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/- This file contains tactics to solve arithmetic goals -/
import Lean
import Lean.Meta.Tactic.Simp
import Init.Data.List.Basic
import Mathlib.Tactic.RunCmd
import Mathlib.Tactic.Linarith
-- TODO: there is no Omega tactic for now - it seems it hasn't been ported yet
--import Mathlib.Tactic.Omega
import Base.Utils
import Base.Arith.Base
namespace Arith
open Utils
-- Remark: I tried a version of the shape `HasScalarProp {a : Type} (x : a)`
-- but the lookup didn't work
class HasIntProp (a : Sort u) where
prop_ty : a → Prop
prop : ∀ x:a, prop_ty x
class PropHasImp (x : Prop) where
concl : Prop
prop : x → concl
-- This also works for `x ≠ y` because this expression reduces to `¬ x = y`
-- and `Ne` is marked as `reducible`
instance (x y : Int) : PropHasImp (¬ x = y) where
concl := x < y ∨ x > y
prop := λ (h:x ≠ y) => ne_is_lt_or_gt h
open Lean Lean.Elab Lean.Meta
-- Explore a term by decomposing the applications (we explore the applied
-- functions and their arguments, but ignore lambdas, forall, etc. -
-- should we go inside?).
partial def foldTermApps (k : α → Expr → MetaM α) (s : α) (e : Expr) : MetaM α := do
-- We do it in a very simpler manner: we deconstruct applications,
-- and recursively explore the sub-expressions. Note that we do
-- not go inside foralls and abstractions (should we?).
e.withApp fun f args => do
let s ← k s f
args.foldlM (foldTermApps k) s
-- Provided a function `k` which lookups type class instances on an expression,
-- collect all the instances lookuped by applying `k` on the sub-expressions of `e`.
def collectInstances
(k : Expr → MetaM (Option Expr)) (s : HashSet Expr) (e : Expr) : MetaM (HashSet Expr) := do
let k s e := do
match ← k e with
| none => pure s
| some i => pure (s.insert i)
foldTermApps k s e
-- Similar to `collectInstances`, but explores all the local declarations in the
-- main context.
def collectInstancesFromMainCtx (k : Expr → MetaM (Option Expr)) : Tactic.TacticM (HashSet Expr) := do
Tactic.withMainContext do
-- Get the local context
let ctx ← Lean.MonadLCtx.getLCtx
-- Just a matter of precaution
let ctx ← instantiateLCtxMVars ctx
-- Initialize the hashset
let hs := HashSet.empty
-- Explore the declarations
let decls ← ctx.getDecls
decls.foldlM (fun hs d => collectInstances k hs d.toExpr) hs
-- Helper
def lookupProp (fName : String) (className : Name) (e : Expr) : MetaM (Option Expr) := do
trace[Arith] fName
-- TODO: do we need Lean.observing?
-- This actually eliminates the error messages
Lean.observing? do
trace[Arith] m!"{fName}: observing"
let ty ← Lean.Meta.inferType e
let hasProp ← mkAppM className #[ty]
let hasPropInst ← trySynthInstance hasProp
match hasPropInst with
| LOption.some i =>
trace[Arith] "Found {fName} instance"
let i_prop ← mkProjection i (Name.mkSimple "prop")
some (← mkAppM' i_prop #[e])
| _ => none
-- Return an instance of `HasIntProp` for `e` if it has some
def lookupHasIntProp (e : Expr) : MetaM (Option Expr) :=
lookupProp "lookupHasIntProp" ``HasIntProp e
-- Collect the instances of `HasIntProp` for the subexpressions in the context
def collectHasIntPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
collectInstancesFromMainCtx lookupHasIntProp
-- Return an instance of `PropHasImp` for `e` if it has some
def lookupPropHasImp (e : Expr) : MetaM (Option Expr) := do
trace[Arith] "lookupPropHasImp"
-- TODO: do we need Lean.observing?
-- This actually eliminates the error messages
Lean.observing? do
trace[Arith] "lookupPropHasImp: observing"
let ty ← Lean.Meta.inferType e
trace[Arith] "lookupPropHasImp: ty: {ty}"
let cl ← mkAppM ``PropHasImp #[ty]
let inst ← trySynthInstance cl
match inst with
| LOption.some i =>
trace[Arith] "Found PropHasImp instance"
let i_prop ← mkProjection i (Name.mkSimple "prop")
some (← mkAppM' i_prop #[e])
| _ => none
-- Collect the instances of `PropHasImp` for the subexpressions in the context
def collectPropHasImpInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
collectInstancesFromMainCtx lookupPropHasImp
elab "display_prop_has_imp_instances" : tactic => do
trace[Arith] "Displaying the PropHasImp instances"
let hs ← collectPropHasImpInstancesFromMainCtx
hs.forM fun e => do
trace[Arith] "+ PropHasImp instance: {e}"
example (x y : Int) (_ : x ≠ y) (_ : ¬ x = y) : True := by
display_prop_has_imp_instances
simp
-- Lookup instances in a context and introduce them with additional declarations.
def introInstances (declToUnfold : Name) (lookup : Expr → MetaM (Option Expr)) : Tactic.TacticM (Array Expr) := do
let hs ← collectInstancesFromMainCtx lookup
hs.toArray.mapM fun e => do
let type ← inferType e
let name ← mkFreshUserName `h
-- Add a declaration
let nval ← Utils.addDeclTac name e type (asLet := false)
-- Simplify to unfold the declaration to unfold (i.e., the projector)
Utils.simpAt [declToUnfold] [] [] (Tactic.Location.targets #[mkIdent name] false)
-- Return the new value
pure nval
def introHasIntPropInstances : Tactic.TacticM (Array Expr) := do
trace[Arith] "Introducing the HasIntProp instances"
introInstances ``HasIntProp.prop_ty lookupHasIntProp
-- Lookup the instances of `HasIntProp for all the sub-expressions in the context,
-- and introduce the corresponding assumptions
elab "intro_has_int_prop_instances" : tactic => do
let _ ← introHasIntPropInstances
-- Lookup the instances of `PropHasImp for all the sub-expressions in the context,
-- and introduce the corresponding assumptions
elab "intro_prop_has_imp_instances" : tactic => do
trace[Arith] "Introducing the PropHasImp instances"
let _ ← introInstances ``PropHasImp.concl lookupPropHasImp
example (x y : Int) (h0 : x ≤ y) (h1 : x ≠ y) : x < y := by
intro_prop_has_imp_instances
rename_i h
split_disj h
. linarith
. linarith
/- Boosting a bit the linarith tac.
We do the following:
- for all the assumptions of the shape `(x : Int) ≠ y` or `¬ (x = y), we
introduce two goals with the assumptions `x < y` and `x > y`
TODO: we could create a PR for mathlib.
-/
def intTacPreprocess (extraPreprocess : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
Tactic.withMainContext do
-- Lookup the instances of PropHasImp (this is how we detect assumptions
-- of the proper shape), introduce assumptions in the context and split
-- on those
-- TODO: get rid of the assumptions that we split
let rec splitOnAsms (asms : List Expr) : Tactic.TacticM Unit :=
match asms with
| [] => pure ()
| asm :: asms =>
let k := splitOnAsms asms
Utils.splitDisjTac asm k k
-- Introduce the scalar bounds
let _ ← introHasIntPropInstances
-- Extra preprocessing, before we split on the disjunctions
extraPreprocess
-- Split
let asms ← introInstances ``PropHasImp.concl lookupPropHasImp
splitOnAsms asms.toList
elab "int_tac_preprocess" : tactic =>
intTacPreprocess (do pure ())
def intTac (extraPreprocess : Tactic.TacticM Unit) : Tactic.TacticM Unit := do
Tactic.withMainContext do
Tactic.focus do
-- Preprocess - wondering if we should do this before or after splitting
-- the goal. I think before leads to a smaller proof term?
Tactic.allGoals (intTacPreprocess extraPreprocess)
-- Split the conjunctions in the goal
Tactic.allGoals (Utils.repeatTac Utils.splitConjTarget)
-- Call linarith
let linarith :=
let cfg : Linarith.LinarithConfig := {
-- We do this with our custom preprocessing
splitNe := false
}
Tactic.liftMetaFinishingTactic <| Linarith.linarith false [] cfg
Tactic.allGoals linarith
elab "int_tac" : tactic =>
intTac (do pure ())
example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
int_tac_preprocess
linarith
linarith
example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
int_tac
-- Checking that things append correctly when there are several disjunctions
example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y := by
int_tac
-- Checking that things append correctly when there are several disjunctions
example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y ∧ x + y ≥ 2 := by
int_tac
end Arith
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