[Merged by Bors] - feat: add exponentiation option back to linear_combination
#7789
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[mathlib3#15428](leanprover-community/mathlib3#15428) added an exponent option to `linear_combination`. The idea is that `linear_combination (exp := 2) ...` will take a linear combination of hypotheses adding up to the *square* of the goal. This is only mildly useful on its own, but it's a very useful certificate syntax for [mathlib3#15425](leanprover-community/mathlib3#15425). Co-authored-by: Rob Lewis <[email protected]>
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) In @hrmacbeth 's tutorial on `polyrith`, there were examples of problems that it could almost solve, but failed. The goal was not expressible as a linear combination of the hypotheses but a power of the goal was. ```lean example (x y z : ℚ) (h : x = y) (h2 : x * y = 0) : x + y*z = 0 := sorry ``` Mathematically, `x+y*z` is in the radical of the ideal generated by `x-y, x*y`. There's a "standard trick" for testing membership in the radical without a search for the proper exponent: see e.g. section 2.2 of [arxiv.org/pdf/1007.3615.pdf](https://arxiv.org/pdf/1007.3615.pdf) or 4.2 Prop 8 of Cox, Little, O'Shea. This PR implements the trick in the Sage call made by `polyrith`. When the power returned is `n > 1`, we use `linear_combination (exp := n)` to check the certificate (#7789 ). The `polyrith` test infrastructure still needs to be ported from mathlib3. All tests in the test file succeed when they are uncommented. A future PR will restore the old test suite. Co-authored-by: Rob Lewis <[email protected]> Co-authored-by: Mario Carneiro <[email protected]>
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) In @hrmacbeth 's tutorial on `polyrith`, there were examples of problems that it could almost solve, but failed. The goal was not expressible as a linear combination of the hypotheses but a power of the goal was. ```lean example (x y z : ℚ) (h : x = y) (h2 : x * y = 0) : x + y*z = 0 := sorry ``` Mathematically, `x+y*z` is in the radical of the ideal generated by `x-y, x*y`. There's a "standard trick" for testing membership in the radical without a search for the proper exponent: see e.g. section 2.2 of [arxiv.org/pdf/1007.3615.pdf](https://arxiv.org/pdf/1007.3615.pdf) or 4.2 Prop 8 of Cox, Little, O'Shea. This PR implements the trick in the Sage call made by `polyrith`. When the power returned is `n > 1`, we use `linear_combination (exp := n)` to check the certificate (#7789 ). The `polyrith` test infrastructure still needs to be ported from mathlib3. All tests in the test file succeed when they are uncommented. A future PR will restore the old test suite. Co-authored-by: Rob Lewis <[email protected]> Co-authored-by: Mario Carneiro <[email protected]>
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mathlib3#15428 added an exponent option to
linear_combination. The idea is thatlinear_combination (exp := 2) ...will take a linear combination of hypotheses adding up to the square of the goal. This is only mildly useful on its own, but it's a very useful certificate syntax for mathlib3#15425.This was merged in mathlib3 but didn't make the port.
I noticed when implementing this that the config object in
linear_combinationis completely unused, so I deleted it, and followed the new syntax for(norm := ...). The current implementation is sensitive to the order of(norm := ...) (exp := ...). I think it will be very uncommon to use these two features together, so it's not a big deal, but the fix would be to return to using aconfigobject.cc @digama0