469 lines
18 KiB
Text
469 lines
18 KiB
Text
/-
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Copyright (c) 2016 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura
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-/
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prelude
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import init.trace init.function init.option
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import init.meta.base_tactic init.meta.environment init.meta.qexpr
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meta_constant tactic_state : Type₁
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namespace tactic_state
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meta_constant env : tactic_state → environment
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meta_constant to_format : tactic_state → format
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/- Format expression with respect to the main goal in the tactic state.
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If the tactic state does not contain any goals, then format expression
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using an empty local context. -/
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meta_constant format_expr : tactic_state → expr → format
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meta_constant get_options : tactic_state → options
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meta_constant set_options : tactic_state → options → tactic_state
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end tactic_state
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meta_definition tactic_state.has_to_format [instance] : has_to_format tactic_state :=
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has_to_format.mk tactic_state.to_format
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meta_definition tactic [reducible] (A : Type) := base_tactic tactic_state A
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meta_definition tactic.format_expr (e : expr) : tactic format :=
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do s ← tactic.read, return (tactic_state.format_expr s e)
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structure has_to_tactic_format [class] (A : Type) :=
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(to_tactic_format : A → tactic format)
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meta_definition expr_has_to_tactic_format [instance] : has_to_tactic_format expr :=
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has_to_tactic_format.mk tactic.format_expr
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meta_definition tactic.pp {A : Type} [has_to_tactic_format A] : A → tactic format :=
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has_to_tactic_format.to_tactic_format
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open tactic format
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meta_definition list_to_tactic_format_aux {A : Type} [has_to_tactic_format A] : bool → list A → tactic format
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| _ [] := return $ to_fmt ""
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| b (x::xs) := do
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f₁ ← pp x,
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f₂ ← list_to_tactic_format_aux ff xs,
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return $ (if b = ff then to_fmt "," ++ line else nil) ++ f₁ ++ f₂
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meta_definition list_to_tactic_format {A : Type} [has_to_tactic_format A] : list A → tactic format
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| [] := return $ to_fmt "[]"
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| (x::xs) := do
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f ← list_to_tactic_format_aux tt (x::xs),
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return $ to_fmt "[" ++ group (nest 1 f) ++ to_fmt "]"
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meta_definition list_has_to_tactic_format [instance] {A : Type} [has_to_tactic_format A] : has_to_tactic_format (list A) :=
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has_to_tactic_format.mk list_to_tactic_format
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meta_definition has_to_format_to_has_to_tactic_format [instance] (A : Type) [has_to_format A] : has_to_tactic_format A :=
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has_to_tactic_format.mk (return ∘ to_fmt)
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namespace tactic
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open tactic_state
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meta_definition get_env : tactic environment :=
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do s ← read,
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return $ env s
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meta_definition get_decl (n : name) : tactic declaration :=
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do s ← read,
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returnex $ environment.get (env s) n
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meta_definition trace {A : Type} [has_to_tactic_format A] (a : A) : tactic unit :=
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do fmt ← pp a,
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return $ _root_.trace_fmt fmt (λ u, ())
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meta_definition trace_state : tactic unit :=
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do s ← read,
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trace $ to_fmt s
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inductive transparency :=
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| all | semireducible | reducible | none
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export transparency (reducible semireducible)
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/- Return the partial term/proof constructed so far. Note that the resultant expression
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may contain variables that are not declarate in the current main goal. -/
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meta_constant result : tactic expr
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/- Display the partial term/proof constructed so far. This tactic is *not* equivalent to
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do { r ← result, s ← read, return (format_expr s r) } because this one will format the result with respect
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to the current goal, and trace_result will do it with respect to the initial goal. -/
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meta_constant format_result : tactic format
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/- Return target type of the main goal. Fail if tactic_state does not have any goal left. -/
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meta_constant target : tactic expr
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meta_constant intro_core : name → tactic expr
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meta_constant intron : nat → tactic unit
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meta_constant rename : name → name → tactic unit
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/- Clear the given local constant. The tactic fails if the given expression is not a local constant. -/
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meta_constant clear : expr → tactic unit
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meta_constant revert_lst : list expr → tactic nat
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meta_constant whnf : expr → tactic expr
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meta_constant unify_core : transparency → expr → expr → tactic unit
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/- Infer the type of the given expression.
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Remark: transparency does not affect type inference -/
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meta_constant infer_type : expr → tactic expr
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meta_constant get_local : name → tactic expr
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/- Return the hypothesis in the main goal. Fail if tactic_state does not have any goal left. -/
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meta_constant local_context : tactic (list expr)
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/- Helper tactic for creating simple applications where some arguments are inferred using
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type inference.
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Example, given
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rel.{l_1 l_2} : Pi (A : Type.{l_1}) (B : A -> Type.{l_2}), (Pi x : A, B x) -> (Pi x : A, B x) -> , Prop
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nat : Type.{1}
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real : Type.{1}
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vec.{l} : Pi (A : Type.{l}) (n : nat), Type.{l1}
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f g : Pi (n : nat), vec real n
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then
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mk_app_core semireducible "rel" [f, g]
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returns the application
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rel.{1 2} nat (fun n : nat, vec real n) f g -/
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meta_constant mk_app_core : transparency → name → list expr → tactic expr
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/- Similar to mk_app, but allows to specify which arguments are explicit/implicit.
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Example, given
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a b : nat
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then
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mk_mapp_core semireducible "ite" [some (a > b), none, none, some a, some b]
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returns the application
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@ite.{1} (a > b) (nat.decidable_gt a b) nat a b -/
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meta_constant mk_mapp_core : transparency → name → list (option expr) → tactic expr
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/- Given a local constant t, if t has type (lhs = rhs) apply susbstitution.
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Otherwise, try to find a local constant that has type of the form (t = t') or (t' = t).
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The tactic fails if the given expression is not a local constant. -/
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meta_constant subst : expr → tactic unit
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meta_constant exact : expr → tactic unit
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/- Elaborate the given quoted expression with respect to the current main goal. -/
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meta_constant to_expr : qexpr → tactic expr
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/- Return true if the given expression is a type class. -/
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meta_constant is_class : expr → tactic bool
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/- Try to create an instance of the given type class. -/
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meta_constant mk_instance : expr → tactic expr
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/- Change the target of the main goal.
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The input expression must be definitionally equal to the current target. -/
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meta_constant change : expr → tactic unit
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/- (assert H T), adds a new goal for T, and the hypothesis (H : T) in the current goal. -/
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meta_constant assert : name → expr → tactic unit
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/- (assertv H T P), adds the hypothesis (H : T) in the current goal if P has type T. -/
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meta_constant assertv : name → expr → expr → tactic unit
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/- (define H T), adds a new goal for T, and the hypothesis (H : T := ?M) in the current goal. -/
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meta_constant define : name → expr → tactic unit
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/- (definev H T P), adds the hypothesis (H : T := P) in the current goal if P has type T. -/
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meta_constant definev : name → expr → expr → tactic unit
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/- rotate goals to the left -/
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meta_constant rotate_left : nat → tactic unit
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meta_constant get_goals : tactic (list expr)
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meta_constant set_goals : list expr → tactic unit
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/- (apply_core t all insts e), apply the expression e to the main goal,
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the unification is performed using the given transparency mode.
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If all is tt, then all unassigned meta-variables are added as new goals.
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If insts is tt, then use type class resolution to instantiate unassigned meta-variables. -/
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meta_constant apply_core : transparency → bool → bool → expr → tactic unit
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/- Create a fresh meta universe variable. -/
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meta_constant mk_meta_univ : tactic level
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/- Create a fresh meta-variable with the given type.
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The scope of the new meta-variable is the local context of the main goal. -/
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meta_constant mk_meta_var : expr → tactic expr
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/- Return the value assigned to the given universe meta-variable.
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Fail if argument is not an universe meta-variable or if it is not assigned. -/
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meta_constant get_univ_assignment : level → tactic level
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/- Return the value assigned to the given meta-variable.
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Fail if argument is not a meta-variable or if it is not assigned. -/
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meta_constant get_assignment : expr → tactic expr
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meta_constant mk_fresh_name : tactic name
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/- Return a hash code for expr that ignores inst_implicit arguments,
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and proofs. -/
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meta_constant abstract_hash : expr → tactic nat
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/- Return the "weight" of the given expr while ignoring inst_implicit arguments,
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and proofs. -/
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meta_constant abstract_weight : expr → tactic nat
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meta_constant abstract_eq : expr → expr → tactic bool
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/- (induction_core m H rec ns) induction on H using recursor rec, names for the new hypotheses
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are retrieved from ns. If ns does not have sufficient names, then use the internal binder names in the recursor. -/
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meta_constant induction_core : transparency → expr → name → list name → tactic unit
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/- (cases_core m H ns) apply cases_on recursor, names for the new hypotheses are retrieved from ns.
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H must be a local constant -/
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meta_constant cases_core : transparency → expr → list name → tactic unit
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/- (generalize_core m e n) -/
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meta_constant generalize_core : transparency → expr → name → tactic unit
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open list nat
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/- Add (H : T := pr) to the current goal -/
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meta_definition note (n : name) (pr : expr) : tactic unit :=
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do t ← infer_type pr,
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definev n t pr
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meta_definition whnf_target : tactic unit :=
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target >>= whnf >>= change
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meta_definition intro (n : name) : tactic expr :=
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do t ← target,
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if expr.is_pi t = tt ∨ expr.is_let t = tt then intro_core n
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else whnf_target >> intro_core n
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meta_definition intro1 : tactic expr :=
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intro `_
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meta_definition intros : tactic (list expr) :=
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do t ← target,
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match t with
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| expr.pi _ _ _ _ := do H ← intro1, Hs ← intros, return (H :: Hs)
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| expr.elet _ _ _ _ := do H ← intro1, Hs ← intros, return (H :: Hs)
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| _ := return []
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end
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meta_definition intro_lst : list name → tactic (list expr)
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| [] := return []
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| (n::ns) := do H ← intro n, Hs ← intro_lst ns, return (H :: Hs)
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meta_definition mk_app : name → list expr → tactic expr :=
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mk_app_core semireducible
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meta_definition mk_mapp : name → list (option expr) → tactic expr :=
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mk_mapp_core semireducible
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meta_definition revert (l : expr) : tactic nat :=
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revert_lst [l]
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meta_definition clear_lst : list name → tactic unit
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| [] := skip
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| (n::ns) := do H ← get_local n, clear H, clear_lst ns
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meta_definition unify : expr → expr → tactic unit :=
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unify_core semireducible
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meta_definition match_not (e : expr) : tactic expr :=
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match expr.is_not e with
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| some a := return a
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| none := fail "expression is not a negation"
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end
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meta_definition match_eq (e : expr) : tactic (expr × expr) :=
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match expr.is_eq e with
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| some (lhs, rhs) := return (lhs, rhs)
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| none := fail "expression is not an equality"
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end
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meta_definition match_ne (e : expr) : tactic (expr × expr) :=
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match expr.is_ne e with
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| some (lhs, rhs) := return (lhs, rhs)
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| none := fail "expression is not a disequality"
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end
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meta_definition match_heq (e : expr) : tactic (expr × expr × expr × expr) :=
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do match expr.is_heq e with
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| some (A, lhs, B, rhs) := return (A, lhs, B, rhs)
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| none := fail "expression is not a heterogeneous equality"
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end
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meta_definition match_refl_app (e : expr) : tactic (name × expr × expr) :=
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do env ← get_env,
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match environment.is_refl_app env e with
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| some (R, lhs, rhs) := return (R, lhs, rhs)
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| none := fail "expression is not an application of a reflexive relation"
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end
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meta_definition get_local_type (n : name) : tactic expr :=
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get_local n >>= infer_type
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meta_definition trace_result : tactic unit :=
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format_result >>= trace
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/- (find_same_type t es) tries to find in es an expression with type definitionally equal to t -/
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meta_definition find_same_type : expr → list expr → tactic expr
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| e [] := failed
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| e (H :: Hs) :=
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do t ← infer_type H,
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(unify e t >> return H) <|> find_same_type e Hs
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meta_definition assumption : tactic unit :=
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do { ctx ← local_context,
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t ← target,
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H ← find_same_type t ctx,
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exact H }
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<|> fail "assumption tactic failed"
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/- Swap first two goals, do nothing if tactic state does not have at least two goals. -/
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meta_definition swap : tactic unit :=
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do gs ← get_goals,
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match gs with
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| g₁ :: g₂ :: rs := set_goals (g₂ :: g₁ :: rs)
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| _ := skip
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end
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/- Return the number of goals that need to be solved -/
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meta_definition num_goals : tactic nat :=
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do gs ← get_goals,
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return (length gs)
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/- We have to provide the instance argument `[has_mod nat]` because
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mod for nat was not defined yet -/
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meta_definition rotate_right (n : nat) [has_mod nat] : tactic unit :=
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do ng ← num_goals,
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if ng = 0 then skip
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else rotate_left (ng - n % ng)
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meta_definition rotate : nat → tactic unit :=
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rotate_left
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/- first [t_1, ..., t_n] applies the first tactic that doesn't fail.
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The tactic fails if all t_i's fail. -/
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meta_definition first : list (tactic unit) → tactic unit
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| [] := fail "first tactic failed, no more alternatives"
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| (t::ts) := t <|> first ts
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/- Applies the given tactic to the main goal and fails if it is not solved. -/
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meta_definition solve1 (tac : tactic unit) : tactic unit :=
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do gs ← get_goals,
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match gs with
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| [] := fail "focus tactic failed, there isn't any goal left to focus"
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| (g::rs) :=
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do set_goals [g],
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tac,
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gs' ← get_goals,
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match gs' with
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| [] := set_goals rs
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| _ := fail "focus tactic failed, focused goal has not been solved"
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end
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end
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/- solve [t_1, ... t_n] applies the first tactic that solves the main goal. -/
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meta_definition solve (ts : list (tactic unit)) : tactic unit :=
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first $ map solve1 ts
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private meta_definition focus_aux : list (tactic unit) → list expr → list expr → tactic unit
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| [] gs rs := set_goals $ gs ++ rs
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| (t::ts) (g::gs) rs := do
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set_goals [g], t, rs' ← get_goals,
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focus_aux ts gs (rs ++ rs')
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| (t::ts) [] rs := fail "focus tactic failed, insufficient number of goals"
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/- focus [t_1, ..., t_n] applies t_i to the i-th goal. Fails if there are less tha n goals. -/
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meta_definition focus (ts : list (tactic unit)) : tactic unit :=
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do gs ← get_goals, focus_aux ts gs []
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private meta_definition all_goals_core : tactic unit → list expr → list expr → tactic unit
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| tac [] ac := set_goals ac
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| tac (g :: gs) ac :=
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do set_goals [g],
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tac,
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new_gs ← get_goals,
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all_goals_core tac gs (ac ++ new_gs)
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/- Apply the given tactic to all goals. -/
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meta_definition all_goals (tac : tactic unit) : tactic unit :=
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do gs ← get_goals,
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all_goals_core tac gs []
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/- LCF-style AND_THEN tactic. It applies tac1, and if succeed applies tac2 to each subgoal produced by tac1 -/
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meta_definition seq (tac1 : tactic unit) (tac2 : tactic unit) : tactic unit :=
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do g::gs ← get_goals | failed,
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set_goals [g],
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tac1, all_goals tac2,
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gs' ← get_goals,
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set_goals (gs' ++ gs)
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meta_definition tactic_has_andthen [instance] : has_andthen (tactic unit) :=
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has_andthen.mk seq
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/- Applies tac if c holds -/
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meta_definition when (c : Prop) [decidable c] (tac : tactic unit) : tactic unit :=
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if c then tac else skip
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meta_constant is_trace_enabled_for : name → bool
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/- Execute tac only if option trace.n is set to true. -/
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meta_definition when_tracing (n : name) (tac : tactic unit) : tactic unit :=
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when (is_trace_enabled_for n = tt) tac
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/- Fail if there are no remaining goals. -/
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meta_definition fail_if_no_goals : tactic unit :=
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do n ← num_goals,
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when (n = 0) (fail "tactic failed, there are no goals to be solved")
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/- Fail if there are unsolved goals. -/
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meta_definition now : tactic unit :=
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do n ← num_goals,
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when (n ≠ 0) (fail "now tactic failed, there are unsolved goals")
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meta_definition apply : expr → tactic unit :=
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apply_core semireducible ff tt
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meta_definition fapply : expr → tactic unit :=
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apply_core semireducible tt tt
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/- Try to solve the main goal using type class resolution. -/
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meta_definition apply_instance : tactic unit :=
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do tgt ← target,
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b ← is_class tgt,
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if b = tt then mk_instance tgt >>= exact
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else fail "apply_instance tactic fail, target is not a type class"
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/- Create a list of universe meta-variables of the given size. -/
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meta_definition mk_num_meta_univs : nat → tactic (list level)
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| 0 := return []
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| (succ n) := do
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l ← mk_meta_univ,
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ls ← mk_num_meta_univs n,
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return (l::ls)
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/- Return (expr.const c [l_1, ..., l_n]) where l_i's are fresh universe meta-variables. -/
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meta_definition mk_const (c : name) : tactic expr :=
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do env ← get_env,
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decl ← returnex (environment.get env c),
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num ← return (length (declaration.univ_params decl)),
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ls ← mk_num_meta_univs num,
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return (expr.const c ls)
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/- Create a fresh universe ?u, a metavariable (?T : Type.{?u}),
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and return metavariable (?M : ?T).
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This action can be used to create a meta-variable when
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we don't know its type at creation time -/
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meta_definition mk_mvar : tactic expr :=
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do u ← mk_meta_univ,
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t ← mk_meta_var (expr.sort u),
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mk_meta_var t
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private meta_definition get_pi_arity_aux : expr → tactic nat
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| (expr.pi n bi d b) :=
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do m ← mk_fresh_name,
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l ← return (expr.local_const m n bi d),
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new_b ← whnf (expr.instantiate_var b l),
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r ← get_pi_arity_aux new_b,
|
||
return (r + 1)
|
||
| _ := return 0
|
||
|
||
/- Compute the arity of the given (Pi-)type -/
|
||
meta_definition get_pi_arity (type : expr) : tactic nat :=
|
||
whnf type >>= get_pi_arity_aux
|
||
|
||
/- Compute the arity of the given function -/
|
||
meta_definition get_arity (fn : expr) : tactic nat :=
|
||
infer_type fn >>= get_pi_arity
|
||
|
||
meta_definition triv : tactic unit := mk_const `trivial >>= exact
|
||
|
||
meta_definition by_contradiction (H : name) : tactic expr :=
|
||
do tgt : expr ← target,
|
||
(match_not tgt >> return ())
|
||
<|>
|
||
(mk_mapp `decidable.by_contradiction [some tgt, none] >>= apply)
|
||
<|>
|
||
fail "tactic by_contradiction failed, target is not a negation nor a decidable proposition (remark: when 'local attribute classical.prop_decidable [instance]' is used all propositions are decidable)",
|
||
intro H
|
||
|
||
meta_definition cases (H : expr) : tactic unit :=
|
||
cases_core semireducible H []
|
||
|
||
meta_definition cases_using : expr → list name → tactic unit :=
|
||
cases_core semireducible
|
||
|
||
meta_definition generalize : expr → name → tactic unit :=
|
||
generalize_core semireducible
|
||
|
||
meta_definition generalizes : list expr → tactic unit
|
||
| [] := skip
|
||
| (e::es) := generalize e `x >> generalizes es
|
||
|
||
end tactic
|