ALMA.Prototype.StandardModel

{-
An attempt to encode the arguments of the paper into Agda code,
before the philosophical thought of the paper fully converges
upon Cosmos as a new mathematical structure.

Retained to reflect the thinking process from attempting to express
philosophical thought in Agda to ultimately converging upon Cosmos.

在论文的哲学思想完全收敛于 Cosmos 这一新的数学结构之前，
将论文各个论证写成 Agda 代码的尝试。
予以保留，以体现从尝试用 Agda 表达哲学思想
并最终收敛于 Cosmos 的思考过程。

Standard Model: Concrete instantiation of the ALMA universe framework
标准模型：ALMA 宇宙框架的具体实例化

Defines basic attribute types, core beings (stable self and random environment),
feedback system frameworks, coherence and consciousness emergence,
transient processes with phase transitions, and the standard model overview
定义了基础属性类型、核心存在者（稳定自我与随机环境）、
反馈系统框架、相干性与意识涌现、
含相变的瞬态过程，以及标准模型总览
-}
{-# OPTIONS --safe --cubical-compatible --exact-split --guardedness --double-check #-}

module ALMA.Prototype.StandardModel where

open import ALMA.Prototype.Universe public

-- Basic types
-- 基础类型
Attr2D : Set
Attr2D = Fin 2 → ℕ
Attr1D : Set
Attr1D = Fin 1 → ℕ
State : Set
State = Attr2D × Attr1D

-- Core being definitions
-- 核心存在者定义
stableSelf : Attr2D
stableSelf i = if toℕ i ≡ᵇ 0 then 1 else 42
private
  self-c : ℕ
  self-c = 0
  self-stream : Stream (Attr2D × ℕ)
  self-stream = constStream (stableSelf , self-c)
  self-essence-const : Always (λ x → proj₂ x ≡ self-c) self-stream
  self-essence-const = constStream-always-gen (stableSelf , self-c) refl
  self-init-consistent : proj₁ (head self-stream) ≡ stableSelf
  self-init-consistent = refl
selfBeing : Process Attr2D ℕ
selfBeing = mkProc stableSelf self-c self-stream self-essence-const self-init-consistent
self-being-weak-hom : WeakHom selfBeing
self-being-weak-hom n m = trans (constStream-obsS-constant (stableSelf , self-c) n)
                                (sym (constStream-obsS-constant (stableSelf , self-c) m))
module _ (init : ℕ) (c : ℕ) where
  private
    mkRandomBeing-coalg : ℕ → StreamF (Attr1D × ℕ) ℕ
    mkRandomBeing-coalg n = ((λ _ → n) , c) , suc n
    mkRandomBeing-stream : Stream (Attr1D × ℕ)
    mkRandomBeing-stream = ana mkRandomBeing-coalg init
    mkRandomBeing-essence-const : Always (λ x → proj₂ x ≡ c) mkRandomBeing-stream
    mkRandomBeing-essence-const = helper init
      where
      helper : ∀ n → Always (λ x → proj₂ x ≡ c) (ana mkRandomBeing-coalg n)
      helper n .Always.head = refl
      helper n .Always.tail = helper (suc n)
    mkRandomBeing-init-consistent : proj₁ (head mkRandomBeing-stream) ≡ (λ _ → init)
    mkRandomBeing-init-consistent = refl
    lemma-random-obs : ∀ k n → 
      proj₁ (head (stream-tail-n (ana mkRandomBeing-coalg k) n)) ≡ (λ _ → k + n)
    lemma-random-obs k zero =
      begin
        proj₁ (head (stream-tail-n (ana mkRandomBeing-coalg k) zero))
          ≡⟨ refl ⟩
        (λ _ → k)
          ≡⟨ cong (λ m → λ _ → m) (sym (+-identityʳ k)) ⟩
        (λ _ → k + zero)
      ∎
    lemma-random-obs k (suc n) =
      begin
        proj₁ (head (stream-tail-n (ana mkRandomBeing-coalg k) (suc n)))
          ≡⟨ refl ⟩
        proj₁ (head (stream-tail-n (Stream.tail (ana mkRandomBeing-coalg k)) n))
          ≡⟨ refl ⟩
        proj₁ (head (stream-tail-n (ana mkRandomBeing-coalg (suc k)) n))
          ≡⟨ lemma-random-obs (suc k) n ⟩
        (λ _ → suc k + n)
          ≡⟨ refl ⟩
        (λ _ → suc (k + n))
          ≡⟨ cong (λ m → λ _ → m) (sym (+-suc k n)) ⟩
        (λ _ → k + suc n)
      ∎
  mkRandomBeing : Process Attr1D ℕ
  mkRandomBeing = mkProc (λ _ → init) c mkRandomBeing-stream mkRandomBeing-essence-const mkRandomBeing-init-consistent
  observe-mkRandomBeing : ∀ n → observe mkRandomBeing n ≡ (λ _ → init + n)
  observe-mkRandomBeing n =
    begin
      observe mkRandomBeing n
        ≡⟨ refl ⟩
      proj₁ (head (stream-tail-n mkRandomBeing-stream n))
        ≡⟨ lemma-random-obs init n ⟩
      (λ _ → init + n)
    ∎
  mkRandomBeing-changes : Changes mkRandomBeing
  mkRandomBeing-changes = 0 , λ eq →
    let eq' : init + 0 ≡ init + 1
        eq' = cong (λ f → f zero) (trans (sym (observe-mkRandomBeing 0)) (trans eq (observe-mkRandomBeing 1)))
        eq1 : init ≡ init + 1
        eq1 = trans (sym (+-identityʳ init)) eq'
        eq2 : init ≡ suc init
        eq2 = trans eq1 (trans (+-suc init 0) (cong suc (+-identityʳ init)))
    in n≢sucn init eq2
randomBeing : Process Attr1D ℕ
randomBeing = mkRandomBeing 0 0
observe-random-n : ∀ n → observe randomBeing n ≡ (λ _ → n)
observe-random-n n = observe-mkRandomBeing 0 0 n
random-being-changes : Changes randomBeing
random-being-changes = mkRandomBeing-changes 0 0
random-being-delta : Δ randomBeing
random-being-delta = 0 , 1 , λ eq →
  zero≢suc (cong (λ f → f zero)
    (trans (sym (observe-random-n 0))
           (trans eq (observe-random-n 1))))
same-essence : core selfBeing ≡ core randomBeing
same-essence = refl
self-random-interact : Process (Attr2D × Attr1D) ℕ
self-random-interact = interact selfBeing randomBeing
interact-essence-conserved : core self-random-interact ≡ core selfBeing
interact-essence-conserved = core-interact selfBeing randomBeing

-- Feedback system framework
-- 反馈系统框架
record FeedbackRules : Set where
  field
    nextSelf   : Attr2D → State → Attr2D
    nextRandom : Attr1D → State → Attr1D
    essence    : ℕ
open FeedbackRules public
mkFeedbackCoalg : FeedbackRules → State → StreamF (State × ℕ) State
mkFeedbackCoalg rules (s , r) =
  let s' = rules .nextSelf s (s , r)
      r' = rules .nextRandom r (s , r)
  in (((s , r) , rules .essence) , (s' , r'))
feedback-essence-const-gen : ∀ (rules : FeedbackRules) (init : State)
                           → Always (λ x → proj₂ x ≡ rules .essence)
                                     (ana (mkFeedbackCoalg rules) init)
feedback-essence-const-gen rules init = helper init
  where
    helper : ∀ state → Always (λ x → proj₂ x ≡ rules .essence)
                              (ana (mkFeedbackCoalg rules) state)
    helper state .Always.head = refl
    helper state .Always.tail = helper (proj₂ (mkFeedbackCoalg rules state))
mkFeedbackProcess : FeedbackRules → State → Process State ℕ
mkFeedbackProcess rules init = mkProc
  init
  (rules .essence)
  (ana (mkFeedbackCoalg rules) init)
  (feedback-essence-const-gen rules init)
  refl
unidirectional-rules : FeedbackRules
unidirectional-rules = record
  { nextSelf   = λ self _ → self
  ; nextRandom = λ env _ → λ _ → suc (env zero)
  ; essence    = 0
  }
bidirectional-rules : FeedbackRules
bidirectional-rules = record
  { nextSelf   = λ self _ → self
  ; nextRandom = λ _ (self , _) → λ _ → suc (self zero)
  ; essence    = 0
  }
initSelf : Attr2D
initSelf = stableSelf
initRandom : Attr1D
initRandom = λ _ → 0
initState : State
initState = (initSelf , initRandom)
feedbackProcess : Process State ℕ
feedbackProcess = mkFeedbackProcess unidirectional-rules initState
bidirectionalProcess : Process State ℕ
bidirectionalProcess = mkFeedbackProcess bidirectional-rules initState
PairAt-gen : ∀ (rules : FeedbackRules) (init : State) → ℕ → State
PairAt-gen rules init n = proj₁ (head (stream-tail-n (ana (mkFeedbackCoalg rules) init) n))
SelfAt-gen : ∀ (rules : FeedbackRules) (init : State) → ℕ → Attr2D
SelfAt-gen rules init n = proj₁ (PairAt-gen rules init n)
RandAt-gen : ∀ (rules : FeedbackRules) (init : State) → ℕ → Attr1D
RandAt-gen rules init n = proj₂ (PairAt-gen rules init n)
bidirPairAt : ℕ → State
bidirPairAt = PairAt-gen bidirectional-rules initState
bidirSelfAt : ℕ → Attr2D
bidirSelfAt = SelfAt-gen bidirectional-rules initState
bidirRandAt : ℕ → Attr1D
bidirRandAt = RandAt-gen bidirectional-rules initState
feedbackPairAt : ℕ → State
feedbackPairAt = PairAt-gen unidirectional-rules initState
feedbackSelfAt : ℕ → Attr2D
feedbackSelfAt = SelfAt-gen unidirectional-rules initState
feedbackRandAt : ℕ → Attr1D
feedbackRandAt = RandAt-gen unidirectional-rules initState
SelfAt-const-gen : ∀ (rules : FeedbackRules) (init : State)
                 → (∀ s st → rules .nextSelf s st ≡ s)
                 → ∀ n → SelfAt-gen rules init n ≡ proj₁ init
SelfAt-const-gen rules init nextSelf-id n = helper init n
  where
    helper : ∀ state n → SelfAt-gen rules state n ≡ proj₁ state
    helper state zero = refl
    helper state (suc n) =
      let next-state = proj₂ (mkFeedbackCoalg rules state)
          next-state-preserves-self : proj₁ next-state ≡ proj₁ state
          next-state-preserves-self = nextSelf-id (proj₁ state) state
      in trans (helper next-state n) next-state-preserves-self
RandAt-follows-self-gen : ∀ (rules : FeedbackRules) (init : State)
                        → (∀ s st → rules .nextSelf s st ≡ s)
                        → (∀ r st → (rules .nextRandom r st) zero ≡ suc (proj₁ st zero))
                        → ∀ n → RandAt-gen rules init (suc n) zero ≡ suc (proj₁ init zero)
RandAt-follows-self-gen rules init nextSelf-id nextRandom-follows-self n = helper init n
  where
    helper : ∀ state n → RandAt-gen rules state (suc n) zero ≡ suc (proj₁ state zero)
    helper state zero = nextRandom-follows-self (proj₂ state) state
    helper state (suc n) =
      let next-state = proj₂ (mkFeedbackCoalg rules state)
          next-state-preserves-self : proj₁ next-state ≡ proj₁ state
          next-state-preserves-self = nextSelf-id (proj₁ state) state
      in trans (helper next-state n) (cong (λ x → suc (x zero)) next-state-preserves-self)
RandAt-strictly-increasing-gen : ∀ (rules : FeedbackRules) (init : State)
                               → (∀ r st → (rules .nextRandom r st) zero ≡ suc (r zero))
                               → ∀ n → RandAt-gen rules init (suc n) zero ≡ suc (RandAt-gen rules init n zero)
RandAt-strictly-increasing-gen rules init nextRandom-inc n = helper init n
  where
    helper : ∀ state n → RandAt-gen rules state (suc n) zero ≡ suc (RandAt-gen rules state n zero)
    helper state zero = nextRandom-inc (proj₂ state) state
    helper state (suc n) = helper (proj₂ (mkFeedbackCoalg rules state)) n
bidirSelfAt-const : ∀ n → bidirSelfAt n ≡ initSelf
bidirSelfAt-const = SelfAt-const-gen bidirectional-rules initState (λ _ _ → refl)
bidirRandAt-follows-self : ∀ n → bidirRandAt (suc n) zero ≡ suc (initSelf zero)
bidirRandAt-follows-self = RandAt-follows-self-gen
  bidirectional-rules
  initState
  (λ _ _ → refl)
  (λ _ _ → refl)
feedbackRandAt-strictly-increasing : ∀ n → feedbackRandAt (suc n) zero ≡ suc (feedbackRandAt n zero)
feedbackRandAt-strictly-increasing = RandAt-strictly-increasing-gen 
  unidirectional-rules 
  initState 
  (λ _ _ → refl)

-- Coherence and consciousness emergence
-- 相干性与意识涌现
countOver : (ℕ → Bool) → ℕ → ℕ
countOver P zero    = 0
countOver P (suc n) = (if P n then 1 else 0) + countOver P n
mkInternalCount : (ℕ → Attr2D) → (ℕ → ℕ)
mkInternalCount selfAt steps = 
  countOver (λ k → does (selfAt k zero ≟ selfAt (suc k) zero)) (steps ∸ 1)
mkExternalCount : (ℕ → Attr2D) → (ℕ → ℕ) → (ℕ → ℕ)
mkExternalCount selfAt externalSource steps = 
  countOver (λ k → does (selfAt k zero ≟ externalSource k)) steps
mkCoherenceProof : (ic ec : ℕ → ℕ)
                 → (steps : ℕ)
                 → (n m : ℕ)
                 → ic steps ≡ n
                 → ec steps ≡ m
                 → n > m
                 → ic steps > ec steps
mkCoherenceProof _ _ _ _ _ ic≡n ec≡m n>m =
  subst-inequality ic≡n ec≡m n>m
mkDissipationProof : (ic ec : ℕ → ℕ)
                   → (steps : ℕ)
                   → (n m : ℕ)
                   → ic steps ≡ n
                   → ec steps ≡ m
                   → m > n
                   → ec steps > ic steps
mkDissipationProof _ _ _ _ _ ic≡n ec≡m m>n =
  subst-inequality ec≡m ic≡n m>n
mkConsciousnessEmergence : (ic ec : ℕ → ℕ)
                         → ic 5 > ec 5
                         → ∃ λ steps → ic steps > ec steps
mkConsciousnessEmergence ic ec proof = 5 , proof
randRandom : ℕ → ℕ
randRandom k = observe randomBeing k zero
internalCount : ℕ → ℕ
internalCount = mkInternalCount feedbackSelfAt
externalCount : ℕ → ℕ
externalCount = mkExternalCount feedbackSelfAt randRandom
internal5 : ℕ
internal5 = internalCount 5
internal5≡4 : internal5 ≡ 4
internal5≡4 = refl
external5 : ℕ
external5 = externalCount 5
external5≡1 : external5 ≡ 1
external5≡1 = refl
coherenceProof : internal5 > external5
coherenceProof = mkCoherenceProof internalCount externalCount 5 4 1 refl refl 
  (s≤s (s≤s (z≤n {2})))
consciousnessEmergence : ∃ λ steps → internalCount steps > externalCount steps
consciousnessEmergence = mkConsciousnessEmergence internalCount externalCount coherenceProof
bidirInternalCount : ℕ → ℕ
bidirInternalCount = mkInternalCount bidirSelfAt
bidirExternalCount : ℕ → ℕ
bidirExternalCount = mkExternalCount bidirSelfAt (λ k → bidirRandAt k zero)
bidirInternal5 : ℕ
bidirInternal5 = bidirInternalCount 5
bidirInternal5≡4 : bidirInternal5 ≡ 4
bidirInternal5≡4 = refl
bidirExternal5 : ℕ
bidirExternal5 = bidirExternalCount 5
bidirExternal5≡0 : bidirExternal5 ≡ 0
bidirExternal5≡0 = refl
bidirCoherenceProof : bidirInternal5 > bidirExternal5
bidirCoherenceProof = mkCoherenceProof bidirInternalCount bidirExternalCount 5 4 0 refl refl 
  (s≤s (z≤n {3}))
bidirConsciousnessEmergence : ∃ λ steps → bidirInternalCount steps > bidirExternalCount steps
bidirConsciousnessEmergence = mkConsciousnessEmergence bidirInternalCount bidirExternalCount bidirCoherenceProof
ConsciousnessTheorem : (Process State ℕ → (ℕ → ℕ)) → Process State ℕ → Set
ConsciousnessTheorem external-source p =
  ∃ λ steps →
    let
      pairAt' n = proj₁ (head (stream-tail-n (stream p) n))
      selfAt' n = proj₁ (pairAt' n)
      internal = countOver (λ k → does (selfAt' k zero ≟ selfAt' (suc k) zero)) (steps ∸ 1)
      external = countOver (λ k → does (selfAt' k zero ≟ external-source p k)) steps
    in internal > external
feedback-external-source : Process State ℕ → (ℕ → ℕ)
feedback-external-source _ k = randRandom k
bidirectional-external-source : Process State ℕ → (ℕ → ℕ)
bidirectional-external-source _ k = bidirRandAt k zero
Theorem-holds-for-feedback : ConsciousnessTheorem feedback-external-source feedbackProcess
Theorem-holds-for-feedback = consciousnessEmergence
Theorem-holds-for-bidirectional : ConsciousnessTheorem bidirectional-external-source bidirectionalProcess
Theorem-holds-for-bidirectional = bidirConsciousnessEmergence

-- Transient processes and phase transitions
-- 瞬态过程与相变
env : ℕ → ℕ
env n = observe randomBeing n zero
env≡n : ∀ n → env n ≡ n
env≡n n = cong (λ f → f zero) (observe-random-n n)
v : ℕ → ℕ
v n = if does (n <? 7) then 0 else n
stage : ℕ → ℕ
stage n = if does (n <? 5) then 0
          else (if does (n <? 7) then 1 else 2)
pair→attr : ℕ × ℕ → Attr2D
pair→attr (val , s) i = if toℕ i ≡ᵇ 0 then val else s
transient-coalg : ℕ → StreamF (Attr2D × ℕ) ℕ
transient-coalg n =
  let state = (v n , stage n)
  in ((pair→attr state , 0) , suc n)
transientStream : Stream (Attr2D × ℕ)
transientStream = ana transient-coalg 0
transient-essence-const : Always (λ x → proj₂ x ≡ 0) transientStream
transient-essence-const = helper 0
  where
    helper : ∀ n → Always (λ x → proj₂ x ≡ 0) (ana transient-coalg n)
    helper n .Always.head = refl
    helper n .Always.tail = helper (suc n)
transientBeing : Process Attr2D ℕ
transientBeing = mkProc
  (pair→attr (v 0 , stage 0))
  0
  transientStream
  transient-essence-const
  refl
transientSelfAt : ℕ → Attr2D
transientSelfAt n = observe transientBeing n
cInternal5 : ℕ
cInternal5 = mkInternalCount transientSelfAt 5
cInternal5≡4 : cInternal5 ≡ 4
cInternal5≡4 = refl
cExternal5 : ℕ
cExternal5 = mkExternalCount transientSelfAt env 5
cExternal5≡1 : cExternal5 ≡ 1
cExternal5≡1 = refl
coherence-phase : cInternal5 > cExternal5
coherence-phase = mkCoherenceProof
  (mkInternalCount transientSelfAt)
  (mkExternalCount transientSelfAt env)
  5 4 1 refl refl
  (s≤s (s≤s (z≤n {2})))
dInternal : ℕ
dInternal = countOver (λ k → v (7 + k) ≡ᵇ v (7 + suc k)) 2
dInternal≡0 : dInternal ≡ 0
dInternal≡0 = refl
dExternal : ℕ
dExternal = countOver (λ k → v (7 + k) ≡ᵇ env (7 + k)) 3
dExternal≡3 : dExternal ≡ 3
dExternal≡3 = refl
dissipation-phase : dExternal > dInternal
dissipation-phase = mkDissipationProof
  (λ steps → countOver (λ k → v (7 + k) ≡ᵇ v (7 + suc k)) (steps ∸ 1))
  (λ steps → countOver (λ k → v (7 + k) ≡ᵇ env (7 + k)) steps)
  3 0 3 refl refl
  (s≤s (z≤n {2}))
∃-coherence : ∃ λ steps → countOver (λ k → v k ≡ᵇ v (suc k)) (steps ∸ 1) > countOver (λ k → v k ≡ᵇ env k) steps
∃-coherence = 5 , subst (λ x → x > cExternal5) (sym cInternal5≡4) coherence-phase
∃-dissipation : ∃ λ steps → countOver (λ k → v (7 + k) ≡ᵇ v (7 + suc k)) (steps ∸ 1) < countOver (λ k → v (7 + k) ≡ᵇ env (7 + k)) steps
∃-dissipation = 3 , subst (λ x → dInternal < x) (sym dExternal≡3) dissipation-phase
phase-transition-exists : ∃ λ (p : Process Attr2D ℕ) →
  (∃ λ steps → countOver (λ k → v k ≡ᵇ v (suc k)) (steps ∸ 1) > countOver (λ k → v k ≡ᵇ env k) steps) ×
  (∃ λ steps → countOver (λ k → v (7 + k) ≡ᵇ v (7 + suc k)) (steps ∸ 1) < countOver (λ k → v (7 + k) ≡ᵇ env (7 + k)) steps)
phase-transition-exists = transientBeing , (∃-coherence , ∃-dissipation)

-- Standard model overview
-- 标准模型总览
record StandardModel : Setω where
  field
    indestructibility : Indestructibility-Theorem
    indestructibility-uniqueness : Indestructibility-Uniqueness
    process-uniqueness       : Process-Uniqueness
    core-conservation        : ∀ {ℓ₁ ℓ₂} {O : Set ℓ₁} {C : Set ℓ₂} (p : Process O C) → core (next p) ≡ core p
    self        : Process Attr2D ℕ
    random      : Process Attr1D ℕ
    nonunif     : ∃ λ n1 → ∃ λ n2 → observe random n1 ≢ observe random n2
    conscious   : ∃ λ steps → internalCount steps > externalCount steps
    consciousBidir : ∃ λ steps → bidirInternalCount steps > bidirExternalCount steps
    coherencePhase   : Σ ℕ (λ steps → cInternal5 > cExternal5)
    dissipationPhase : Σ ℕ (λ steps → dExternal > dInternal)
standardModel : StandardModel
standardModel = record
  { indestructibility = indestructibility-proof
  ; indestructibility-uniqueness = indestructibility-uniqueness-proof
  ; process-uniqueness       = process-uniqueness-proof
  ; core-conservation        = c-conservation
  ; self            = selfBeing
  ; random          = randomBeing
  ; nonunif         = 0 , 1 , λ p → zero≢suc (cong (λ f → f zero) p)
  ; conscious       = consciousnessEmergence
  ; consciousBidir  = bidirConsciousnessEmergence
  ; coherencePhase  = 5 , coherence-phase
  ; dissipationPhase = 3 , dissipation-phase
  }