ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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A dimension-two Halpern–Läuchli word rearrangement

Example

For $d = 2$ , the two endpoint words admit the following complete derivation:

$\forall a_{1} \forall a_{2} \exists x_{1} \exists x_{2} ⊢_{2} \exists A_{1} \exists A_{2} \forall x_{1} \forall x_{2} .$

Facts & Assumptions

Given: Dimension $d = 2$ and the endpoint words above.

[F1]

The preceding definition gives all legal Rule 1, Rule 2, and Rule 3 moves in $L_{2}$ . The finite word calculus for the Halpern–Läuchli argument

[F2]

The general endpoint rearrangement holds for every positive dimension. Finite word-calculus rearrangement

Verification

1.1F1given

Start with $W_{0} = \forall a_{1} \forall a_{2} \exists x_{1} \exists x_{2} \in L_{2}$ .

2.1F1step 1.1

Commute the universal symbols by Rule 1: $W_{0} ⊢_{2} W_{1} = \forall a_{2} \forall a_{1} \exists x_{1} \exists x_{2}$ .

3.1F1step 2.1

Apply Rule 2 to the adjacent coordinate-1 pair: $W_{1} ⊢_{2} W_{2} = \forall a_{2} \exists A_{1} \forall x_{1} \exists x_{2}$ .

4.1F1step 3.1

Apply Rule 3 with $r = 1$ and permutation $σ = (2, 1)$ : $W_{2} ⊢_{2} W_{3} = \exists A_{1} \forall a_{2} \forall x_{1} \exists x_{2}$ .

5.1F1step 4.1

Commute the adjacent universal symbols by Rule 1: $W_{3} ⊢_{2} W_{4} = \exists A_{1} \forall x_{1} \forall a_{2} \exists x_{2}$ .

6.1F1step 5.1

Apply the reverse direction of Rule 2 to coordinate 1: $W_{4} ⊢_{2} W_{5} = \forall a_{1} \exists x_{1} \forall a_{2} \exists x_{2}$ .

7.1F1step 6.1

Apply Rule 2 to coordinate 2: $W_{5} ⊢_{2} W_{6} = \forall a_{1} \exists x_{1} \exists A_{2} \forall x_{2}$ .

8.1F1step 7.1

Commute the adjacent existential symbols by Rule 1: $W_{6} ⊢_{2} W_{7} = \forall a_{1} \exists A_{2} \exists x_{1} \forall x_{2}$ .

9.1F1step 8.1

Apply Rule 3 with $r = 1$ and $σ = (1, 2)$ : $W_{7} ⊢_{2} W_{8} = \exists A_{2} \forall a_{1} \exists x_{1} \forall x_{2}$ .

10.1F1F2step 9.1∎

Apply Rule 2 to coordinate 1 and then commute the two existential symbols by Rule 1: $W_{8} ⊢_{2} \exists A_{2} \exists A_{1} \forall x_{1} \forall x_{2} ⊢_{2} \exists A_{1} \exists A_{2} \forall x_{1} \forall x_{2}$ . Every displayed word contains, for each coordinate, exactly one legal ordered pair, so all lie in $L_{2}$ ; this is the $d = 2$ instance of F2.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Halpern–Läuchli, A partition theorem (1966), Lemma 1 specialized to d=2, pp. 364–365 (standard reference, not scraped)