linear_combination Tactic

In this file, the linear_combination tactic is created. This tactic, which works over CommRings, attempts to simplify the target by creating a linear combination of a list of equalities and subtracting it from the target. A Syntax.Tactic object can also be passed into the tactic, allowing the user to specify a normalization tactic.

Implementation Notes

This tactic works by creating a weighted sum of the given equations with the given coefficients. Then, it subtracts the right side of the weighted sum from the left side so that the right side equals 0, and it does the same with the target. Afterwards, it sets the goal to be the equality between the lefthand side of the new goal and the lefthand side of the new weighted sum. Lastly, calls a normalization tactic on this target.

References

• https://leanprover.zulipchat.com/#narrow/stream/239415-metaprogramming-.2F.20tactics/topic/Linear.20algebra.20tactic/near/213928196

theorem Mathlib.Tactic.LinearCombination.pf_add_c {α : Type u_1} {a : α} {b : α} [Add α] (p : a = b) (c : α) : a + c = b + c

theorem Mathlib.Tactic.LinearCombination.c_add_pf {α : Type u_1} {b : α} {c : α} [Add α] (p : b = c) (a : α) : a + b = a + c

theorem Mathlib.Tactic.LinearCombination.add_pf {α : Type u_1} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} [Add α] (p₁ : a₁ = b₁) (p₂ : a₂ = b₂) : a₁ + a₂ = b₁ + b₂

theorem Mathlib.Tactic.LinearCombination.pf_sub_c {α : Type u_1} {a : α} {b : α} [Sub α] (p : a = b) (c : α) : a - c = b - c

theorem Mathlib.Tactic.LinearCombination.c_sub_pf {α : Type u_1} {b : α} {c : α} [Sub α] (p : b = c) (a : α) : a - b = a - c

theorem Mathlib.Tactic.LinearCombination.pf_mul_c {α : Type u_1} {a : α} {b : α} [Mul α] (p : a = b) (c : α) : a * c = b * c

theorem Mathlib.Tactic.LinearCombination.c_mul_pf {α : Type u_1} {b : α} {c : α} [Mul α] (p : b = c) (a : α) : a * b = a * c

theorem Mathlib.Tactic.LinearCombination.mul_pf {α : Type u_1} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} [Mul α] (p₁ : a₁ = b₁) (p₂ : a₂ = b₂) : a₁ * a₂ = b₁ * b₂

theorem Mathlib.Tactic.LinearCombination.inv_pf {α : Type u_1} {a : α} {b : α} [Inv α] (p : a = b) : a⁻¹ = b⁻¹

theorem Mathlib.Tactic.LinearCombination.pf_div_c {α : Type u_1} {a : α} {b : α} [Div α] (p : a = b) (c : α) : a / c = b / c

theorem Mathlib.Tactic.LinearCombination.c_div_pf {α : Type u_1} {b : α} {c : α} [Div α] (p : b = c) (a : α) : a / b = a / c

theorem Mathlib.Tactic.LinearCombination.div_pf {α : Type u_1} {a₁ : α} {a₂ : α} {b₁ : α} {b₂ : α} [Div α] (p₁ : a₁ = b₁) (p₂ : a₂ = b₂) : a₁ / a₂ = b₁ / b₂

inductive Mathlib.Tactic.LinearCombination.Expanded : Type

Result of expandLinearCombo, either an equality proof or a value.

• proof: Lean.Term → Mathlib.Tactic.LinearCombination.Expanded

  A proof of a = b.

• const: Lean.Term → Mathlib.Tactic.LinearCombination.Expanded

  A value, equivalently a proof of c = c.

partial def Mathlib.Tactic.LinearCombination.expandLinearCombo (ty : Lean.Expr) (stx : Lean.Term) : Lean.Elab.TermElabM Mathlib.Tactic.LinearCombination.Expanded

Performs macro expansion of a linear combination expression, using +/-/*// on equations and values.

• .proof p means that p is a syntax corresponding to a proof of an equation. For example, if h : a = b then expandLinearCombo (2 * h) returns .proof (c_add_pf 2 h) which is a proof of 2 * a = 2 * b.
• .const c means that the input expression is not an equation but a value.

theorem Mathlib.Tactic.LinearCombination.eq_of_sub {α : Type u_1} {a : α} {a' : α} {b : α} {b' : α} [AddGroup α] (p : a = b) (H : a' - b' - (a - b) = 0) : a' = b'

theorem Mathlib.Tactic.LinearCombination.eq_of_add {α : Type u_1} {a : α} {a' : α} {b : α} {b' : α} [Add α] [IsRightCancelAdd α] (p : a = b) (H : a' + b = b' + a) : a' = b'

theorem Mathlib.Tactic.LinearCombination.eq_of_add_pow {α : Type u_1} {a : α} {a' : α} {b : α} {b' : α} [Ring α] [NoZeroDivisors α] (n : ℕ) (p : a = b) (H : (a' - b') ^ n - (a - b) = 0) : a' = b'

def Mathlib.Tactic.LinearCombination.elabLinearCombination (tk : Lean.Syntax) (norm? : Option Lean.Syntax.Tactic) (exp? : Option Lean.NumLit) (input : Option Lean.Term) : Lean.Elab.Tactic.TacticM Unit

Implementation of linear_combination.

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def Mathlib.Tactic.LinearCombination.normStx : Lean.ParserDescr

The (norm := $tac) syntax says to use tac as a normalization postprocessor for linear_combination. The default normalizer is ring1, but you can override it with ring_nf to get subgoals from linear_combination or with skip to disable normalization.

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def Mathlib.Tactic.LinearCombination.expStx : Lean.ParserDescr

The (exp := n) syntax for linear_combination says to take the goal to the nth power before subtracting the given combination of hypotheses.

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• One or more equations did not get rendered due to their size.

def Mathlib.Tactic.LinearCombination.linearCombination : Lean.ParserDescr

linear_combination attempts to simplify the target by creating a linear combination of a list of equalities and subtracting it from the target. The tactic will create a linear combination by adding the equalities together from left to right, so the order of the input hypotheses does matter. If the normalize field of the configuration is set to false, then the tactic will simply set the user up to prove their target using the linear combination instead of normalizing the subtraction.

Note: The left and right sides of all the equalities should have the same type α, and the coefficients should also have type α. For full functionality α should be a commutative ring -- strictly speaking, a commutative semiring with "cancellative" addition (in the semiring case, negation and subtraction will be handled "formally" as if operating in the enveloping ring). If a nonstandard normalization is used (for example abel or skip), the tactic will work over types α with less algebraic structure: the minimum is instances of [Add α] [IsRightCancelAdd α] together with instances of whatever operations are used in the tactic call.

• The input e in linear_combination e is a linear combination of proofs of equalities, given as a sum/difference of coefficients multiplied by expressions. The coefficients may be arbitrary expressions. The expressions can be arbitrary proof terms proving equalities. Most commonly they are hypothesis names h1, h2, ....
• linear_combination (norm := tac) e runs the "normalization tactic" tac on the subgoal(s) after constructing the linear combination.
  • The default normalization tactic is ring1, which closes the goal or fails.
  • To get a subgoal in the case that it is not immediately provable, use ring_nf as the normalization tactic.
  • To avoid normalization entirely, use skip as the normalization tactic.
• linear_combination (exp := n) e will take the goal to the nth power before subtracting the combination e. In other words, if the goal is t1 = t2, linear_combination (exp := n) e will change the goal to (t1 - t2)^n = 0 before proceeding as above.

Example Usage:

example (x y : ℤ) (h1 : x*y + 2*x = 1) (h2 : x = y) : x*y = -2*y + 1 := by
  linear_combination 1*h1 - 2*h2

example (x y : ℤ) (h1 : x*y + 2*x = 1) (h2 : x = y) : x*y = -2*y + 1 := by
  linear_combination h1 - 2*h2

example (x y : ℤ) (h1 : x*y + 2*x = 1) (h2 : x = y) : x*y = -2*y + 1 := by
  linear_combination (norm := ring_nf) -2*h2
  /- Goal: x * y + x * 2 - 1 = 0 -/

example (x y z : ℝ) (ha : x + 2*y - z = 4) (hb : 2*x + y + z = -2)
    (hc : x + 2*y + z = 2) :
    -3*x - 3*y - 4*z = 2 := by
  linear_combination ha - hb - 2*hc

example (x y : ℚ) (h1 : x + y = 3) (h2 : 3*x = 7) :
    x*x*y + y*x*y + 6*x = 3*x*y + 14 := by
  linear_combination x*y*h1 + 2*h2

example (x y : ℤ) (h1 : x = -3) (h2 : y = 10) : 2*x = -6 := by
  linear_combination (norm := skip) 2*h1
  simp

axiom qc : ℚ
axiom hqc : qc = 2*qc

example (a b : ℚ) (h : ∀ p q : ℚ, p = q) : 3*a + qc = 3*b + 2*qc := by
  linear_combination 3 * h a b + hqc

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• One or more equations did not get rendered due to their size.