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import OAI.Geometry.ProjectionBody.ProductFacetIndices
namespace OAI
noncomputable section
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
lemma firstBlock_productFacetLinear_first (i : Option (Fin 10)) (x : E 19) :
firstBlock (productFacetLinear (false, i) x) =
simplexFacetLinear 9 i ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).1 := by
change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (false, i) x)).1 = _
simp [productFacetLinear, LinearMap.prodMap_apply]
lemma secondBlock_productFacetLinear_first (i : Option (Fin 10)) (x : E 19) :
secondBlock (productFacetLinear (false, i) x) =
((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).2 := by
change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (false, i) x)).2 = _
simp [productFacetLinear, LinearMap.prodMap_apply]
lemma firstBlock_productFacetLinear_second (i : Option (Fin 10)) (x : E 19) :
firstBlock (productFacetLinear (true, i) x) =
((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).1 := by
change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (true, i) x)).1 = _
simp [productFacetLinear, LinearMap.prodMap_apply]
lemma secondBlock_productFacetLinear_second (i : Option (Fin 10)) (x : E 19) :
secondBlock (productFacetLinear (true, i) x) =
simplexFacetLinear 9 i ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).2 := by
change ((@EuclideanSpace.finAddEquivProd ℝ _ 10 10) (productFacetLinear (true, i) x)).2 = _
simp [productFacetLinear, LinearMap.prodMap_apply]
lemma productFacetLinear_first_coordinate (i : Fin 10) :
productFacetLinear (false, some i) = (insertZero (i.castAdd 10)).toLinearMap := by
ext x j
induction j using (i.castAdd 10).succAboveCases
· change firstBlock (productFacetLinear (false, some i) x) i = _
rw [firstBlock_productFacetLinear_first]
simp [simplexFacetLinear]
· rename_i j
simp only [LinearIsometry.coe_toLinearMap]
rw [insertZero_succAbove]
induction j using Fin.addCases (m := 9) (n := 10) with
| left j =>
rw [first_succAbove_first]
change firstBlock (productFacetLinear (false, some i) x) (i.succAbove j) = _
rw [firstBlock_productFacetLinear_first]
simp only [simplexFacetLinear, LinearIsometry.coe_toLinearMap, insertZero_succAbove]
rfl
| right j =>
rw [first_succAbove_second]
change secondBlock (productFacetLinear (false, some i) x) j = _
rw [secondBlock_productFacetLinear_first]
rfl
lemma productFacetLinear_second_coordinate (i : Fin 10) :
productFacetLinear (true, some i) = (insertZero (i.natAdd 10)).toLinearMap := by
ext x j
induction j using (i.natAdd 10).succAboveCases
· change secondBlock (productFacetLinear (true, some i) x) i = _
rw [secondBlock_productFacetLinear_second]
simp [simplexFacetLinear]
· rename_i j
simp only [LinearIsometry.coe_toLinearMap]
rw [insertZero_succAbove]
induction j using Fin.addCases (m := 10) (n := 9) with
| left j =>
rw [second_succAbove_first]
change firstBlock (productFacetLinear (true, some i) x) j = _
rw [firstBlock_productFacetLinear_second]
rfl
| right j =>
rw [second_succAbove_second]
change secondBlock (productFacetLinear (true, some i) x) (i.succAbove j) = _
rw [secondBlock_productFacetLinear_second]
simp only [simplexFacetLinear, LinearIsometry.coe_toLinearMap, insertZero_succAbove]
rfl
/-- Graph coefficient vector for the first block's sum facet. -/
def firstSumGraph : E 19 := (@EuclideanSpace.finAddEquivProd ℝ _ 9 10).symm (allOnes 9, 0)
/-- Graph coefficient vector for the second block's sum facet. -/
def secondSumGraph : E 19 := (@EuclideanSpace.finAddEquivProd ℝ _ 10 9).symm (0, allOnes 9)
lemma inner_firstSumGraph (x : E 19) :
⟪firstSumGraph, x⟫ = ⟪allOnes 9, ((@EuclideanSpace.finAddEquivProd ℝ _ 9 10) x).1⟫ := by
rw [DiagonalZonotope.inner_finAdd 9 10]
have h : (@EuclideanSpace.finAddEquivProd ℝ _ 9 10) firstSumGraph =
(allOnes 9, 0) := (@EuclideanSpace.finAddEquivProd ℝ _ 9 10).apply_symm_apply _
rw [h]
simp only [inner_zero_left, add_zero]
lemma inner_secondSumGraph (x : E 19) :
⟪secondSumGraph, x⟫ = ⟪allOnes 9, ((@EuclideanSpace.finAddEquivProd ℝ _ 10 9) x).2⟫ := by
rw [DiagonalZonotope.inner_finAdd 10 9]
have h : (@EuclideanSpace.finAddEquivProd ℝ _ 10 9) secondSumGraph =
(0, allOnes 9) := (@EuclideanSpace.finAddEquivProd ℝ _ 10 9).apply_symm_apply _
rw [h]
simp only [inner_zero_left, zero_add]
lemma productFacetLinear_first_sum :
productFacetLinear (false, none) = graphLinear ((Fin.last 9).castAdd 10) firstSumGraph := by
ext x j
induction j using ((Fin.last 9).castAdd 10).succAboveCases
· rw [graphLinear_same, inner_firstSumGraph]
change firstBlock (productFacetLinear (false, none) x) (Fin.last 9) = _
rw [firstBlock_productFacetLinear_first]
simp [simplexFacetLinear]
· rename_i j
rw [graphLinear_succAbove]
induction j using Fin.addCases (m := 9) (n := 10) with
| left j =>
rw [first_succAbove_first]
change firstBlock (productFacetLinear (false, none) x) ((Fin.last 9).succAbove j) = _
rw [firstBlock_productFacetLinear_first]
simp only [simplexFacetLinear, graphLinear_succAbove]
rfl
| right j =>
rw [first_succAbove_second]
change secondBlock (productFacetLinear (false, none) x) j = _
rw [secondBlock_productFacetLinear_first]
rfl
lemma productFacetLinear_second_sum :
productFacetLinear (true, none) = graphLinear ((Fin.last 9).natAdd 10) secondSumGraph := by
ext x j
induction j using ((Fin.last 9).natAdd 10).succAboveCases
· rw [graphLinear_same, inner_secondSumGraph]
change secondBlock (productFacetLinear (true, none) x) (Fin.last 9) = _
rw [secondBlock_productFacetLinear_second]
simp [simplexFacetLinear]
· rename_i j
rw [graphLinear_succAbove]
induction j using Fin.addCases (m := 10) (n := 9) with
| left j =>
rw [second_succAbove_first]
change firstBlock (productFacetLinear (true, none) x) j = _
rw [firstBlock_productFacetLinear_second]
rfl
| right j =>
rw [second_succAbove_second]
change secondBlock (productFacetLinear (true, none) x) ((Fin.last 9).succAbove j) = _
rw [secondBlock_productFacetLinear_second]
simp only [simplexFacetLinear, graphLinear_succAbove]
rfl
@[simp] lemma blockJoin_first {n m : ℕ} (x : E n) (y : E m) (i : Fin n) :
(@EuclideanSpace.finAddEquivProd ℝ _ n m).symm (x, y) (i.castAdd m) = x i := by
exact congrArg (fun z : E n × E m => z.1 i)
((@EuclideanSpace.finAddEquivProd ℝ _ n m).apply_symm_apply (x, y))
@[simp] lemma blockJoin_second {n m : ℕ} (x : E n) (y : E m) (i : Fin m) :
(@EuclideanSpace.finAddEquivProd ℝ _ n m).symm (x, y) (i.natAdd n) = y i := by
exact congrArg (fun z : E n × E m => z.2 i)
((@EuclideanSpace.finAddEquivProd ℝ _ n m).apply_symm_apply (x, y))
lemma graphNormal_succAbove {n : ℕ} (i : Fin (n + 1)) (a : E n) (j : Fin n) :
graphNormal i a (i.succAbove j) = a j := by
simp [graphNormal]
lemma firstSumGraph_normal :
graphNormal ((Fin.last 9).castAdd 10) firstSumGraph =
(@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm (allOnes 10, 0) := by
ext j
induction j using ((Fin.last 9).castAdd 10).succAboveCases
· rw [graphNormal_same, blockJoin_first]
rfl
· rename_i j
rw [graphNormal_succAbove]
induction j using Fin.addCases (m := 9) (n := 10) with
| left j =>
rw [first_succAbove_first (Fin.last 9) j, firstSumGraph, blockJoin_first, blockJoin_first]
rfl
| right j =>
rw [first_succAbove_second (Fin.last 9) j, firstSumGraph, blockJoin_second, blockJoin_second]
lemma secondSumGraph_normal :
graphNormal ((Fin.last 9).natAdd 10) secondSumGraph =
(@EuclideanSpace.finAddEquivProd ℝ _ 10 10).symm (0, allOnes 10) := by
ext j
induction j using ((Fin.last 9).natAdd 10).succAboveCases
· rw [graphNormal_same, blockJoin_second]
rfl
· rename_i j
rw [graphNormal_succAbove]
induction j using Fin.addCases (m := 10) (n := 9) with
| left j =>
rw [second_succAbove_first (Fin.last 9) j, secondSumGraph, blockJoin_first, blockJoin_first]
| right j =>
rw [second_succAbove_second (Fin.last 9) j, secondSumGraph, blockJoin_second, blockJoin_second]
rfl
lemma firstSumGraph_normal_inner (u : E 20) :
⟪graphNormal ((Fin.last 9).castAdd 10) firstSumGraph, u⟫ =
simplexFunctional 10 none (firstBlock u) := by
rw [firstSumGraph_normal, DiagonalZonotope.inner_finAdd 10 10,
ContinuousLinearEquiv.apply_symm_apply]
simp only [inner_zero_left, add_zero, inner_allOnes, firstBlock_finAdd, simplexFunctional_none]
lemma secondSumGraph_normal_inner (u : E 20) :
⟪graphNormal ((Fin.last 9).natAdd 10) secondSumGraph, u⟫ =
simplexFunctional 10 none (secondBlock u) := by
rw [secondSumGraph_normal, DiagonalZonotope.inner_finAdd 10 10,
ContinuousLinearEquiv.apply_symm_apply]
simp only [inner_zero_left, zero_add, inner_allOnes, secondBlock_finAdd, simplexFunctional_none]
end ProjectionCounterexample
end
end OAI
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