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import Base.Arith.Int
import Base.Primitives.Scalar
/- Automation for scalars - TODO: not sure it is worth having two files (Int.lean and Scalar.lean) -/
namespace Arith
open Lean Lean.Elab Lean.Meta
open Primitives
def scalarTacExtraPreprocess : Tactic.TacticM Unit := do
Tactic.withMainContext do
-- Inroduce the bounds for the isize/usize types
let add (e : Expr) : Tactic.TacticM Unit := do
let ty ← inferType e
let _ ← Utils.addDeclTac (← Utils.mkFreshAnonPropUserName) e ty (asLet := false)
add (← mkAppM ``Scalar.cMin_bound #[.const ``ScalarTy.Isize []])
add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Usize []])
add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Isize []])
-- Reveal the concrete bounds, simplify calls to [ofInt]
Utils.simpAt true {}
-- Unfoldings
[``Scalar.min, ``Scalar.max, ``Scalar.cMin, ``Scalar.cMax,
``I8.min, ``I16.min, ``I32.min, ``I64.min, ``I128.min,
``I8.max, ``I16.max, ``I32.max, ``I64.max, ``I128.max,
``U8.min, ``U16.min, ``U32.min, ``U64.min, ``U128.min,
``U8.max, ``U16.max, ``U32.max, ``U64.max, ``U128.max,
``Usize.min
]
-- Simp lemmas
[``Scalar.ofInt_val_eq, ``Scalar.neq_to_neq_val,
``Scalar.lt_equiv, ``Scalar.le_equiv, ``Scalar.eq_equiv]
-- Hypotheses
[] .wildcard
elab "scalar_tac_preprocess" : tactic =>
intTacPreprocess scalarTacExtraPreprocess
-- A tactic to solve linear arithmetic goals in the presence of scalars
def scalarTac (splitGoalConjs : Bool) : Tactic.TacticM Unit := do
intTac "scalar_tac" splitGoalConjs scalarTacExtraPreprocess
elab "scalar_tac" : tactic =>
scalarTac false
-- For termination proofs
syntax "scalar_decr_tac" : tactic
macro_rules
| `(tactic| scalar_decr_tac) =>
`(tactic|
simp_wf;
-- TODO: don't use a macro (namespace problems)
(first | apply Arith.to_int_to_nat_lt
| apply Arith.to_int_sub_to_nat_lt) <;>
simp_all <;> scalar_tac)
instance (ty : ScalarTy) : HasIntProp (Scalar ty) where
-- prop_ty is inferred
prop := λ x => And.intro x.hmin x.hmax
example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
intro_has_int_prop_instances
simp [*]
example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
scalar_tac
-- Checking that we explore the goal *and* projectors correctly
example (x : U32 × U32) : 0 ≤ x.fst.val := by
scalar_tac
-- Checking that we properly handle [ofInt]
example : U32.ofInt 1 ≤ U32.max := by
scalar_tac
example (x : Int) (h0 : 0 ≤ x) (h1 : x ≤ U32.max) :
U32.ofIntCore x (by constructor <;> scalar_tac) ≤ U32.max := by
scalar_tac
-- Not equal
example (x : U32) (h0 : ¬ x = U32.ofInt 0) : 0 < x.val := by
scalar_tac
/- See this: https://aeneas-verif.zulipchat.com/#narrow/stream/349819-general/topic/U64.20trouble/near/444049757
We solved it by removing the instance `OfNat` for `Scalar`.
Note however that we could also solve it with a simplification lemma.
However, after testing, we noticed we could only apply such a lemma with
the rewriting tactic (not the simplifier), probably because of the use
of typeclasses. -/
example {u: U64} (h1: (u : Int) < 2): (u : Int) = 0 ∨ (u : Int) = 1 := by
scalar_tac
end Arith
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