Project-declaredLean 4.32.1 · mathlib@520045ab

Small satisfiable of consistent

Theory.small_satisfiable_of_consistent

formal logic metatheory proof theory counter model

Project documentation

Completeness theorem (I)

Exact Lean statement

theorem small_satisfiable_of_consistent :
    Consistent T → Satisfiable T

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

theorem small_satisfiable_of_consistent :    Consistent T → Satisfiable T := fun consistent ↦ compact.mpr fun u hu ↦ by  let σ := ⋀u.toList  have : T ⊢ σ := Conj₂!_intro fun φ hφ ↦ by_axm <| hu (by simpa using hφ)  have : 𝐋𝐊¹ ⊬ ∼(σ : Proposition L) := fun h ↦    have : T ⊢ ∼σ := Theory.Proof.of_LK_provable (φ := ∼σ) (by simpa using h)    have : T ⊢ ⊥ := neg_mdp this (by assumption)    consistent_iff_unprovable_bot.mp consistent this  have : Satisfiable {σ} := LK.satisfiable_of_irrefutable σ this  simpa [σ] using this

Project: Foundation
License: Apache-2.0
Commit: 8dcdb3196454
Source: Foundation/FirstOrder/Completeness/CounterModel.lean:220-229

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