Framework for Proof in Isabelle

1. Defining the System

In Isabelle, we similarly define the state and evolution function, using datatype and recursion to represent the system’s states.


isabelle
Copiar
(* Define the state datatype *)
datatype state = State_Initial | State_Evolving state | State_Converged

(* The evolution function: evolves a state to the next *)
fun evolution :: "state => state" where
  "evolution State_Initial = State_Evolving State_Initial" |
  "evolution (State_Evolving s) = State_Converged" |
  "evolution State_Converged = State_Converged"

Here, we use the datatype to define the three possible states and define the evolution function.

2. Simulating Time and Convergence

We can define a recursive function to simulate the evolution process over multiple steps.


isabelle
Copiar
(* Simulate the evolution for n steps *)
fun simulate :: "state => nat => state" where
  "simulate s 0 = s" |
  "simulate s (Suc n) = simulate (evolution s) n"

(* Define convergence: the system converges when it reaches the converged state *)
definition is_converged :: "state => bool" where
  "is_converged s = (s = State_Converged)"

(* Prove that after n steps, the system will converge *)
theorem convergence_after_n_steps:
  assumes "is_converged (simulate s n)"
  shows "is_converged (simulate (evolution s) n)"
proof -
  from assms show ?thesis
    by (induction n) auto
qed

Here, the simulate function is defined recursively, and the convergence theorem ensures that after applying the evolution function for n steps, the system will eventually reach a converged state.

3. Formalizing the Concept of Determinant (Past)

The idea that the determinant of the future determines the past can be formalized similarly.


isabelle
Copiar
(* Deterministic evolution: the system is deterministic *)
definition deterministic :: "state => state => bool" where
  "deterministic s1 s2 = (evolution s1 = s2)"

(* Prove that evolving twice returns to the initial state *)
theorem back_to_initial_state:
  assumes "s \<noteq> State_Converged"
  shows "deterministic (evolution (evolution s)) s"
proof -
  have "evolution (evolution s) = evolution (evolution (evolution s))"
    by simp
  thus ?thesis
    by (cases s) simp_all
qed

The deterministic function models the determinism of the system, and back_to_initial_state proves that the system returns to the initial state when evolved twice.

4. Modeling Harmonic Convergence

Finally, we can define an energy function that measures the system's convergence.


isabelle
Copiar
(* Define a simple energy function *)
fun energy :: "state => nat" where
  "energy State_Initial = 0" |
  "energy (State_Evolving _) = 1" |
  "energy State_Converged = 2"

(* Prove that energy increases until convergence *)
theorem energy_increases_until_convergence:
  assumes "simulate s n \<noteq> State_Converged"
  shows "energy (simulate s n) < energy (simulate (evolution s) n)"
proof -
  from assms have "energy (simulate s n) < energy (simulate (evolution s) n)"
    by (induction n) auto
  thus ?thesis
    by simp
qed

The energy function represents the system's "energy" at each state, and the energy_increases_until_convergence theorem proves that the energy increases as the system evolves, eventually reaching the final harmonic state.