#[local] Infix " \in " := E.In.
#[local] Infix " \subseteq " := E.isSubsetOf.
#[local] Notation In := List.In.
#[local] Hint Resolve E.insert_monotonic : core.
Proof with (simpl in *; tauto) || eauto with *.
ii. eapply ENTAILS1; done!.
Qed.
Proof with eauto.
induction INFERS.
- eapply ByAssumption_preserves with (C := C)...
- eapply ContradictionI_preserves with (A := A)...
- eapply ContradictionE_preserves with (A := A)...
- eapply NegationI_preserves with (A := A)...
- eapply NegationE_preserves with (A := A)...
- eapply ConjunctionI_preserves with (A := A) (B := B)...
- eapply ConjunctionE1_preserves with (A := A) (B := B)...
- eapply ConjunctionE2_preserves with (A := A) (B := B)...
- eapply DisjunctionI1_preserves with (A := A) (B := B)...
- eapply DisjunctionI2_preserves with (A := A) (B := B)...
- eapply DisjunctionE_preserves with (A := A) (B := B) (C := C)...
- eapply ImplicationI_preserves with (A := A) (B := B)...
- eapply ImplicationE_preserves with (A := A) (B := B)...
- eapply BiconditionalI_preserves with (A := A) (B := B)...
- eapply BiconditionalE1_preserves with (A := A) (B := B)...
- eapply BiconditionalE2_preserves with (A := A) (B := B)...
Qed.
Proof with first [eassumption | trivial]. (* Infinitely grateful for Sohn's advice! *)
ii. set (X_dagger := MaximallyConsistentSet X).
pose proof (theorem_of_1_3_10 X) as [? ? ? ? ?].
fold X_dagger in SUBSET, EQUICONSISTENT, CLOSED_infers, META_DN, IMPLICATION_FAITHFUL.
pose proof (theorem_of_1_2_14 (Th X) (lemma1_of_1_3_8 X)) as [SUBSET' IS_FILTER' COMPLETE' EQUICONSISTENT'].
fold (MaximallyConsistentSet X) in SUBSET', IS_FILTER', COMPLETE', EQUICONSISTENT'.
fold X_dagger in SUBSET', IS_FILTER', COMPLETE', EQUICONSISTENT'.
pose proof (claim1 := Th_isSubsetOf_cl X).
pose proof (claim2 := cl_isSubsetOf_Th X).
assert (cl_eq_Th : cl X == Th X) by ss!.
assert (claim3 : equiconsistent (cl X) X_dagger).
{ transitivity (Th X); trivial. now rewrite cl_eq_Th. }
assert (claim4 : ~ inconsistent X_dagger).
{ intros INCONSISTENT. contradiction CONSISTENT. eapply inconsistent_cl_iff, claim3... }
assert (claim5 : ~ inconsistent (cl X_dagger)).
{ intros INCONSISTENT. contradiction claim4. pose proof (fact5_of_1_2_8 X_dagger IS_FILTER').
rewrite -> filters_is_inconsistent_iff... eapply extend_infers with (Gamma := cl X_dagger)...
rewrite <- filters_is_inconsistent_iff... eapply fact1_of_1_2_8...
}
assert (
forall i : propLetter,
AtomF i \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) (AtomF i)
) as caseAtomF.
{ ii. change (AtomF i \in X_dagger <-> i \in E.preimage AtomF X_dagger). s!.
rewrite CLOSED_infers. split.
- intros INFERS. exists (AtomF i). split... rewrite CLOSED_infers...
- intros [? [-> IN]]. rewrite <- CLOSED_infers...
}
assert (
ContradictionF \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) ContradictionF
) as caseContradictionF.
{ simpl. rewrite CLOSED_infers, <- inconsistent_cl_iff. tauto. }
assert (
forall p1 : formula,
forall IH1 : p1 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p1,
NegationF p1 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) (NegationF p1)
) as caseNegationF.
{ ii. simpl. rewrite <- IH1, CLOSED_infers. split.
- intros INFERS H_in. contradiction claim5.
eapply inconsistent_cl_iff. eapply ContradictionI with (A := p1)...
eapply CLOSED_infers...
- intros H_not_in.
eapply CLOSED_infers, META_DN. unnw. intros H_in.
eapply CLOSED_infers. eapply ContradictionI with (A := NegationF p1).
+ enough (claim6 : MaximallyConsistentSet X ⊢ ImplicationF p1 ContradictionF).
{ eapply NegationI. eapply ImplicationE with (A := p1).
- eapply extend_infers... ss!.
- eapply ByAssumption... ss!.
}
eapply CLOSED_infers, IMPLICATION_FAITHFUL. tauto.
+ eapply ByAssumption...
}
assert (
forall p1 : formula,
forall p2 : formula,
forall IH1 : p1 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p1,
forall IH2 : p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p2,
ConjunctionF p1 p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) (ConjunctionF p1 p2)
) as caseConjunctionF.
{ ii. simpl. rewrite <- IH1, <- IH2. split.
- intros H_in. split.
+ eapply CLOSED_infers, ConjunctionE1, CLOSED_infers...
+ eapply CLOSED_infers, ConjunctionE2, CLOSED_infers...
- intros [H_in1 H_in2].
eapply CLOSED_infers, ConjunctionI; eapply CLOSED_infers...
}
assert (
forall p1 : formula,
forall p2 : formula,
forall IH1 : p1 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p1,
forall IH2 : p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p2,
DisjunctionF p1 p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) (DisjunctionF p1 p2)
) as caseDisjunctionF.
{ ii. simpl. rewrite <- IH1, <- IH2. split.
- intros H_in. pose proof (classic (X_dagger ⊢ p1)) as [H_yes | H_no].
+ left. eapply CLOSED_infers...
+ right. eapply CLOSED_infers.
eapply ImplicationE with (A := NegationF p1).
{ eapply DisjunctionE with (A := p1) (B := p2) (C := ImplicationF (NegationF p1) p2).
- eapply CLOSED_infers...
- eapply ImplicationI, ContradictionE. eapply ContradictionI with (A := p1).
+ eapply ByAssumption. right; left...
+ eapply ByAssumption. left...
- eapply ImplicationI, ByAssumption. right; left...
}
{ eapply CLOSED_infers, caseNegationF...
simpl. rewrite <- IH1. intros H_false.
apply CLOSED_infers in H_false... ss!.
}
- intros [H_in | H_in].
+ eapply CLOSED_infers, DisjunctionI1, CLOSED_infers...
+ eapply CLOSED_infers, DisjunctionI2, CLOSED_infers...
}
assert (
forall p1 : formula,
forall p2 : formula,
forall IH1 : p1 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p1,
forall IH2 : p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p2,
ImplicationF p1 p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) (ImplicationF p1 p2)
) as caseImplicationF.
{ ii. rewrite IMPLICATION_FAITHFUL. simpl. unnw. tauto. }
assert (
forall p1 : formula,
forall p2 : formula,
forall IH1 : p1 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p1,
forall IH2 : p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) p2,
BiconditionalF p1 p2 \in X_dagger <-> evalFormula (E.preimage AtomF X_dagger) (BiconditionalF p1 p2)
) as caseBiconditionalF.
{ ii. simpl. transitivity (ImplicationF p1 p2 \in X_dagger /\ ImplicationF p2 p1 \in X_dagger).
{ split.
- intros H_in. split.
{ eapply CLOSED_infers, ImplicationI. eapply BiconditionalE1 with (A := p1) (B := p2).
- eapply extend_infers with (Gamma := X_dagger)...
eapply CLOSED_infers... ss!.
- eapply ByAssumption. left...
}
{ eapply CLOSED_infers, ImplicationI. eapply BiconditionalE2 with (A := p1) (B := p2).
- eapply extend_infers with (Gamma := X_dagger)...
eapply CLOSED_infers... ss!.
- eapply ByAssumption. left...
}
- intros [H_in1 H_in2].
eapply CLOSED_infers, BiconditionalI.
{ eapply ImplicationE with (A := p1).
- eapply extend_infers with (Gamma := X_dagger)...
eapply CLOSED_infers... ss!.
- eapply ByAssumption. left...
}
{ eapply ImplicationE with (A := p2).
- eapply extend_infers with (Gamma := X_dagger)...
eapply CLOSED_infers... ss!.
- eapply ByAssumption. left...
}
}
{ split.
- intros [H_in1 H_in2]. eapply caseImplicationF in H_in1, H_in2... ss!.
- intros [H_in1 H_in2]. eapply caseImplicationF in H_in1, H_in2... ss!.
}
}
split.
{ transitivity (Th X)... ii. econstructor. eapply ByAssumption... }
{ unfold is_structure. induction p; done!. }
Qed.
Proof with eauto with *.
eapply NNPP. intros it_is_false_that_Gamma_infers_C.
set (X := E.insert (NegationF C) Gamma).
assert (CONSISTENT : X ⊬ ContradictionF).
{ intros INCONSISTENT. contradiction it_is_false_that_Gamma_infers_C. eapply NegationE... }
pose proof (theorem_of_1_2_14 (Th X) (lemma1_of_1_3_8 X)) as [SUBSET' IS_FILTER' COMPLETE' EQUICONSISTENT'].
fold (MaximallyConsistentSet X) in SUBSET', IS_FILTER', COMPLETE', EQUICONSISTENT'.
pose proof (hasModel_ifConsistent X CONSISTENT) as [INCL IS_STRUCTURE].
unfold is_structure in IS_STRUCTURE.
pose proof (theorem_of_1_3_10 Gamma) as [? ? ? ? ?]; unnw.
contradiction it_is_false_that_Gamma_infers_C.
eapply completeness_theorem_prototype with (env := E.preimage AtomF (MaximallyConsistentSet X)); trivial.
- unfold equiconsistent in *.
transitivity (inconsistent (MaximallyConsistentSet X))...
split; intros [botB [botB_in botB_eq_falseB]].
+ exists botB. split... eapply IS_STRUCTURE...
+ exists botB. split... eapply IS_STRUCTURE...
- transitivity (MaximallyConsistentSet X)...
ii. eapply IS_STRUCTURE...
- eapply isFilter_compatWith_eqProp...
Qed.

Corollary the_propositional_compactness_theorem (Gamma : ensemble formula) (C : formula)
: Gamma ⊨ C <-> << FINITE_ENTAILS : exists xs, exists X, L.is_finsubset_of xs Gamma /\ L.is_listrep_of xs X /\ X ⊨ C >>.