300 lines
10 KiB
Text
300 lines
10 KiB
Text
/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.Tree.Basic
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import Lean.Elab.Term
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/-!
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## Elaboration of tensor trees
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This file turns tensor expressions into tensor trees.
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-/
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open Lean
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open Lean.Elab.Term
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open Lean
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open Lean.Meta
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open Lean.Elab
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open Lean.Elab.Term
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open Lean Meta Elab Tactic
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open IndexNotation
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/-!
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## Indexies
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-/
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/-- A syntax category for indices of tensor expressions. -/
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declare_syntax_cat indexExpr
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/-- A basic index is a ident. -/
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syntax ident : indexExpr
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/-- An index can be a num, which will be used to evaluate the tensor. -/
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syntax num : indexExpr
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/-- Notation to discribe the jiggle of a tensor index. -/
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syntax "τ(" ident ")" : indexExpr
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/-- Bool which is ture if an index is a num. -/
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def indexExprIsNum (stx : Syntax) : Bool :=
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match stx with
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| `(indexExpr|$_:num) => true
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| _ => false
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/-- If an index is a num - the undelrying natural number. -/
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def indexToNum (stx : Syntax) : TermElabM Nat :=
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match stx with
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| `(indexExpr|$a:num) =>
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match a.raw.isNatLit? with
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| some n => return n
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| none => throwError "Expected a natural number literal."
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| _ =>
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throwError "Unsupported tensor expression syntax in indexToNum: {stx}"
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/-- When an index is not a num, the corresponding ident. -/
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def indexToIdent (stx : Syntax) : TermElabM Ident :=
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match stx with
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| `(indexExpr|$a:ident) => return a
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| `(indexExpr| τ($a:ident)) => return a
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| _ =>
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throwError "Unsupported tensor expression syntax in indexToIdent: {stx}"
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/-- Takes a pair ``a b : ℕ × TSyntax `indexExpr``. If `a.1 < b.1` and `a.2 = b.2` then
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outputs `some (a.1, b.1)`, otherwise `none`. -/
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def indexPosEq (a b : ℕ × TSyntax `indexExpr) : TermElabM (Option (ℕ × ℕ)) := do
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let a' ← indexToIdent a.2
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let b' ← indexToIdent b.2
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if a.1 < b.1 ∧ Lean.TSyntax.getId a' = Lean.TSyntax.getId b' then
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return some (a.1, b.1)
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else
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return none
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/-- Bool which is true if an index is of the form τ(i) that is, to be dualed. -/
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def indexToDual (stx : Syntax) : Bool :=
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match stx with
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| `(indexExpr| τ($_)) => true
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| _ => false
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/-!
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## Tensor expressions
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-/
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/-- A syntax category for tensor expressions. -/
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declare_syntax_cat tensorExpr
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/-- The syntax for a tensor node. -/
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syntax term "|" (ppSpace indexExpr)* : tensorExpr
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/-- The syntax for tensor prod two tensor nodes. -/
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syntax tensorExpr "⊗" tensorExpr : tensorExpr
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/-- The syntax for tensor addition. -/
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syntax tensorExpr "+" tensorExpr : tensorExpr
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/-- Allowing brackets to be used in a tensor expression. -/
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syntax "(" tensorExpr ")" : tensorExpr
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namespace TensorNode
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/-!
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## For tensor nodes.
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The operations are done in the following order:
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- evaluation.
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- dualization.
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- contraction.
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We also want to ensure the number of indices is correct.
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-/
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/-- The indices of a tensor node. Before contraction, dualisation, and evaluation. -/
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partial def getIndices (stx : Syntax) : TermElabM (List (TSyntax `indexExpr)) := do
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match stx with
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| `(tensorExpr| $_:term | $[$args]*) => do
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let indices ← args.toList.mapM fun arg => do
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match arg with
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| `(indexExpr|$t:indexExpr) => pure t
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return indices
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| _ =>
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throwError "Unsupported tensor expression syntax in getIndicesNode: {stx}"
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/-- Uses the structure of the tensor to get the number of indices. -/
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def getNoIndicesExact (stx : Syntax) : TermElabM ℕ := do
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let expr ← elabTerm stx none
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let type ← inferType expr
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match type with
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| Expr.app _ (Expr.app _ (Expr.app _ c)) =>
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let typeC ← inferType c
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match typeC with
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| Expr.forallE _ (Expr.app _ (Expr.app (Expr.app _ (Expr.lit (Literal.natVal n))) _)) _ _ =>
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return n
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| _ => throwError "Could not extract number of indices from tensor (getNoIndicesExact). "
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| _ =>
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throwError "Could not extract number of indices from tensor (getNoIndicesExact)."
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/-- The positions in getIndicesNode which get evaluated, and the value they take. -/
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partial def getEvalPos (stx : Syntax) : TermElabM (List (ℕ × ℕ)) := do
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let ind ← getIndices stx
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let indEnum := ind.enum
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let evals := indEnum.filter (fun x => indexExprIsNum x.2)
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let evals2 ← (evals.mapM (fun x => indexToNum x.2))
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return List.zip (evals.map (fun x => x.1)) evals2
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/-- For each element of `l : List (ℕ × ℕ)` applies `TensorTree.eval` to the given term. -/
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def evalSyntax (l : List (ℕ × ℕ)) (T : Term) : Term :=
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l.foldl (fun T' (x1, x2) => Syntax.mkApp (mkIdent ``TensorTree.eval)
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#[Syntax.mkNumLit (toString x1), Syntax.mkNumLit (toString x2), T']) T
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/-- The positions in getIndicesNode which get dualized. -/
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partial def getDualPos (stx : Syntax) : TermElabM (List ℕ) := do
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let ind ← getIndices stx
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let indFilt : List (TSyntax `indexExpr) := ind.filter (fun x => ¬ indexExprIsNum x)
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let indEnum := indFilt.enum
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let duals := indEnum.filter (fun x => indexToDual x.2)
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return duals.map (fun x => x.1)
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/-- For each element of `l : List ℕ` applies `TensorTree.jiggle` to the given term. -/
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def dualSyntax (l : List ℕ) (T : Term) : Term :=
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l.foldl (fun T' x => Syntax.mkApp (mkIdent ``TensorTree.jiggle)
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#[Syntax.mkNumLit (toString x), T']) T
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/-- The pairs of positions in getIndicesNode which get contracted. -/
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partial def getContrPos (stx : Syntax) : TermElabM (List (ℕ × ℕ)) := do
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let ind ← getIndices stx
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let indFilt : List (TSyntax `indexExpr) := ind.filter (fun x => ¬ indexExprIsNum x)
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let indEnum := indFilt.enum
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let bind := List.bind indEnum (fun a => indEnum.map (fun b => (a, b)))
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let filt ← bind.filterMapM (fun x => indexPosEq x.1 x.2)
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if ¬ ((filt.map Prod.fst).Nodup ∧ (filt.map Prod.snd).Nodup) then
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throwError "To many contractions"
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return filt
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/-- The list of indices after contraction. -/
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def withoutContr (stx : Syntax) : TermElabM (List (TSyntax `indexExpr)) := do
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let ind ← getIndices stx
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let indFilt : List (TSyntax `indexExpr) := ind.filter (fun x => ¬ indexExprIsNum x)
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return ind.filter (fun x => indFilt.count x ≤ 1)
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/-- For each element of `l : List (ℕ × ℕ)` applies `TensorTree.contr` to the given term. -/
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def contrSyntax (l : List (ℕ × ℕ)) (T : Term) : Term :=
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l.foldl (fun T' (x0, x1) => Syntax.mkApp (mkIdent ``TensorTree.contr)
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#[Syntax.mkNumLit (toString x1), Syntax.mkNumLit (toString x0), T']) T
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/-- Creates the syntax associated with a tensor node. -/
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def syntaxFull (stx : Syntax) : TermElabM Term := do
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match stx with
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| `(tensorExpr| $T:term | $[$args]*) => do
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let indices ← getIndices stx
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let rawIndex ← getNoIndicesExact T
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if indices.length ≠ rawIndex then
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throwError "The number of indices does not match the tensor {T}."
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let tensorNodeSyntax := Syntax.mkApp (mkIdent ``TensorTree.tensorNode) #[T]
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let evalSyntax := evalSyntax (← getEvalPos stx) tensorNodeSyntax
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let dualSyntax := dualSyntax (← getDualPos stx) evalSyntax
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let contrSyntax := contrSyntax (← getContrPos stx) dualSyntax
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return contrSyntax
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| _ =>
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throwError "Unsupported tensor expression syntax in elaborateTensorNode: {stx}"
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end TensorNode
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namespace ProdNode
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/-!
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## For product nodes.
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For a product node we can take the tensor product, and then contract the indices.
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-/
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/-- Gets the indices associated with a product node. -/
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partial def getIndices (stx : Syntax) : TermElabM (List (TSyntax `indexExpr)) := do
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match stx with
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| `(tensorExpr| $_:term | $[$args]*) => do
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return (← TensorNode.withoutContr stx)
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| `(tensorExpr| $a:tensorExpr ⊗ $b:tensorExpr) => do
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let indicesA ← getIndices a
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let indicesB ← getIndices b
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return indicesA ++ indicesB
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| `(tensorExpr| ($a:tensorExpr)) => do
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return (← getIndices a)
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| _ =>
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throwError "Unsupported tensor expression syntax in getIndicesProd: {stx}"
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/-- The pairs of positions in getIndicesNode which get contracted. -/
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partial def getContrPos (stx : Syntax) : TermElabM (List (ℕ × ℕ)) := do
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let ind ← getIndices stx
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let indFilt : List (TSyntax `indexExpr) := ind.filter (fun x => ¬ indexExprIsNum x)
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let indEnum := indFilt.enum
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let bind := List.bind indEnum (fun a => indEnum.map (fun b => (a, b)))
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let filt ← bind.filterMapM (fun x => indexPosEq x.1 x.2)
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if ¬ ((filt.map Prod.fst).Nodup ∧ (filt.map Prod.snd).Nodup) then
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throwError "To many contractions"
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return filt
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/-- The list of indices after contraction. -/
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def withoutContr (stx : Syntax) : TermElabM (List (TSyntax `indexExpr)) := do
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let ind ← getIndices stx
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let indFilt : List (TSyntax `indexExpr) := ind.filter (fun x => ¬ indexExprIsNum x)
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return ind.filter (fun x => indFilt.count x ≤ 1)
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/-- For each element of `l : List (ℕ × ℕ)` applies `TensorTree.contr` to the given term. -/
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def contrSyntax (l : List (ℕ × ℕ)) (T : Term) : Term :=
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l.foldl (fun T' (x0, x1) => Syntax.mkApp (mkIdent ``TensorTree.contr)
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#[Syntax.mkNumLit (toString x1), Syntax.mkNumLit (toString x0), T']) T
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/-- The syntax associated with a product of tensors. -/
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def prodSyntax (T1 T2 : Term) : Term :=
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Syntax.mkApp (mkIdent ``TensorTree.prod) #[T1, T2]
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/-- The full term taking tensor syntax into a term for products and single tensor nodes. -/
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partial def syntaxFull (stx : Syntax) : TermElabM Term := do
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match stx with
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| `(tensorExpr| $_:term | $[$args]*) => TensorNode.syntaxFull stx
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| `(tensorExpr| $a:tensorExpr ⊗ $b:tensorExpr) => do
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let prodSyntax := prodSyntax (← syntaxFull a) (← syntaxFull b)
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let contrSyntax := contrSyntax (← getContrPos stx) prodSyntax
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return contrSyntax
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| `(tensorExpr| ($a:tensorExpr)) => do
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return (← syntaxFull a)
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| _ =>
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throwError "Unsupported tensor expression syntax in elaborateTensorNode: {stx}"
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/-- An elaborator for tensor nodes. This is to be generalized. -/
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def elaborateTensorNode (stx : Syntax) : TermElabM Expr := do
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let tensorExpr ← elabTerm (← syntaxFull stx) none
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return tensorExpr
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/-- Syntax turning a tensor expression into a term. -/
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syntax (name := tensorExprSyntax) "{" tensorExpr "}ᵀ" : term
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elab_rules (kind:=tensorExprSyntax) : term
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| `(term| {$e:tensorExpr}ᵀ) => do
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let tensorTree ← elaborateTensorNode e
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return tensorTree
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variable {S : TensorStruct} {c4 : Fin 4 → S.C} (T4 : S.F.obj (OverColor.mk c4))
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{c5 : Fin 5 → S.C} (T5 : S.F.obj (OverColor.mk c5))
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/-!
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# Checks
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-/
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/-
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example : {T4 | i j k m}ᵀ = TensorTree.tensorNode T4 := by rfl
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#check {T4 | i j l d ⊗ T5 | i j k a b}ᵀ
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#check {(T4 | i j l a ⊗ T5 | i j k c d) ⊗ T5 | i1 i2 i3 e f}ᵀ
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-/
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end ProdNode
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