Lnal Instr Cost Instr Cost

The instruction cost table

In a machine-checked library of formal statements, the declaration instrCost defines a function from primitive instructions to integers. The function assigns a cost to each instruction, where the cost is a plain number in dimensionless units. The definition is a table: a FOLD with direction 0 costs 0, any other FOLD costs 1, a BRAID or MERGE with a delta token costs that delta, a SEED with a set token costs the given cost, and everything else costs 0. The table is a definitional choice, not a proven statement.

The name comes from the framework's idea of a ledger, a discrete record of events. In that picture, each primitive instruction is a recognition event, and the cost is the price the ledger pays to recognize it. The table sets those prices. It does not derive them from anything else; it simply chooses them. The declaration is a model of how costs attach to instructions, not a claim about what costs must be.

What the table does not claim is as important as what it does. It does not claim that these costs are the only possible costs, or that they are optimal, or that they match any measured physical quantity. It does not claim that the costs are derived from the forcing chain that produces constants like the golden ratio. The declaration is a local definition, used as a building block for later statements about instruction sequences. Its role is to give a concrete cost function, so that other statements can talk about total costs of programs.

In plain terms: the declaration says "here is how much each instruction costs," and nothing more. It is a starting point, not a conclusion. A reader who wants to know why these costs, or what they imply, must look at the statements that use this table, not at the table itself.

/-- J-cost rate for a primitive instruction (dimensionless units). -/
def instrCost : LInstr → Int
  | ⟨Opcode.FOLD, OpcodeArg.fold dir⟩ => if dir = 0 then 0 else 1
  | ⟨Opcode.FOLD, _⟩ => 1
  | ⟨Opcode.BRAID, OpcodeArg.token (TokenAction.delta d)⟩ => d
  | ⟨Opcode.MERGE, OpcodeArg.token (TokenAction.delta d)⟩ => d
  | ⟨Opcode.SEED, OpcodeArg.token (TokenAction.set _ cost)⟩ => cost
  | _ => 0

/-- J-cost rate for a primitive instruction (dimensionless units). -/
def instrCost : LInstr → Int
  | ⟨Opcode.FOLD, OpcodeArg.fold dir⟩ => if dir = 0 then 0 else 1
  | ⟨Opcode.FOLD, _⟩ => 1
  | ⟨Opcode.BRAID, OpcodeArg.token (TokenAction.delta d)⟩ => d
  | ⟨Opcode.MERGE, OpcodeArg.token (TokenAction.delta d)⟩ => d
  | ⟨Opcode.SEED, OpcodeArg.token (TokenAction.set _ cost)⟩ => cost
  | _ => 0

/-- J-cost rate for a primitive instruction (dimensionless units). -/
def instrCost : LInstr → Int
  | ⟨Opcode.FOLD, OpcodeArg.fold dir⟩ => if dir = 0 then 0 else 1
  | ⟨Opcode.FOLD, _⟩ => 1
  | ⟨Opcode.BRAID, OpcodeArg.token (TokenAction.delta d)⟩ => d
  | ⟨Opcode.MERGE, OpcodeArg.token (TokenAction.delta d)⟩ => d
  | ⟨Opcode.SEED, OpcodeArg.token (TokenAction.set _ cost)⟩ => cost
  | _ => 0

What this page does not claim

The costs are not derived from the forcing chain or any other statement. The costs are not claimed to be optimal or unique. The costs are not claimed to match any measured physical quantity.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/LNAL/InstrCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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