structure
definition
TruthEvaluableComparison
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IndisputableMonolith.Foundation.LogicAsFunctionalEquation.RealityStructure on GitHub at line 24.
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depends on
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nontrivial -
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ComparisonOperator -
NonTrivial -
ScaleInvariant -
FinitePairwisePolynomialClosure -
identity -
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self -
identity
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formal source
21/-- A truth-evaluable comparison is the minimal structure needed to evaluate
22statements about positive-ratio comparisons. The four fields are stated in
23semantic language; the lemmas below translate them into (L1)--(L4). -/
24structure TruthEvaluableComparison (C : ComparisonOperator) : Prop where
25 self_evaluable : ∀ x : ℝ, 0 < x → C x x = 0
26 reorder_single_valued : ∀ x y : ℝ, 0 < x → 0 < y → C x y = C y x
27 determinate_continuous :
28 ContinuousOn (Function.uncurry C) (Set.Ioi (0 : ℝ) ×ˢ Set.Ioi (0 : ℝ))
29 composite_determinate : FinitePairwisePolynomialClosure C
30 scale_free : ScaleInvariant C
31 nontrivial : NonTrivial C
32
33/-- Truth-evaluability of self-statements gives identity. -/
34theorem truth_eval_implies_identity
35 (C : ComparisonOperator)
36 (hT : TruthEvaluableComparison C) :
37 Identity C :=
38 hT.self_evaluable
39
40/-- Truth-evaluability of reordered pair-statements gives non-contradiction. -/
41theorem truth_eval_implies_non_contradiction
42 (C : ComparisonOperator)
43 (hT : TruthEvaluableComparison C) :
44 NonContradiction C :=
45 hT.reorder_single_valued
46
47/-- Truth-evaluability of every positive pair gives totality/continuity on the
48open positive quadrant. -/
49theorem truth_eval_implies_totality
50 (C : ComparisonOperator)
51 (hT : TruthEvaluableComparison C) :
52 ExcludedMiddle C :=
53 hT.determinate_continuous
54