Project-declaredLean 4.32.1 · mathlib@520045ab

Fvar Vec val eq

LO.FirstOrder.Arithmetic.Bootstrapping.fvarVec_val_eq

formal logic metatheory proof theory induction scheme delta1

Plain-language statement

fvarVec is the code of the typed substitution vector fun i ↦ ^&i (over a standard length).

Exact Lean statement

lemma fvarVec_val_eq (m : ℕ) :
    fvarVec ((m : ℕ) : V)
      = SemitermVec.val (fun i : Fin m ↦ (Semiterm.fvar (↑(i : ℕ)) : Bootstrapping.Semiterm V ℒₒᵣ 0))

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma fvarVec_val_eq (m : ℕ) :    fvarVec ((m : ℕ) : V)      = SemitermVec.val (fun i : Fin m ↦ (Semiterm.fvar (↑(i : ℕ)) : Bootstrapping.Semiterm V ℒₒᵣ 0)) := by  apply nth_ext (by simp)  intro i hi  rw [len_fvarVec] at hi  obtain ⟨j, rfl⟩ := eq_nat_of_lt_nat hi  have hj : j < m := by exact_mod_cast hi  rw [nth_fvarVec _ _ hi, show ((j : ℕ) : V) = ((⟨j, hj⟩ : Fin m) : ℕ) from rfl]  rw [SemitermVec.val_nth_eq (fun i : Fin m ↦ (Semiterm.fvar (↑(i : ℕ)) : Bootstrapping.Semiterm V ℒₒᵣ 0)) ⟨j, hj⟩]  simp

Project: Foundation
License: Apache-2.0
Commit: 8dcdb3196454
Source: Foundation/FirstOrder/Incompleteness/InductionSchemeDelta1.lean:419-429

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