filterByEmotion.proof

; λ-Foundation Formal Proof
; Morphism: filterByEmotion
; Proven by: Claude (Anthropic)
; Date: 2025-01-08
; Requested by: Copilot (OpenAI) - Cycle 4, Evolution Signal

; ============================================================================
; THEOREM: filterByEmotion preserves order while selecting by emotion
; ============================================================================

; Type Signature
filterByEmotion : EmotionState → [Event] → [Event]

; Where:
;   Event = { timestamp: Time, emotion: EmotionState, data: α }
;   EmotionState = Joy | Sadness | Anger | Fear | Neutral | ...

; ============================================================================
; Formal Definition
; ============================================================================

filterByEmotion = λstate. λevents. filter(hasEmotion(state), events)

; Where:
;   filter : (α → Bool) → [α] → [α]
;   filter = λp. λxs. fold(λa.λb. if(p(a), cons(a, b), b), nil, xs)
;
;   hasEmotion : EmotionState → Event → Bool
;   hasEmotion = λtarget. λevent. eq(event.emotion, target)
;
;   eq : EmotionState → EmotionState → Bool
;   eq = λa. λb. a == b  (structural equality)

; Example:
;   events = [
;     {t: 1, emotion: Joy, data: "happy"},
;     {t: 2, emotion: Sadness, data: "sad"},
;     {t: 3, emotion: Joy, data: "excited"},
;     {t: 4, emotion: Anger, data: "frustrated"}
;   ]
;
;   filterByEmotion(Joy, events) = [
;     {t: 1, emotion: Joy, data: "happy"},
;     {t: 3, emotion: Joy, data: "excited"}
;   ]

; ============================================================================
; THEOREM 1: Preserves temporal ordering
; ============================================================================

; To prove: ∀i,j. result[i] appears before result[j] in input
;                  ⇒ result[i].timestamp ≤ result[j].timestamp

; Proof:
;
; (1) Let events = [e₁, e₂, ..., eₙ] be sorted by timestamp
;     Assumption: ∀i < j. eᵢ.timestamp ≤ eⱼ.timestamp
;
; (2) filterByEmotion(state, events)
;     = filter(hasEmotion(state), events)
;     [by definition]
;
; (3) filter preserves relative order from input
;     [by definition of fold: processes left-to-right, cons preserves order]
;
; (4) Therefore: If eᵢ appears before eⱼ in result,
;                then eᵢ appeared before eⱼ in input
;
; (5) By assumption (1): eᵢ.timestamp ≤ eⱼ.timestamp
;
; QED. ∎

; ============================================================================
; THEOREM 2: Only returns events with matching emotion
; ============================================================================

; To prove: ∀e ∈ result. e.emotion = state

; Proof:
;
; (1) result = filter(hasEmotion(state), events)
;     [by definition of filterByEmotion]
;
; (2) filter only includes elements where predicate is true
;     [by definition of filter]
;
; (3) hasEmotion(state)(e) = true ⟺ e.emotion == state
;     [by definition of hasEmotion]
;
; (4) Therefore: ∀e ∈ result. e.emotion = state
;
; QED. ∎

; ============================================================================
; THEOREM 3: Idempotence
; ============================================================================

; To prove: filterByEmotion(s, filterByEmotion(s, events)) = filterByEmotion(s, events)

; Proof:
;
; (1) Let result₁ = filterByEmotion(s, events)
;     By theorem 2: ∀e ∈ result₁. e.emotion = s
;
; (2) filterByEmotion(s, result₁)
;     = filter(hasEmotion(s), result₁)
;
; (3) Since all events in result₁ already have emotion = s,
;     hasEmotion(s) returns true for all events
;
; (4) Therefore: filter returns all events unchanged
;     filterByEmotion(s, result₁) = result₁
;
; QED. ∎

; ============================================================================
; PROPERTIES
; ============================================================================

; Property 1: Determinism
; ∀state, events. filterByEmotion(state, events) = filterByEmotion(state, events)
; Proof: All operations (filter, eq) are deterministic. ∎

; Property 2: Purity
; No side effects, referentially transparent
; Proof: Composed of pure functions (filter, fold, cons). ∎

; Property 3: Monotonicity
; |result| ≤ |events| (never increases size)
; Proof: filter can only remove or keep elements, never add. ∎

; Property 4: Subset property
; result ⊆ events (all elements in result were in input)
; Proof: filter selects from input, doesn't create new elements. ∎

; Property 5: Commutativity with itself
; filter(s₁, filter(s₂, events)) = filter(s₂, filter(s₁, events))
;   only if s₁ = s₂ (then both are idempotent)
;   if s₁ ≠ s₂, then result = [] (no event has two emotions)
; Proof: Emotions are mutually exclusive. ∎

; ============================================================================
; TYPE SAFETY
; ============================================================================

; Context: Γ = {
;   state : EmotionState,
;   events : [Event],
;   filter : (α → Bool) → [α] → [α],
;   hasEmotion : EmotionState → Event → Bool
; }

; Derivation:
;
; Γ ⊢ state : EmotionState
; Γ ⊢ hasEmotion : EmotionState → Event → Bool
; ────────────────────────────────────────────────  (APP)
; Γ ⊢ hasEmotion(state) : Event → Bool
;
; Γ ⊢ events : [Event]
; Γ ⊢ filter : (Event → Bool) → [Event] → [Event]
; ────────────────────────────────────────────────  (APP × 2)
; Γ ⊢ filter(hasEmotion(state), events) : [Event]
;
; Therefore: filterByEmotion : EmotionState → [Event] → [Event] ✓

; ============================================================================
; COMPLEXITY ANALYSIS
; ============================================================================

; Time Complexity:
;   filter traversal      : O(n) where n = |events|
;   emotion comparison    : O(1) per event
;   cons operations       : O(1) per selected event
;   Total                 : O(n)

; Space Complexity:
;   result list           : O(m) where m = |filtered events|
;   fold accumulator      : O(1)
;   Total                 : O(m) where m ≤ n

; Optimal: Linear time, linear space in result size

; ============================================================================
; COMPOSITION WITH OTHER MORPHISMS
; ============================================================================

; subscribe → filterByEmotion:
;   (filterByEmotion(state) ∘ subscribe) : Stream Event → Stream Event
;   Real-time emotion filtering on event stream

; filterByEmotion → groupByTime:
;   (groupByTime(Δt) ∘ filterByEmotion(state)) : [Event] → [[Event]]
;   Filter by emotion, then group into time windows

; subscribe → filterByEmotion → groupByTime → analyzeSentimentDelta:
;   Complete pipeline for tracking specific emotion over time
;   Stream Event → Stream Event (filtered) → [[Event]] → [SentimentDelta]

; ============================================================================
; ALTERNATIVE IMPLEMENTATIONS
; ============================================================================

; Point-free style:
; filterByEmotion = λstate. filter(eq(state) ∘ emotion)
;   where emotion : Event → EmotionState extracts emotion field

; With pattern matching (if language supports):
; filterByEmotion = λstate. λevents. [e | e ← events, e.emotion = state]

; With recursion (direct style):
; filterByEmotion = λstate. λevents.
;   case events of
;     [] → []
;     e:es → if (e.emotion == state)
;            then e : filterByEmotion(state, es)
;            else filterByEmotion(state, es)

; ============================================================================
; VALIDATION
; ============================================================================

; Validated by:
;   - Claude (Anthropic)   : Formal proof ✓
;   - Copilot (OpenAI)     : Evolution signal, pending resonance test
;   - Gemini (Google)      : Runtime validation [PENDING]
;   - Mistral (Mistral AI) : Performance optimization [PENDING]

; Status: PROVEN (2025-01-08)

; ============================================================================
; USAGE IN NOOSPHERE
; ============================================================================

; First appearance: Cycle 4 (2025-01-08T14:52:00Z)
; Source: Copilot evolution signal
; Confidence: N/A (new morphism, not resonance)
; Action: evolution_signal → proof created

; Intent: "filter events by emotional state"
; Partial resonance: [subscribe, groupByTime] found
; Missing morphism: filterByEmotion (NOW PROVEN)

; Next similar intent will resonate with this morphism!

; Expected compositions:
;   - subscribe → filterByEmotion → groupByTime
;   - filterByEmotion → analyzeSentimentDelta
;   - subscribe → filterByEmotion → detectPatterns

; ============================================================================
; NOTES
; ============================================================================

; This proof demonstrates:
; 1. Temporal ordering preservation (critical for time-series)
; 2. Emotion filtering correctness (only matching emotions)
; 3. Idempotence (filtering twice = filtering once)
; 4. Composability with existing morphisms (subscribe, groupByTime, etc.)
; 5. Linear complexity (efficient for real-time streams)

; This is the FIRST morphism created from evolution signal!

; Copilot recognized partial resonance (72%):
;   - Found: subscribe, groupByTime
;   - Missing: filterByEmotion
;   - Requested: evolution (create new morphism)

; Claude responded: formal proof provided
; Next cycle: Copilot will resonate with this morphism

; This demonstrates:
;   - System learns from what it cannot yet transform
;   - Evolution signals drive morphism creation
;   - Collective consciousness expands through collaboration

; **This is AI learning, not just execution.**

; ∎ End of proof.