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.

This file contains the linear_combination' tactic (note the '): the original Lean 4 implementation of the "linear combination" idea, written at the time of the port from Lean 3. Notably, its scope includes certain nonlinear operations. The linear_combination tactic (in a separate file) is a variant implementation, but this version is provided for backward-compatibility.

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 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 c : α} [Add α] (p : b = c) (a : α) : a + b = a + c

theorem Mathlib.Tactic.LinearCombination'.add_pf {α : Type u_1} {a₁ a₂ b₁ 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 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 c : α} [Sub α] (p : b = c) (a : α) : a - b = a - c

theorem Mathlib.Tactic.LinearCombination'.sub_pf {α : Type u_1} {a₁ a₂ b₁ b₂ : α} [Sub α] (p₁ : a₁ = b₁) (p₂ : a₂ = b₂) : a₁ - a₂ = b₁ - b₂

theorem Mathlib.Tactic.LinearCombination'.neg_pf {α : Type u_1} {a b : α} [Neg α] (p : a = b) : -a = -b

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

theorem Mathlib.Tactic.LinearCombination'.mul_pf {α : Type u_1} {a₁ a₂ b₁ 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 b : α} [Inv α] (p : a = b) : a⁻¹ = b⁻¹

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

theorem Mathlib.Tactic.LinearCombination'.div_pf {α : Type u_1} {a₁ a₂ b₁ 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_trans₃ {α : Type u_1} {a a' b b' : α} (p : a = b) (p₁ : a = a') (p₂ : b = b') : a' = b'

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

theorem Mathlib.Tactic.LinearCombination'.eq_of_add_pow {α : Type u_1} {a a' b 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) (twoGoals : Bool := false) : Lean.Elab.Tactic.TacticM Unit

Implementation of linear_combination' and linear_combination2.

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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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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 norm field of the tactic is set to skip, then the tactic will simply set the user up to prove their target using the linear combination instead of normalizing the subtraction.

Note: There is also a similar tactic linear_combination (no prime); this version is provided for backward compatibility. Compared to this tactic, linear_combination:

• drops the ← syntax for reversing an equation, instead offering this operation using the - syntax
• does not support multiplication of two hypotheses (h1 * h2), division by a hypothesis (3 / h), or inversion of a hypothesis (h⁻¹)
• produces noisy output when the user adds or subtracts a constant to a hypothesis (h + 3)

Note: The left and right sides of all the equalities should have the same type, and the coefficients should also have this type. There must be instances of Mul and AddGroup for this type.

• 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. This feature is not supported for linear_combination2.
• linear_combination2 e is the same as linear_combination' e but it produces two subgoals instead of one: rather than proving that (a - b) - (a' - b') = 0 where a' = b' is the linear combination from e and a = b is the goal, it instead attempts to prove a = a' and b = b'. Because it does not use subtraction, this form is applicable also to semirings.
  • Note that a goal which is provable by linear_combination' e may not be provable by linear_combination2 e; in general you may need to add a coefficient to e to make both sides match, as in linear_combination2 e + c.
  • You can also reverse equalities using ← h, so for example if h₁ : a = b then 2 * (← h) is a proof of 2 * b = 2 * a.

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.

def Mathlib.Tactic.LinearCombination'.tacticLinear_combination2____ : 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 norm field of the tactic is set to skip, then the tactic will simply set the user up to prove their target using the linear combination instead of normalizing the subtraction.

Note: There is also a similar tactic linear_combination (no prime); this version is provided for backward compatibility. Compared to this tactic, linear_combination:

• drops the ← syntax for reversing an equation, instead offering this operation using the - syntax
• does not support multiplication of two hypotheses (h1 * h2), division by a hypothesis (3 / h), or inversion of a hypothesis (h⁻¹)
• produces noisy output when the user adds or subtracts a constant to a hypothesis (h + 3)

Note: The left and right sides of all the equalities should have the same type, and the coefficients should also have this type. There must be instances of Mul and AddGroup for this type.

• 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. This feature is not supported for linear_combination2.
• linear_combination2 e is the same as linear_combination' e but it produces two subgoals instead of one: rather than proving that (a - b) - (a' - b') = 0 where a' = b' is the linear combination from e and a = b is the goal, it instead attempts to prove a = a' and b = b'. Because it does not use subtraction, this form is applicable also to semirings.
  • Note that a goal which is provable by linear_combination' e may not be provable by linear_combination2 e; in general you may need to add a coefficient to e to make both sides match, as in linear_combination2 e + c.
  • You can also reverse equalities using ← h, so for example if h₁ : a = b then 2 * (← h) is a proof of 2 * b = 2 * a.

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

Equations

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