Project-declaredLean 4.32.1 · mathlib@520045ab

Quote subst fvar fixitr

LO.FirstOrder.Arithmetic.Bootstrapping.quote_subst_fvar_fixitr

formal logic metatheory proof theory induction scheme delta1

Plain-language statement

Closure inversion at the code level. Substituting the free-variable atoms &0 … &(m-1) back into the fixitr-image recovers ⌜φ⌝. This is the DECODE direction: the recognizer can recover ⌜succInd ψ⌝ (hence ψ) from the freevar-free closure body using the already-proven internal subst, with no need for an internal fixitr. Meta witness: `sub...

Exact Lean statement

lemma quote_subst_fvar_fixitr (φ : _root_.LO.FirstOrder.Semiformula L ℕ 0) :
    (⌜(Rew.fixitr 0 φ.fvSup ▹ φ : _root_.LO.FirstOrder.Semiformula L ℕ (0 + φ.fvSup))
        ⇜ (fun x : Fin (0 + φ.fvSup) ↦ (&↑x : SyntacticTerm L))⌝ : V) = ⌜φ⌝

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma quote_subst_fvar_fixitr (φ : _root_.LO.FirstOrder.Semiformula L ℕ 0) :    (⌜(Rew.fixitr 0 φ.fvSup ▹ φ : _root_.LO.FirstOrder.Semiformula L ℕ (0 + φ.fvSup))        ⇜ (fun x : Fin (0 + φ.fvSup) ↦ (&↑x : SyntacticTerm L))⌝ : V) = ⌜φ⌝ := by  rw [show (Rew.fixitr 0 φ.fvSup ▹ φ : _root_.LO.FirstOrder.Semiformula L ℕ (0 + φ.fvSup))        ⇜ (fun x : Fin (0 + φ.fvSup) ↦ (&↑x : SyntacticTerm L)) = φ from by    have := Semiformula.subst_comp_fixitr (L := L) φ    convert this using 2]

Project: Foundation
License: Apache-2.0
Commit: 8dcdb3196454
Source: Foundation/FirstOrder/Incompleteness/InductionSchemeDelta1.lean:172-178

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Source project: Foundation

Person-level attribution pending.

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Project-declaredLean 4.32.1

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Plain-language statement

bv-pin bridge (over ℕ): bv ⌜fixitr 0 (fvSup χ) ▹ χ⌝ = fvSup χ. - ≤ is immediate from quote_univCl_eq + bv_qqAlls (closing fvSup quantifiers reaches a sentence, whose bv is 0). - ≥ is by level-factoring: were the body an IsSemiformula j for some j < fvSup, IsSemiformula.sound + castLE-invariance would re-express χ as `γ ⇜ ![...

formal logicmetatheoryproof theory

Source project: Foundation

Person-level attribution pending.

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Project-declaredLean 4.32.1

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Plain-language statement

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formal logicmetatheoryproof theory

Source project: Foundation

Person-level attribution pending.

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