diff options
Diffstat (limited to 'backends/lean')
| -rw-r--r-- | backends/lean/Base/Primitives/Scalar.lean | 396 |
1 files changed, 198 insertions, 198 deletions
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean index b11bd2a1..285bc7fb 100644 --- a/backends/lean/Base/Primitives/Scalar.lean +++ b/backends/lean/Base/Primitives/Scalar.lean @@ -271,6 +271,9 @@ structure Scalar (ty : ScalarTy) where hmax : val ≤ Scalar.max ty deriving Repr +instance (ty : ScalarTy) : CoeOut (Scalar ty) Int where + coe := λ v => v.val + theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) : Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty -> Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty @@ -535,9 +538,9 @@ instance {ty} : HAnd (Scalar ty) (Scalar ty) (Scalar ty) where -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.add_spec {ty} {x y : Scalar ty} - (hmin : Scalar.min ty ≤ x.val + y.val) - (hmax : x.val + y.val ≤ Scalar.max ty) : - (∃ z, x + y = ret z ∧ z.val = x.val + y.val) := by + (hmin : Scalar.min ty ≤ ↑x + y.val) + (hmax : ↑x + ↑y ≤ Scalar.max ty) : + (∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y) := by -- Applying the unfoldings only on the left conv => congr; ext; lhs; unfold HAdd.hAdd instHAddScalarResult; simp [add, tryMk] split @@ -545,75 +548,75 @@ theorem Scalar.add_spec {ty} {x y : Scalar ty} . tauto theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty} - (hmax : x.val + y.val ≤ Scalar.max ty) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by - have hmin : Scalar.min ty ≤ x.val + y.val := by + (hmax : ↑x + ↑y ≤ Scalar.max ty) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by + have hmin : Scalar.min ty ≤ ↑x + ↑y := by have hx := x.hmin have hy := y.hmin cases ty <;> simp [min, ScalarTy.isSigned] at * <;> linarith apply add_spec <;> assumption /- Fine-grained theorems -/ -@[pspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by +@[pspec] theorem Usize.add_spec {x y : Usize} (hmax : ↑x + ↑y ≤ Usize.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by apply Scalar.add_unsigned_spec <;> simp [ScalarTy.isSigned, Scalar.max, *] -@[pspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by +@[pspec] theorem U8.add_spec {x y : U8} (hmax : ↑x + ↑y ≤ U8.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by apply Scalar.add_unsigned_spec <;> simp [ScalarTy.isSigned, Scalar.max, *] -@[pspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by +@[pspec] theorem U16.add_spec {x y : U16} (hmax : ↑x + ↑y ≤ U16.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by apply Scalar.add_unsigned_spec <;> simp [ScalarTy.isSigned, Scalar.max, *] -@[pspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by +@[pspec] theorem U32.add_spec {x y : U32} (hmax : ↑x + ↑y ≤ U32.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by apply Scalar.add_unsigned_spec <;> simp [ScalarTy.isSigned, Scalar.max, *] -@[pspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by +@[pspec] theorem U64.add_spec {x y : U64} (hmax : ↑x + ↑y ≤ U64.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by apply Scalar.add_unsigned_spec <;> simp [ScalarTy.isSigned, Scalar.max, *] -@[pspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by +@[pspec] theorem U128.add_spec {x y : U128} (hmax : ↑x + ↑y ≤ U128.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := by apply Scalar.add_unsigned_spec <;> simp [ScalarTy.isSigned, Scalar.max, *] @[pspec] theorem Isize.add_spec {x y : Isize} - (hmin : Isize.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ Isize.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := + (hmin : Isize.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ Isize.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I8.add_spec {x y : I8} - (hmin : I8.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I8.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := + (hmin : I8.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I8.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I16.add_spec {x y : I16} - (hmin : I16.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I16.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := + (hmin : I16.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I16.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I32.add_spec {x y : I32} - (hmin : I32.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I32.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := + (hmin : I32.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I32.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I64.add_spec {x y : I64} - (hmin : I64.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I64.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := + (hmin : I64.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I64.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I128.add_spec {x y : I128} - (hmin : I128.min ≤ x.val + y.val) (hmax : x.val + y.val ≤ I128.max) : - ∃ z, x + y = ret z ∧ z.val = x.val + y.val := + (hmin : I128.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I128.max) : + ∃ z, x + y = ret z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.sub_spec {ty} {x y : Scalar ty} - (hmin : Scalar.min ty ≤ x.val - y.val) - (hmax : x.val - y.val ≤ Scalar.max ty) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by + (hmin : Scalar.min ty ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ Scalar.max ty) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by conv => congr; ext; lhs; simp [HSub.hSub, sub, tryMk, Sub.sub] split . simp [pure] @@ -621,9 +624,9 @@ theorem Scalar.sub_spec {ty} {x y : Scalar ty} . tauto theorem Scalar.sub_unsigned_spec {ty : ScalarTy} (s : ¬ ty.isSigned) - {x y : Scalar ty} (hmin : Scalar.min ty ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by - have : x.val - y.val ≤ Scalar.max ty := by + {x y : Scalar ty} (hmin : Scalar.min ty ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by + have : ↑x - ↑y ≤ Scalar.max ty := by have hx := x.hmin have hxm := x.hmax have hy := y.hmin @@ -632,65 +635,65 @@ theorem Scalar.sub_unsigned_spec {ty : ScalarTy} (s : ¬ ty.isSigned) apply sub_spec <;> assumption /- Fine-grained theorems -/ -@[pspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by +@[pspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by apply Scalar.sub_unsigned_spec <;> simp_all [Scalar.min, ScalarTy.isSigned] -@[pspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by +@[pspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by apply Scalar.sub_unsigned_spec <;> simp_all [Scalar.min, ScalarTy.isSigned] -@[pspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by +@[pspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by apply Scalar.sub_unsigned_spec <;> simp_all [Scalar.min, ScalarTy.isSigned] -@[pspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by +@[pspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by apply Scalar.sub_unsigned_spec <;> simp_all [Scalar.min, ScalarTy.isSigned] -@[pspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by +@[pspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by apply Scalar.sub_unsigned_spec <;> simp_all [Scalar.min, ScalarTy.isSigned] -@[pspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by +@[pspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ ↑x - ↑y) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := by apply Scalar.sub_unsigned_spec <;> simp_all [Scalar.min, ScalarTy.isSigned] -@[pspec] theorem Isize.sub_spec {x y : Isize} (hmin : Isize.min ≤ x.val - y.val) - (hmax : x.val - y.val ≤ Isize.max) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := +@[pspec] theorem Isize.sub_spec {x y : Isize} (hmin : Isize.min ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ Isize.max) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax -@[pspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ x.val - y.val) - (hmax : x.val - y.val ≤ I8.max) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := +@[pspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ I8.max) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax -@[pspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ x.val - y.val) - (hmax : x.val - y.val ≤ I16.max) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := +@[pspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ I16.max) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax -@[pspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ x.val - y.val) - (hmax : x.val - y.val ≤ I32.max) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := +@[pspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ I32.max) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax -@[pspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ x.val - y.val) - (hmax : x.val - y.val ≤ I64.max) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := +@[pspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ I64.max) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax -@[pspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ x.val - y.val) - (hmax : x.val - y.val ≤ I128.max) : - ∃ z, x - y = ret z ∧ z.val = x.val - y.val := +@[pspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ ↑x - ↑y) + (hmax : ↑x - ↑y ≤ I128.max) : + ∃ z, x - y = ret z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax -- Generic theorem - shouldn't be used much theorem Scalar.mul_spec {ty} {x y : Scalar ty} - (hmin : Scalar.min ty ≤ x.val * y.val) - (hmax : x.val * y.val ≤ Scalar.max ty) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by + (hmin : Scalar.min ty ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ Scalar.max ty) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by conv => congr; ext; lhs; simp [HMul.hMul] simp [mul, tryMk] split @@ -699,91 +702,91 @@ theorem Scalar.mul_spec {ty} {x y : Scalar ty} . tauto theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty} - (hmax : x.val * y.val ≤ Scalar.max ty) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by - have : Scalar.min ty ≤ x.val * y.val := by + (hmax : ↑x * ↑y ≤ Scalar.max ty) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by + have : Scalar.min ty ≤ ↑x * ↑y := by have hx := x.hmin have hy := y.hmin cases ty <;> simp [ScalarTy.isSigned] at * <;> apply mul_nonneg hx hy apply mul_spec <;> assumption /- Fine-grained theorems -/ -@[pspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by +@[pspec] theorem Usize.mul_spec {x y : Usize} (hmax : ↑x * ↑y ≤ Usize.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by apply Scalar.mul_unsigned_spec <;> simp_all [Scalar.max, ScalarTy.isSigned] -@[pspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by +@[pspec] theorem U8.mul_spec {x y : U8} (hmax : ↑x * ↑y ≤ U8.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by apply Scalar.mul_unsigned_spec <;> simp_all [Scalar.max, ScalarTy.isSigned] -@[pspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by +@[pspec] theorem U16.mul_spec {x y : U16} (hmax : ↑x * ↑y ≤ U16.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by apply Scalar.mul_unsigned_spec <;> simp_all [Scalar.max, ScalarTy.isSigned] -@[pspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by +@[pspec] theorem U32.mul_spec {x y : U32} (hmax : ↑x * ↑y ≤ U32.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by apply Scalar.mul_unsigned_spec <;> simp_all [Scalar.max, ScalarTy.isSigned] -@[pspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by +@[pspec] theorem U64.mul_spec {x y : U64} (hmax : ↑x * ↑y ≤ U64.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by apply Scalar.mul_unsigned_spec <;> simp_all [Scalar.max, ScalarTy.isSigned] -@[pspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by +@[pspec] theorem U128.mul_spec {x y : U128} (hmax : ↑x * ↑y ≤ U128.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := by apply Scalar.mul_unsigned_spec <;> simp_all [Scalar.max, ScalarTy.isSigned] -@[pspec] theorem Isize.mul_spec {x y : Isize} (hmin : Isize.min ≤ x.val * y.val) - (hmax : x.val * y.val ≤ Isize.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := +@[pspec] theorem Isize.mul_spec {x y : Isize} (hmin : Isize.min ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ Isize.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax -@[pspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ x.val * y.val) - (hmax : x.val * y.val ≤ I8.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := +@[pspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ I8.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax -@[pspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ x.val * y.val) - (hmax : x.val * y.val ≤ I16.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := +@[pspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ I16.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax -@[pspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ x.val * y.val) - (hmax : x.val * y.val ≤ I32.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := +@[pspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ I32.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax -@[pspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ x.val * y.val) - (hmax : x.val * y.val ≤ I64.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := +@[pspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ I64.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax -@[pspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ x.val * y.val) - (hmax : x.val * y.val ≤ I128.max) : - ∃ z, x * y = ret z ∧ z.val = x.val * y.val := +@[pspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ ↑x * ↑y) + (hmax : ↑x * ↑y ≤ I128.max) : + ∃ z, x * y = ret z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.div_spec {ty} {x y : Scalar ty} - (hnz : y.val ≠ 0) - (hmin : Scalar.min ty ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ Scalar.max ty) : - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := by + (hnz : ↑y ≠ (0 : Int)) + (hmin : Scalar.min ty ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ Scalar.max ty) : + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := by simp [HDiv.hDiv, div, Div.div] simp [tryMk, *] simp [pure] rfl theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty} - (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by + (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by have h : Scalar.min ty = 0 := by cases ty <;> simp [ScalarTy.isSigned, min] at * have hx := x.hmin have hy := y.hmin simp [h] at hx hy - have hmin : 0 ≤ x.val / y.val := Int.ediv_nonneg hx hy - have hmax : x.val / y.val ≤ Scalar.max ty := by - have := Int.ediv_le_self y.val hx + have hmin : 0 ≤ ↑x / ↑y := Int.ediv_nonneg hx hy + have hmax : ↑x / ↑y ≤ Scalar.max ty := by + have := Int.ediv_le_self ↑y hx have := x.hmax linarith have hs := @div_spec ty x y hnz @@ -791,158 +794,158 @@ theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S apply hs /- Fine-grained theorems -/ -@[pspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by +@[pspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by apply Scalar.div_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by +@[pspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by apply Scalar.div_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by +@[pspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by apply Scalar.div_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by +@[pspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by apply Scalar.div_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by +@[pspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by apply Scalar.div_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) : - ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by +@[pspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x / y = ret z ∧ (↑z : Int) = ↑x / ↑y := by apply Scalar.div_unsigned_spec <;> simp [ScalarTy.isSigned, *] @[pspec] theorem Isize.div_spec (x : Isize) {y : Isize} - (hnz : y.val ≠ 0) - (hmin : Isize.min ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ Isize.max): - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : Isize.min ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ Isize.max): + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I8.div_spec (x : I8) {y : I8} - (hnz : y.val ≠ 0) - (hmin : I8.min ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ I8.max): - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I8.min ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ I8.max): + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I16.div_spec (x : I16) {y : I16} - (hnz : y.val ≠ 0) - (hmin : I16.min ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ I16.max): - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I16.min ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ I16.max): + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I32.div_spec (x : I32) {y : I32} - (hnz : y.val ≠ 0) - (hmin : I32.min ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ I32.max): - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I32.min ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ I32.max): + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I64.div_spec (x : I64) {y : I64} - (hnz : y.val ≠ 0) - (hmin : I64.min ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ I64.max): - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I64.min ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ I64.max): + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I128.div_spec (x : I128) {y : I128} - (hnz : y.val ≠ 0) - (hmin : I128.min ≤ scalar_div x.val y.val) - (hmax : scalar_div x.val y.val ≤ I128.max): - ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I128.min ≤ scalar_div ↑x ↑y) + (hmax : scalar_div ↑x ↑y ≤ I128.max): + ∃ z, x / y = ret z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.rem_spec {ty} {x y : Scalar ty} - (hnz : y.val ≠ 0) - (hmin : Scalar.min ty ≤ scalar_rem x.val y.val) - (hmax : scalar_rem x.val y.val ≤ Scalar.max ty) : - ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by + (hnz : ↑y ≠ (0 : Int)) + (hmin : Scalar.min ty ≤ scalar_rem ↑x ↑y) + (hmax : scalar_rem ↑x ↑y ≤ Scalar.max ty) : + ∃ z, x % y = ret z ∧ (↑z : Int) = scalar_rem ↑x ↑y := by simp [HMod.hMod, rem] simp [tryMk, *] simp [pure] rfl theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty} - (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by + (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by have h : Scalar.min ty = 0 := by cases ty <;> simp [ScalarTy.isSigned, min] at * have hx := x.hmin have hy := y.hmin simp [h] at hx hy - have hmin : 0 ≤ x.val % y.val := Int.emod_nonneg x.val hnz - have hmax : x.val % y.val ≤ Scalar.max ty := by - have h : 0 < y.val := by int_tac - have h := Int.emod_lt_of_pos x.val h + have hmin : (0 : Int) ≤ x % y := Int.emod_nonneg ↑x hnz + have hmax : ↑x % ↑y ≤ Scalar.max ty := by + have h : (0 : Int) < y := by int_tac + have h := Int.emod_lt_of_pos ↑x h have := y.hmax linarith have hs := @rem_spec ty x y hnz simp [*] at hs simp [*] -@[pspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by +@[pspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by apply Scalar.rem_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by +@[pspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by apply Scalar.rem_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by +@[pspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by apply Scalar.rem_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by +@[pspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by apply Scalar.rem_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by +@[pspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by apply Scalar.rem_unsigned_spec <;> simp [ScalarTy.isSigned, *] -@[pspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) : - ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by +@[pspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : ↑y ≠ (0 : Int)) : + ∃ z, x % y = ret z ∧ (↑z : Int) = ↑x % ↑y := by apply Scalar.rem_unsigned_spec <;> simp [ScalarTy.isSigned, *] @[pspec] theorem I8.rem_spec (x : I8) {y : I8} - (hnz : y.val ≠ 0) - (hmin : I8.min ≤ scalar_rem x.val y.val) - (hmax : scalar_rem x.val y.val ≤ I8.max): - ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I8.min ≤ scalar_rem ↑x ↑y) + (hmax : scalar_rem ↑x ↑y ≤ I8.max): + ∃ z, x % y = ret z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I16.rem_spec (x : I16) {y : I16} - (hnz : y.val ≠ 0) - (hmin : I16.min ≤ scalar_rem x.val y.val) - (hmax : scalar_rem x.val y.val ≤ I16.max): - ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I16.min ≤ scalar_rem ↑x ↑y) + (hmax : scalar_rem ↑x ↑y ≤ I16.max): + ∃ z, x % y = ret z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I32.rem_spec (x : I32) {y : I32} - (hnz : y.val ≠ 0) - (hmin : I32.min ≤ scalar_rem x.val y.val) - (hmax : scalar_rem x.val y.val ≤ I32.max): - ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I32.min ≤ scalar_rem ↑x ↑y) + (hmax : scalar_rem ↑x ↑y ≤ I32.max): + ∃ z, x % y = ret z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I64.rem_spec (x : I64) {y : I64} - (hnz : y.val ≠ 0) - (hmin : I64.min ≤ scalar_rem x.val y.val) - (hmax : scalar_rem x.val y.val ≤ I64.max): - ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I64.min ≤ scalar_rem ↑x ↑y) + (hmax : scalar_rem ↑x ↑y ≤ I64.max): + ∃ z, x % y = ret z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I128.rem_spec (x : I128) {y : I128} - (hnz : y.val ≠ 0) - (hmin : I128.min ≤ scalar_rem x.val y.val) - (hmax : scalar_rem x.val y.val ≤ I128.max): - ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := + (hnz : ↑y ≠ (0 : Int)) + (hmin : I128.min ≤ scalar_rem ↑x ↑y) + (hmax : scalar_rem ↑x ↑y ≤ I128.max): + ∃ z, x % y = ret z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax -- ofIntCore @@ -1039,24 +1042,24 @@ instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val -- Not marking this one with @[simp] on purpose theorem Scalar.eq_equiv {ty : ScalarTy} (x y : Scalar ty) : - x = y ↔ x.val = y.val := by + x = y ↔ (↑x : Int) = ↑y := by cases x; cases y; simp_all -- This is sometimes useful when rewriting the goal with the local assumptions @[simp] theorem Scalar.eq_imp {ty : ScalarTy} (x y : Scalar ty) : - x = y → x.val = y.val := (eq_equiv x y).mp + x = y → (↑x : Int) = ↑y := (eq_equiv x y).mp theorem Scalar.lt_equiv {ty : ScalarTy} (x y : Scalar ty) : - x < y ↔ x.val < y.val := by simp [LT.lt] + x < y ↔ (↑x : Int) < ↑y := by simp [LT.lt] @[simp] theorem Scalar.lt_imp {ty : ScalarTy} (x y : Scalar ty) : - x < y → x.val < y.val := (lt_equiv x y).mp + x < y → (↑x : Int) < ↑y := (lt_equiv x y).mp theorem Scalar.le_equiv {ty : ScalarTy} (x y : Scalar ty) : - x ≤ y ↔ x.val ≤ y.val := by simp [LE.le] + x ≤ y ↔ (↑x : Int) ≤ ↑y := by simp [LE.le] @[simp] theorem Scalar.le_imp {ty : ScalarTy} (x y : Scalar ty) : - x ≤ y → x.val ≤ y.val := (le_equiv x y).mp + x ≤ y → (↑x : Int) ≤ ↑y := (le_equiv x y).mp instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt .. instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe .. @@ -1076,9 +1079,6 @@ instance (ty : ScalarTy) : DecidableEq (Scalar ty) := | isTrue h => isTrue (Scalar.eq_of_val_eq h) | isFalse h => isFalse (Scalar.ne_of_val_ne h) -instance (ty : ScalarTy) : CoeOut (Scalar ty) Int where - coe := λ v => v.val - @[simp] theorem Scalar.neq_to_neq_val {ty} : ∀ {i j : Scalar ty}, (¬ i = j) ↔ ¬ i.val = j.val := by intro i j; cases i; cases j; simp |