diff options
| -rw-r--r-- | doc/sphinx/proof-engine/detailed-tactic-examples.rst | 378 | ||||
| -rw-r--r-- | doc/sphinx/proof-engine/ltac.rst | 407 | ||||
| -rw-r--r-- | doc/sphinx/proof-engine/tactics.rst | 53 |
3 files changed, 414 insertions, 424 deletions
diff --git a/doc/sphinx/proof-engine/detailed-tactic-examples.rst b/doc/sphinx/proof-engine/detailed-tactic-examples.rst index b629d15b11..0ace9ef5b9 100644 --- a/doc/sphinx/proof-engine/detailed-tactic-examples.rst +++ b/doc/sphinx/proof-engine/detailed-tactic-examples.rst @@ -396,381 +396,3 @@ the optional tactic of the ``Hint Rewrite`` command. .. coqtop:: none Qed. - -Using the tactic language -------------------------- - - -About the cardinality of the set of natural numbers -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -The first example which shows how to use pattern matching over the -proof context is a proof of the fact that natural numbers have more -than two elements. This can be done as follows: - -.. coqtop:: in reset - - Lemma card_nat : - ~ exists x : nat, exists y : nat, forall z:nat, x = z \/ y = z. - Proof. - -.. coqtop:: in - - red; intros (x, (y, Hy)). - -.. coqtop:: in - - elim (Hy 0); elim (Hy 1); elim (Hy 2); intros; - - match goal with - | _ : ?a = ?b, _ : ?a = ?c |- _ => - cut (b = c); [ discriminate | transitivity a; auto ] - end. - -.. coqtop:: in - - Qed. - -We can notice that all the (very similar) cases coming from the three -eliminations (with three distinct natural numbers) are successfully -solved by a match goal structure and, in particular, with only one -pattern (use of non-linear matching). - - -Permutations of lists -~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -A more complex example is the problem of permutations of -lists. The aim is to show that a list is a permutation of -another list. - -.. coqtop:: in reset - - Section Sort. - -.. coqtop:: in - - Variable A : Set. - -.. coqtop:: in - - Inductive perm : list A -> list A -> Prop := - | perm_refl : forall l, perm l l - | perm_cons : forall a l0 l1, perm l0 l1 -> perm (a :: l0) (a :: l1) - | perm_append : forall a l, perm (a :: l) (l ++ a :: nil) - | perm_trans : forall l0 l1 l2, perm l0 l1 -> perm l1 l2 -> perm l0 l2. - -.. coqtop:: in - - End Sort. - -First, we define the permutation predicate as shown above. - -.. coqtop:: none - - Require Import List. - - -.. coqtop:: in - - Ltac perm_aux n := - match goal with - | |- (perm _ ?l ?l) => apply perm_refl - | |- (perm _ (?a :: ?l1) (?a :: ?l2)) => - let newn := eval compute in (length l1) in - (apply perm_cons; perm_aux newn) - | |- (perm ?A (?a :: ?l1) ?l2) => - match eval compute in n with - | 1 => fail - | _ => - let l1' := constr:(l1 ++ a :: nil) in - (apply (perm_trans A (a :: l1) l1' l2); - [ apply perm_append | compute; perm_aux (pred n) ]) - end - end. - -Next we define an auxiliary tactic ``perm_aux`` which takes an argument -used to control the recursion depth. This tactic behaves as follows. If -the lists are identical (i.e. convertible), it concludes. Otherwise, if -the lists have identical heads, it proceeds to look at their tails. -Finally, if the lists have different heads, it rotates the first list by -putting its head at the end if the new head hasn't been the head previously. To check this, we keep track of the -number of performed rotations using the argument ``n``. We do this by -decrementing ``n`` each time we perform a rotation. It works because -for a list of length ``n`` we can make exactly ``n - 1`` rotations -to generate at most ``n`` distinct lists. Notice that we use the natural -numbers of Coq for the rotation counter. From :ref:`ltac-syntax` we know -that it is possible to use the usual natural numbers, but they are only -used as arguments for primitive tactics and they cannot be handled, so, -in particular, we cannot make computations with them. Thus the natural -choice is to use Coq data structures so that Coq makes the computations -(reductions) by ``eval compute in`` and we can get the terms back by match. - -.. coqtop:: in - - Ltac solve_perm := - match goal with - | |- (perm _ ?l1 ?l2) => - match eval compute in (length l1 = length l2) with - | (?n = ?n) => perm_aux n - end - end. - -The main tactic is ``solve_perm``. It computes the lengths of the two lists -and uses them as arguments to call ``perm_aux`` if the lengths are equal (if they -aren't, the lists cannot be permutations of each other). Using this tactic we -can now prove lemmas as follows: - -.. coqtop:: in - - Lemma solve_perm_ex1 : - perm nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil). - Proof. solve_perm. Qed. - -.. coqtop:: in - - Lemma solve_perm_ex2 : - perm nat - (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil) - (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil). - Proof. solve_perm. Qed. - -Deciding intuitionistic propositional logic -~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ - -Pattern matching on goals allows a powerful backtracking when returning tactic -values. An interesting application is the problem of deciding intuitionistic -propositional logic. Considering the contraction-free sequent calculi LJT* of -Roy Dyckhoff :cite:`Dyc92`, it is quite natural to code such a tactic using the -tactic language as shown below. - -.. coqtop:: in reset - - Ltac basic := - match goal with - | |- True => trivial - | _ : False |- _ => contradiction - | _ : ?A |- ?A => assumption - end. - -.. coqtop:: in - - Ltac simplify := - repeat (intros; - match goal with - | H : ~ _ |- _ => red in H - | H : _ /\ _ |- _ => - elim H; do 2 intro; clear H - | H : _ \/ _ |- _ => - elim H; intro; clear H - | H : ?A /\ ?B -> ?C |- _ => - cut (A -> B -> C); - [ intro | intros; apply H; split; assumption ] - | H: ?A \/ ?B -> ?C |- _ => - cut (B -> C); - [ cut (A -> C); - [ intros; clear H - | intro; apply H; left; assumption ] - | intro; apply H; right; assumption ] - | H0 : ?A -> ?B, H1 : ?A |- _ => - cut B; [ intro; clear H0 | apply H0; assumption ] - | |- _ /\ _ => split - | |- ~ _ => red - end). - -.. coqtop:: in - - Ltac my_tauto := - simplify; basic || - match goal with - | H : (?A -> ?B) -> ?C |- _ => - cut (B -> C); - [ intro; cut (A -> B); - [ intro; cut C; - [ intro; clear H | apply H; assumption ] - | clear H ] - | intro; apply H; intro; assumption ]; my_tauto - | H : ~ ?A -> ?B |- _ => - cut (False -> B); - [ intro; cut (A -> False); - [ intro; cut B; - [ intro; clear H | apply H; assumption ] - | clear H ] - | intro; apply H; red; intro; assumption ]; my_tauto - | |- _ \/ _ => (left; my_tauto) || (right; my_tauto) - end. - -The tactic ``basic`` tries to reason using simple rules involving truth, falsity -and available assumptions. The tactic ``simplify`` applies all the reversible -rules of Dyckhoff’s system. Finally, the tactic ``my_tauto`` (the main -tactic to be called) simplifies with ``simplify``, tries to conclude with -``basic`` and tries several paths using the backtracking rules (one of the -four Dyckhoff’s rules for the left implication to get rid of the contraction -and the right ``or``). - -Having defined ``my_tauto``, we can prove tautologies like these: - -.. coqtop:: in - - Lemma my_tauto_ex1 : - forall A B : Prop, A /\ B -> A \/ B. - Proof. my_tauto. Qed. - -.. coqtop:: in - - Lemma my_tauto_ex2 : - forall A B : Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B. - Proof. my_tauto. Qed. - - -Deciding type isomorphisms -~~~~~~~~~~~~~~~~~~~~~~~~~~ - -A more tricky problem is to decide equalities between types modulo -isomorphisms. Here, we choose to use the isomorphisms of the simply -typed λ-calculus with Cartesian product and unit type (see, for -example, :cite:`RC95`). The axioms of this λ-calculus are given below. - -.. coqtop:: in reset - - Open Scope type_scope. - -.. coqtop:: in - - Section Iso_axioms. - -.. coqtop:: in - - Variables A B C : Set. - -.. coqtop:: in - - Axiom Com : A * B = B * A. - - Axiom Ass : A * (B * C) = A * B * C. - - Axiom Cur : (A * B -> C) = (A -> B -> C). - - Axiom Dis : (A -> B * C) = (A -> B) * (A -> C). - - Axiom P_unit : A * unit = A. - - Axiom AR_unit : (A -> unit) = unit. - - Axiom AL_unit : (unit -> A) = A. - -.. coqtop:: in - - Lemma Cons : B = C -> A * B = A * C. - - Proof. - - intro Heq; rewrite Heq; reflexivity. - - Qed. - -.. coqtop:: in - - End Iso_axioms. - -.. coqtop:: in - - Ltac simplify_type ty := - match ty with - | ?A * ?B * ?C => - rewrite <- (Ass A B C); try simplify_type_eq - | ?A * ?B -> ?C => - rewrite (Cur A B C); try simplify_type_eq - | ?A -> ?B * ?C => - rewrite (Dis A B C); try simplify_type_eq - | ?A * unit => - rewrite (P_unit A); try simplify_type_eq - | unit * ?B => - rewrite (Com unit B); try simplify_type_eq - | ?A -> unit => - rewrite (AR_unit A); try simplify_type_eq - | unit -> ?B => - rewrite (AL_unit B); try simplify_type_eq - | ?A * ?B => - (simplify_type A; try simplify_type_eq) || - (simplify_type B; try simplify_type_eq) - | ?A -> ?B => - (simplify_type A; try simplify_type_eq) || - (simplify_type B; try simplify_type_eq) - end - with simplify_type_eq := - match goal with - | |- ?A = ?B => try simplify_type A; try simplify_type B - end. - -.. coqtop:: in - - Ltac len trm := - match trm with - | _ * ?B => let succ := len B in constr:(S succ) - | _ => constr:(1) - end. - -.. coqtop:: in - - Ltac assoc := repeat rewrite <- Ass. - -.. coqtop:: in - - Ltac solve_type_eq n := - match goal with - | |- ?A = ?A => reflexivity - | |- ?A * ?B = ?A * ?C => - apply Cons; let newn := len B in solve_type_eq newn - | |- ?A * ?B = ?C => - match eval compute in n with - | 1 => fail - | _ => - pattern (A * B) at 1; rewrite Com; assoc; solve_type_eq (pred n) - end - end. - -.. coqtop:: in - - Ltac compare_structure := - match goal with - | |- ?A = ?B => - let l1 := len A - with l2 := len B in - match eval compute in (l1 = l2) with - | ?n = ?n => solve_type_eq n - end - end. - -.. coqtop:: in - - Ltac solve_iso := simplify_type_eq; compare_structure. - -The tactic to judge equalities modulo this axiomatization is shown above. -The algorithm is quite simple. First types are simplified using axioms that -can be oriented (this is done by ``simplify_type`` and ``simplify_type_eq``). -The normal forms are sequences of Cartesian products without Cartesian product -in the left component. These normal forms are then compared modulo permutation -of the components by the tactic ``compare_structure``. If they have the same -lengths, the tactic ``solve_type_eq`` attempts to prove that the types are equal. -The main tactic that puts all these components together is called ``solve_iso``. - -Here are examples of what can be solved by ``solve_iso``. - -.. coqtop:: in - - Lemma solve_iso_ex1 : - forall A B : Set, A * unit * B = B * (unit * A). - Proof. - intros; solve_iso. - Qed. - -.. coqtop:: in - - Lemma solve_iso_ex2 : - forall A B C : Set, - (A * unit -> B * (C * unit)) = - (A * unit -> (C -> unit) * C) * (unit -> A -> B). - Proof. - intros; solve_iso. - Qed. diff --git a/doc/sphinx/proof-engine/ltac.rst b/doc/sphinx/proof-engine/ltac.rst index d3562b52c5..d35e4ab782 100644 --- a/doc/sphinx/proof-engine/ltac.rst +++ b/doc/sphinx/proof-engine/ltac.rst @@ -3,12 +3,25 @@ Ltac ==== -This chapter gives a compact documentation of |Ltac|, the tactic language -available in |Coq|. We start by giving the syntax, and next, we present the -informal semantics. If you want to know more regarding this language and -especially about its foundations, you can refer to :cite:`Del00`. Chapter -:ref:`detailedexamplesoftactics` is devoted to giving small but nontrivial -use examples of this language. +This chapter documents the tactic language |Ltac|. + +We start by giving the syntax, and next, we present the informal +semantics. If you want to know more regarding this language and +especially about its foundations, you can refer to :cite:`Del00`. + +.. example:: + + Here are some examples of the kind of tactic macros that this + language allows to write. + + .. coqdoc:: + + Ltac reduce_and_try_to_solve := simpl; intros; auto. + + Ltac destruct_bool_and_rewrite b H1 H2 := + destruct b; [ rewrite H1; eauto | rewrite H2; eauto ]. + + Some more advanced examples are given in Section :ref:`ltac-examples`. .. _ltac-syntax: @@ -1160,6 +1173,388 @@ Printing |Ltac| tactics This command displays a list of all user-defined tactics, with their arguments. + +.. _ltac-examples: + +Examples of using |Ltac| +------------------------- + + +About the cardinality of the set of natural numbers +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The first example which shows how to use pattern matching over the +proof context is a proof of the fact that natural numbers have more +than two elements. This can be done as follows: + +.. coqtop:: in reset + + Lemma card_nat : + ~ exists x : nat, exists y : nat, forall z:nat, x = z \/ y = z. + Proof. + +.. coqtop:: in + + red; intros (x, (y, Hy)). + +.. coqtop:: in + + elim (Hy 0); elim (Hy 1); elim (Hy 2); intros; + + match goal with + | _ : ?a = ?b, _ : ?a = ?c |- _ => + cut (b = c); [ discriminate | transitivity a; auto ] + end. + +.. coqtop:: in + + Qed. + +We can notice that all the (very similar) cases coming from the three +eliminations (with three distinct natural numbers) are successfully +solved by a match goal structure and, in particular, with only one +pattern (use of non-linear matching). + + +Permutations of lists +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A more complex example is the problem of permutations of +lists. The aim is to show that a list is a permutation of +another list. + +.. coqtop:: in reset + + Section Sort. + +.. coqtop:: in + + Variable A : Set. + +.. coqtop:: in + + Inductive perm : list A -> list A -> Prop := + | perm_refl : forall l, perm l l + | perm_cons : forall a l0 l1, perm l0 l1 -> perm (a :: l0) (a :: l1) + | perm_append : forall a l, perm (a :: l) (l ++ a :: nil) + | perm_trans : forall l0 l1 l2, perm l0 l1 -> perm l1 l2 -> perm l0 l2. + +.. coqtop:: in + + End Sort. + +First, we define the permutation predicate as shown above. + +.. coqtop:: none + + Require Import List. + + +.. coqtop:: in + + Ltac perm_aux n := + match goal with + | |- (perm _ ?l ?l) => apply perm_refl + | |- (perm _ (?a :: ?l1) (?a :: ?l2)) => + let newn := eval compute in (length l1) in + (apply perm_cons; perm_aux newn) + | |- (perm ?A (?a :: ?l1) ?l2) => + match eval compute in n with + | 1 => fail + | _ => + let l1' := constr:(l1 ++ a :: nil) in + (apply (perm_trans A (a :: l1) l1' l2); + [ apply perm_append | compute; perm_aux (pred n) ]) + end + end. + +Next we define an auxiliary tactic ``perm_aux`` which takes an argument +used to control the recursion depth. This tactic behaves as follows. If +the lists are identical (i.e. convertible), it concludes. Otherwise, if +the lists have identical heads, it proceeds to look at their tails. +Finally, if the lists have different heads, it rotates the first list by +putting its head at the end if the new head hasn't been the head previously. To check this, we keep track of the +number of performed rotations using the argument ``n``. We do this by +decrementing ``n`` each time we perform a rotation. It works because +for a list of length ``n`` we can make exactly ``n - 1`` rotations +to generate at most ``n`` distinct lists. Notice that we use the natural +numbers of Coq for the rotation counter. From :ref:`ltac-syntax` we know +that it is possible to use the usual natural numbers, but they are only +used as arguments for primitive tactics and they cannot be handled, so, +in particular, we cannot make computations with them. Thus the natural +choice is to use Coq data structures so that Coq makes the computations +(reductions) by ``eval compute in`` and we can get the terms back by match. + +.. coqtop:: in + + Ltac solve_perm := + match goal with + | |- (perm _ ?l1 ?l2) => + match eval compute in (length l1 = length l2) with + | (?n = ?n) => perm_aux n + end + end. + +The main tactic is ``solve_perm``. It computes the lengths of the two lists +and uses them as arguments to call ``perm_aux`` if the lengths are equal (if they +aren't, the lists cannot be permutations of each other). Using this tactic we +can now prove lemmas as follows: + +.. coqtop:: in + + Lemma solve_perm_ex1 : + perm nat (1 :: 2 :: 3 :: nil) (3 :: 2 :: 1 :: nil). + Proof. solve_perm. Qed. + +.. coqtop:: in + + Lemma solve_perm_ex2 : + perm nat + (0 :: 1 :: 2 :: 3 :: 4 :: 5 :: 6 :: 7 :: 8 :: 9 :: nil) + (0 :: 2 :: 4 :: 6 :: 8 :: 9 :: 7 :: 5 :: 3 :: 1 :: nil). + Proof. solve_perm. Qed. + +Deciding intuitionistic propositional logic +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Pattern matching on goals allows a powerful backtracking when returning tactic +values. An interesting application is the problem of deciding intuitionistic +propositional logic. Considering the contraction-free sequent calculi LJT* of +Roy Dyckhoff :cite:`Dyc92`, it is quite natural to code such a tactic using the +tactic language as shown below. + +.. coqtop:: in reset + + Ltac basic := + match goal with + | |- True => trivial + | _ : False |- _ => contradiction + | _ : ?A |- ?A => assumption + end. + +.. coqtop:: in + + Ltac simplify := + repeat (intros; + match goal with + | H : ~ _ |- _ => red in H + | H : _ /\ _ |- _ => + elim H; do 2 intro; clear H + | H : _ \/ _ |- _ => + elim H; intro; clear H + | H : ?A /\ ?B -> ?C |- _ => + cut (A -> B -> C); + [ intro | intros; apply H; split; assumption ] + | H: ?A \/ ?B -> ?C |- _ => + cut (B -> C); + [ cut (A -> C); + [ intros; clear H + | intro; apply H; left; assumption ] + | intro; apply H; right; assumption ] + | H0 : ?A -> ?B, H1 : ?A |- _ => + cut B; [ intro; clear H0 | apply H0; assumption ] + | |- _ /\ _ => split + | |- ~ _ => red + end). + +.. coqtop:: in + + Ltac my_tauto := + simplify; basic || + match goal with + | H : (?A -> ?B) -> ?C |- _ => + cut (B -> C); + [ intro; cut (A -> B); + [ intro; cut C; + [ intro; clear H | apply H; assumption ] + | clear H ] + | intro; apply H; intro; assumption ]; my_tauto + | H : ~ ?A -> ?B |- _ => + cut (False -> B); + [ intro; cut (A -> False); + [ intro; cut B; + [ intro; clear H | apply H; assumption ] + | clear H ] + | intro; apply H; red; intro; assumption ]; my_tauto + | |- _ \/ _ => (left; my_tauto) || (right; my_tauto) + end. + +The tactic ``basic`` tries to reason using simple rules involving truth, falsity +and available assumptions. The tactic ``simplify`` applies all the reversible +rules of Dyckhoff’s system. Finally, the tactic ``my_tauto`` (the main +tactic to be called) simplifies with ``simplify``, tries to conclude with +``basic`` and tries several paths using the backtracking rules (one of the +four Dyckhoff’s rules for the left implication to get rid of the contraction +and the right ``or``). + +Having defined ``my_tauto``, we can prove tautologies like these: + +.. coqtop:: in + + Lemma my_tauto_ex1 : + forall A B : Prop, A /\ B -> A \/ B. + Proof. my_tauto. Qed. + +.. coqtop:: in + + Lemma my_tauto_ex2 : + forall A B : Prop, (~ ~ B -> B) -> (A -> B) -> ~ ~ A -> B. + Proof. my_tauto. Qed. + + +Deciding type isomorphisms +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A more tricky problem is to decide equalities between types modulo +isomorphisms. Here, we choose to use the isomorphisms of the simply +typed λ-calculus with Cartesian product and unit type (see, for +example, :cite:`RC95`). The axioms of this λ-calculus are given below. + +.. coqtop:: in reset + + Open Scope type_scope. + +.. coqtop:: in + + Section Iso_axioms. + +.. coqtop:: in + + Variables A B C : Set. + +.. coqtop:: in + + Axiom Com : A * B = B * A. + + Axiom Ass : A * (B * C) = A * B * C. + + Axiom Cur : (A * B -> C) = (A -> B -> C). + + Axiom Dis : (A -> B * C) = (A -> B) * (A -> C). + + Axiom P_unit : A * unit = A. + + Axiom AR_unit : (A -> unit) = unit. + + Axiom AL_unit : (unit -> A) = A. + +.. coqtop:: in + + Lemma Cons : B = C -> A * B = A * C. + + Proof. + + intro Heq; rewrite Heq; reflexivity. + + Qed. + +.. coqtop:: in + + End Iso_axioms. + +.. coqtop:: in + + Ltac simplify_type ty := + match ty with + | ?A * ?B * ?C => + rewrite <- (Ass A B C); try simplify_type_eq + | ?A * ?B -> ?C => + rewrite (Cur A B C); try simplify_type_eq + | ?A -> ?B * ?C => + rewrite (Dis A B C); try simplify_type_eq + | ?A * unit => + rewrite (P_unit A); try simplify_type_eq + | unit * ?B => + rewrite (Com unit B); try simplify_type_eq + | ?A -> unit => + rewrite (AR_unit A); try simplify_type_eq + | unit -> ?B => + rewrite (AL_unit B); try simplify_type_eq + | ?A * ?B => + (simplify_type A; try simplify_type_eq) || + (simplify_type B; try simplify_type_eq) + | ?A -> ?B => + (simplify_type A; try simplify_type_eq) || + (simplify_type B; try simplify_type_eq) + end + with simplify_type_eq := + match goal with + | |- ?A = ?B => try simplify_type A; try simplify_type B + end. + +.. coqtop:: in + + Ltac len trm := + match trm with + | _ * ?B => let succ := len B in constr:(S succ) + | _ => constr:(1) + end. + +.. coqtop:: in + + Ltac assoc := repeat rewrite <- Ass. + +.. coqtop:: in + + Ltac solve_type_eq n := + match goal with + | |- ?A = ?A => reflexivity + | |- ?A * ?B = ?A * ?C => + apply Cons; let newn := len B in solve_type_eq newn + | |- ?A * ?B = ?C => + match eval compute in n with + | 1 => fail + | _ => + pattern (A * B) at 1; rewrite Com; assoc; solve_type_eq (pred n) + end + end. + +.. coqtop:: in + + Ltac compare_structure := + match goal with + | |- ?A = ?B => + let l1 := len A + with l2 := len B in + match eval compute in (l1 = l2) with + | ?n = ?n => solve_type_eq n + end + end. + +.. coqtop:: in + + Ltac solve_iso := simplify_type_eq; compare_structure. + +The tactic to judge equalities modulo this axiomatization is shown above. +The algorithm is quite simple. First types are simplified using axioms that +can be oriented (this is done by ``simplify_type`` and ``simplify_type_eq``). +The normal forms are sequences of Cartesian products without Cartesian product +in the left component. These normal forms are then compared modulo permutation +of the components by the tactic ``compare_structure``. If they have the same +lengths, the tactic ``solve_type_eq`` attempts to prove that the types are equal. +The main tactic that puts all these components together is called ``solve_iso``. + +Here are examples of what can be solved by ``solve_iso``. + +.. coqtop:: in + + Lemma solve_iso_ex1 : + forall A B : Set, A * unit * B = B * (unit * A). + Proof. + intros; solve_iso. + Qed. + +.. coqtop:: in + + Lemma solve_iso_ex2 : + forall A B C : Set, + (A * unit -> B * (C * unit)) = + (A * unit -> (C -> unit) * C) * (unit -> A -> B). + Proof. + intros; solve_iso. + Qed. + + Debugging |Ltac| tactics ------------------------ diff --git a/doc/sphinx/proof-engine/tactics.rst b/doc/sphinx/proof-engine/tactics.rst index 0f78a9b84a..c728b925ac 100644 --- a/doc/sphinx/proof-engine/tactics.rst +++ b/doc/sphinx/proof-engine/tactics.rst @@ -3561,7 +3561,7 @@ Automation .. tacn:: autorewrite with {+ @ident} :name: autorewrite - This tactic [4]_ carries out rewritings according to the rewriting rule + This tactic carries out rewritings according to the rewriting rule bases :n:`{+ @ident}`. Each rewriting rule from the base :n:`@ident` is applied to the main subgoal until @@ -4661,9 +4661,12 @@ Non-logical tactics .. example:: - .. coqtop:: all reset + .. coqtop:: none reset Parameter P : nat -> Prop. + + .. coqtop:: all abort + Goal P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5. repeat split. all: cycle 2. @@ -4679,9 +4682,8 @@ Non-logical tactics .. example:: - .. coqtop:: reset all + .. coqtop:: all abort - Parameter P : nat -> Prop. Goal P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5. repeat split. all: swap 1 3. @@ -4694,9 +4696,8 @@ Non-logical tactics .. example:: - .. coqtop:: all reset + .. coqtop:: all abort - Parameter P : nat -> Prop. Goal P 1 /\ P 2 /\ P 3 /\ P 4 /\ P 5. repeat split. all: revgoals. @@ -4717,7 +4718,7 @@ Non-logical tactics .. example:: - .. coqtop:: all reset + .. coqtop:: all abort Goal exists n, n=0. refine (ex_intro _ _ _). @@ -4746,39 +4747,6 @@ Non-logical tactics The ``give_up`` tactic can be used while editing a proof, to choose to write the proof script in a non-sequential order. -Simple tactic macros -------------------------- - -A simple example has more value than a long explanation: - -.. example:: - - .. coqtop:: reset all - - Ltac Solve := simpl; intros; auto. - - Ltac ElimBoolRewrite b H1 H2 := - elim b; [ intros; rewrite H1; eauto | intros; rewrite H2; eauto ]. - -The tactics macros are synchronous with the Coq section mechanism: a -tactic definition is deleted from the current environment when you -close the section (see also :ref:`section-mechanism`) where it was -defined. If you want that a tactic macro defined in a module is usable in the -modules that require it, you should put it outside of any section. - -:ref:`ltac` gives examples of more complex -user-defined tactics. - -.. [1] Actually, only the second subgoal will be generated since the - other one can be automatically checked. -.. [2] This corresponds to the cut rule of sequent calculus. -.. [3] Reminder: opaque constants will not be expanded by δ reductions. -.. [4] The behavior of this tactic has changed a lot compared to the - versions available in the previous distributions (V6). This may cause - significant changes in your theories to obtain the same result. As a - drawback of the re-engineering of the code, this tactic has also been - completely revised to get a very compact and readable version. - Delaying solving unification constraints ---------------------------------------- @@ -4917,3 +4885,8 @@ Performance-oriented tactic variants Goal False. native_cast_no_check I. Fail Qed. + +.. [1] Actually, only the second subgoal will be generated since the + other one can be automatically checked. +.. [2] This corresponds to the cut rule of sequent calculus. +.. [3] Reminder: opaque constants will not be expanded by δ reductions. |