Documentation

Foundation.Propositional.Classical.Basic.Calculus

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@[reducible, inline]
abbrev LO.Propositional.Classical.Sequent (α : Type u_2) :
Type u_2
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    inductive LO.Propositional.Classical.Derivation {α : Type u_1} (T : LO.Propositional.Classical.Theory α) :
    LO.Propositional.Classical.Sequent α → Type u_1
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      instance LO.Propositional.Classical.instOneSidedFormulaTheory {α : Type u_1} :
      LO.OneSided (LO.Propositional.Classical.Formula α) (LO.Propositional.Classical.Theory α)
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      • LO.Propositional.Classical.instOneSidedFormulaTheory = { Derivation := LO.Propositional.Classical.Derivation }
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      def LO.Propositional.Classical.Derivation.length {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {Δ : LO.Propositional.Classical.Sequent α} :
      T ⟹ Δ → ℕ
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        def LO.Propositional.Classical.Derivation.cast {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {Δ : LO.Propositional.Classical.Sequent α} {Γ : LO.Propositional.Classical.Sequent α} (d : T ⟹ Δ) (e : Δ = Γ) :
        T ⟹ Γ
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          @[simp]
          theorem LO.Propositional.Classical.Derivation.length_cast {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {Δ : LO.Propositional.Classical.Sequent α} {Γ : LO.Propositional.Classical.Sequent α} (d : T ⟹ Δ) (e : Δ = Γ) :
          LO.Propositional.Classical.Derivation.length (LO.Propositional.Classical.Derivation.cast d e) = LO.Propositional.Classical.Derivation.length d
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          def LO.Propositional.Classical.Derivation.verum' {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {Δ : LO.Propositional.Classical.Sequent α} (h : ⊤ ∈ Δ) :
          T ⟹ Δ
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            def LO.Propositional.Classical.Derivation.axL' {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {Δ : LO.Propositional.Classical.Sequent α} (a : α) (h : LO.Propositional.Classical.Formula.atom a ∈ Δ) (hn : LO.Propositional.Classical.Formula.natom a ∈ Δ) :
            T ⟹ Δ
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              def LO.Propositional.Classical.Derivation.em {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {p : LO.Propositional.Classical.Formula α} {Δ : LO.Propositional.Classical.Sequent α} (hpos : p ∈ Δ) (hneg : ∼p ∈ Δ) :
              T ⟹ Δ
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              • One or more equations did not get rendered due to their size.
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                instance LO.Propositional.Classical.Derivation.instTaitFormulaTheory {α : Type u_1} :
                LO.Tait (LO.Propositional.Classical.Formula α) (LO.Propositional.Classical.Theory α)
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                • One or more equations did not get rendered due to their size.
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                instance LO.Propositional.Classical.Derivation.instCutFormulaTheory {α : Type u_1} :
                LO.Tait.Cut (LO.Propositional.Classical.Formula α) (LO.Propositional.Classical.Theory α)
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                • One or more equations did not get rendered due to their size.
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                def LO.Propositional.Classical.Derivation.trans {α : Type u_1} {T : LO.Propositional.Classical.Theory α} {U : LO.Propositional.Classical.Theory α} (F : U ⊢* T) {Γ : LO.Propositional.Classical.Sequent α} :
                T ⟹ Γ → U ⟹ Γ
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                  instance LO.Propositional.Classical.Derivation.instAxiomatizedFormulaTheory {α : Type u_1} :
                  LO.Tait.Axiomatized (LO.Propositional.Classical.Formula α) (LO.Propositional.Classical.Theory α)
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                  • One or more equations did not get rendered due to their size.
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                  def LO.Propositional.Classical.Derivation.compact {α : Type u_1} {T : LO.Propositional.Classical.Theory α} [DecidableEq α] {Γ : LO.Propositional.Classical.Sequent α} :
                  T ⟹ Γ → (s : { s : Finset (LO.Propositional.Classical.Formula α) // ↑s ⊆ T }) × ↑↑s ⟹ Γ
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                    instance LO.Propositional.Classical.Derivation.instCompactFormulaTheory {α : Type u_1} [DecidableEq α] :
                    LO.System.Compact (LO.Propositional.Classical.Theory α)
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                    • One or more equations did not get rendered due to their size.
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                    def LO.Propositional.Classical.Derivation.deductionAux {α : Type u_1} {T : LO.Propositional.Classical.Theory α} [DecidableEq α] {Γ : LO.Propositional.Classical.Sequent α} {p : LO.Propositional.Classical.Formula α} :
                    T ⟹ Γ → T \ {p} ⟹ ∼p :: Γ
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                      def LO.Propositional.Classical.Derivation.deduction {α : Type u_1} {T : LO.Propositional.Classical.Theory α} [DecidableEq α] {Γ : LO.Propositional.Classical.Sequent α} {p : LO.Propositional.Classical.Formula α} (d : insert p T ⟹ Γ) :
                      T ⟹ ∼p :: Γ
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                        theorem LO.Propositional.Classical.Derivation.inconsistent_iff_provable {α : Type u_1} {T : LO.Propositional.Classical.Theory α} [DecidableEq α] {p : LO.Propositional.Classical.Formula α} :
                        LO.System.Inconsistent (insert p T) ↔ T ⊢! ∼p
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                        theorem LO.Propositional.Classical.Derivation.consistent_iff_unprovable {α : Type u_1} {T : LO.Propositional.Classical.Theory α} [DecidableEq α] {p : LO.Propositional.Classical.Formula α} :
                        LO.System.Consistent (insert p T) ↔ T ⊬ ∼p
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                        @[simp]
                        theorem LO.Propositional.Classical.Derivation.inconsistent_theory_iff {α : Type u_1} {T : LO.Propositional.Classical.Theory α} :
                        LO.System.Inconsistent (LO.System.theory T) ↔ LO.System.Inconsistent T
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                        @[simp]
                        theorem LO.Propositional.Classical.Derivation.consistent_theory_iff {α : Type u_1} {T : LO.Propositional.Classical.Theory α} :
                        LO.System.Consistent (LO.System.theory T) ↔ LO.System.Consistent T