Plain-English statement
- The prefix integral is continuous in its upper time argument.
Read the mathematics first, then descend into Lean
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Proof architecture
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- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
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Syntax used on this page
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Lean statement
theorem continuous_prefixIntegral
{f : ℝ≥0 → ℝ} {T : ℝ≥0}
(hf : Integrable f (TimeMeasure.upTo T)) :
Continuous (prefixIntegral f T) := by
rw [continuous_iff_continuousAt]
intro t0
let g : ℝ≥0 → ℝ≥0 → ℝ := fun t s =>
if s < min t T then f s else 0
have hmeas : ∀ᶠ t in 𝓝 t0,
AEStronglyMeasurable (g t) (TimeMeasure.upTo T) := by
filter_upwards [] with t
have hind := hf.1.indicator
(measurableSet_Iio : MeasurableSet (Iio (min t T)))
simpa only [g, prefixIntegrand_eq_indicator] using hind
have hbound : ∀ᶠ t in 𝓝 t0,
∀ᵐ s ∂(TimeMeasure.upTo T), ‖g t s‖ ≤ ‖f s‖ := by
filter_upwards [] with t
filter_upwards [] with s
by_cases hs : s < min t T
· simp only [g, if_pos hs]
exact le_rfl
· simp only [g, if_neg hs, norm_zero]
exact norm_nonneg _
have hneq : ∀ᵐ s ∂(TimeMeasure.upTo T), s ≠ min t0 T := by
rw [ae_iff]
simpa using TimeMeasure.upTo_singleton T (min t0 T)
have hb : Tendsto (fun t : ℝ≥0 => min t T) (𝓝 t0) (𝓝 (min t0 T)) :=
continuousAt_id.min continuousAt_const
have hpoint : ∀ᵐ s ∂(TimeMeasure.upTo T),
Tendsto (fun t => g t s) (𝓝 t0) (𝓝 (g t0 s)) := by
filter_upwards [hneq] with s hs
by_cases hslt : s < min t0 T
· have hev : ∀ᶠ t in 𝓝 t0, s < min t T :=
(tendsto_order.1 hb).1 s hslt
have heq : (fun t => g t s) =ᶠ[𝓝 t0]
(fun _ : ℝ≥0 => f s) := by
filter_upwards [hev] with t ht
simp only [g, if_pos ht]
rw [show g t0 s = f s by simp only [g, if_pos hslt]]
exact (tendsto_congr' heq).2 tendsto_const_nhds
· have hle : min t0 T ≤ s := le_of_not_gt hslt
have hlt : min t0 T < s := lt_of_le_of_ne hle hs.symm
have hev : ∀ᶠ t in 𝓝 t0, min t T < s :=
(tendsto_order.1 hb).2 s hlt
have heq : (fun t => g t s) =ᶠ[𝓝 t0]
(fun _ : ℝ≥0 => 0) := by
filter_upwards [hev] with t ht
have hnot : ¬s < min t T := not_lt_of_ge ht.le
simp only [g, if_neg hnot]
rw [show g t0 s = 0 by simp only [g, if_neg hslt]]
exact (tendsto_congr' heq).2 tendsto_const_nhds
change Tendsto (fun t => ∫ s, g t s ∂(TimeMeasure.upTo T))
(𝓝 t0) (𝓝 (∫ s, g t0 s ∂(TimeMeasure.upTo T)))
exact tendsto_integral_filter_of_dominated_convergence
(fun s => ‖f s‖) hmeas hbound hf.norm hpoint
/-- Prefix integration is monotone in time for pointwise nonnegative
integrands. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/PrefixIntegral.lean:64published source at 7bcd37294df1
Proof architecture
supply the analytic continuity theorem for accumulated square energy
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `intro` introduces quantified hypotheses into the local proof context.
- `have` creates a named intermediate mathematical fact.
- `rw` rewrites by an established identity.
- `simp` normalizes through registered definitional and theorem rewrites.
- `exact` closes the current goal with an already typed term.
- `simpa` closes the goal after a controlled simplification of a typed result.
- `filter_upwards` moves an almost-everywhere or eventual statement into a pointwise local context.
Why the statement has this shape
The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.
Hidden assumptions and non-claims
- Measurability is represented explicitly or must be supplied by a dependency.
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Almost-everywhere hypotheses depend on the stated measure and representative.