Succ Above of eq zero
TensorSpecies.Tensor.IsReindexing.succAbove_of_eq_zero
Plain-language statement
Given a reindexing of c by c1 via σ for which the index i is sent to 0, removing the i-th entry of c1 and the first entry of c yields a reindexing of c ∘ Fin.succ by c1 ∘ i.succAbove via the map sending j to the predecessor of σ (i.succAbove j).
Exact Lean statement
lemma succAbove_of_eq_zero {n n1 : ℕ} {c : Fin (n + 1) → C} {c1 : Fin (n1 + 1) → C}
{σ : Fin (n1 + 1) → Fin (n + 1)} (i : Fin (n1 + 1))
(h : IsReindexing c c1 σ) (hi : σ i = 0) :
IsReindexing (c ∘ Fin.succ) (c1 ∘ i.succAbove)
(fun j => (σ (i.succAbove j)).pred (by simp [← hi, h.injective.eq_iff]))Formal artifact
Lean source
lemma succAbove_of_eq_zero {n n1 : ℕ} {c : Fin (n + 1) → C} {c1 : Fin (n1 + 1) → C} {σ : Fin (n1 + 1) → Fin (n + 1)} (i : Fin (n1 + 1)) (h : IsReindexing c c1 σ) (hi : σ i = 0) : IsReindexing (c ∘ Fin.succ) (c1 ∘ i.succAbove) (fun j => (σ (i.succAbove j)).pred (by simp [← hi, h.injective.eq_iff])) := by refine ⟨⟨?_, ?_⟩, ?_⟩ · intro x1 x2 h1 simpa [h.injective.eq_iff] using h1 · intro k suffices ha : ∃ a, σ (i.succAbove a) = k.succ by obtain ⟨a, ha⟩ := ha use a simp [ha] obtain ⟨j, hj⟩ := h.surjective k.succ simp only [← hj, h.injective.eq_iff, Fin.exists_succAbove_eq_iff, ne_eq] grind · intro x simp [h.preserve_color]- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/Relativity/Tensors/Reindexing.lean:295-312
Reuse this declaration
Bring the exact result into your workflow
The import identifies the source module. Your project still needs the pinned package dependency shown on this page.
What this badge means
This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.
Continue in this project
Related declarations
Adiabatic relation log
adiabatic_relation_log
Plain-language statement
Adiabatic relation in logarithmic form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then c * log (Ua/Ub) + log (Va/Vb) = 0.
Source project: Physlib
Person-level attribution pending.
Adiabatic relation Ua Ub Va Vb
adiabatic_relation_UaUbVaVb
Plain-language statement
Adiabatic relation in product form: If S(Ua,Va,N) = S(Ub,Vb,N) with N fixed, then (Ua/Ub)^c * (Va/Vb) = 1.
Source project: Physlib
Person-level attribution pending.
Deriv Within mean Energy Beta eq neg variance
CanonicalEnsemble.derivWithin_meanEnergy_Beta_eq_neg_variance
Plain-language statement
(∂U/∂β) = -Var(E) for finite systems.
Source project: Physlib
Person-level attribution pending.