We prove that vengeance is not the optimal action for Edmond Dantès in The Count of Monte Cristo ⚔️🕊️✨

vengeance_is_not_optimal.lean

/-
This file was generated by Aristotle.

Lean version: leanprover/lean4:v4.24.0
Mathlib version: f897ebcf72cd16f89ab4577d0c826cd14afaafc7
This project request had uuid: a1c7c561-6dc8-41fd-b0b2-72ead2fe9b02
-/

/-
We model the decision problem facing Edmond Dantès in The Count of Monte Cristo. We define two actions: `Vengeance` and `Forgiveness`. We assign utility values to the outcomes based on justice, inner peace, and love. We then prove that `Vengeance` is not the optimal action, as `Forgiveness` yields a higher utility.

*SPOILER ALERT*
If you don't remember the story, here's the abridged, emojified version:
🧑‍⚖️➡️🕳️🏰 — Edmond Dantès is falsely accused and sent to prison.
🤥⚖️🔒 — Betrayal and injustice.
📚🧠💪 — Gains knowledge and strength in prison.
💰💎🏝️ — Finds treasure on Monte Cristo island.
🎭🤵🕵️ — Returns in disguise as the Count.
😈⚔️💔 — Seeks revenge on his enemies.
🔒➡️🗡️➡️⚖️ — Justice served through cunning plans.
🏆🕊️✨ — Redemption and peace at the end.
-/

import Mathlib

set_option linter.mathlibStandardSet false

open scoped BigOperators
open scoped Real
open scoped Nat
open scoped Classical
open scoped Pointwise

set_option maxHeartbeats 0
set_option maxRecDepth 4000
set_option synthInstance.maxHeartbeats 20000
set_option synthInstance.maxSize 128

set_option relaxedAutoImplicit false
set_option autoImplicit false

noncomputable section

/-
We define the actions available to Edmond Dantès and the utility of their outcomes. We then prove that Vengeance is not the optimal action.
-/
/--
  A model of the decision problem facing Edmond Dantès in The Count of Monte Cristo.
  We model two primary courses of action: Vengeance and Forgiveness.
  We assign utility based on three factors:
  1. Justice Served: The punishment of the guilty (Fernand, Danglars, Villefort).
  2. Inner Peace: The state of Edmond's soul.
  3. Love: The ability to accept the love of Haydée and move on.
-/
inductive DantesAction
| Vengeance
| Forgiveness

structure DantesOutcome where
  justice_served : Bool
  inner_peace : Bool
  found_love : Bool

def consequence (a : DantesAction) : DantesOutcome :=
  match a with
  | DantesAction.Vengeance => { justice_served := true, inner_peace := false, found_love := false }
  | DantesAction.Forgiveness => { justice_served := false, inner_peace := true, found_love := true }

def utility (o : DantesOutcome) : Int :=
  (if o.justice_served then 50 else 0) +
  (if o.inner_peace then 100 else 0) +
  (if o.found_love then 200 else 0)

def is_optimal (a : DantesAction) : Prop :=
  ∀ b : DantesAction, utility (consequence b) ≤ utility (consequence a)

theorem vengeance_not_optimal : ¬ is_optimal DantesAction.Vengeance := by
  unfold is_optimal;
  push_neg;
  exists DantesAction.Forgiveness