PhysLean/HepLean/Mathematics/Fin.lean

/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.LinearAlgebra.PiTensorProduct
import Mathlib.Tactic.Polyrith
import Mathlib.Tactic.Linarith
/-!
# Fin lemmas

The purpose of this file is to define some results Fin currently
in Mathlib.

At some point these should either be up-streamed to Mathlib or replaced with definitions already
in Mathlib.

-/
namespace HepLean.Fin

open Fin
variable {n : Nat}

def predAboveI (i x : Fin n.succ.succ) : Fin n.succ :=
  if h : x.val < i.val then
    ⟨x.val, by omega⟩
  else
    ⟨x.val - 1, by
      by_cases hx : x = 0
      · omega
      · omega⟩

lemma predAboveI_self  (i : Fin n.succ.succ) : predAboveI i i = ⟨i.val - 1, by omega⟩ := by
  simp [predAboveI]

@[simp]
lemma predAboveI_succAbove (i : Fin n.succ.succ) (x : Fin n.succ) :
    predAboveI i (Fin.succAbove i x) = x := by
  simp [predAboveI, Fin.succAbove, Fin.ext_iff]
  split_ifs
  · rfl
  · rename_i h1 h2
    simp [Fin.lt_def] at h1 h2
    omega
  · rfl
lemma succsAbove_predAboveI {i x : Fin n.succ.succ} (h : i ≠ x) :
    Fin.succAbove i (predAboveI i x) = x := by
  simp [Fin.succAbove, predAboveI, Fin.ext_iff]
  split_ifs
  · rfl
  · rename_i h1 h2
    rw [Fin.lt_def] at h1 h2
    simp
    simp at h2
    rw [Fin.le_def] at h2
    omega
  · rename_i h1 h2
    simp at h1
    rw [Fin.le_def] at h1
    rw [Fin.lt_def] at h2
    simp at h2
    omega
  · rename_i h1 h2
    simp
    simp at h1
    rw [Fin.le_def] at h1
    omega


lemma predAbove_eq_iff {i x : Fin n.succ.succ} (h : i ≠ x) (y : Fin n.succ) :
    y = predAboveI i x  ↔ i.succAbove y  = x := by
  apply Iff.intro
  · intro h
    subst h
    rw [succsAbove_predAboveI h]
  · intro h
    rw [← h]
    simp

lemma predAboveI_lt {i x : Fin n.succ.succ} (h : x.val < i.val) :
    predAboveI i x = ⟨x.val, by omega⟩ := by
  simp [predAboveI, h]

lemma predAboveI_ge {i x : Fin n.succ.succ} (h : i.val < x.val) :
    predAboveI i x = ⟨x.val - 1, by omega⟩ := by
  simp [predAboveI, h]
  omega

/-- 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 :=
  (finCongr (by omega : n.succ = i + 1 + (n - i))).trans <|
  finSumFinEquiv.symm.trans <|
  (Equiv.sumCongr (finSumFinEquiv.symm.trans (Equiv.sumComm (Fin i) (Fin 1)))
    (Equiv.refl (Fin (n-i)))).trans <|
  (Equiv.sumAssoc (Fin 1) (Fin i) (Fin (n - i))).trans <|
  Equiv.sumCongr (Equiv.refl (Fin 1)) (finSumFinEquiv.trans (finCongr (by omega)))

lemma finExtractOne_apply_eq {n : ℕ} (i : Fin n.succ) :
    finExtractOne i i = Sum.inl 0 := by
  simp [finExtractOne]
  have h1 : Fin.cast (finExtractOne.proof_1 i) i = Fin.castAdd ((n - ↑i) ) ⟨i.1, lt_add_one i.1⟩ := by
    simp [Fin.ext_iff]
  rw [h1, finSumFinEquiv_symm_apply_castAdd]
  simp
  have h2 : @Fin.mk (↑i + 1) ↑i (lt_add_one i.1) = Fin.natAdd i.val 1 := by
    simp [Fin.ext_iff]
  rw [h2, finSumFinEquiv_symm_apply_natAdd]
  rfl

lemma finExtractOne_symm_inr {n : ℕ} (i : Fin n.succ) :
    (finExtractOne i).symm ∘ Sum.inr = i.succAbove := by
  ext x
  simp only [Nat.succ_eq_add_one, finExtractOne, Function.comp_apply, Equiv.symm_trans_apply,
    finCongr_symm, Equiv.symm_symm, Equiv.sumCongr_symm, Equiv.refl_symm, Equiv.sumCongr_apply,
    Equiv.coe_refl, Sum.map_inr, finCongr_apply, Fin.coe_cast]
  change (finSumFinEquiv
    (Sum.map (⇑(finSumFinEquiv.symm.trans (Equiv.sumComm (Fin ↑i) (Fin 1))).symm) id
    ((Equiv.sumAssoc (Fin 1) (Fin ↑i) (Fin (n - i))).symm
    (Sum.inr (finSumFinEquiv.symm (Fin.cast (finExtractOne.proof_2 i).symm x)))))).val = _
  by_cases hi : x.1 < i.1
  · have h1 : (finSumFinEquiv.symm (Fin.cast (finExtractOne.proof_2 i).symm x)) =
        Sum.inl ⟨x, hi⟩ := by
      rw [← finSumFinEquiv_symm_apply_castAdd]
      apply congrArg
      ext
      simp
    rw [h1]
    simp only [Nat.succ_eq_add_one, Equiv.sumAssoc_symm_apply_inr_inl, Sum.map_inl,
      Equiv.symm_trans_apply, Equiv.symm_symm, Equiv.sumComm_symm, Equiv.sumComm_apply,
      Sum.swap_inr, finSumFinEquiv_apply_left, Fin.castAdd_mk]
    rw [Fin.succAbove]
    split
    · rfl
    rename_i hn
    simp_all only [Nat.succ_eq_add_one, not_lt, Fin.le_def, Fin.coe_castSucc, Fin.val_succ,
      self_eq_add_right, one_ne_zero]
    omega
  · have h1 : (finSumFinEquiv.symm (Fin.cast (finExtractOne.proof_2 i).symm x)) =
        Sum.inr ⟨x - i, by omega⟩ := by
      rw [← finSumFinEquiv_symm_apply_natAdd]
      apply congrArg
      ext
      simp only [Nat.succ_eq_add_one, Fin.coe_cast, Fin.natAdd_mk]
      omega
    rw [h1, Fin.succAbove]
    split
    · rename_i hn
      simp_all [Fin.lt_def]
    simp only [Nat.succ_eq_add_one, Equiv.sumAssoc_symm_apply_inr_inr, Sum.map_inr, id_eq,
      finSumFinEquiv_apply_right, Fin.natAdd_mk, Fin.val_succ]
    omega

@[simp]
lemma finExtractOne_symm_inr_apply {n : ℕ} (i : Fin n.succ) (x : Fin n) :
    (finExtractOne i).symm (Sum.inr x) = i.succAbove x := calc
  _ = ((finExtractOne i).symm ∘ Sum.inr) x := rfl
  _ = i.succAbove x := by rw [finExtractOne_symm_inr]

@[simp]
lemma finExtractOne_symm_inl_apply {n : ℕ} (i : Fin n.succ) :
    (finExtractOne i).symm (Sum.inl 0) = i := by
  simp only [Nat.succ_eq_add_one, finExtractOne, Fin.isValue, Equiv.symm_trans_apply, finCongr_symm,
    Equiv.symm_symm, Equiv.sumCongr_symm, Equiv.refl_symm, Equiv.sumCongr_apply, Equiv.coe_refl,
    Sum.map_inl, id_eq, Equiv.sumAssoc_symm_apply_inl, Equiv.sumComm_symm, Equiv.sumComm_apply,
    Sum.swap_inl, finSumFinEquiv_apply_right, finSumFinEquiv_apply_left, finCongr_apply]
  ext
  rfl


def finExtractOnPermHom (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ) :
   Fin n.succ → Fin n.succ := fun x => predAboveI (σ i) (σ ((finExtractOne i).symm (Sum.inr x)))

lemma finExtractOnPermHom_inv (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ) :
  (finExtractOnPermHom (σ i) σ.symm) ∘ (finExtractOnPermHom i σ)  = id := by
  funext x
  simp [finExtractOnPermHom]
  by_cases h : σ (i.succAbove x) < σ i
  · rw [predAboveI_lt h, Fin.succAbove_of_castSucc_lt]
    · simp
    · simp_all [Fin.lt_def]
  have hσ : σ (i.succAbove x) ≠ σ i := by
    simp only [Nat.succ_eq_add_one, ne_eq, EmbeddingLike.apply_eq_iff_eq]
    exact Fin.succAbove_ne i x
  have hn : σ i < σ (i.succAbove x) := by omega
  rw [predAboveI_ge hn]
  rw [Fin.succAbove_of_le_castSucc]
  · simp
    trans  predAboveI i (σ.symm (σ (i.succAbove x)))
    · congr
      exact Nat.sub_add_cancel (Fin.lt_of_le_of_lt (Fin.zero_le (σ i)) hn)
    simp
  rw [Fin.le_def]
  simp
  omega

def finExtractOnePerm (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ) :
    Fin n.succ ≃ Fin n.succ where
  toFun x :=  finExtractOnPermHom i σ x
  invFun x := finExtractOnPermHom (σ i) σ.symm x
  left_inv x := by
    simpa using congrFun (finExtractOnPermHom_inv i σ) x
  right_inv x := by
    simpa using congrFun (finExtractOnPermHom_inv (σ i) σ.symm) x


/-- The equivalence of types `Fin n.succ.succ ≃ (Fin 1 ⊕ Fin 1) ⊕ Fin n` extracting
  the `i` and `(i.succAbove j)`. -/
def finExtractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
    Fin n.succ.succ ≃ (Fin 1 ⊕ Fin 1) ⊕ Fin n :=
  (finExtractOne i).trans <|
  (Equiv.sumCongr (Equiv.refl (Fin 1)) (finExtractOne j)).trans <|
  (Equiv.sumAssoc (Fin 1) (Fin 1) (Fin n)).symm

@[simp]
lemma finExtractTwo_apply_fst {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
    finExtractTwo i j i = Sum.inl (Sum.inl 0) := by
  simp [finExtractTwo]
  simp [finExtractOne_apply_eq]

lemma finExtractTwo_symm_inr {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
    (finExtractTwo i j).symm ∘ Sum.inr = i.succAbove ∘ j.succAbove := by
  rw [finExtractTwo]
  ext1 x
  simp

@[simp]
lemma finExtractTwo_symm_inr_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) (x : Fin n) :
    (finExtractTwo i j).symm (Sum.inr x) = i.succAbove (j.succAbove x) := by
  rw [finExtractTwo]
  simp

@[simp]
lemma finExtractTwo_symm_inl_inr_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
    (finExtractTwo i j).symm (Sum.inl (Sum.inr 0)) = i.succAbove j := by
  rw [finExtractTwo]
  simp

@[simp]
lemma finExtractTwo_symm_inl_inl_apply {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
    (finExtractTwo i j).symm (Sum.inl (Sum.inl 0)) = i := by
  rw [finExtractTwo]
  simp

end HepLean.Fin