`phi_approx`

proved

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module: IndisputableMonolith.Constants
domain: Constants
line: 507 · github
papers citing: none yet

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IndisputableMonolith.Constants on GitHub at line 507.

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depends on

phi_lt_onePointSixTwo IndisputableMonolith.Constants

formal source

 504lemma phi_gt_one : 1 < phi := one_lt_phi
 505
 506/-- φ ≈ 1.618 (coarse upper bound used in some modules). -/
 507lemma phi_approx : phi < (1.62 : ℝ) := phi_lt_onePointSixTwo
 508
 509/-- J(φ) = φ - 3/2 (exact, using φ² = φ + 1). -/
 510lemma Jcost_phi_val : Cost.Jcost phi = phi - 3 / 2 := by
 511  rw [Cost.Jcost_eq_sq phi_ne_zero]
 512  have hphi_sq : phi ^ 2 = phi + 1 := phi_sq_eq
 513  rw [div_eq_iff (by linarith [phi_pos] : 2 * phi ≠ 0)]
 514  nlinarith [phi_pos, hphi_sq]
 515
 516/-- J(φ) > 0 -/
 517lemma Jcost_phi_pos : 0 < Cost.Jcost phi :=
 518  Cost.Jcost_pos_of_ne_one _ phi_pos phi_ne_one
 519
 520end Constants
 521end IndisputableMonolith

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