move_add a tactic for moving summands in expressions

The tactic move_add rearranges summands in expressions.

The tactic takes as input a list of terms, each one optionally preceded by ←. A term preceded by ← gets moved to the left, while a term without ← gets moved to the right.

• Empty input: move_add []

  In this case, the effect of move_add [] is equivalent to simp only [← add_assoc]: essentially the tactic removes all visible parentheses.

• Singleton input: move_add [a] and move_add [← a]

  If ⊢ b + a + c is (a summand in) the goal, then

  • move_add [← a] changes the goal to a + b + c (effectively, a moved to the left).
  • move_add [a] changes the goal to b + c + a (effectively, a moved to the right);

  The tactic reorders all sub-expressions of the target at the same same. For instance, if ⊢ 0 < if b + a < b + a + c then a + b else b + a is the goal, then

  • move_add [a] changes the goal to 0 < if b + a < b + c + a then b + a else b + a (a moved to the right in three sums);
  • move_add [← a] changes the goal to 0 < if a + b < a + b + c then a + b else a + b (a again moved to the left in three sums).

• Longer inputs: move_add [..., a, ..., ← b, ...]

  If the list contains more than one term, the tactic effectively tries to move each term preceded by ← to the left, each term not preceded by ← to the right maintaining the relative order in the call. Thus, applying move_add [a, b, c, ← d, ← e] returns summands of the form d + e + [...] + a + b + c, i.e. d and e have the same relative position in the input list and in the final rearrangement (and similarly for a, b, c). In particular, move_add [a, b] likely has the same effect as move_add [a]; move_add [b]: first, we move a to the right, then we move b also to the right, after a. However, if the terms matched by a and b do not overlap, then move_add [← a, ← b] has the same effect as move_add [b]; move_add [a]: first, we move b to the left, then we move a also to the left, before a. The behaviour in the situation where a and b overlap is unspecified: move_add will descend into subexpressions, but the order in which they are visited depends on which rearrangements have already happened. Also note, though, that move_add [a, b] may differ move_add [a]; move_add [b], for instance when a and b are DefEq.

• Unification of inputs and repetitions: move_add [_, ← _, a * _]

  The matching of the user inputs with the atoms of the summands in the target expression is performed via checking DefEq and selecting the first, still available match. Thus, if a sum in the target is 2 * 3 + 4 * (5 + 6) + 4 * 7 + 10 * 10, then move_add [4 * _] moves the summand 4 * (5 + 6) to the right.

  The unification of later terms only uses the atoms in the target that have not yet been unified. Thus, if again the target contains 2 * 3 + 4 * (5 + 6) + 4 * 7 + 10 * 10, then move_add [_, ← _, 4 * _] matches

  • the first input (_) with 2 * 3;
  • the second input (_) with 4 * (5 + 6);
  • the third input (4 * _) with 4 * 7.

  The resulting permutation therefore places 2 * 3 and 4 * 7 to the left (in this order) and 4 * (5 + 6) to the right: 2 * 3 + 4 * 7 + 10 * 10 + 4 * (5 + 6).

For the technical description, look at Mathlib.MoveAdd.weight and Mathlib.MoveAdd.reorderUsing.

move_add is the specialization of a more general move_oper tactic that takes a binary, associative, commutative operation and a list of "operand atoms" and rearranges the operation.

Extension notes

To work with a general associative, commutative binary operation, move_oper needs to have inbuilt the lemmas asserting the analogues of add_comm, add_assoc, add_left_comm for the new operation. Currently, move_oper supports HAdd.hAdd, HMul.hMul, And, Or, Max.max, Min.min.

These lemmas should be added to Mathlib.MoveAdd.move_oper_simpCtx.

See test/MoveAdd.lean for sample usage of move_oper.

Implementation notes

The main driver behind the tactic is Mathlib.MoveAdd.reorderAndSimp.

The tactic takes the target, replaces the maximal subexpressions whose head symbol is the given operation and replaces them by their reordered versions. Once that is done, it tries to replace the initial goal with the permuted one by using simp.

Currently, no attempt is made at guiding simp by doing a congr-like destruction of the goal. This will be the content of a later PR.

def Lean.Expr.getExprInputs : Lean.Expr → Array Lean.Expr

getExprInputs e inspects the outermost constructor of e and returns the array of all the arguments to that constructor that are themselves Expressions.

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partial def Lean.Expr.size (e : Lean.Expr) : ℕ

size e returns the number of subexpressions of e.

Reordering the variables

This section produces the permutations of the variables for move_add.

The user controls the final order by passing a list of terms to the tactic. Each term can be preceded by ← or not. In the final ordering,

• terms preceded by ← appear first,
• terms not preceded by ← appear last,
• all remaining terms remain in their current relative order.

def Mathlib.MoveAdd.uniquify {α : Type u_1} [BEq α] : List α → List (α × ℕ)

uniquify L takes a list L : List α as input and it returns a list L' : List (α × ℕ). The two lists L and L'.map Prod.fst coincide. The second component of each entry (a, n) in L' is the number of times that a appears in L before the current location.

The resulting list of pairs has no duplicates.

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• Mathlib.MoveAdd.uniquify [] = []

def Mathlib.MoveAdd.weight {α : Type u_1} [BEq α] (L : List (α × Bool)) (a : α) : ℤ

Return a sorting key so that all (a, true)s are in the list's order and sorted before all (a, false)s, which are also in the list's order. Although weight does not require this, we use weight in the case where the list obtained from L by only keeping the first component (i.e. L.map Prod.fst) has no duplicates. The properties that we mention here assume that this is the case.

Thus, weight L is a function α → ℤ with the following properties:

• if (a, true) ∈ L, then weight L a is strictly negative;
• if (a, false) ∈ L, then weight L a is strictly positive;
• if neither (a, true) nor (a, false) is in L, then weight L a = 0.

Moreover, the function weight L is strictly monotone increasing on both {a : α | (a, true) ∈ L} and {a : α | (a, false) ∈ L}, in the sense that if a' = (a, true) and b' = (b, true) are in L, then a' appears before b' in L if and only if weight L a < weight L b and similarly for the pairs with second coordinate equal to false.

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def Mathlib.MoveAdd.reorderUsing {α : Type u_1} [BEq α] (toReorder : List α) (instructions : List (α × Bool)) : List α

reorderUsing toReorder instructions produces a reordering of toReorder : List α, following the requirements imposed by instructions : List (α × Bool).

These are the requirements:

• elements of toReorder that appear with true in instructions appear at the beginning of the reordered list, in the order in which they appear in instructions;
• similarly, elements of toReorder that appear with false in instructions appear at the end of the reordered list, in the order in which they appear in instructions;
• finally, elements of toReorder that do not appear in instructions appear "in the middle" with the order that they had in toReorder.

For example,

• reorderUsing [0, 1, 2] [(0, false)] = [1, 2, 0],
• reorderUsing [0, 1, 2] [(1, true)] = [1, 0, 2],
• reorderUsing [0, 1, 2] [(1, true), (0, false)] = [1, 2, 0].

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def Mathlib.MoveAdd.prepareOp (sum : Lean.Expr) : Lean.Expr

prepareOp sum takes an Expression as input. It assumes that sum is a well-formed term representing a repeated application of a binary operation and that the summands are the last two arguments passed to the operation. It returns the expression consisting of the operation with all its arguments already applied, except for the last two. This is similar to Lean.Meta.mkAdd, Lean.Meta.mkMul, except that the resulting operation is primed to work with operands of the same type as the ones already appearing in sum.

This is useful to rearrange the operands.

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• Mathlib.MoveAdd.prepareOp sum = let opargs := sum.getAppArgs; List.foldl (fun (x y : Lean.Expr) => x.app y) sum.getAppFn (List.take (opargs.size - 2) opargs.toList)

partial def Mathlib.MoveAdd.sumList (prepOp : Lean.Expr) (left_assoc? : Bool) : List Lean.Expr → Lean.Expr

sumList prepOp left_assoc? exs assumes that prepOp is an Expression representing a binary operation already fully applied up until its last two arguments and assumes that the last two arguments are the operands of the operation. Such an expression is the result of prepareOp.

If exs is the list [e₁, e₂, ..., eₙ] of Expressions, then sumList prepOp left_assoc? exs returns

• prepOp (prepOp( ... prepOp (prepOp e₁ e₂) e₃) ... eₙ), if left_assoc? is false, and
• prepOp e₁ (prepOp e₂ (... prepOp (prepOp eₙ₋₁ eₙ)), if left_assoc? is true.

partial def Mathlib.MoveAdd.getAddends (op : Lean.Name) (R : Lean.Expr) (sum : Lean.Expr) : Lean.MetaM (Array Lean.Expr)

If sum is an expression consisting of repeated applications of op, then getAddends returns the Array of those recursively determined arguments whose type is DefEq to R.

partial def Mathlib.MoveAdd.getOps (op : Lean.Name) (sum : Lean.Expr) : Lean.MetaM (Array (Array Lean.Expr × Lean.Expr))

Recursively compute the Array of getAddends Arrays by recursing into the expression sum looking for instance of the operation op.

Possibly returns duplicates!

def Mathlib.MoveAdd.rankSums (op : Lean.Name) (tgt : Lean.Expr) (instructions : List (Lean.Expr × Bool)) : Lean.MetaM (List (Lean.Expr × Lean.Expr))

rankSums op tgt instructions takes as input

• the name op of a binary operation,
• an Expression tgt,
• a list instructions of pair (expression, boolean).

It extracts the maximal subexpressions of tgt whose head symbol is op (i.e. the maximal subexpressions that consist only of applications of the binary operation op), it rearranges the operands of such subexpressions following the order implied by instructions (as in reorderUsing), it returns the list of pairs of expressions (old_sum, new_sum), for which old_sum ≠ new_sum sorted by decreasing value of Lean.Expr.size. In particular, a subexpression of an old_sum can only appear after its over-expression.

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def Mathlib.MoveAdd.permuteExpr (op : Lean.Name) (tgt : Lean.Expr) (instructions : List (Lean.Expr × Bool)) : Lean.MetaM Lean.Expr

permuteExpr op tgt instructions takes the same input as rankSums and returns the expression obtained from tgt by replacing all old_sums by the corresponding new_sum. If there were no required changes, then permuteExpr reports this in its second factor.

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def Mathlib.MoveAdd.pairUp : List (Lean.Expr × Bool × Lean.Syntax) → List Lean.Expr → Lean.MetaM (List (Lean.Expr × Bool) × List (Lean.Expr × Bool × Lean.Syntax))

pairUp L R takes to lists L R : List Expr as inputs. It scans the elements of L, looking for a corresponding DefEq Expression in R. If it finds one such element d, then it sets the element d : R aside, removing it from R, and it continues with the matching on the remainder of L and on R.erase d.

At the end, it returns the sublist of R of the elements that were matched to some element of R, in the order in which they appeared in L, as well as the sublist of L of elements that were not matched, also in the order in which they appeared in L.

Example:

#eval do
  let L := [mkNatLit 0, (← mkFreshExprMVar (some (mkConst ``Nat))), mkNatLit 0] -- i.e. [0, _, 0]
  let R := [mkNatLit 0, mkNatLit 0,                                 mkNatLit 1] -- i.e. [0, 1]
  dbg_trace f!"{(← pairUp L R)}"
/- output:
`([0, 0], [0])`
the output LHS list `[0, 0]` consists of the first `0` and the `MVarId`.
the output RHS list `[0]` corresponds to the last `0` in `L`.
-/

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• Mathlib.MoveAdd.pairUp x✝ x = pure ([], [])

def Mathlib.MoveAdd.move_oper_simpCtx : Lean.MetaM Lean.Meta.Simp.Context

move_oper_simpCtx is the Simp.Context for the reordering internal to move_oper. To support a new binary operation, extend the list in this definition, so that it contains enough lemmas to allow simp to close a generic permutation goal for the new binary operation.

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def Mathlib.MoveAdd.reorderAndSimp (op : Lean.Name) (mv : Lean.MVarId) (instr : List (Lean.Expr × Bool)) : Lean.MetaM (List Lean.MVarId)

reorderAndSimp mv op instr takes as input an MVarId mv, the name op of a binary operation and a list of "instructions" instr that it passes to permuteExpr.

• It creates a version permuted_mv of mv with subexpressions representing op-sums reordered following instructions.
• It produces 2 temporary goals by applying Eq.mpr and unifying the resulting meta-variable with permuted_mv: [⊢ mv = permuted_mv, ⊢ permuted_mv].
• It tries to solve the goal mv = permuted_mv by a simple-minded simp call, using the op-analogues of add_comm, add_assoc, add_left_comm.

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def Mathlib.MoveAdd.unifyMovements (op : Lean.Name) (data : Array (Lean.Expr × Bool × Lean.Syntax)) (tgt : Lean.Expr) : Lean.MetaM (List (Lean.Expr × Bool) × (List Lean.MessageData × List Lean.Syntax) × Array Lean.MessageData)

unifyMovements takes as input

• an array of Expr × Bool × Syntax, as in the output of parseArrows,
• the Name op of a binary operation,
• an Expression tgt. It unifies each Expression appearing as a first factor of the array with the atoms for the operation op in the expression tgt, returning
• the lists of pairs of a matched subexpression with the corresponding Boolean;
• a pair of a list of error messages and the corresponding list of Syntax terms where the error should be thrown;
• an array of debugging messages.

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def Mathlib.MoveAdd.parseArrows : Lean.TSyntax `Lean.Parser.Tactic.rwRuleSeq → Lean.Elab.TermElabM (Array (Lean.Expr × Bool × Lean.Syntax))

parseArrows parses an input of the form [a, ← b, _ * (1 : ℤ)], consisting of a list of terms, each optionally preceded by the arrow ←. It returns an array of triples consisting of

• the Expression corresponding to the parsed term,
• the Boolean true if the arrow is present in front of the term,
• the underlying Syntax of the given term.

E.g. convert [a, ← b, _ * (1 : ℤ)] to [(a, false, `(a)), (b, true, `(b)), (_ * 1, false, `(_ * 1))].

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def Mathlib.MoveAdd.moveOperTac : Lean.ParserDescr

The tactic move_add rearranges summands of expressions. Calling move_add [a, ← b, ...] matches a, b,... with summands in the main goal. It then moves a to the far right and b to the far left of each addition in which they appear. The side to which the summands are moved is determined by the presence or absence of the arrow ←.

The inputs a, b,... can be any terms, also with underscores. The tactic uses the first "new" summand that unifies with each one of the given inputs.

There is a multiplicative variant, called move_mul.

There is also a general tactic for a "binary associative commutative operation": move_oper. In this case the syntax requires providing first a term whose head symbol is the operation. E.g. move_oper HAdd.hAdd [...] is the same as move_add, while move_oper Max.max [...] rearranges maxs.

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def Mathlib.MoveAdd.tacticMove_add_ : Lean.ParserDescr

The tactic move_add rearranges summands of expressions. Calling move_add [a, ← b, ...] matches a, b,... with summands in the main goal. It then moves a to the far right and b to the far left of each addition in which they appear. The side to which the summands are moved is determined by the presence or absence of the arrow ←.

The inputs a, b,... can be any terms, also with underscores. The tactic uses the first "new" summand that unifies with each one of the given inputs.

There is a multiplicative variant, called move_mul.

There is also a general tactic for a "binary associative commutative operation": move_oper. In this case the syntax requires providing first a term whose head symbol is the operation. E.g. move_oper HAdd.hAdd [...] is the same as move_add, while move_oper Max.max [...] rearranges maxs.

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def Mathlib.MoveAdd.tacticMove_mul_ : Lean.ParserDescr

The tactic move_add rearranges summands of expressions. Calling move_add [a, ← b, ...] matches a, b,... with summands in the main goal. It then moves a to the far right and b to the far left of each addition in which they appear. The side to which the summands are moved is determined by the presence or absence of the arrow ←.

The inputs a, b,... can be any terms, also with underscores. The tactic uses the first "new" summand that unifies with each one of the given inputs.

There is a multiplicative variant, called move_mul.

There is also a general tactic for a "binary associative commutative operation": move_oper. In this case the syntax requires providing first a term whose head symbol is the operation. E.g. move_oper HAdd.hAdd [...] is the same as move_add, while move_oper Max.max [...] rearranges maxs.

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