import Mathlib.Data.Fintype.Card
import Mathlib.Data.Fin.Basic

/-!
# Human Neural Compression Verification

This module records the hard finite-state obstruction behind any claimed
lossless compression of arbitrary neural snapshots: a smaller finite code space
cannot injectively represent a larger finite source space.

The result is intentionally modest. It does not rule out useful neural
compression; it rules out universal lossless compression unless the model
restricts the admissible state space, permits bounded loss, or uses an explicit
probabilistic guarantee.
-/

namespace Semantics.HumanNeuralCompressionVerification

/-! ## Published budget constants -/

/-- Nominal uncompressed human neural snapshot size in gigabytes: 1 PB. -/
def uncompressedStateGb : Nat := 1000000

/-- Conservative practical target size in gigabytes. -/
def targetMaxGb : Nat := 800

/-- Optimistic practical target size in gigabytes. -/
def targetMinGb : Nat := 300

/-- Minimum nominal ratio needed to map 1 PB into 800 GB. -/
def minimumCompressionRatio : Nat := uncompressedStateGb / targetMaxGb

/-- Idealized ratio needed to map 1 PB into 300 GB. -/
def idealCompressionRatio : Nat := uncompressedStateGb / targetMinGb

/-- Sparse active-neuron estimate used in the exploratory budget model. -/
def activeRatioPercent : Nat := 15

/-- Effective uncompressed budget after the active-neuron estimate. -/
def effectiveUncompressedGb : Nat :=
  (uncompressedStateGb * activeRatioPercent) / 100

/-- Sparsity-adjusted minimum ratio for an 800 GB target. -/
def effectiveMinimumRatio : Nat := effectiveUncompressedGb / targetMaxGb

theorem minimumCompressionRatio_eq : minimumCompressionRatio = 1250 := by
  native_decide

theorem idealCompressionRatio_eq : idealCompressionRatio = 3333 := by
  native_decide

theorem effectiveUncompressedGb_eq : effectiveUncompressedGb = 150000 := by
  native_decide

theorem effectiveMinimumRatio_eq : effectiveMinimumRatio = 187 := by
  native_decide

/-! ## Byte-budget facts -/

/-- 1 PB in decimal bytes. -/
def uncompressedBytes : Nat := 1000000000000000

/-- 800 GB in decimal bytes. -/
def compressedBytes : Nat := 800000000000

/-- The proposed compressed budget is strictly smaller than the source budget. -/
theorem byte_budget_strictly_smaller : compressedBytes < uncompressedBytes := by
  native_decide

/-! ## Finite lossless compression obstruction -/

/--
A witness for a lossless compression scheme from `sourceCodes` possible source
states into `compressedCodes` possible code words.

Injectivity is the key property: two source states may not share a code word if
decoding is required to be universally lossless.
-/
structure LosslessCompressionWitness (sourceCodes compressedCodes : Nat) where
  encode : Fin sourceCodes → Fin compressedCodes
  injective : Function.Injective encode

/-- Any injective finite encoding requires at least as many code words as inputs. -/
theorem lossless_witness_requires_capacity
    {sourceCodes compressedCodes : Nat}
    (w : LosslessCompressionWitness sourceCodes compressedCodes) :
    sourceCodes ≤ compressedCodes := by
  simpa using Fintype.card_le_of_injective w.encode w.injective

/--
Pigeonhole form: there is no injective encoding from a larger finite type into a
smaller finite type.
-/
theorem no_injective_compression_to_smaller_fintype
    {α β : Type} [Fintype α] [Fintype β]
    (h : Fintype.card β < Fintype.card α) :
    ¬ ∃ encode : α → β, Function.Injective encode := by
  rintro ⟨encode, hInjective⟩
  exact Nat.not_le_of_gt h (Fintype.card_le_of_injective encode hInjective)

/-- Concrete finite-code version of the same obstruction. -/
theorem no_lossless_universal_compression
    {sourceCodes compressedCodes : Nat}
    (h : compressedCodes < sourceCodes) :
    ¬ ∃ encode : Fin sourceCodes → Fin compressedCodes, Function.Injective encode := by
  exact no_injective_compression_to_smaller_fintype
    (α := Fin sourceCodes) (β := Fin compressedCodes) (by simpa using h)

/--
Witness form: a claimed universal lossless compressor is impossible whenever the
compressed code space has lower cardinality than the source space.
-/
theorem arbitrary_lossless_compression_impossible
    {sourceCodes compressedCodes : Nat}
    (h : compressedCodes < sourceCodes) :
    ¬ Nonempty (LosslessCompressionWitness sourceCodes compressedCodes) := by
  rintro ⟨w⟩
  exact Nat.not_le_of_gt h (lossless_witness_requires_capacity w)

/--
Research gate for the 1 PB → 800 GB target: because the byte budget is smaller,
any successful rigorous model must discharge one of the missing assumptions:
restricted admissible source states, bounded lossy reconstruction error, or an
explicit stochastic confidence theorem.
-/
theorem onePbTo800Gb_needs_extra_model_structure :
    compressedBytes < uncompressedBytes :=
  byte_budget_strictly_smaller

end Semantics.HumanNeuralCompressionVerification
