] to apply [ring], via [Hint Extern]. We should
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* try this step for any equality goal. *)
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* try this step for any equality goal. *)
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@ -597,7 +597,7 @@ Proof.
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simplify; subst; auto.
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simplify; subst; auto.
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Qed.
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Qed.
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Hint Resolve EvalConst' EvalVar' : core.
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Local Hint Resolve EvalConst' EvalVar' : core.
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(* Next, we prove a few hints that feel a bit like cheating, as they telegraph
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(* Next, we prove a few hints that feel a bit like cheating, as they telegraph
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* the procedure for choosing values of [k] and [n]. Nonetheless, with these
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* the procedure for choosing values of [k] and [n]. Nonetheless, with these
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@ -673,14 +673,14 @@ Ltac robust_ring_simplify :=
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(* This tactic is pretty expensive, but let's try it eventually whenever the
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(* This tactic is pretty expensive, but let's try it eventually whenever the
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* goal is an equality. *)
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* goal is an equality. *)
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Hint Extern 5 (_ = _) => robust_ring_simplify : core.
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Local Hint Extern 5 (_ = _) => robust_ring_simplify : core.
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(* The only other missing ingredient is priming Coq with some good ideas for
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(* The only other missing ingredient is priming Coq with some good ideas for
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* instantiating existential quantifiers. These will all be tried in some
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* instantiating existential quantifiers. These will all be tried in some
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* order, in a particular proof search. *)
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* order, in a particular proof search. *)
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Hint Extern 1 (exists n : nat, _) => exists 0 : core.
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Local Hint Extern 1 (exists n : nat, _) => exists 0 : core.
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Hint Extern 1 (exists n : nat, _) => exists 1 : core.
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Local Hint Extern 1 (exists n : nat, _) => exists 1 : core.
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Hint Extern 1 (exists n : nat, _) => eexists (_ + _) : core.
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Local Hint Extern 1 (exists n : nat, _) => eexists (_ + _) : core.
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(* Note how this last hint uses [eexists] to provide an instantiation with
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(* Note how this last hint uses [eexists] to provide an instantiation with
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* wildcards inside it. Each underscore is replaced with a fresh unification
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* wildcards inside it. Each underscore is replaced with a fresh unification
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* variable. *)
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* variable. *)
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@ -695,9 +695,9 @@ Qed.
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(* Here's a quick tease using a feature that we'll explore fully in a later
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(* Here's a quick tease using a feature that we'll explore fully in a later
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* class. Let's use a mysterious construct [sigT] instead of [exists]. *)
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* class. Let's use a mysterious construct [sigT] instead of [exists]. *)
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Hint Extern 1 (sigT (fun n : nat => _)) => exists 0 : core.
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Local Hint Extern 1 (sigT (fun n : nat => _)) => exists 0 : core.
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Hint Extern 1 (sigT (fun n : nat => _)) => exists 1 : core.
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Local Hint Extern 1 (sigT (fun n : nat => _)) => exists 1 : core.
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Hint Extern 1 (sigT (fun n : nat => _)) => eexists (_ + _) : core.
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Local Hint Extern 1 (sigT (fun n : nat => _)) => eexists (_ + _) : core.
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Theorem linear_computable : forall e, sigT (fun k => sigT (fun n =>
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Theorem linear_computable : forall e, sigT (fun k => sigT (fun n =>
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forall var, eval var e (k * var + n))).
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forall var, eval var e (k * var + n))).
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@ -814,7 +814,7 @@ Abort.
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* restatement of the theorem we mean to prove. Luckily, a simpler form
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* restatement of the theorem we mean to prove. Luckily, a simpler form
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* suffices, by appealing to the [equality] tactic. *)
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* suffices, by appealing to the [equality] tactic. *)
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Hint Extern 1 (_ <> _) => equality : core.
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Local Hint Extern 1 (_ <> _) => equality : core.
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Theorem bool_neq : true <> false.
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Theorem bool_neq : true <> false.
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Proof.
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Proof.
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@ -856,7 +856,7 @@ End forall_and.
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(* After our success on this example, we might get more ambitious and seek to
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(* After our success on this example, we might get more ambitious and seek to
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* generalize the hint to all possible predicates [P]. *)
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* generalize the hint to all possible predicates [P]. *)
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Fail Hint Extern 1 (?P ?X) =>
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Fail Local Hint Extern 1 (?P ?X) =>
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match goal with
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match goal with
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| [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
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| [ H : forall x, P x /\ _ |- _ ] => apply (proj1 (H X))
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end : core.
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end : core.
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@ -871,7 +871,7 @@ Fail Hint Extern 1 (?P ?X) =>
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* leave out the pattern to the left of the [=>], incorporating the
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* leave out the pattern to the left of the [=>], incorporating the
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* corresponding logic into the Ltac script. *)
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* corresponding logic into the Ltac script. *)
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Hint Extern 1 =>
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Local Hint Extern 1 =>
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match goal with
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match goal with
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| [ H : forall x, ?P x /\ _ |- ?P ?X ] => apply (proj1 (H X))
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| [ H : forall x, ?P x /\ _ |- ?P ?X ] => apply (proj1 (H X))
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end : core.
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end : core.
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@ -173,7 +173,7 @@ Inductive eval (var : nat) : exp -> nat -> Prop :=
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-> eval var e2 n2
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-> eval var e2 n2
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-> eval var (Plus e1 e2) (n1 + n2).
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-> eval var (Plus e1 e2) (n1 + n2).
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Hint Constructors eval : core.
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Local Hint Constructors eval : core.
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Example eval1 : forall var, eval var (Plus Var (Plus (Const 8) Var)) (var + (8 + var)).
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Example eval1 : forall var, eval var (Plus Var (Plus (Const 8) Var)) (var + (8 + var)).
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Proof.
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Proof.
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