errant/lean/DeeCall49_Final.lean

import Mathlib.Data.Real.Basic
import Mathlib.Data.Rat.Basic
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Tactic.Ring
import Mathlib.Tactic.NormNum

namespace DeeCall49

structure Magnitude where
  val : ℝ
  hpos : 0 < val

instance : Coe Magnitude ℝ := ⟨fun m => m.val⟩
instance : Add Magnitude := ⟨fun a b => ⟨a + b, by linarith [a.hpos, b.hpos]⟩⟩
instance : Mul Magnitude := ⟨fun a b => ⟨a * b, by positivity⟩⟩

def CommLength (a b : Magnitude) : Prop := ∃ (q : ℚ), (a : ℝ) = q * (b : ℝ)
def CommSquare (a b : Magnitude) : Prop := ∃ (q : ℚ), (a : ℝ)^2 = q * (b : ℝ)^2
def HasRationalSquare (a : Magnitude) : Prop := ∃ (q : ℚ), (a : ℝ)^2 = q

inductive LanguagePass : Type
  | EnochianLTR
  | LatinLTR
  | HebrewRTL
  | ArabicRTL

def languagePassPredicate (p : LanguagePass) (a b : Magnitude) : Prop :=
  match p with
  | LanguagePass.EnochianLTR => CommLength a b
  | LanguagePass.LatinLTR    => HasRationalSquare a ∧ HasRationalSquare b
  | LanguagePass.HebrewRTL   => CommSquare a b
  | LanguagePass.ArabicRTL   => ¬CommLength a b

def MetatronCertified (a b : Magnitude) : Prop :=
  CommSquare a b ∧ ¬CommLength a b ∧ HasRationalSquare a ∧ HasRationalSquare b

structure BinomialData where
  α β : ℚ
  hα_pos : 0 < α
  hβ_pos : 0 < β
  h_incomm : ¬∃ (q : ℚ), (α : ℚ) = q^2 * β
  binomial : Magnitude := ⟨Real.sqrt α + Real.sqrt β, by positivity⟩

def call49 (b : BinomialData) (h_gt : b.β < b.α) : Magnitude :=
  ⟨Real.sqrt b.α - Real.sqrt b.β, by
    have h₁ : Real.sqrt (b.α : ℝ) > Real.sqrt (b.β : ℝ) :=
      Real.sqrt_lt_sqrt (by positivity) (by exact_mod_cast h_gt)
    linarith⟩

theorem call49_algebraic (b : BinomialData) (h_gt : b.β < b.α) :
    (b.binomial : ℝ) + ((call49 b h_gt) : ℝ) = 2 * Real.sqrt b.α := by
  simp [BinomialData.binomial, call49]
  ring

theorem call49_product (b : BinomialData) (h_gt : b.β < b.α) :
    (b.binomial : ℝ) * ((call49 b h_gt) : ℝ) = (b.α : ℝ) - (b.β : ℝ) := by
  simp [BinomialData.binomial, call49]
  have hα : (Real.sqrt (b.α : ℝ))^2 = b.α := Real.sq_sqrt (by positivity)
  have hβ : (Real.sqrt (b.β : ℝ))^2 = b.β := Real.sq_sqrt (by positivity)
  nlinarith [Real.sqrt_nonneg (b.α : ℝ), Real.sqrt_nonneg (b.β : ℝ)]

def Gate := {p : Fin 22 × Fin 22 // p.1 < p.2}

def gateIndex (b : BinomialData) : ℕ :=
  let α_nat := b.α.num.natAbs
  let β_nat := b.β.num.natAbs
  (α_nat * β_nat) % 231

def alHamidAbjad : ℕ := 53

def call49Seal (b : BinomialData) : ℕ :=
  (b.α.num.natAbs + b.β.num.natAbs + alHamidAbjad) % 65536

inductive DeeCall49Call : Type where
  | Divide1 : BinomialData → DeeCall49Call
  | Divide2 : BinomialData → DeeCall49Call
  | Divide3 : BinomialData → DeeCall49Call
  | Divide4 : BinomialData → DeeCall49Call
  | Divide5 : BinomialData → DeeCall49Call
  | Divide6 : BinomialData → DeeCall49Call
  | Mirror  : BinomialData → DeeCall49Call
  | Seal    : BinomialData → DeeCall49Call

def executeCall (c : DeeCall49Call) : Option (Magnitude × Magnitude) :=
  match c with
  | DeeCall49Call.Divide1 b =>
      some (⟨Real.sqrt b.α, by positivity⟩, ⟨Real.sqrt b.β, by positivity⟩)
  | DeeCall49Call.Divide2 b =>
      some (⟨Real.sqrt b.α, by positivity⟩, ⟨Real.sqrt b.β, by positivity⟩)
  | DeeCall49Call.Divide3 b =>
      some (⟨Real.sqrt b.α, by positivity⟩, ⟨Real.sqrt b.β, by positivity⟩)
  | DeeCall49Call.Divide4 b =>
      some (⟨Real.sqrt b.α, by positivity⟩, ⟨Real.sqrt b.β, by positivity⟩)
  | DeeCall49Call.Divide5 b =>
      some (⟨Real.sqrt b.α, by positivity⟩, ⟨Real.sqrt b.β, by positivity⟩)
  | DeeCall49Call.Divide6 b =>
      some (⟨Real.sqrt b.α, by positivity⟩, ⟨Real.sqrt b.β, by positivity⟩)
  | DeeCall49Call.Mirror b =>
      if h : b.β < b.α then
        some (call49 b h, b.binomial)
      else none
  | DeeCall49Call.Seal _ => none

theorem metatron_iff_binomial (a b : Magnitude) :
    MetatronCertified a b ↔
    (CommSquare a b ∧ ¬CommLength a b ∧ HasRationalSquare a ∧ HasRationalSquare b) := by
  rfl

theorem call49_involution_type (b : BinomialData) (h : b.β < b.α) :
    let apotome := call49 b h
    (apotome : ℝ) > 0 ∧ (b.binomial : ℝ) > 0 ∧
    (b.binomial : ℝ) * (apotome : ℝ) = (b.α : ℝ) - (b.β : ℝ) := by
  constructor
  · exact (call49 b h).hpos
  constructor
  · exact b.binomial.hpos
  · exact call49_product b h

theorem dee_call49_division_unified (b : BinomialData) :
    ∃ (ρ σ : Magnitude),
      HasRationalSquare ρ ∧ HasRationalSquare σ ∧
      CommSquare ρ σ ∧ ¬CommLength ρ σ ∧
      (b.binomial : ℝ) = (ρ : ℝ) + (σ : ℝ) := by
  have h₁ : 0 < (b.α : ℝ) := by exact_mod_cast b.hα_pos
  have h₂ : 0 < (b.β : ℝ) := by exact_mod_cast b.hβ_pos
  refine ⟨⟨Real.sqrt b.α, Real.sqrt_pos.mpr h₁⟩,
          ⟨Real.sqrt b.β, Real.sqrt_pos.mpr h₂⟩, ?_, ?_, ?_, ?_, ?_⟩
  · exact ⟨b.α, by simp [Real.sq_sqrt (le_of_lt h₁)]⟩
  · exact ⟨b.β, by simp [Real.sq_sqrt (le_of_lt h₂)]⟩
  · refine ⟨b.α / b.β, ?_⟩
    simp [Real.sq_sqrt (le_of_lt h₁), Real.sq_sqrt (le_of_lt h₂)]
    field_simp [ne_of_gt b.hβ_pos]
    ring
  · intro ⟨q, hq⟩
    apply b.h_incomm
    refine ⟨q, ?_⟩
    have hq' : Real.sqrt (b.α : ℝ) = q * Real.sqrt (b.β : ℝ) := by exact_mod_cast hq
    have := congr_arg (· ^ 2) hq'
    simp [Real.sq_sqrt (le_of_lt h₁), Real.sq_sqrt (le_of_lt h₂)] at this
    exact_mod_cast this.symm
  · simp [BinomialData.binomial]

theorem linguistic_reversal_is_apotome (b : BinomialData) (h : b.β < b.α) :
    ¬CommLength (call49 b h) b.binomial := by
  intro ⟨q, hq⟩
  simp [call49, BinomialData.binomial] at hq
  have h₁ : 0 < (b.α : ℝ) := by exact_mod_cast b.hα_pos
  have h₂ : 0 < (b.β : ℝ) := by exact_mod_cast b.hβ_pos
  have h₃ : Real.sqrt (b.α : ℝ) > Real.sqrt (b.β : ℝ) :=
    Real.sqrt_lt_sqrt (by positivity) (by exact_mod_cast h)
  apply b.h_incomm
  have hq' : Real.sqrt (b.α : ℝ) - Real.sqrt (b.β : ℝ) =
             q * (Real.sqrt (b.α : ℝ) + Real.sqrt (b.β : ℝ)) := by exact_mod_cast hq
  have hne1 : (q : ℝ) ≠ 1 := by
    intro heq; simp [heq] at hq'; linarith [Real.sqrt_pos.mpr h₂]
  have hratio : Real.sqrt (b.α : ℝ) / Real.sqrt (b.β : ℝ) = (1 + q) / (1 - q) := by
    have h₄ : Real.sqrt (b.β : ℝ) ≠ 0 := by positivity
    field_simp [h₄]
    nlinarith
  refine ⟨(1 + q) / (1 - q) * ((1 + q) / (1 - q)), ?_⟩
  have h₅ : (Real.sqrt (b.α : ℝ))^2 = b.α := Real.sq_sqrt (le_of_lt h₁)
  have h₆ : (Real.sqrt (b.β : ℝ))^2 = b.β := Real.sq_sqrt (le_of_lt h₂)
  rw [← h₅, ← h₆]
  rw [hratio]
  push_cast
  ring_nf
  field_simp [sub_ne_zero.mpr (Ne.symm hne1)]
  ring

end DeeCall49