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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import OAI.Computability.WitnessedChoice.ChoiceSeparation
namespace OAI
namespace WitnessedChoice
noncomputable section
open Classical WitnessedSeparation WitnessedSeparation.Grid BGS
theorem main : BGS.MainStatement := by
let φ := SourceProgram.program.sentence SourceProgram.polynomial
refine ⟨φ,SourceProgram.one_occurrence,
SourceProgram.boolean_all SourceProgram.polynomial SourceProgram.resource_basic,?_⟩
intro ψ hψ heq
obtain ⟨n,hn,hh⟩ := ψ.agrees_on_some_grid_pair hψ
let vstar : Vertex n := (0,0,0)
have hzero : φ.eval (gridInput (0 : Vertex n → WitnessedSeparation.Scalar)) = some true := by
change SourceProgram.program.formula.eval (atomInput (0 : Vertex n → WitnessedSeparation.Scalar))
SourceProgram.polynomial Fin.elim0 = _
rw [QuotedGrid.formula_value _ (gridResources hn) hn]
simp
have hone : φ.eval (gridInput (Pi.single vstar 1)) = some false := by
change SourceProgram.program.formula.eval (atomInput (Pi.single vstar 1))
SourceProgram.polynomial Fin.elim0 = _
rw [QuotedGrid.formula_value _ (gridResources hn) hn]
simp
have hmem : gridInput (0 : Vertex n → WitnessedSeparation.Scalar) ∈ ψ.TrueModels := by
rw [←heq]
exact hzero
have hmem' : gridInput (Pi.single vstar 1) ∈ φ.TrueModels := by
rw [heq]
exact (hh vstar).symm.trans hmem
change φ.eval (gridInput (Pi.single vstar 1)) = some true at hmem'
rw [hone] at hmem'
contradiction
end
end WitnessedChoice
end OAI
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