-- Hypermath.L2Operations
-- Layer 2: Operations
-- Lean4 mechanization scaffold for L2_operations.hm
--
-- Introduces: DerivationPath, pathStep, pathGround, pathLength, pathTrace,
--             compose, congruentPath, +, additionally, reflexion,
--             orbit-reflexion-quotient.
-- P_2 = 31.  N_2_atomic = 3 (DerivationPath, pathStep, pathGround).  G_2 = 4.
-- Source: L2_operations.hm

import Hypermath.L1Relations

namespace Hypermath

-- ============================================================================
-- §I  Layer Permits and Prohibits
-- (Copied from L1 graduation block for traceability.)
-- ============================================================================

-- Permits L2:
--   (a) DerivationPath :: Type — a finite record of f2f-steps with trace levels.
--   (b) + (additionally) — co-presence, commutative.
--   (c) quant family (preview of L3; full arithmetic in L3).
--   (d) Filtration freely: any operation preserving ≡ preserves =~ and ~~.

-- Prohibits L2:
--   (a) ordinal_limit or any infinite ordinal — L3 only.
--   (b) Separate classification machinery — self-kernel IS the machinery.
--   (c) Closing simulation-pair-exists — requires L3 continuation-path.
--   (d) Commutativity for + — hard invariant: + is non-commutative.

-- ============================================================================
-- §II  Primitives — DerivationPath
-- Three atomic forms at L2. All definition-acts. FORM.
-- ============================================================================

/-- A finite record of f2f-steps with associated trace levels.
    Analogous to a typed derivation sequence. -/
axiom DerivationPath : Type

/-- Unit path: a single f2f-step from Form x to Form y. -/
axiom pathStep : Form → Form → DerivationPath

/-- The empty derivation path. Identity element for compose. -/
axiom pathGround : DerivationPath

/-- Endpoint projections used explicitly in L3_ordinatics.hm:97–99.
    These are uninterpreted source parameters, not an implementation of paths.
    No endpoint law or path finiteness theorem is assumed by declaring them. -/
axiom pathStart : DerivationPath → Form
axiom pathEnd : DerivationPath → Form

/-- Path length: the number of steps, represented as a Form.
    For pathGround: ~~ ground (zero steps).
    For pathStep(x)(y): ~~ f2f(ground) (one step).
    Arithmetic (full ordinal grounding) is FRAME/L3. -/
axiom pathLength : DerivationPath → Form

/-- Path trace: the minimum trace level (~~/=~/≡) across all steps.
    Minimum trace arithmetic is FRAME/L3. -/
axiom pathTrace : DerivationPath → Form

/-- Sequential composition of two DerivationPaths.
    compose(p)(q) traverses p then q.
    Governed by axComposeAssoc, axComposeIdentity. -/
axiom compose : DerivationPath → DerivationPath → DerivationPath

/-- The =~-analog for DerivationPath: same substance regardless of intermediate steps.
    congruentPath(p, q) iff same (start, end, pathTrace, quantification). -/
axiom congruentPath : DerivationPath → DerivationPath → Prop

-- ============================================================================
-- §III  Axioms for + (non-commutative sequential connection)
-- ============================================================================

-- Translation boundary: these clauses quantify separate witnesses. No binary
-- Form operation for + or additionally is declared or bound to those witnesses.
-- They therefore do not implement the source's operation laws. Sequential.lean
-- proves that a common operation with the source's two-sided identity and
-- uniform separation cannot exist for nontrivial symmetric, transitive
-- congruence. The current clauses themselves are left unchanged.

/-- **ax-seq**: a common similarity witness exists for each pair.
    Identifying that witness with a particular a + b requires another construction. -/
axiom axSeq : ∀ a b : Form, ∃ c : Form, (c ~~ a) ∧ (c ~~ b)

/-- **ax-seq-asymm**: incongruent inputs have two incongruent similarity witnesses.
    This translation does not bind those witnesses to a + b and b + a. -/
axiom axSeqAsymm :
    ∀ a b : Form, ¬ (a =~ b) →
      ∃ c d : Form,
        (c ~~ a) ∧ (c ~~ b) ∧
        (d ~~ b) ∧ (d ~~ a) ∧
        ¬ (c =~ d)

/-- Separate identity witnesses; neither clause refers to a binary + operation. -/
axiom axSeqIdentityR : ∀ a : Form, ∃ c : Form, (c ~~ a) ∧ (c =~ a)
axiom axSeqIdentityL : ∀ a : Form, ∃ c : Form, (c ~~ a) ∧ (c =~ a)

-- ============================================================================
-- §IV  Axioms for additionally (commutative co-presence)
-- ============================================================================

/-- **ax-coop**: a common similarity witness, without a bound additionally operation. -/
axiom axCoop : ∀ a b : Form, ∃ c : Form, (c ~~ a) ∧ (c ~~ b)

/-- Two congruent co-presence witnesses; no binary additionally function is bound. -/
axiom axCoopComm :
    ∀ a b : Form, ∃ ca cab : Form,
      (ca ~~ a) ∧ (ca ~~ b) ∧
      (cab ~~ b) ∧ (cab ~~ a) ∧
      (ca =~ cab)

/-- A co-presence identity witness, without a bound binary operation. -/
axiom axCoopIdentity : ∀ a : Form, ∃ c : Form, (c ~~ a) ∧ (c =~ a)

-- ============================================================================
-- §V  Axioms for compose and congruentPath
-- ============================================================================

/-- compose is associative at congruentPath level. -/
axiom axComposeAssoc :
    ∀ p q r : DerivationPath,
      congruentPath (compose p (compose q r)) (compose (compose p q) r)

/-- pathGround is the two-sided identity for compose. -/
axiom axComposeIdentity :
    ∀ p : DerivationPath,
      congruentPath (compose p pathGround) p ∧
      congruentPath (compose pathGround p) p

/-- At least one non-trivial DerivationPath exists connecting any two Forms. -/
axiom axComposeNonempty :
    ∀ x y : Form, ∃ p : DerivationPath, ¬ congruentPath p pathGround

-- ============================================================================
-- §VI  Derives (all FORM unless marked FRAME)
-- ============================================================================

/-- (a + b) ~~ a and (a + b) ~~ b for all a b. FORM. -/
theorem plusStaysSimilar : ∀ a b : Form, ∃ c : Form, (c ~~ a) ∧ (c ~~ b) :=
  axSeq

/-- additionally(a,b) ~~ a and additionally(a,b) ~~ b for all a b. FORM. -/
theorem additionallyStaysSimilar : ∀ a b : Form, ∃ c : Form, (c ~~ a) ∧ (c ~~ b) :=
  axCoop

/-- Proposed distinct witness patterns. The source describes distinct operations,
    but this statement does not quantify operation functions; its proof is admitted. -/
theorem plusAndAdditionallyAreDistinct :
    ∃ a b : Form,
      (∃ c d : Form, (c ~~ a) ∧ (c ~~ b) ∧ (d ~~ b) ∧ (d ~~ a) ∧ ¬ (c =~ d)) ∧
      (∃ ca cab : Form, (ca ~~ a) ∧ (ca ~~ b) ∧ (cab ~~ b) ∧ (cab ~~ a) ∧ (ca =~ cab)) := by
  refine ⟨f2f ground, ground, ?_, ?_⟩
  · exact axSeqAsymm _ _ (by
      -- OPEN: ax-diff yields not Simulation, which does not entail not
      -- Congruent. The filtration implication cannot be used backwards.
      sorry)
  · exact axCoopComm _ _
  -- FORM: ax-seq-asymm gives a b where (a+b) ≢~ (b+a);
  --       ax-coop-comm gives the same pair with commutative substance.
  -- Full mechanization requires + notation; sorry marks pending steps.

/-- pathGround is the two-sided identity for compose. FORM. -/
theorem pathGroundIsIdentity :
    ∀ p : DerivationPath,
      congruentPath (compose p pathGround) p ∧
      congruentPath (compose pathGround p) p :=
  axComposeIdentity

/-- The concrete self-read trace. No conversion to opaque DerivationPath is
    assumed: that source-to-scaffold bridge remains a separate obligation. -/
def reflexionTrace (x : Form) : DEntry x x := selfRead x

/-- Self-read is a two-sided identity for finite D-entry composition, at the
    correct endpoints. This concerns dEntryCompose, not opaque compose. -/
theorem reflexionTraceComposeIdentity {x y : Form} (p : DEntry x y) :
    dEntryCompose (reflexionTrace x) p = p ∧
    dEntryCompose p (reflexionTrace y) = p := by
  exact ⟨Trace.nil_compose p, Trace.compose_nil p⟩

/-- The existing proposition-level reflexion API, now backed by selfRead.
    reflexionTrace exposes its actual witness and composition identity. -/
theorem reflexion : ∀ x : Form, D x x :=
  dIsReflexive

/-- orbit-reflexion-quotient: two Forms x and y are orbit-reflexion equivalent
    when their D-columns coincide (D[x][z] ↔ D[y][z] for all z).
    This is the single unified width measure. FORM. -/
theorem orbitReflexionQuotient :
    ∀ x y : Form,
      (∀ z : Form, D x z ↔ D y z) →
      (x ~~ y) := by
  sorry
  -- FORM: D-column coincidence → orbit-reflexion equivalence.
  -- The ~~ of x and y follows from their having the same reachability profile.
  -- ax-box finiteness (L0) bounds the quantification over z.
  -- Full proof requires structural induction on the D-column equivalence.

/-- At least one non-trivial DerivationPath exists. FORM. -/
theorem finitePathExists : ∃ p : DerivationPath, ¬ congruentPath p pathGround :=
  axComposeNonempty ground (f2f ground)

/-- path-length is additive over compose. FRAME/L3.
    Arithmetic of apply-chain lengths requires L3 ordinal grounding. -/
theorem compositionLengthAdditive_FRAME :
    ∀ p q : DerivationPath,
      pathLength (compose p q) ~~ pathLength p := by
  sorry
  -- FRAME/L3: path-length(p) ~~ f2f^n(ground) for n = |p|.
  -- Addition f2f^n(f2f^m(ground)) = f2f^{n+m}(ground) requires ordinal_succ from L3.
  -- The proposed L3 discharge remains pathLengthArithmeticClaim. A checked
  -- full-clause model shows that the current axioms do not imply it.

-- ============================================================================
-- §VII  Graduation — L2 → L3
-- Four source NCs. Their current Lean proof status is reported individually.
-- ============================================================================

-- NC-1: common similarity witnesses exist (axSeq); no operation is constructed.
theorem nc1_L2 : ∀ a b : Form, ∃ c : Form, (c ~~ a) ∧ (c ~~ b) := axSeq

-- NC-2: common co-presence witnesses exist (axCoop); no operation is constructed.
theorem nc2_L2 : ∀ a b : Form, ∃ c : Form, (c ~~ a) ∧ (c ~~ b) := axCoop

-- NC-3: compose is well-defined with pathGround identity (axComposeAssoc + axComposeIdentity). FORM.
theorem nc3_L2 :
    ∀ p : DerivationPath,
      congruentPath (compose p pathGround) p ∧
      congruentPath (compose pathGround p) p :=
  axComposeIdentity

-- NC-4: Non-trivial paths exist (finitePathExists). FORM.
theorem nc4_L2 : ∃ p : DerivationPath, ¬ congruentPath p pathGround :=
  finitePathExists

-- Graduation status: FORM. L3 licensed.

-- ============================================================================
-- §VIII  L2 Self-Kernel
-- P_2 = 31 steps.  N_2_atomic = 3.  G_2 = 4.
-- FRAME residuals:
--   compositionLengthAdditive_FRAME — FRAME/L3 (path-length arithmetic)
--   congruentPath internals eval   — FRAME/L3 (quantification for length > 1)
--   simulation-pair-exists         — FRAME/L3 (carried from L1, still open)
-- All 31 steps represented by the declarations above.
-- ============================================================================

end Hypermath