PvsNP.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 6 contain sorry, 0 declared as axioms

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
the P vs BPP problem

References:

Imports (1)

FormalConjecturesUtil

theorem ComplexityTheory.P_ne_NP P is not equal to NP sorry line 45 source
Source documentation
P ≠ NP:
The conjecture that the complexity classes P and NP are not equal.
L4DEL explanation
The complexity classes P and NP differ; the file marks this as open.
1. The statement is the inequality of two language classes, P and NP, defined elsewhere in the repository's utility library.
2. The attribute reads category research open and the proof body is sorry.
See also: ComplexityTheory.NP_ne_coNP
Sources and attribution: Pinned Lean source (1646ca16afd6)
depends on sorry what this means
Statement uses: Ne, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.P, ComplexityTheory.NP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem P_ne_NP : P ≠ NP := by sorry
…
```
Mentions: ComplexityTheory.P, ComplexityTheory.NP
theorem ComplexityTheory.NP_ne_coNP NP ≠ coNP: The conjecture that the complexity classes NP and coNP are not equal sorry line 53 source
Source documentation
NP ≠ coNP:
The conjecture that the complexity classes NP and coNP are not equal.

depends on sorry what this means
Statement uses: Ne, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.NP, ComplexityTheory.coNP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem NP_ne_coNP : NP ≠ coNP := by sorry
…
```
Mentions: ComplexityTheory.NP, ComplexityTheory.coNP
theorem ComplexityTheory.P_eq_BPP P = BPP: The conjecture that the complexity classes P and BPP are equal sorry line 61 source
Source documentation
P = BPP:
The conjecture that the complexity classes P and BPP are equal.

depends on sorry what this means
Statement uses: Eq, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.P, ComplexityTheory.BPP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem P_eq_BPP : P = BPP := by sorry
…
```
Mentions: ComplexityTheory.P, ComplexityTheory.BPP
theorem ComplexityTheory.coP_eq_P The theorem that the set of complements of languages in P is itself P sorry line 70 source
Source documentation
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.

depends on sorry what this means
Statement uses: Eq, Set, ComplexityTheory.DecisionProblem, Set.ofPred, Membership.mem, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.P, Compl.compl, List, Bool
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem coP_eq_P :
    { L | Lᶜ ∈ P } = P := by
…
```
Mentions: ComplexityTheory.P
theorem ComplexityTheory.P_subset_NP sorry line 83 source
depends on sorry what this means
Statement uses: LE.le, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.DecisionProblem, ComplexityTheory.P, ComplexityTheory.NP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem ComplexityTheory.P_subset_coNP The theorem that P is a subset of coNP line 91 source
Source documentation
The theorem that P is a subset of coNP.

depends on sorry what this means
Statement uses: LE.le, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.DecisionProblem, ComplexityTheory.P, ComplexityTheory.coNP
Proof uses: Eq.mpr, Set.ofPred, Membership.mem, ComplexityTheory.NP, Compl.compl, List, Bool, id, Eq, congrArg, ComplexityTheory.coNP.eq_1, Set, Eq.symm, ComplexityTheory.coP_eq_P, ComplexityTheory.P_subset_NP
7 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem P_subset_coNP :
    P ⊆ coNP := by
…
```
Mentions: ComplexityTheory.P, ComplexityTheory.coNP
theorem ComplexityTheory.P_subset_BPP The theorem that P is a subset of BPP sorry line 106 source
Source documentation
The theorem that P is a subset of BPP.
This can be proven by observing that for any language in P, we can construct a randomized decider that ignores the random bits and simply runs the poly-time decider for the language.

depends on sorry what this means
Statement uses: LE.le, ComplexityTheory.DecisionComplexityClass, ComplexityTheory.DecisionProblem, ComplexityTheory.P, ComplexityTheory.BPP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem P_subset_BPP :
    P ⊆ BPP := by
…
```
Mentions: ComplexityTheory.P, ComplexityTheory.BPP