import Lean import Lean.Meta.Tactic.Simp import Mathlib.Tactic.Linarith import Base.Primitives.Base import Base.Diverge.Base import Base.Progress.Base import Base.Arith.Int namespace Primitives ---------------------- -- MACHINE INTEGERS -- ---------------------- -- We redefine our machine integers types. -- For Isize/Usize, we reuse `getNumBits` from `USize`. You cannot reduce `getNumBits` -- using the simplifier, meaning that proofs do not depend on the compile-time value of -- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really support, at -- least officially, 16-bit microcontrollers, so this seems like a fine design decision -- for now.) -- Note from Chris Bailey: "If there's more than one salient property of your -- definition then the subtyping strategy might get messy, and the property part -- of a subtype is less discoverable by the simplifier or tactics like -- library_search." So, we will not add refinements on the return values of the -- operations defined on Primitives, but will rather rely on custom lemmas to -- invert on possible return values of the primitive operations. -- Machine integer constants, done via `ofNatCore`, which requires a proof that -- the `Nat` fits within the desired integer type. We provide a custom tactic. open Result Error open System.Platform.getNumBits -- TODO: is there a way of only importing System.Platform.getNumBits? -- @[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val -- Remark: Lean seems to use < for the comparisons with the upper bounds by convention. -- The "structured" bounds def Isize.smin : Int := - (HPow.hPow 2 (size_num_bits - 1)) def Isize.smax : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1 def I8.smin : Int := - (HPow.hPow 2 7) def I8.smax : Int := HPow.hPow 2 7 - 1 def I16.smin : Int := - (HPow.hPow 2 15) def I16.smax : Int := HPow.hPow 2 15 - 1 def I32.smin : Int := -(HPow.hPow 2 31) def I32.smax : Int := HPow.hPow 2 31 - 1 def I64.smin : Int := -(HPow.hPow 2 63) def I64.smax : Int := HPow.hPow 2 63 - 1 def I128.smin : Int := -(HPow.hPow 2 127) def I128.smax : Int := HPow.hPow 2 127 - 1 def Usize.smin : Int := 0 def Usize.smax : Int := HPow.hPow 2 size_num_bits - 1 def U8.smin : Int := 0 def U8.smax : Int := HPow.hPow 2 8 - 1 def U16.smin : Int := 0 def U16.smax : Int := HPow.hPow 2 16 - 1 def U32.smin : Int := 0 def U32.smax : Int := HPow.hPow 2 32 - 1 def U64.smin : Int := 0 def U64.smax : Int := HPow.hPow 2 64 - 1 def U128.smin : Int := 0 def U128.smax : Int := HPow.hPow 2 128 - 1 -- The "normalized" bounds, that we use in practice def I8.min : Int := -128 def I8.max : Int := 127 def I16.min : Int := -32768 def I16.max : Int := 32767 def I32.min : Int := -2147483648 def I32.max : Int := 2147483647 def I64.min : Int := -9223372036854775808 def I64.max : Int := 9223372036854775807 def I128.min : Int := -170141183460469231731687303715884105728 def I128.max : Int := 170141183460469231731687303715884105727 @[simp] def U8.min : Int := 0 def U8.max : Int := 255 @[simp] def U16.min : Int := 0 def U16.max : Int := 65535 @[simp] def U32.min : Int := 0 def U32.max : Int := 4294967295 @[simp] def U64.min : Int := 0 def U64.max : Int := 18446744073709551615 @[simp] def U128.min : Int := 0 def U128.max : Int := 340282366920938463463374607431768211455 @[simp] def Usize.min : Int := 0 def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } := ⟨ Isize.smin, by simp [Isize.smin] cases System.Platform.numBits_eq <;> unfold System.Platform.numBits at * <;> simp [*] <;> decide ⟩ def Isize.refined_max : { n:Int // n = I32.max ∨ n = I64.max } := ⟨ Isize.smax, by simp [Isize.smax] cases System.Platform.numBits_eq <;> unfold System.Platform.numBits at * <;> simp [*] <;> decide ⟩ def Usize.refined_max : { n:Int // n = U32.max ∨ n = U64.max } := ⟨ Usize.smax, by simp [Usize.smax] cases System.Platform.numBits_eq <;> unfold System.Platform.numBits at * <;> simp [*] <;> decide ⟩ def Isize.min := Isize.refined_min.val def Isize.max := Isize.refined_max.val def Usize.max := Usize.refined_max.val inductive ScalarTy := | Isize | I8 | I16 | I32 | I64 | I128 | Usize | U8 | U16 | U32 | U64 | U128 def ScalarTy.isSigned (ty : ScalarTy) : Bool := match ty with | Isize | I8 | I16 | I32 | I64 | I128 => true | Usize | U8 | U16 | U32 | U64 | U128 => false def Scalar.smin (ty : ScalarTy) : Int := match ty with | .Isize => Isize.smin | .I8 => I8.smin | .I16 => I16.smin | .I32 => I32.smin | .I64 => I64.smin | .I128 => I128.smin | .Usize => Usize.smin | .U8 => U8.smin | .U16 => U16.smin | .U32 => U32.smin | .U64 => U64.smin | .U128 => U128.smin def Scalar.smax (ty : ScalarTy) : Int := match ty with | .Isize => Isize.smax | .I8 => I8.smax | .I16 => I16.smax | .I32 => I32.smax | .I64 => I64.smax | .I128 => I128.smax | .Usize => Usize.smax | .U8 => U8.smax | .U16 => U16.smax | .U32 => U32.smax | .U64 => U64.smax | .U128 => U128.smax def Scalar.min (ty : ScalarTy) : Int := match ty with | .Isize => Isize.min | .I8 => I8.min | .I16 => I16.min | .I32 => I32.min | .I64 => I64.min | .I128 => I128.min | .Usize => Usize.min | .U8 => U8.min | .U16 => U16.min | .U32 => U32.min | .U64 => U64.min | .U128 => U128.min def Scalar.max (ty : ScalarTy) : Int := match ty with | .Isize => Isize.max | .I8 => I8.max | .I16 => I16.max | .I32 => I32.max | .I64 => I64.max | .I128 => I128.max | .Usize => Usize.max | .U8 => U8.max | .U16 => U16.max | .U32 => U32.max | .U64 => U64.max | .U128 => U128.max def Scalar.smin_eq (ty : ScalarTy) : Scalar.min ty = Scalar.smin ty := by cases ty <;> rfl def Scalar.smax_eq (ty : ScalarTy) : Scalar.max ty = Scalar.smax ty := by cases ty <;> rfl -- "Conservative" bounds -- We use those because we can't compare to the isize bounds (which can't -- reduce at compile-time). Whenever we perform an arithmetic operation like -- addition we need to check that the result is in bounds: we first compare -- to the conservative bounds, which reduce, then compare to the real bounds. -- This is useful for the various #asserts that we want to reduce at -- type-checking time. def Scalar.cMin (ty : ScalarTy) : Int := match ty with | .Isize => Scalar.min .I32 | _ => Scalar.min ty def Scalar.cMax (ty : ScalarTy) : Int := match ty with | .Isize => Scalar.max .I32 | .Usize => Scalar.max .U32 | _ => Scalar.max ty theorem Scalar.min_lt_max (ty : ScalarTy) : Scalar.min ty < Scalar.max ty := by cases ty <;> simp [Scalar.min, Scalar.max] <;> try decide . simp [Isize.min, Isize.max] have h1 := Isize.refined_min.property have h2 := Isize.refined_max.property cases h1 <;> cases h2 <;> simp [*] <;> decide . simp [Usize.max] have h := Usize.refined_max.property cases h <;> simp [*] <;> decide theorem Scalar.min_le_max (ty : ScalarTy) : Scalar.min ty ≤ Scalar.max ty := by have := Scalar.min_lt_max ty int_tac theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by cases ty <;> (simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *; try decide) have h := Isize.refined_min.property cases h <;> simp [*, Isize.min] decide theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by cases ty <;> (simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *; try decide) . have h := Isize.refined_max.property cases h <;> simp [*, Isize.max]; decide . have h := Usize.refined_max.property cases h <;> simp [*, Usize.max]; decide theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by have := Scalar.cMin_bound ty linarith theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by have := Scalar.cMax_bound ty linarith /-- The scalar type. We could use a subtype, but it using a custom structure type allows us to have more control over the coercions and the simplifications (we tried using a subtype and it caused issues especially as we had to make the Scalar type non-reducible, so that we could have more control, but leading to some natural equalities not being obvious to the simplifier anymore). -/ structure Scalar (ty : ScalarTy) where val : Int hmin : Scalar.min ty ≤ val hmax : val ≤ Scalar.max ty deriving Repr instance (ty : ScalarTy) : CoeOut (Scalar ty) Int where coe := λ v => v.val /- Activate the ↑ notation -/ attribute [coe] Scalar.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 := λ h => by apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith /- [match_pattern] attribute: allows to us `Scalar.ofIntCore` inside of patterns. This is particularly useful once we introduce notations like `#u32` (which desugards to `Scalar.ofIntCore`) as it allows to write expressions like this: Example: ``` match x with | 0#u32 => ... | 1#u32 => ... | ... ``` -/ @[match_pattern] def Scalar.ofIntCore {ty : ScalarTy} (x : Int) (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : Scalar ty := { val := x, hmin := h.left, hmax := h.right } -- The definitions below are used later to introduce nice syntax for constants, -- like `1#u32`. We are reusing the technique described here: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Different.20elaboration.20inside.2Foutside.20of.20match.20patterns/near/425455284 class InBounds (ty : ScalarTy) (x : Int) := hInBounds : Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty -- This trick to trigger reduction for decidable propositions comes from -- here: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/instance.20with.20tactic.20autoparam/near/343495807 class Decide (p : Prop) [Decidable p] : Prop where isTrue : p instance : @Decide p (.isTrue h) := @Decide.mk p (_) h instance [Decide (Scalar.cMin ty ≤ v ∧ v ≤ Scalar.cMax ty)] : InBounds ty v where hInBounds := Decide.isTrue @[reducible, match_pattern] def Scalar.ofInt {ty : ScalarTy} (x : Int) [InBounds ty x] : Scalar ty := Scalar.ofIntCore x (Scalar.bound_suffices ty x InBounds.hInBounds) @[simp] abbrev Scalar.in_bounds (ty : ScalarTy) (x : Int) : Prop := Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty @[simp] abbrev Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool := (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty) theorem Scalar.check_bounds_imp_in_bounds {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) : Scalar.in_bounds ty x := by simp at * have ⟨ hmin, hmax ⟩ := h have hbmin := Scalar.cMin_bound ty have hbmax := Scalar.cMax_bound ty cases hmin <;> cases hmax <;> apply And.intro <;> linarith theorem Scalar.check_bounds_eq_in_bounds (ty : ScalarTy) (x : Int) : Scalar.check_bounds ty x ↔ Scalar.in_bounds ty x := by constructor <;> intro h . apply (check_bounds_imp_in_bounds h) . simp_all -- Further thoughts: look at what has been done here: -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean -- and -- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean -- which both contain a fair amount of reasoning already! def Scalar.tryMkOpt (ty : ScalarTy) (x : Int) : Option (Scalar ty) := if h:Scalar.check_bounds ty x then -- If we do: -- ``` -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_imp_in_bounds h) -- Scalar.ofIntCore x hmin hmax -- ``` -- then normalization blocks (for instance, some proofs which use reflexivity fail). -- However, the version below doesn't block reduction (TODO: investigate): some (Scalar.ofIntCore x (Scalar.check_bounds_imp_in_bounds h)) else none def Result.ofOption {a : Type u} (x : Option a) (e : Error) : Result a := match x with | some x => ok x | none => fail e def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) := Result.ofOption (tryMkOpt ty x) integerOverflow theorem Scalar.tryMk_eq (ty : ScalarTy) (x : Int) : match tryMk ty x with | ok y => y.val = x ∧ in_bounds ty x | fail _ => ¬ (in_bounds ty x) | _ => False := by simp [tryMk, ofOption, tryMkOpt, ofIntCore] have h := check_bounds_eq_in_bounds ty x split_ifs <;> simp_all @[simp] theorem Scalar.tryMk_eq_div (ty : ScalarTy) (x : Int) : tryMk ty x = div ↔ False := by simp [tryMk, ofOption, tryMkOpt] split_ifs <;> simp def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val) -- Our custom remainder operation, which satisfies the semantics of Rust -- TODO: is there a better way? def scalar_rem (x y : Int) : Int := if 0 ≤ x then x % y else - (|x| % |y|) @[simp] def scalar_rem_nonneg {x y : Int} (hx : 0 ≤ x) : scalar_rem x y = x % y := by intros simp [*, scalar_rem] -- Our custom division operation, which satisfies the semantics of Rust -- TODO: is there a better way? def scalar_div (x y : Int) : Int := if 0 ≤ x && 0 ≤ y then x / y else if 0 ≤ x && y < 0 then - (|x| / |y|) else if x < 0 && 0 ≤ y then - (|x| / |y|) else |x| / |y| @[simp] def scalar_div_nonneg {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : scalar_div x y = x / y := by intros simp [*, scalar_div] -- Checking that the remainder operation is correct #assert scalar_rem 1 2 = 1 #assert scalar_rem (-1) 2 = -1 #assert scalar_rem 1 (-2) = 1 #assert scalar_rem (-1) (-2) = -1 #assert scalar_rem 7 3 = (1:Int) #assert scalar_rem (-7) 3 = -1 #assert scalar_rem 7 (-3) = 1 #assert scalar_rem (-7) (-3) = -1 -- Checking that the division operation is correct #assert scalar_div 3 2 = 1 #assert scalar_div (-3) 2 = -1 #assert scalar_div 3 (-2) = -1 #assert scalar_div (-3) (-2) = 1 #assert scalar_div 7 3 = 2 #assert scalar_div (-7) 3 = -2 #assert scalar_div 7 (-3) = -2 #assert scalar_div (-7) (-3) = 2 def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) := if y.val != 0 then Scalar.tryMk ty (scalar_div x.val y.val) else fail divisionByZero def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) := if y.val != 0 then Scalar.tryMk ty (scalar_rem x.val y.val) else fail divisionByZero def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (x.val + y.val) def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (x.val - y.val) def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (x.val * y.val) -- TODO: shift left def Scalar.shiftl {ty0 ty1 : ScalarTy} (x : Scalar ty0) (y : Scalar ty1) : Result (Scalar ty0) := sorry -- TODO: shift right def Scalar.shiftr {ty0 ty1 : ScalarTy} (x : Scalar ty0) (y : Scalar ty1) : Result (Scalar ty0) := sorry -- TODO: xor def Scalar.xor {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Scalar ty := sorry -- TODO: and def Scalar.and {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Scalar ty := sorry -- TODO: or def Scalar.or {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Scalar ty := sorry -- Cast an integer from a [src_ty] to a [tgt_ty] -- TODO: double-check the semantics of casts in Rust def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) := Scalar.tryMk tgt_ty x.val -- This can't fail, but for now we make all casts faillible (easier for the translation) def Scalar.cast_bool (tgt_ty : ScalarTy) (x : Bool) : Result (Scalar tgt_ty) := Scalar.tryMk tgt_ty (if x then 1 else 0) -- The scalar types -- We declare the definitions as reducible so that Lean can unfold them (useful -- for type class resolution for instance). @[reducible] def Isize := Scalar .Isize @[reducible] def I8 := Scalar .I8 @[reducible] def I16 := Scalar .I16 @[reducible] def I32 := Scalar .I32 @[reducible] def I64 := Scalar .I64 @[reducible] def I128 := Scalar .I128 @[reducible] def Usize := Scalar .Usize @[reducible] def U8 := Scalar .U8 @[reducible] def U16 := Scalar .U16 @[reducible] def U32 := Scalar .U32 @[reducible] def U64 := Scalar .U64 @[reducible] def U128 := Scalar .U128 instance (ty : ScalarTy) : Inhabited (Scalar ty) := by constructor; cases ty <;> apply (Scalar.ofInt 0) -- TODO: reducible? @[reducible] def core_isize_min : Isize := Scalar.ofIntCore Isize.min (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Isize)) @[reducible] def core_isize_max : Isize := Scalar.ofIntCore Isize.max (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Isize)) @[reducible] def core_i8_min : I8 := Scalar.ofInt I8.min @[reducible] def core_i8_max : I8 := Scalar.ofInt I8.max @[reducible] def core_i16_min : I16 := Scalar.ofInt I16.min @[reducible] def core_i16_max : I16 := Scalar.ofInt I16.max @[reducible] def core_i32_min : I32 := Scalar.ofInt I32.min @[reducible] def core_i32_max : I32 := Scalar.ofInt I32.max @[reducible] def core_i64_min : I64 := Scalar.ofInt I64.min @[reducible] def core_i64_max : I64 := Scalar.ofInt I64.max @[reducible] def core_i128_min : I128 := Scalar.ofInt I128.min @[reducible] def core_i128_max : I128 := Scalar.ofInt I128.max -- TODO: reducible? @[reducible] def core_usize_min : Usize := Scalar.ofIntCore Usize.min (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Usize)) @[reducible] def core_usize_max : Usize := Scalar.ofIntCore Usize.max (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Usize)) @[reducible] def core_u8_min : U8 := Scalar.ofInt U8.min @[reducible] def core_u8_max : U8 := Scalar.ofInt U8.max @[reducible] def core_u16_min : U16 := Scalar.ofInt U16.min @[reducible] def core_u16_max : U16 := Scalar.ofInt U16.max @[reducible] def core_u32_min : U32 := Scalar.ofInt U32.min @[reducible] def core_u32_max : U32 := Scalar.ofInt U32.max @[reducible] def core_u64_min : U64 := Scalar.ofInt U64.min @[reducible] def core_u64_max : U64 := Scalar.ofInt U64.max @[reducible] def core_u128_min : U128 := Scalar.ofInt U128.min @[reducible] def core_u128_max : U128 := Scalar.ofInt U128.max -- TODO: below: not sure this is the best way. -- Should we rather overload operations like +, -, etc.? -- Also, it is possible to automate the generation of those definitions -- with macros (but would it be a good idea? It would be less easy to -- read the file, which is not supposed to change a lot) -- Negation /-- Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce one here. The notation typeclass for heterogeneous negation. -/ class HNeg (α : Type u) (β : outParam (Type v)) where /-- `- a` computes the negation of `a`. The meaning of this notation is type-dependent. -/ hNeg : α → β /- Notation for heterogeneous negation. We initially used the notation "-" but it conflicted with the homogeneous negation too much. In particular, it made terms like `-10` ambiguous, and seemingly caused to backtracking in elaboration, leading to definitions like arrays of constants to take an unreasonable time to get elaborated and type-checked. TODO: PR to replace Neg with HNeg in Lean? -/ prefix:75 "-." => HNeg.hNeg /- We need this, otherwise we break pattern matching like in: ``` def is_minus_one (x : Int) : Bool := match x with | -1 => true | _ => false ``` -/ attribute [match_pattern] HNeg.hNeg instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x -- Addition instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where hAdd x y := Scalar.add x y -- Substraction instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where hSub x y := Scalar.sub x y -- Multiplication instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where hMul x y := Scalar.mul x y -- Division instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where hDiv x y := Scalar.div x y -- Remainder instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where hMod x y := Scalar.rem x y -- Shift left instance {ty0 ty1} : HShiftLeft (Scalar ty0) (Scalar ty1) (Result (Scalar ty0)) where hShiftLeft x y := Scalar.shiftl x y -- Shift right instance {ty0 ty1} : HShiftRight (Scalar ty0) (Scalar ty1) (Result (Scalar ty0)) where hShiftRight x y := Scalar.shiftr x y -- Xor instance {ty} : HXor (Scalar ty) (Scalar ty) (Scalar ty) where hXor x y := Scalar.xor x y -- Or instance {ty} : HOr (Scalar ty) (Scalar ty) (Scalar ty) where hOr x y := Scalar.or x y -- And instance {ty} : HAnd (Scalar ty) (Scalar ty) (Scalar ty) where hAnd x y := Scalar.and x y -- core checked arithmetic operations /- A helper function that converts failure to none and success to some TODO: move up to Base module? -/ def Option.ofResult {a : Type u} (x : Result a) : Option a := match x with | ok x => some x | _ => none /- [core::num::{T}::checked_add] -/ def core.num.checked_add (x y : Scalar ty) : Option (Scalar ty) := Option.ofResult (x + y) def U8.checked_add (x y : U8) : Option U8 := core.num.checked_add x y def U16.checked_add (x y : U16) : Option U16 := core.num.checked_add x y def U32.checked_add (x y : U32) : Option U32 := core.num.checked_add x y def U64.checked_add (x y : U64) : Option U64 := core.num.checked_add x y def U128.checked_add (x y : U128) : Option U128 := core.num.checked_add x y def Usize.checked_add (x y : Usize) : Option Usize := core.num.checked_add x y def I8.checked_add (x y : I8) : Option I8 := core.num.checked_add x y def I16.checked_add (x y : I16) : Option I16 := core.num.checked_add x y def I32.checked_add (x y : I32) : Option I32 := core.num.checked_add x y def I64.checked_add (x y : I64) : Option I64 := core.num.checked_add x y def I128.checked_add (x y : I128) : Option I128 := core.num.checked_add x y def Isize.checked_add (x y : Isize) : Option Isize := core.num.checked_add x y /- [core::num::{T}::checked_sub] -/ def core.num.checked_sub (x y : Scalar ty) : Option (Scalar ty) := Option.ofResult (x - y) def U8.checked_sub (x y : U8) : Option U8 := core.num.checked_sub x y def U16.checked_sub (x y : U16) : Option U16 := core.num.checked_sub x y def U32.checked_sub (x y : U32) : Option U32 := core.num.checked_sub x y def U64.checked_sub (x y : U64) : Option U64 := core.num.checked_sub x y def U128.checked_sub (x y : U128) : Option U128 := core.num.checked_sub x y def Usize.checked_sub (x y : Usize) : Option Usize := core.num.checked_sub x y def I8.checked_sub (x y : I8) : Option I8 := core.num.checked_sub x y def I16.checked_sub (x y : I16) : Option I16 := core.num.checked_sub x y def I32.checked_sub (x y : I32) : Option I32 := core.num.checked_sub x y def I64.checked_sub (x y : I64) : Option I64 := core.num.checked_sub x y def I128.checked_sub (x y : I128) : Option I128 := core.num.checked_sub x y def Isize.checked_sub (x y : Isize) : Option Isize := core.num.checked_sub x y /- [core::num::{T}::checked_mul] -/ def core.num.checked_mul (x y : Scalar ty) : Option (Scalar ty) := Option.ofResult (x * y) def U8.checked_mul (x y : U8) : Option U8 := core.num.checked_mul x y def U16.checked_mul (x y : U16) : Option U16 := core.num.checked_mul x y def U32.checked_mul (x y : U32) : Option U32 := core.num.checked_mul x y def U64.checked_mul (x y : U64) : Option U64 := core.num.checked_mul x y def U128.checked_mul (x y : U128) : Option U128 := core.num.checked_mul x y def Usize.checked_mul (x y : Usize) : Option Usize := core.num.checked_mul x y def I8.checked_mul (x y : I8) : Option I8 := core.num.checked_mul x y def I16.checked_mul (x y : I16) : Option I16 := core.num.checked_mul x y def I32.checked_mul (x y : I32) : Option I32 := core.num.checked_mul x y def I64.checked_mul (x y : I64) : Option I64 := core.num.checked_mul x y def I128.checked_mul (x y : I128) : Option I128 := core.num.checked_mul x y def Isize.checked_mul (x y : Isize) : Option Isize := core.num.checked_mul x y /- [core::num::{T}::checked_div] -/ def core.num.checked_div (x y : Scalar ty) : Option (Scalar ty) := Option.ofResult (x / y) def U8.checked_div (x y : U8) : Option U8 := core.num.checked_div x y def U16.checked_div (x y : U16) : Option U16 := core.num.checked_div x y def U32.checked_div (x y : U32) : Option U32 := core.num.checked_div x y def U64.checked_div (x y : U64) : Option U64 := core.num.checked_div x y def U128.checked_div (x y : U128) : Option U128 := core.num.checked_div x y def Usize.checked_div (x y : Usize) : Option Usize := core.num.checked_div x y def I8.checked_div (x y : I8) : Option I8 := core.num.checked_div x y def I16.checked_div (x y : I16) : Option I16 := core.num.checked_div x y def I32.checked_div (x y : I32) : Option I32 := core.num.checked_div x y def I64.checked_div (x y : I64) : Option I64 := core.num.checked_div x y def I128.checked_div (x y : I128) : Option I128 := core.num.checked_div x y def Isize.checked_div (x y : Isize) : Option Isize := core.num.checked_div x y /- [core::num::{T}::checked_rem] -/ def core.num.checked_rem (x y : Scalar ty) : Option (Scalar ty) := Option.ofResult (x % y) def U8.checked_rem (x y : U8) : Option U8 := core.num.checked_rem x y def U16.checked_rem (x y : U16) : Option U16 := core.num.checked_rem x y def U32.checked_rem (x y : U32) : Option U32 := core.num.checked_rem x y def U64.checked_rem (x y : U64) : Option U64 := core.num.checked_rem x y def U128.checked_rem (x y : U128) : Option U128 := core.num.checked_rem x y def Usize.checked_rem (x y : Usize) : Option Usize := core.num.checked_rem x y def I8.checked_rem (x y : I8) : Option I8 := core.num.checked_rem x y def I16.checked_rem (x y : I16) : Option I16 := core.num.checked_rem x y def I32.checked_rem (x y : I32) : Option I32 := core.num.checked_rem x y def I64.checked_rem (x y : I64) : Option I64 := core.num.checked_rem x y def I128.checked_rem (x y : I128) : Option I128 := core.num.checked_rem x y def Isize.checked_rem (x y : Isize) : Option Isize := core.num.checked_rem x y theorem Scalar.add_equiv {ty} {x y : Scalar ty} : match x + y with | ok z => Scalar.in_bounds ty (↑x + ↑y) ∧ (↑z : Int) = ↑x + ↑y | fail _ => ¬ (Scalar.in_bounds ty (↑x + ↑y)) | _ => ⊥ := by -- Applying the unfoldings only inside the match conv in _ + _ => unfold HAdd.hAdd instHAddScalarResult; simp [add] have h := tryMk_eq ty (↑x + ↑y) simp [in_bounds] at h split at h <;> simp_all [check_bounds_eq_in_bounds] -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.add_spec {ty} {x y : Scalar ty} (hmin : Scalar.min ty ≤ ↑x + y.val) (hmax : ↑x + ↑y ≤ Scalar.max ty) : (∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y) := by have h := @add_equiv ty x y split at h <;> simp_all apply h theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty} (hmax : ↑x + ↑y ≤ Scalar.max ty) : ∃ z, x + y = ok 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 + ↑y ≤ Usize.max) : ∃ z, x + y = ok 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 + ↑y ≤ U8.max) : ∃ z, x + y = ok 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 + ↑y ≤ U16.max) : ∃ z, x + y = ok 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 + ↑y ≤ U32.max) : ∃ z, x + y = ok 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 + ↑y ≤ U64.max) : ∃ z, x + y = ok 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 + ↑y ≤ U128.max) : ∃ z, x + y = ok 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 + ↑y) (hmax : ↑x + ↑y ≤ Isize.max) : ∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I8.add_spec {x y : I8} (hmin : I8.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I8.max) : ∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I16.add_spec {x y : I16} (hmin : I16.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I16.max) : ∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I32.add_spec {x y : I32} (hmin : I32.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I32.max) : ∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I64.add_spec {x y : I64} (hmin : I64.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I64.max) : ∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax @[pspec] theorem I128.add_spec {x y : I128} (hmin : I128.min ≤ ↑x + ↑y) (hmax : ↑x + ↑y ≤ I128.max) : ∃ z, x + y = ok z ∧ (↑z : Int) = ↑x + ↑y := Scalar.add_spec hmin hmax theorem core.num.checked_add_spec {ty} {x y : Scalar ty} : match core.num.checked_add x y with | some z => Scalar.in_bounds ty (↑x + ↑y) ∧ ↑z = (↑x + ↑y : Int) | none => ¬ (Scalar.in_bounds ty (↑x + ↑y)) := by have h := Scalar.tryMk_eq ty (↑x + ↑y) simp only [checked_add, Option.ofResult] cases heq: x + y <;> simp_all <;> simp [HAdd.hAdd, Scalar.add] at heq <;> simp [Add.add] at heq <;> simp_all theorem Scalar.sub_equiv {ty} {x y : Scalar ty} : match x - y with | ok z => Scalar.in_bounds ty (↑x - ↑y) ∧ (↑z : Int) = ↑x - ↑y | fail _ => ¬ (Scalar.in_bounds ty (↑x - ↑y)) | _ => ⊥ := by -- Applying the unfoldings only inside the match conv in _ - _ => unfold HSub.hSub instHSubScalarResult; simp [sub] have h := tryMk_eq ty (↑x - ↑y) simp [in_bounds] at h split at h <;> simp_all [check_bounds_eq_in_bounds] -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.sub_spec {ty} {x y : Scalar ty} (hmin : Scalar.min ty ≤ ↑x - ↑y) (hmax : ↑x - ↑y ≤ Scalar.max ty) : ∃ z, x - y = ok z ∧ (↑z : Int) = ↑x - ↑y := by have h := @sub_equiv ty x y split at h <;> simp_all apply h theorem Scalar.sub_unsigned_spec {ty : ScalarTy} (s : ¬ ty.isSigned) {x y : Scalar ty} (hmin : Scalar.min ty ≤ ↑x - ↑y) : ∃ z, x - y = ok 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 cases ty <;> simp [min, max, ScalarTy.isSigned] at * <;> linarith intros apply sub_spec <;> assumption /- Fine-grained theorems -/ @[pspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ ↑x - ↑y) : ∃ z, x - y = ok 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 - ↑y) : ∃ z, x - y = ok 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 - ↑y) : ∃ z, x - y = ok 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 - ↑y) : ∃ z, x - y = ok 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 - ↑y) : ∃ z, x - y = ok 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 - ↑y) : ∃ z, x - y = ok 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 - ↑y) (hmax : ↑x - ↑y ≤ Isize.max) : ∃ z, x - y = ok z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax @[pspec] theorem I8.sub_spec {x y : I8} (hmin : I8.min ≤ ↑x - ↑y) (hmax : ↑x - ↑y ≤ I8.max) : ∃ z, x - y = ok z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax @[pspec] theorem I16.sub_spec {x y : I16} (hmin : I16.min ≤ ↑x - ↑y) (hmax : ↑x - ↑y ≤ I16.max) : ∃ z, x - y = ok z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax @[pspec] theorem I32.sub_spec {x y : I32} (hmin : I32.min ≤ ↑x - ↑y) (hmax : ↑x - ↑y ≤ I32.max) : ∃ z, x - y = ok z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax @[pspec] theorem I64.sub_spec {x y : I64} (hmin : I64.min ≤ ↑x - ↑y) (hmax : ↑x - ↑y ≤ I64.max) : ∃ z, x - y = ok z ∧ (↑z : Int) = ↑x - ↑y := Scalar.sub_spec hmin hmax @[pspec] theorem I128.sub_spec {x y : I128} (hmin : I128.min ≤ ↑x - ↑y) (hmax : ↑x - ↑y ≤ I128.max) : ∃ z, x - y = ok 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 * ↑y) (hmax : ↑x * ↑y ≤ Scalar.max ty) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := by conv => congr; ext; lhs; simp [HMul.hMul] simp [mul, tryMk, tryMkOpt, ofOption] split_ifs . simp [pure] rfl . tauto theorem core.num.checked_sub_spec {ty} {x y : Scalar ty} : match core.num.checked_sub x y with | some z => Scalar.in_bounds ty (↑x - ↑y) ∧ ↑z = (↑x - ↑y : Int) | none => ¬ (Scalar.in_bounds ty (↑x - ↑y)) := by have h := Scalar.tryMk_eq ty (↑x - ↑y) simp only [checked_sub, Option.ofResult] have add_neg_eq : x.val + (-y.val) = x.val - y.val := by omega -- TODO: why do we need this?? cases heq: x - y <;> simp_all <;> simp only [HSub.hSub, Scalar.sub, Sub.sub, Int.sub] at heq <;> simp_all theorem Scalar.mul_equiv {ty} {x y : Scalar ty} : match x * y with | ok z => Scalar.in_bounds ty (↑x * ↑y) ∧ (↑z : Int) = ↑x * ↑y | fail _ => ¬ (Scalar.in_bounds ty (↑x * ↑y)) | _ => ⊥ := by -- Applying the unfoldings only inside the match conv in _ * _ => unfold HMul.hMul instHMulScalarResult; simp [mul] have h := tryMk_eq ty (↑x * ↑y) simp [in_bounds] at h split at h <;> simp_all [check_bounds_eq_in_bounds] theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty} (hmax : ↑x * ↑y ≤ Scalar.max ty) : ∃ z, x * y = ok 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 * ↑y ≤ Usize.max) : ∃ z, x * y = ok 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 * ↑y ≤ U8.max) : ∃ z, x * y = ok 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 * ↑y ≤ U16.max) : ∃ z, x * y = ok 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 * ↑y ≤ U32.max) : ∃ z, x * y = ok 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 * ↑y ≤ U64.max) : ∃ z, x * y = ok 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 * ↑y ≤ U128.max) : ∃ z, x * y = ok 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 * ↑y) (hmax : ↑x * ↑y ≤ Isize.max) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax @[pspec] theorem I8.mul_spec {x y : I8} (hmin : I8.min ≤ ↑x * ↑y) (hmax : ↑x * ↑y ≤ I8.max) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax @[pspec] theorem I16.mul_spec {x y : I16} (hmin : I16.min ≤ ↑x * ↑y) (hmax : ↑x * ↑y ≤ I16.max) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax @[pspec] theorem I32.mul_spec {x y : I32} (hmin : I32.min ≤ ↑x * ↑y) (hmax : ↑x * ↑y ≤ I32.max) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax @[pspec] theorem I64.mul_spec {x y : I64} (hmin : I64.min ≤ ↑x * ↑y) (hmax : ↑x * ↑y ≤ I64.max) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax @[pspec] theorem I128.mul_spec {x y : I128} (hmin : I128.min ≤ ↑x * ↑y) (hmax : ↑x * ↑y ≤ I128.max) : ∃ z, x * y = ok z ∧ (↑z : Int) = ↑x * ↑y := Scalar.mul_spec hmin hmax theorem core.num.checked_mul_spec {ty} {x y : Scalar ty} : match core.num.checked_mul x y with | some z => Scalar.in_bounds ty (↑x * ↑y) ∧ ↑z = (↑x * ↑y : Int) | none => ¬ (Scalar.in_bounds ty (↑x * ↑y)) := by have h := Scalar.tryMk_eq ty (↑x * ↑y) simp only [checked_mul, Option.ofResult] have : Int.mul ↑x ↑y = ↑x * ↑y := by simp -- TODO: why do we need this?? cases heq: x * y <;> simp_all <;> simp only [HMul.hMul, Scalar.mul, Mul.mul] at heq <;> simp_all theorem Scalar.div_equiv {ty} {x y : Scalar ty} : match x / y with | ok z => y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_div ↑x ↑y) ∧ (↑z : Int) = scalar_div ↑x ↑y | fail _ => ¬ (y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_div ↑x ↑y)) | _ => ⊥ := by -- Applying the unfoldings only inside the match conv in _ / _ => unfold HDiv.hDiv instHDivScalarResult; simp [div] have h := tryMk_eq ty (scalar_div ↑x ↑y) simp [in_bounds] at h split_ifs <;> simp <;> split at h <;> simp_all [check_bounds_eq_in_bounds] -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.div_spec {ty} {x y : Scalar ty} (hnz : ↑y ≠ (0 : Int)) (hmin : Scalar.min ty ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ Scalar.max ty) : ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := by simp [HDiv.hDiv, div, Div.div] simp [tryMk, tryMkOpt, ofOption, *] rfl theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty} (hnz : ↑y ≠ (0 : Int)) : ∃ z, x / y = ok 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 / ↑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 simp [*] at hs apply hs /- Fine-grained theorems -/ @[pspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : ↑y ≠ (0 : Int)) : ∃ z, x / y = ok 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 ≠ (0 : Int)) : ∃ z, x / y = ok 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 ≠ (0 : Int)) : ∃ z, x / y = ok 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 ≠ (0 : Int)) : ∃ z, x / y = ok 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 ≠ (0 : Int)) : ∃ z, x / y = ok 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 ≠ (0 : Int)) : ∃ z, x / y = ok 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 ≠ (0 : Int)) (hmin : Isize.min ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ Isize.max): ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I8.div_spec (x : I8) {y : I8} (hnz : ↑y ≠ (0 : Int)) (hmin : I8.min ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ I8.max): ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I16.div_spec (x : I16) {y : I16} (hnz : ↑y ≠ (0 : Int)) (hmin : I16.min ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ I16.max): ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I32.div_spec (x : I32) {y : I32} (hnz : ↑y ≠ (0 : Int)) (hmin : I32.min ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ I32.max): ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I64.div_spec (x : I64) {y : I64} (hnz : ↑y ≠ (0 : Int)) (hmin : I64.min ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ I64.max): ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax @[pspec] theorem I128.div_spec (x : I128) {y : I128} (hnz : ↑y ≠ (0 : Int)) (hmin : I128.min ≤ scalar_div ↑x ↑y) (hmax : scalar_div ↑x ↑y ≤ I128.max): ∃ z, x / y = ok z ∧ (↑z : Int) = scalar_div ↑x ↑y := Scalar.div_spec hnz hmin hmax theorem core.num.checked_div_spec {ty} {x y : Scalar ty} : match core.num.checked_div x y with | some z => y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_div ↑x ↑y) ∧ ↑z = (scalar_div ↑x ↑y : Int) | none => ¬ (y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_div ↑x ↑y)) := by have h := Scalar.tryMk_eq ty (scalar_div ↑x ↑y) simp only [checked_div, Option.ofResult] cases heq0: (y.val = 0 : Bool) <;> cases heq1: x / y <;> simp_all <;> simp only [HDiv.hDiv, Scalar.div, Div.div] at heq1 <;> simp_all theorem Scalar.rem_equiv {ty} {x y : Scalar ty} : match x % y with | ok z => y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_rem ↑x ↑y) ∧ (↑z : Int) = scalar_rem ↑x ↑y | fail _ => ¬ (y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_rem ↑x ↑y)) | _ => ⊥ := by -- Applying the unfoldings only inside the match conv in _ % _ => unfold HMod.hMod instHModScalarResult; simp [rem] have h := tryMk_eq ty (scalar_rem ↑x ↑y) simp [in_bounds] at h split_ifs <;> simp <;> split at h <;> simp_all [check_bounds_eq_in_bounds] -- Generic theorem - shouldn't be used much @[pspec] theorem Scalar.rem_spec {ty} {x y : Scalar ty} (hnz : ↑y ≠ (0 : Int)) (hmin : Scalar.min ty ≤ scalar_rem ↑x ↑y) (hmax : scalar_rem ↑x ↑y ≤ Scalar.max ty) : ∃ z, x % y = ok z ∧ (↑z : Int) = scalar_rem ↑x ↑y := by simp [HMod.hMod, rem] simp [tryMk, tryMkOpt, ofOption, *] rfl theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty} (hnz : ↑y ≠ (0 : Int)) : ∃ z, x % y = ok 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 : 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 ≠ (0 : Int)) : ∃ z, x % y = ok 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 ≠ (0 : Int)) : ∃ z, x % y = ok 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 ≠ (0 : Int)) : ∃ z, x % y = ok 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 ≠ (0 : Int)) : ∃ z, x % y = ok 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 ≠ (0 : Int)) : ∃ z, x % y = ok 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 ≠ (0 : Int)) : ∃ z, x % y = ok 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 ≠ (0 : Int)) (hmin : I8.min ≤ scalar_rem ↑x ↑y) (hmax : scalar_rem ↑x ↑y ≤ I8.max): ∃ z, x % y = ok z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I16.rem_spec (x : I16) {y : I16} (hnz : ↑y ≠ (0 : Int)) (hmin : I16.min ≤ scalar_rem ↑x ↑y) (hmax : scalar_rem ↑x ↑y ≤ I16.max): ∃ z, x % y = ok z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I32.rem_spec (x : I32) {y : I32} (hnz : ↑y ≠ (0 : Int)) (hmin : I32.min ≤ scalar_rem ↑x ↑y) (hmax : scalar_rem ↑x ↑y ≤ I32.max): ∃ z, x % y = ok z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I64.rem_spec (x : I64) {y : I64} (hnz : ↑y ≠ (0 : Int)) (hmin : I64.min ≤ scalar_rem ↑x ↑y) (hmax : scalar_rem ↑x ↑y ≤ I64.max): ∃ z, x % y = ok z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax @[pspec] theorem I128.rem_spec (x : I128) {y : I128} (hnz : ↑y ≠ (0 : Int)) (hmin : I128.min ≤ scalar_rem ↑x ↑y) (hmax : scalar_rem ↑x ↑y ≤ I128.max): ∃ z, x % y = ok z ∧ (↑z : Int) = scalar_rem ↑x ↑y := Scalar.rem_spec hnz hmin hmax theorem core.num.checked_rem_spec {ty} {x y : Scalar ty} : match core.num.checked_rem x y with | some z => y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_rem ↑x ↑y) ∧ ↑z = (scalar_rem ↑x ↑y : Int) | none => ¬ (y.val ≠ 0 ∧ Scalar.in_bounds ty (scalar_rem ↑x ↑y)) := by have h := Scalar.tryMk_eq ty (scalar_rem ↑x ↑y) simp only [checked_rem, Option.ofResult] cases heq0: (y.val = 0 : Bool) <;> cases heq1: x % y <;> simp_all <;> simp only [HMod.hMod, Scalar.rem, Mod.mod] at heq1 <;> simp_all -- ofIntCore -- TODO: typeclass? def Isize.ofIntCore := @Scalar.ofIntCore .Isize def I8.ofIntCore := @Scalar.ofIntCore .I8 def I16.ofIntCore := @Scalar.ofIntCore .I16 def I32.ofIntCore := @Scalar.ofIntCore .I32 def I64.ofIntCore := @Scalar.ofIntCore .I64 def I128.ofIntCore := @Scalar.ofIntCore .I128 def Usize.ofIntCore := @Scalar.ofIntCore .Usize def U8.ofIntCore := @Scalar.ofIntCore .U8 def U16.ofIntCore := @Scalar.ofIntCore .U16 def U32.ofIntCore := @Scalar.ofIntCore .U32 def U64.ofIntCore := @Scalar.ofIntCore .U64 def U128.ofIntCore := @Scalar.ofIntCore .U128 -- ofInt -- TODO: typeclass? @[match_pattern] abbrev Isize.ofInt := @Scalar.ofInt .Isize @[match_pattern] abbrev I8.ofInt := @Scalar.ofInt .I8 @[match_pattern] abbrev I16.ofInt := @Scalar.ofInt .I16 @[match_pattern] abbrev I32.ofInt := @Scalar.ofInt .I32 @[match_pattern] abbrev I64.ofInt := @Scalar.ofInt .I64 @[match_pattern] abbrev I128.ofInt := @Scalar.ofInt .I128 @[match_pattern] abbrev Usize.ofInt := @Scalar.ofInt .Usize @[match_pattern] abbrev U8.ofInt := @Scalar.ofInt .U8 @[match_pattern] abbrev U16.ofInt := @Scalar.ofInt .U16 @[match_pattern] abbrev U32.ofInt := @Scalar.ofInt .U32 @[match_pattern] abbrev U64.ofInt := @Scalar.ofInt .U64 @[match_pattern] abbrev U128.ofInt := @Scalar.ofInt .U128 postfix:max "#isize" => Isize.ofInt postfix:max "#i8" => I8.ofInt postfix:max "#i16" => I16.ofInt postfix:max "#i32" => I32.ofInt postfix:max "#i64" => I64.ofInt postfix:max "#i128" => I128.ofInt postfix:max "#usize" => Usize.ofInt postfix:max "#u8" => U8.ofInt postfix:max "#u16" => U16.ofInt postfix:max "#u32" => U32.ofInt postfix:max "#u64" => U64.ofInt postfix:max "#u128" => U128.ofInt /- Testing the notations -/ example := 0#u32 example := 1#u32 example := 1#i32 example := 0#isize example := (-1)#isize example (x : U32) : Bool := match x with | 0#u32 => true | _ => false example (x : U32) : Bool := match x with | 1#u32 => true | _ => false example (x : I32) : Bool := match x with | (-1)#i32 => true | _ => false -- Notation for pattern matching -- We make the precedence looser than the negation. notation:70 a:70 "#scalar" => Scalar.mk (a) _ _ example {ty} (x : Scalar ty) : ℤ := match x with | v#scalar => v example {ty} (x : Scalar ty) : Bool := match x with | 1#scalar => true | _ => false example {ty} (x : Scalar ty) : Bool := match x with | -(1 : Int)#scalar => true | _ => false -- Testing the notations example : Result Usize := 0#usize + 1#usize -- TODO: factor those lemmas out @[simp] theorem Scalar.ofInt_val_eq {ty} (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : (Scalar.ofIntCore x h).val = x := by simp [Scalar.ofInt, Scalar.ofIntCore] @[simp] theorem Isize.ofInt_val_eq (h : Scalar.min ScalarTy.Isize ≤ x ∧ x ≤ Scalar.max ScalarTy.Isize) : (Isize.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem I8.ofInt_val_eq (h : Scalar.min ScalarTy.I8 ≤ x ∧ x ≤ Scalar.max ScalarTy.I8) : (I8.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem I16.ofInt_val_eq (h : Scalar.min ScalarTy.I16 ≤ x ∧ x ≤ Scalar.max ScalarTy.I16) : (I16.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem I32.ofInt_val_eq (h : Scalar.min ScalarTy.I32 ≤ x ∧ x ≤ Scalar.max ScalarTy.I32) : (I32.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem I64.ofInt_val_eq (h : Scalar.min ScalarTy.I64 ≤ x ∧ x ≤ Scalar.max ScalarTy.I64) : (I64.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem I128.ofInt_val_eq (h : Scalar.min ScalarTy.I128 ≤ x ∧ x ≤ Scalar.max ScalarTy.I128) : (I128.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem Usize.ofInt_val_eq (h : Scalar.min ScalarTy.Usize ≤ x ∧ x ≤ Scalar.max ScalarTy.Usize) : (Usize.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem U8.ofInt_val_eq (h : Scalar.min ScalarTy.U8 ≤ x ∧ x ≤ Scalar.max ScalarTy.U8) : (U8.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem U16.ofInt_val_eq (h : Scalar.min ScalarTy.U16 ≤ x ∧ x ≤ Scalar.max ScalarTy.U16) : (U16.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem U32.ofInt_val_eq (h : Scalar.min ScalarTy.U32 ≤ x ∧ x ≤ Scalar.max ScalarTy.U32) : (U32.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem U64.ofInt_val_eq (h : Scalar.min ScalarTy.U64 ≤ x ∧ x ≤ Scalar.max ScalarTy.U64) : (U64.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h @[simp] theorem U128.ofInt_val_eq (h : Scalar.min ScalarTy.U128 ≤ x ∧ x ≤ Scalar.max ScalarTy.U128) : (U128.ofIntCore x h).val = x := by apply Scalar.ofInt_val_eq h -- Comparisons instance {ty} : LT (Scalar ty) where lt a b := LT.lt a.val b.val 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 : 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 : Int) = ↑y → x = y := (eq_equiv x y).mpr theorem Scalar.lt_equiv {ty : ScalarTy} (x y : Scalar ty) : x < y ↔ (↑x : Int) < ↑y := by simp [LT.lt] @[simp] theorem Scalar.lt_imp {ty : ScalarTy} (x y : Scalar ty) : (↑x : Int) < (↑y) → x < y := (lt_equiv x y).mpr theorem Scalar.le_equiv {ty : ScalarTy} (x y : Scalar ty) : x ≤ y ↔ (↑x : Int) ≤ ↑y := by simp [LE.le] @[simp] theorem Scalar.le_imp {ty : ScalarTy} (x y : Scalar ty) : (↑x : Int) ≤ ↑y → x ≤ y := (le_equiv x y).mpr 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 .. theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val := h ▸ rfl theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) := fun h' => absurd (val_eq_of_eq h') h instance (ty : ScalarTy) : DecidableEq (Scalar ty) := fun i j => match decEq i.val j.val with | isTrue h => isTrue (Scalar.eq_of_val_eq h) | isFalse h => isFalse (Scalar.ne_of_val_ne h) @[simp] theorem Scalar.neq_to_neq_val {ty} : ∀ {i j : Scalar ty}, (¬ i = j) ↔ ¬ i.val = j.val := by simp [eq_equiv] end Primitives