Conjectures in Complexity Theory
Conjectures in Complexity Theory
Inferred from the community field; finer classification has not been reviewed.
Problem statement
P_ne_NP
P ≠ NP:
The conjecture that the complexity classes P and NP are not equal.
NP_ne_coNP
NP ≠ coNP:
The conjecture that the complexity classes NP and coNP are not equal.
Lean 4
Formal statement
Lean 4/-
Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
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See the License for the specific language governing permissions and
limitations under the License.
-/
import FormalConjecturesUtil
/-!
# Conjectures in Complexity Theory
This file contains formal statements of some of the main open conjectures
in complexity theory, including
- the P vs NP problem
- the NP vs coNP problem
*References:*
- [Wikipedia](https://en.wikipedia.org/wiki/P_versus_NP_problem)
- [The Clay Institute](https://www.claymath.org/millennium/p-vs-np/)
-/
namespace ComplexityTheory
/--
**P ≠ NP**:
The conjecture that the complexity classes P and NP are not equal.
-/
@[category research open, AMS 68]
theorem P_ne_NP : P ≠ NP := by sorry
/--
**NP ≠ coNP**:
The conjecture that the complexity classes NP and coNP are not equal.
-/
@[category research open, AMS 68]
theorem NP_ne_coNP : NP ≠ coNP := by sorry
/--
The theorem that the set of complements of languages in P is itself P.
This can be proven by observing that the boolean negation function is computable in polynomial time,
and that compositions of poly-time computable functions are also poly-time computable.
-/
@[category textbook, AMS 68]
theorem coP_eq_P :
{ L | Lᶜ ∈ P } = P := by
sorry
/--
The theorem that P is a subset of NP.
This can be proven by observing that for any language in P,
we can construct a verifier that ignores the witness and simply runs the poly-time decider for the
language.
-/
@[category textbook, AMS 68]
theorem P_subset_NP :
P ⊆ NP := by
sorry
/--
The theorem that P is a subset of coNP.
-/
@[category textbook, AMS 68]
theorem P_subset_coNP :
P ⊆ coNP := by
rw [coNP, ← coP_eq_P]
simp only [Set.ofPred_subset_ofPred]
intros L hL
exact P_subset_NP hL
end ComplexityTheory
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Proofs & verification
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 2 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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