Documentation

Foundation.Propositional.Decidable

@[simp]
theorem LO.Propositional.NNFormula.ne_neg {α : Type u_1} (φ : NNFormula α) :
φ φ
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    theorem LO.Propositional.NNFormula.weight_atom {α : Type u_1} (a : α) :
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    theorem LO.Propositional.NNFormula.weight_and {α : Type u_1} (φ ψ : NNFormula α) :
    (φ ψ).weight = φ.weight + ψ.weight + 1
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    theorem LO.Propositional.NNFormula.weight_or {α : Type u_1} (φ ψ : NNFormula α) :
    (φ ψ).weight = φ.weight + ψ.weight + 1
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    theorem LO.Propositional.Sequent.weight_cons {α : Type u_1} (φ : NNFormula α) (Γ : Sequent α) :
    weight (φ :: Γ) = φ.weight + Γ.weight
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    theorem LO.Propositional.Sequent.weight_append {α : Type u_1} (Γ Δ : Sequent α) :
    (Γ ++ Δ).weight = Γ.weight + Δ.weight
    theorem LO.Propositional.Sequent.weight_le_weight_of_mem {α : Type u_1} {φ : NNFormula α} {Γ : Sequent α} (h : φ Γ) :
    theorem LO.Propositional.Sequent.weight_remove_le_of_mem {α : Type u_1} [DecidableEq α] {φ : NNFormula α} {Γ : Sequent α} (h : φ Γ) :
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        @[reducible, inline]
        abbrev LO.Propositional.Sequents (α : Type u_2) :
        Type u_2
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          theorem LO.Propositional.Sequents.weight_eq_succ_iff {α : Type u_1} {n : } {S : Sequents α} :
          S.weight = n + 1 ΓS, Γ.weight = n ΔS, Δ.weight n
          theorem LO.Propositional.Sequents.weight_lt_weight_of {α : Type u_1} {S₁ S₂ : Sequents α} {Γ : Sequent α} ( : Γ S₂) (h : ΔS₁, Δ.weight < Γ.weight) :
          S₁.weight < S₂.weight
          @[reducible, inline]
          abbrev LO.Propositional.Derivations {α : Type u_1} (T : Theory α) (S : Sequents α) :
          Type u_1
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            @[reducible, inline]
            abbrev LO.Propositional.Derivations! {α : Type u_1} (T : Theory α) (S : Sequents α) :
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              def LO.Propositional.Derivations.ofSubset {α : Type u_1} {T : Theory α} {S₁ S₂ : Sequents α} (ss : S₁ S₂) :
              Derivations T S₂Derivations T S₁
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                  @[simp]
                  theorem LO.Propositional.Derivations!.nil {α✝ : Type u_2} {T : Theory α✝} :
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                    theorem LO.Propositional.Sequent.IsClosed.cons_iff {α : Type u_1} {φ : NNFormula α} {Γ : Sequent α} :
                    IsClosed (φ :: Γ) φ Γ Γ.IsClosed
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                    def LO.Propositional.Sequent.IsClosed.choose {α : Type u_1} [DecidableEq α] (Γ : Sequent α) :
                    Γ.IsClosed{ φ : NNFormula α // φ Γ φ Γ }
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                      @[reducible, inline]
                      abbrev LO.Propositional.Sequent.IsOpen {α : Type u_1} (Γ : Sequent α) :
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                            theorem LO.Propositional.Sequent.reduction_and {α : Type u_1} {Γ : Sequent α} [DecidableEq α] {φ ψ : NNFormula α} {h : ¬Γ.IsAtomic} (H : Γ.chooseNonAtomic h = φ ψ) :
                            reduction h = [(List.remove (φ ψ) Γ).concat φ, (List.remove (φ ψ) Γ).concat ψ]
                            theorem LO.Propositional.Sequent.reduction_or {α : Type u_1} {Γ : Sequent α} [DecidableEq α] {φ ψ : NNFormula α} {h : ¬Γ.IsAtomic} (H : Γ.chooseNonAtomic h = φ ψ) :
                            reduction h = [((List.remove (φ ψ) Γ).concat φ).concat ψ]
                            def LO.Propositional.Derivation.ofReduction {α : Type u_1} [DecidableEq α] {T : Theory α} {Γ : Sequent α} { : ¬Γ.IsAtomic} (d : Derivations T (Sequent.reduction )) :
                            T Γ
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                              theorem LO.Propositional.Derivation.toReduction {α : Type u_1} [DecidableEq α] {T : Theory α} {Γ : Sequent α} ( : ¬Γ.IsAtomic) (d : T ⟹! Γ) :
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                                theorem LO.Propositional.Sequents.Refuted.not_iff {α : Type u_1} {S : Sequents α} :
                                ¬S.Refuted ΓS, Γ.IsAtomicΓ.IsClosed
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                                  theorem LO.Propositional.Sequent.reduction_sublist {α : Type u_1} [DecidableEq α] {Γ : Sequent α} ( : ¬Γ.IsAtomic) {S : Sequents α} (h : Γ S) :
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                                    inductive LO.Propositional.Derivations.Dec {α : Type u_1} :
                                    Sequents αType u_1
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