cd3eb9125c
This PR implements the normalizer for linear integer arithmetic expressions. It is not connect to `simp +arith` yet because of some spurious `[simp]` attributes.
87 lines
2.8 KiB
Text
87 lines
2.8 KiB
Text
import Lean
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open Int.Linear
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-- Convenient RArray literals
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elab tk:"#R[" ts:term,* "]" : term => do
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let ts : Array Lean.Syntax := ts
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let es ← ts.mapM fun stx => Lean.Elab.Term.elabTerm stx none
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if h : 0 < es.size then
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return (Lean.RArray.toExpr (← Lean.Meta.inferType es[0]!) id (Lean.RArray.ofArray es h))
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else
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throwErrorAt tk "RArray cannot be empty"
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example (x₁ x₂ : Int) :
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Expr.denote #R[x₁, x₂] (.add (.add (.var 0) (.var 1)) (.num 3))
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=
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x₁ + x₂ + 3 :=
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rfl
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example (x₁ x₂ : Int) :
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Poly.denote #R[x₁, x₂] (.add 1 0 (.add 3 1 (.num 4)))
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=
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1 * x₁ + ((3 * x₂) + 4) :=
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rfl
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example (x₁ x₂ : Int) :
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Expr.denote #R[x₁, x₂] (.sub (.add (.mulR (.var 0) 4) (.mulL 2 (.var 1))) (.num 3))
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=
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(x₁*4) + 2*x₂ - 3 :=
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rfl
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example :
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Expr.toPoly (.add (.add (.var 1) (.var 1)) (.num 3))
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=
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Expr.toPoly (.add (.num 3) (.mulL 2 (.var 1))) :=
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rfl
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example :
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Expr.toPoly (.add (.add (.add (.var 1) (.var 1)) (.num 3)) (.var 2))
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=
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Expr.toPoly (.add (.add (.num 3) (.var 2)) (.mulL 2 (.var 1))) :=
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rfl
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example (x₁ x₂ x₃ : Int) :
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ExprCnstr.denote #R[x₁, x₂, x₃] (.eq (.sub (.add (.mulR (.var 0) 4) (.mulL 2 (.var 1))) (.num 3)) (.sub (.var 1) (.var 2)))
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=
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((x₁*4) + 2*x₂ - 3 = x₂ - x₃) :=
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rfl
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example :
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ExprCnstr.toPoly (.eq (.sub (.add (.mulR (.var 0) 4) (.mulL 2 (.var 1))) (.num 3)) (.sub (.var 1) (.var 2)))
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=
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ExprCnstr.toPoly (.eq (.add (.var 2) (.add (.var 1) (.add (.mulL 4 (.var 0)) (.num (-3))))) (.num 0)) :=
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rfl
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example (x₁ x₂ x₃ : Int) : (x₁ + x₂) + (x₂ + x₃) = x₃ + 2*x₂ + x₁ :=
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Expr.eq_of_toPoly_eq #R[x₁, x₂, x₃]
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(Expr.add (Expr.add (Expr.var 0) (Expr.var 1)) (Expr.add (Expr.var 1) (Expr.var 2)))
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(Expr.add (Expr.add (Expr.var 2) (Expr.mulL 2 (Expr.var 1))) (Expr.var 0))
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rfl
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example :
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ExprCnstr.toPoly
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(.eq (Expr.add (Expr.add (Expr.var 0) (Expr.var 1)) (Expr.add (Expr.var 1) (Expr.var 2)))
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(Expr.add (Expr.var 2) (Expr.var 1)))
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=
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ExprCnstr.toPoly
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(.eq (Expr.add (Expr.var 0) (Expr.var 1))
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(Expr.num 0))
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:=
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rfl
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example (x₁ x₂ x₃ : Int) : ((x₁ + x₂) + (x₂ + x₃) = x₃ + x₂) = (x₁ + x₂ = 0) :=
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ExprCnstr.eq_of_toPoly_eq #R[x₁, x₂, x₃]
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(.eq (Expr.add (Expr.add (Expr.var 0) (Expr.var 1)) (Expr.add (Expr.var 1) (Expr.var 2)))
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(Expr.add (Expr.var 2) (Expr.var 1)))
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(.eq (Expr.add (Expr.var 0) (Expr.var 1))
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(Expr.num 0))
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rfl
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example (x₁ x₂ x₃ : Int) : ((x₁ + x₂) + (x₂ + x₃) ≤ x₃ + x₂) = (x₁ + x₂ ≤ 0) :=
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ExprCnstr.eq_of_toPoly_eq #R[x₁, x₂, x₃]
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(.le (Expr.add (Expr.add (Expr.var 0) (Expr.var 1)) (Expr.add (Expr.var 1) (Expr.var 2)))
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(Expr.add (Expr.var 2) (Expr.var 1)))
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(.le (Expr.add (Expr.var 0) (Expr.var 1))
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(Expr.num 0))
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rfl
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