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feat: more work on node identities

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jstoobysmith 2024-10-18 16:08:17 +00:00
parent d2d75e4d36
commit 0bbc3f4019
6 changed files with 710 additions and 272 deletions
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78
HepLean/Mathematics/Fin.lean
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@ -6,6 +6,7 @@ Authors: Joseph Tooby-Smith
import Mathlib.LinearAlgebra.PiTensorProduct
import Mathlib.Tactic.Polyrith
import Mathlib.Tactic.Linarith
import LLMLean
/-!
# Fin lemmas
@ -86,6 +87,83 @@ lemma predAboveI_ge {i x : Fin n.succ.succ} (h : i.val < x.val) :
simp [predAboveI, h]
omega
lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x : Fin n) :
i.succAbove (j.succAbove x) =
(i.succAbove j).succAbove ((predAboveI (i.succAbove j) i).succAbove x) := by
by_cases h1 : j.castSucc < i
· have hx := Fin.succAbove_of_castSucc_lt _ _ h1
rw [hx]
rw [predAboveI_ge h1]
by_cases hx1 : x.castSucc < j
· rw [Fin.succAbove_of_castSucc_lt _ _ hx1]
rw [Fin.succAbove_of_castSucc_lt _ _]
· nth_rewrite 2 [Fin.succAbove_of_castSucc_lt _ _]
· rw [Fin.succAbove_of_castSucc_lt _ _]
exact hx1
· rw [Fin.lt_def] at h1 hx1 ⊢
simp_all
omega
· exact Nat.lt_trans hx1 h1
· simp at hx1
rw [Fin.le_def] at hx1
rw [Fin.lt_def] at h1
rw [Fin.succAbove_of_le_castSucc _ _ hx1]
by_cases hx2 : x.succ.castSucc < i
· rw [Fin.succAbove_of_castSucc_lt _ _ hx2]
nth_rewrite 2 [Fin.succAbove_of_castSucc_lt _ _]
· rw [Fin.succAbove_of_le_castSucc]
· simp [Fin.ext_iff]
· exact hx1
· rw [Fin.lt_def] at hx2 ⊢
simp_all
omega
· simp at hx2
rw [Fin.succAbove_of_le_castSucc _ _ hx2]
nth_rewrite 2 [Fin.succAbove_of_le_castSucc]
· rw [Fin.succAbove_of_le_castSucc]
rw [Fin.le_def]
exact Nat.le_succ_of_le hx1
· rw [Fin.le_def] at hx2 ⊢
simp_all
· simp at h1
have hx := Fin.succAbove_of_le_castSucc _ _ h1
rw [hx]
rw [predAboveI_lt (Nat.lt_add_one_of_le h1)]
by_cases hx1 : j ≤ x.castSucc
· rw [Fin.succAbove_of_le_castSucc _ _ hx1]
rw [Fin.succAbove_of_le_castSucc _ _]
· nth_rewrite 2 [Fin.succAbove_of_le_castSucc _ _]
· rw [Fin.succAbove_of_le_castSucc _ _]
rw [Fin.le_def] at hx1 ⊢
simp_all
· rw [Fin.le_def] at h1 hx1 ⊢
simp_all
omega
· rw [Fin.le_def] at hx1 h1 ⊢
simp_all
omega
· simp at hx1
rw [Fin.lt_def] at hx1
rw [Fin.le_def] at h1
rw [Fin.succAbove_of_castSucc_lt _ _ hx1]
by_cases hx2 : x.castSucc.castSucc < i
· rw [Fin.succAbove_of_castSucc_lt _ _ hx2]
nth_rewrite 2 [Fin.succAbove_of_castSucc_lt _ _]
· rw [Fin.succAbove_of_castSucc_lt _ _]
rw [Fin.lt_def] at hx2 ⊢
simp_all
omega
· rw [Fin.lt_def] at hx2 ⊢
simp_all
· simp at hx2
rw [Fin.succAbove_of_le_castSucc _ _ hx2]
nth_rewrite 2 [Fin.succAbove_of_le_castSucc]
· rw [Fin.succAbove_of_castSucc_lt]
· simp [Fin.ext_iff]
exact Fin.castSucc_lt_succ_iff.mpr hx1
· rw [Fin.le_def] at hx2 ⊢
simp_all
/-- The equivalence between `Fin n.succ` and `Fin 1 ⊕ Fin n` extracting the
`i`th component. -/
def finExtractOne {n : ℕ} (i : Fin n.succ) : Fin n.succ ≃ Fin 1 ⊕ Fin n :=