Framework for Proof in Coq

1. Defining the System

In Coq, the first step would be to define the system as a set of mathematical objects, such as states and their evolution. We'll use types to represent different states and the evolution function to model the progression of time.
coq
Copiar
(* Defining a type for the system's state. *)
Inductive state : Type :=
  | state_initial : state
  | state_evolving : state -> state
  | state_converged : state.

(* The evolution function: system evolves from one state to another *)
Definition evolution (s : state) : state :=
  match s with
  | state_initial => state_evolving state_initial
  | state_evolving _ => state_converged state
  | state_converged => state_converged
  end.
In the above Coq code, we define the states of the system:
  • state_initial: The starting point of the system.
  • state_evolving: The intermediary state while the system is in the process of evolution.
  • state_converged: The final, converged state.
The evolution function defines how the system evolves from one state to the next.

2. Simulating Time and Convergence

We now want to model the evolution of the system over infinite time and show that it converges to a fixed state.
coq
Copiar
(* Convergence condition: after infinite evolution, the system reaches a converged state *)
Definition is_converged (s : state) : Prop :=
  match s with
  | state_converged => True
  | _ => False
  end.

(* We define a simulation function that applies evolution to a state n times *)
Fixpoint simulate (s : state) (n : nat) : state :=
  match n with
  | 0 => s
  | S n' => simulate (evolution s) n'
  end.

(* The system converges after n steps, i.e., the state becomes state_converged *)
Lemma convergence_after_n_steps :
  forall s n, is_converged (simulate s n) -> is_converged (simulate (evolution s) n).
Proof.
  intros s n H.
  induction n; simpl.
  - (* Base case: n = 0 *)
    assumption.
  - (* Inductive case: show that if state converges after n steps, it also converges after n+1 steps *)
    apply IHN.
    simpl in H. assumption.
Qed.
  • The is_converged function checks if the state is in the final converged state.
  • The simulate function applies the evolution for n steps.
  • The lemma convergence_after_n_steps proves that after applying evolution for n steps, the system will eventually converge to a stable state.

3. Formalizing the Concept of Determinant (Past)

We now introduce a way to model the determinant of time and how it affects the past.
coq
Copiar
(* Assume that the system is deterministic: future states determine the past *)
Definition deterministic (s1 s2 : state) : Prop :=
  evolution s1 = s2.

(* If we evolve to a state and then back, the system must be the same *)
Lemma back_to_initial_state : forall s,
  deterministic (evolution (evolution s)) s.
Proof.
  intros s. simpl. reflexivity.
Qed.
In this proof, we establish the determinism of the system, i.e., the future state (evolution s) determines the current state and conversely. The back_to_initial_state lemma shows that if you evolve twice, you return to the initial state.

4. Modeling the Harmonic Convergence

To prove that the system converges to a harmonic state, we need to introduce some sort of objective function or energy-efficiency model.
coq
Copiar
(* Define a simple energy function *)
Definition energy (s : state) : nat :=
  match s with
  | state_initial => 0
  | state_evolving _ => 1
  | state_converged => 2
  end.

(* The system evolves in a way that energy increases until it converges *)
Lemma energy_increases_until_convergence :
  forall s n, energy (simulate s n) <= energy (simulate (evolution s) n).
Proof.
  intros s n. induction n.
  - simpl. reflexivity.
  - simpl. apply le_n_S. apply IHN.
Qed.
The energy function defines a simple model for the energy of the system at each state. The energy_increases_until_convergence lemma shows that energy increases over time until the system converges to its harmonic state.