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misc lemmas min/max/clamp real numbers#1957

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malarbol wants to merge 3 commits into
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malarbol:lemmas-min/max/clamp
Open

misc lemmas min/max/clamp real numbers#1957
malarbol wants to merge 3 commits into
UniMath:masterfrom
malarbol:lemmas-min/max/clamp

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This PR introduces a few lemmas on the min/max/clamp functions on real numbers:

  • for any x : ℝ, the functions max-ℝ x and min-ℝ x are increasing;
  • for any closed interval I, the clamping map clamp-I is increasing and strictly increasing on I;

We also prove the following location property for proper closed intervals: for any proper closed interval [a, b] and rational numbers p < q, then either:

  • p < a;
  • q < b;
  • there exists (s r : ℚ) such that a < s < r < b and p < s < r < q.

Comment thread src/real-numbers/proper-closed-intervals-real-numbers.lagda.md
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