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import Mathlib.Analysis.Normed.Field.Basic | ||
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variable {R : Type*} [SeminormedRing R] {a b c : R} | ||
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lemma norm_one_sub_mul (ha : ‖a‖ ≤ 1) : ‖c - a * b‖ ≤ ‖c - a‖ + ‖1 - b‖ := | ||
calc | ||
_ ≤ ‖c - a‖ + ‖a * (1 - b)‖ := by | ||
simpa [mul_one_sub] using norm_sub_le_norm_sub_add_norm_sub c a (a * b) | ||
_ ≤ ‖c - a‖ + ‖a‖ * ‖1 - b‖ := by gcongr; exact norm_mul_le .. | ||
_ ≤ ‖c - a‖ + 1 * ‖1 - b‖ := by gcongr | ||
_ = ‖c - a‖ + ‖1 - b‖ := by simp | ||
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lemma norm_one_sub_mpul' (hb : ‖b‖ ≤ 1) : ‖c - a * b‖ ≤ ‖1 - a‖ + ‖c - b‖ := by | ||
rw [add_comm]; exact norm_one_sub_mul (R := Rᵐᵒᵖ) hb | ||
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lemma nnnorm_one_sub_mul (ha : ‖a‖₊ ≤ 1) : ‖c - a * b‖₊ ≤ ‖c - a‖₊ + ‖1 - b‖₊ := norm_one_sub_mul ha | ||
lemma nnnorm_one_sub_mul' (hb : ‖b‖₊ ≤ 1) : ‖c - a * b‖₊ ≤ ‖1 - a‖₊ + ‖c - b‖₊ := | ||
norm_one_sub_mul' hb |
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import Mathlib.Algebra.Module.Basic | ||
import Mathlib.Data.ENNReal.Operations | ||
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open scoped NNReal | ||
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namespace ENNReal | ||
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lemma nnreal_smul_lt_top {x : ℝ≥0} {y : ℝ≥0∞} (hy : y < ⊤) : x • y < ⊤ := mul_lt_top (by simp) hy | ||
lemma nnreal_smul_ne_top {x : ℝ≥0} {y : ℝ≥0∞} (hy : y ≠ ⊤) : x • y ≠ ⊤ := mul_ne_top (by simp) hy | ||
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lemma nnreal_smul_ne_top_iff {x : ℝ≥0} {y : ℝ≥0∞} (hx : x ≠ 0) : x • y ≠ ⊤ ↔ y ≠ ⊤ := | ||
⟨by rintro h rfl; simp [smul_top, hx] at h, nnreal_smul_ne_top⟩ | ||
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lemma nnreal_smul_lt_top_iff {x : ℝ≥0} {y : ℝ≥0∞} (hx : x ≠ 0) : x • y < ⊤ ↔ y < ⊤ := by | ||
rw [lt_top_iff_ne_top, lt_top_iff_ne_top, nnreal_smul_ne_top_iff hx] | ||
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end ENNReal |
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LeanAPAP/Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
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import Mathlib.Order.ConditionallyCompleteLattice.Basic | ||
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variable {α β : Type*} [CompleteLattice β] {S : Set α} {f : α → β} | ||
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lemma iInf_lt_top : ⨅ i ∈ S, f i < ⊤ ↔ ∃ i ∈ S, f i < ⊤ := by simp [lt_top_iff_ne_top] |
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