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https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | case axiom_ h1_phi h1_1 =>
apply IsDeduct.axiom_
exact h1_1 | F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | case assume_ h1_phi h1_1 => exact h2 h1_phi h1_1 | F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi : Formula
h1_1 : h1_phi ∈ Γ
⊢ IsDeduct Δ h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | case mp_ h1_phi h1_psi _ _ h1_ih_1 h1_ih_2 =>
exact IsDeduct.mp_ h1_phi h1_psi h1_ih_1 h1_ih_2 | F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Γ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Γ h1_phi
h1_ih_1 : IsDeduct Δ (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct Δ h1_phi
⊢ IsDeduct Δ h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | apply IsDeduct.axiom_ | F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsDeduct Δ h1_phi | case a
F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | exact h1_1 | case a
F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi : Formula
h1_1 : IsAxiom h1_phi
⊢ IsAxiom h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | exact h2 h1_phi h1_1 | F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi : Formula
h1_1 : h1_phi ∈ Γ
⊢ IsDeduct Δ h1_phi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.T_14_16 | [537, 1] | [550, 53] | exact IsDeduct.mp_ h1_phi h1_psi h1_ih_1 h1_ih_2 | F : Formula
Δ Γ : Set Formula
h2 : ∀ H ∈ Γ, IsDeduct Δ H
h1_phi h1_psi : Formula
a✝¹ : IsDeduct Γ (h1_phi.imp_ h1_psi)
a✝ : IsDeduct Γ h1_phi
h1_ih_1 : IsDeduct Δ (h1_phi.imp_ h1_psi)
h1_ih_2 : IsDeduct Δ h1_phi
⊢ IsDeduct Δ h1_psi | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.C_14_17 | [553, 1] | [563, 28] | simp only [IsProof] at h2 | Q : Formula
Γ : Set Formula
h1 : IsDeduct Γ Q
h2 : ∀ P ∈ Γ, IsProof P
⊢ IsProof Q | Q : Formula
Γ : Set Formula
h1 : IsDeduct Γ Q
h2 : ∀ P ∈ Γ, IsDeduct ∅ P
⊢ IsProof Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.C_14_17 | [553, 1] | [563, 28] | simp only [IsProof] | Q : Formula
Γ : Set Formula
h1 : IsDeduct Γ Q
h2 : ∀ P ∈ Γ, IsDeduct ∅ P
⊢ IsProof Q | Q : Formula
Γ : Set Formula
h1 : IsDeduct Γ Q
h2 : ∀ P ∈ Γ, IsDeduct ∅ P
⊢ IsDeduct ∅ Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.C_14_17 | [553, 1] | [563, 28] | exact T_14_16 Q ∅ Γ h1 h2 | Q : Formula
Γ : Set Formula
h1 : IsDeduct Γ Q
h2 : ∀ P ∈ Γ, IsDeduct ∅ P
⊢ IsDeduct ∅ Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.eval_not | [566, 1] | [572, 32] | simp only [Formula.evalPrime] | P : Formula
V : VarBoolAssignment
⊢ evalPrime V P.not_ ↔ ¬evalPrime V P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.eval_imp | [575, 1] | [581, 32] | simp only [Formula.evalPrime] | P Q : Formula
V : VarBoolAssignment
⊢ evalPrime V (P.imp_ Q) ↔ evalPrime V P → evalPrime V Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.eval_false | [584, 1] | [589, 32] | simp only [Formula.evalPrime] | V : VarBoolAssignment
⊢ evalPrime V false_ ↔ False | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.eval_and | [592, 1] | [598, 32] | simp only [Formula.evalPrime] | P Q : Formula
V : VarBoolAssignment
⊢ evalPrime V (P.and_ Q) ↔ evalPrime V P ∧ evalPrime V Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.eval_or | [601, 1] | [607, 32] | simp only [Formula.evalPrime] | P Q : Formula
V : VarBoolAssignment
⊢ evalPrime V (P.or_ Q) ↔ evalPrime V P ∨ evalPrime V Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.eval_iff | [610, 1] | [616, 32] | simp only [Formula.evalPrime] | P Q : Formula
V : VarBoolAssignment
⊢ evalPrime V (P.iff_ Q) ↔ (evalPrime V P ↔ evalPrime V Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_true | [619, 1] | [624, 7] | simp only [Formula.IsTautoPrime] | ⊢ true_.IsTautoPrime | ⊢ ∀ (V : VarBoolAssignment), evalPrime V true_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_true | [619, 1] | [624, 7] | simp only [Formula.evalPrime] | ⊢ ∀ (V : VarBoolAssignment), evalPrime V true_ | ⊢ VarBoolAssignment → True |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_true | [619, 1] | [624, 7] | simp | ⊢ VarBoolAssignment → True | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_1 | [627, 1] | [632, 8] | simp only [Formula.IsTautoPrime] | P Q : Formula
⊢ (P.imp_ (Q.imp_ P)).IsTautoPrime | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V (P.imp_ (Q.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_1 | [627, 1] | [632, 8] | tauto | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V (P.imp_ (Q.imp_ P)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_2 | [635, 1] | [640, 8] | simp only [Formula.IsTautoPrime] | P Q R : Formula
⊢ ((P.imp_ (Q.imp_ R)).imp_ ((P.imp_ Q).imp_ (P.imp_ R))).IsTautoPrime | P Q R : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.imp_ (Q.imp_ R)).imp_ ((P.imp_ Q).imp_ (P.imp_ R))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_2 | [635, 1] | [640, 8] | tauto | P Q R : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.imp_ (Q.imp_ R)).imp_ ((P.imp_ Q).imp_ (P.imp_ R))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_3 | [643, 1] | [649, 8] | simp only [Formula.IsTautoPrime] | P Q : Formula
⊢ ((P.not_.imp_ Q.not_).imp_ (Q.imp_ P)).IsTautoPrime | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.not_.imp_ Q.not_).imp_ (Q.imp_ P)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_3 | [643, 1] | [649, 8] | simp only [eval_not, eval_imp] | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.not_.imp_ Q.not_).imp_ (Q.imp_ P)) | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), (¬evalPrime V P → ¬evalPrime V Q) → evalPrime V Q → evalPrime V P |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_prop_3 | [643, 1] | [649, 8] | tauto | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), (¬evalPrime V P → ¬evalPrime V Q) → evalPrime V Q → evalPrime V P | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_mp | [652, 1] | [663, 8] | simp only [Formula.IsTautoPrime] at h1 | P Q : Formula
h1 : (P.imp_ Q).IsTautoPrime
h2 : P.IsTautoPrime
⊢ Q.IsTautoPrime | P Q : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V (P.imp_ Q)
h2 : P.IsTautoPrime
⊢ Q.IsTautoPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_mp | [652, 1] | [663, 8] | simp only [eval_imp] at h1 | P Q : Formula
h1 : ∀ (V : VarBoolAssignment), evalPrime V (P.imp_ Q)
h2 : P.IsTautoPrime
⊢ Q.IsTautoPrime | P Q : Formula
h2 : P.IsTautoPrime
h1 : ∀ (V : VarBoolAssignment), evalPrime V P → evalPrime V Q
⊢ Q.IsTautoPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_mp | [652, 1] | [663, 8] | simp only [Formula.IsTautoPrime] at h2 | P Q : Formula
h2 : P.IsTautoPrime
h1 : ∀ (V : VarBoolAssignment), evalPrime V P → evalPrime V Q
⊢ Q.IsTautoPrime | P Q : Formula
h2 : ∀ (V : VarBoolAssignment), evalPrime V P
h1 : ∀ (V : VarBoolAssignment), evalPrime V P → evalPrime V Q
⊢ Q.IsTautoPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_mp | [652, 1] | [663, 8] | tauto | P Q : Formula
h2 : ∀ (V : VarBoolAssignment), evalPrime V P
h1 : ∀ (V : VarBoolAssignment), evalPrime V P → evalPrime V Q
⊢ Q.IsTautoPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_false | [666, 1] | [671, 8] | simp only [Formula.IsTautoPrime] | ⊢ (false_.iff_ true_.not_).IsTautoPrime | ⊢ ∀ (V : VarBoolAssignment), evalPrime V (false_.iff_ true_.not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_false | [666, 1] | [671, 8] | simp only [eval_not, eval_iff] | ⊢ ∀ (V : VarBoolAssignment), evalPrime V (false_.iff_ true_.not_) | ⊢ ∀ (V : VarBoolAssignment), evalPrime V false_ ↔ ¬evalPrime V true_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_false | [666, 1] | [671, 8] | tauto | ⊢ ∀ (V : VarBoolAssignment), evalPrime V false_ ↔ ¬evalPrime V true_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_and | [673, 1] | [679, 8] | simp only [Formula.IsTautoPrime] | P Q : Formula
⊢ ((P.and_ Q).iff_ (P.imp_ Q.not_).not_).IsTautoPrime | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.and_ Q).iff_ (P.imp_ Q.not_).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_and | [673, 1] | [679, 8] | simp only [eval_and, eval_not, eval_imp, eval_iff] | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.and_ Q).iff_ (P.imp_ Q.not_).not_) | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V P ∧ evalPrime V Q ↔ ¬(evalPrime V P → ¬evalPrime V Q) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_and | [673, 1] | [679, 8] | tauto | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V P ∧ evalPrime V Q ↔ ¬(evalPrime V P → ¬evalPrime V Q) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_or | [681, 1] | [687, 8] | simp only [Formula.IsTautoPrime] | P Q : Formula
⊢ ((P.or_ Q).iff_ (P.not_.imp_ Q)).IsTautoPrime | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.or_ Q).iff_ (P.not_.imp_ Q)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_or | [681, 1] | [687, 8] | simp only [eval_or, eval_not, eval_imp, eval_iff] | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V ((P.or_ Q).iff_ (P.not_.imp_ Q)) | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V P ∨ evalPrime V Q ↔ ¬evalPrime V P → evalPrime V Q |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_or | [681, 1] | [687, 8] | tauto | P Q : Formula
⊢ ∀ (V : VarBoolAssignment), evalPrime V P ∨ evalPrime V Q ↔ ¬evalPrime V P → evalPrime V Q | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_iff | [689, 1] | [695, 8] | simp only [Formula.IsTautoPrime] | P Q : Formula
⊢ (((P.iff_ Q).imp_ ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.iff_ Q)).not_).not_.IsTautoPrime | P Q : Formula
⊢ ∀ (V : VarBoolAssignment),
evalPrime V
(((P.iff_ Q).imp_ ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.iff_ Q)).not_).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_iff | [689, 1] | [695, 8] | simp only [eval_iff, eval_not, eval_imp] | P Q : Formula
⊢ ∀ (V : VarBoolAssignment),
evalPrime V
(((P.iff_ Q).imp_ ((P.imp_ Q).imp_ (Q.imp_ P).not_).not_).imp_
(((P.imp_ Q).imp_ (Q.imp_ P).not_).not_.imp_ (P.iff_ Q)).not_).not_ | P Q : Formula
⊢ ∀ (V : VarBoolAssignment),
¬(((evalPrime V P ↔ evalPrime V Q) → ¬((evalPrime V P → evalPrime V Q) → ¬(evalPrime V Q → evalPrime V P))) →
¬(¬((evalPrime V P → evalPrime V Q) → ¬(evalPrime V Q → evalPrime V P)) → (evalPrime V P ↔ evalPrime V Q))) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.is_tauto_def_iff | [689, 1] | [695, 8] | tauto | P Q : Formula
⊢ ∀ (V : VarBoolAssignment),
¬(((evalPrime V P ↔ evalPrime V Q) → ¬((evalPrime V P → evalPrime V Q) → ¬(evalPrime V Q → evalPrime V P))) →
¬(¬((evalPrime V P → evalPrime V Q) → ¬(evalPrime V Q → evalPrime V P)) → (evalPrime V P ↔ evalPrime V Q))) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | induction F | F F' : Formula
h1 : F' ∈ F.primeSet
⊢ F'.IsPrime | case pred_const_
F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' ∈ (pred_const_ a✝¹ a✝).primeSet
⊢ F'.IsPrime
case pred_var_
F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' ∈ (pred_var_ a✝¹ a✝).primeSet
⊢ F'.IsPrime
case eq_
F' : Formula
a✝¹ a✝ : VarName
h1 : F' ∈ (eq_ a✝¹ a✝).primeSet
⊢ F'.IsPrime
case true_
F' : Formula
h1 : F' ∈ true_.primeSet
⊢ F'.IsPrime
case false_
F' : Formula
h1 : F' ∈ false_.primeSet
⊢ F'.IsPrime
case not_
F' a✝ : Formula
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ a✝.not_.primeSet
⊢ F'.IsPrime
case imp_
F' a✝¹ a✝ : Formula
a_ih✝¹ : F' ∈ a✝¹.primeSet → F'.IsPrime
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ (a✝¹.imp_ a✝).primeSet
⊢ F'.IsPrime
case and_
F' a✝¹ a✝ : Formula
a_ih✝¹ : F' ∈ a✝¹.primeSet → F'.IsPrime
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ (a✝¹.and_ a✝).primeSet
⊢ F'.IsPrime
case or_
F' a✝¹ a✝ : Formula
a_ih✝¹ : F' ∈ a✝¹.primeSet → F'.IsPrime
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ (a✝¹.or_ a✝).primeSet
⊢ F'.IsPrime
case iff_
F' a✝¹ a✝ : Formula
a_ih✝¹ : F' ∈ a✝¹.primeSet → F'.IsPrime
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ (a✝¹.iff_ a✝).primeSet
⊢ F'.IsPrime
case forall_
F' : Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ (forall_ a✝¹ a✝).primeSet
⊢ F'.IsPrime
case exists_
F' : Formula
a✝¹ : VarName
a✝ : Formula
a_ih✝ : F' ∈ a✝.primeSet → F'.IsPrime
h1 : F' ∈ (exists_ a✝¹ a✝).primeSet
⊢ F'.IsPrime
case def_
F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' ∈ (def_ a✝¹ a✝).primeSet
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case pred_const_ | pred_var_ =>
simp only [Formula.primeSet] at h1
simp at h1
subst h1
simp only [Formula.IsPrime] | F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' ∈ (pred_var_ a✝¹ a✝).primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case true_ | false_ =>
simp only [Formula.primeSet] at h1
simp at h1 | F' : Formula
h1 : F' ∈ false_.primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case eq_ x y =>
simp only [Formula.primeSet] at h1
simp at h1
subst h1
simp only [Formula.IsPrime] | F' : Formula
x y : VarName
h1 : F' ∈ (eq_ x y).primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case not_ phi phi_ih =>
simp only [Formula.primeSet] at h1
exact phi_ih h1 | F' phi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
h1 : F' ∈ phi.not_.primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case
imp_ phi psi phi_ih psi_ih
| and_ phi psi phi_ih psi_ih
| or_ phi psi phi_ih psi_ih
| iff_ phi psi phi_ih psi_ih =>
simp only [Formula.primeSet] at h1
simp at h1
tauto | F' phi psi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
psi_ih : F' ∈ psi.primeSet → F'.IsPrime
h1 : F' ∈ (phi.iff_ psi).primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case forall_ x phi | exists_ x phi =>
simp only [Formula.primeSet] at h1
simp at h1
subst h1
simp only [Formula.IsPrime] | F' : Formula
a✝ : VarName
x : Formula
phi : F' ∈ x.primeSet → F'.IsPrime
h1 : F' ∈ (exists_ a✝ x).primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | case def_ =>
simp only [Formula.primeSet] at h1
simp at h1
subst h1
simp only [Formula.IsPrime] | F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' ∈ (def_ a✝¹ a✝).primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' ∈ (pred_var_ a✝¹ a✝).primeSet
⊢ F'.IsPrime | F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' ∈ {pred_var_ a✝¹ a✝}
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp at h1 | F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' ∈ {pred_var_ a✝¹ a✝}
⊢ F'.IsPrime | F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' = pred_var_ a✝¹ a✝
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | subst h1 | F' : Formula
a✝¹ : PredName
a✝ : List VarName
h1 : F' = pred_var_ a✝¹ a✝
⊢ F'.IsPrime | a✝¹ : PredName
a✝ : List VarName
⊢ (pred_var_ a✝¹ a✝).IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.IsPrime] | a✝¹ : PredName
a✝ : List VarName
⊢ (pred_var_ a✝¹ a✝).IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' : Formula
h1 : F' ∈ false_.primeSet
⊢ F'.IsPrime | F' : Formula
h1 : F' ∈ ∅
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp at h1 | F' : Formula
h1 : F' ∈ ∅
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' : Formula
x y : VarName
h1 : F' ∈ (eq_ x y).primeSet
⊢ F'.IsPrime | F' : Formula
x y : VarName
h1 : F' ∈ {eq_ x y}
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp at h1 | F' : Formula
x y : VarName
h1 : F' ∈ {eq_ x y}
⊢ F'.IsPrime | F' : Formula
x y : VarName
h1 : F' = eq_ x y
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | subst h1 | F' : Formula
x y : VarName
h1 : F' = eq_ x y
⊢ F'.IsPrime | x y : VarName
⊢ (eq_ x y).IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.IsPrime] | x y : VarName
⊢ (eq_ x y).IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' phi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
h1 : F' ∈ phi.not_.primeSet
⊢ F'.IsPrime | F' phi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
h1 : F' ∈ phi.primeSet
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | exact phi_ih h1 | F' phi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
h1 : F' ∈ phi.primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' phi psi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
psi_ih : F' ∈ psi.primeSet → F'.IsPrime
h1 : F' ∈ (phi.iff_ psi).primeSet
⊢ F'.IsPrime | F' phi psi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
psi_ih : F' ∈ psi.primeSet → F'.IsPrime
h1 : F' ∈ phi.primeSet ∪ psi.primeSet
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp at h1 | F' phi psi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
psi_ih : F' ∈ psi.primeSet → F'.IsPrime
h1 : F' ∈ phi.primeSet ∪ psi.primeSet
⊢ F'.IsPrime | F' phi psi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
psi_ih : F' ∈ psi.primeSet → F'.IsPrime
h1 : F' ∈ phi.primeSet ∨ F' ∈ psi.primeSet
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | tauto | F' phi psi : Formula
phi_ih : F' ∈ phi.primeSet → F'.IsPrime
psi_ih : F' ∈ psi.primeSet → F'.IsPrime
h1 : F' ∈ phi.primeSet ∨ F' ∈ psi.primeSet
⊢ F'.IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' : Formula
a✝ : VarName
x : Formula
phi : F' ∈ x.primeSet → F'.IsPrime
h1 : F' ∈ (exists_ a✝ x).primeSet
⊢ F'.IsPrime | F' : Formula
a✝ : VarName
x : Formula
phi : F' ∈ x.primeSet → F'.IsPrime
h1 : F' ∈ {exists_ a✝ x}
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp at h1 | F' : Formula
a✝ : VarName
x : Formula
phi : F' ∈ x.primeSet → F'.IsPrime
h1 : F' ∈ {exists_ a✝ x}
⊢ F'.IsPrime | F' : Formula
a✝ : VarName
x : Formula
phi : F' ∈ x.primeSet → F'.IsPrime
h1 : F' = exists_ a✝ x
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | subst h1 | F' : Formula
a✝ : VarName
x : Formula
phi : F' ∈ x.primeSet → F'.IsPrime
h1 : F' = exists_ a✝ x
⊢ F'.IsPrime | a✝ : VarName
x : Formula
phi : exists_ a✝ x ∈ x.primeSet → (exists_ a✝ x).IsPrime
⊢ (exists_ a✝ x).IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.IsPrime] | a✝ : VarName
x : Formula
phi : exists_ a✝ x ∈ x.primeSet → (exists_ a✝ x).IsPrime
⊢ (exists_ a✝ x).IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.primeSet] at h1 | F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' ∈ (def_ a✝¹ a✝).primeSet
⊢ F'.IsPrime | F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' ∈ {def_ a✝¹ a✝}
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp at h1 | F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' ∈ {def_ a✝¹ a✝}
⊢ F'.IsPrime | F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' = def_ a✝¹ a✝
⊢ F'.IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | subst h1 | F' : Formula
a✝¹ : DefName
a✝ : List VarName
h1 : F' = def_ a✝¹ a✝
⊢ F'.IsPrime | a✝¹ : DefName
a✝ : List VarName
⊢ (def_ a✝¹ a✝).IsPrime |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.mem_primeSet_isPrime | [731, 1] | [770, 32] | simp only [Formula.IsPrime] | a✝¹ : DefName
a✝ : List VarName
⊢ (def_ a✝¹ a✝).IsPrime | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | subst h2 | F F' : Formula
Δ_U : Set Formula
V : VarBoolAssignment
Δ_U' : Set Formula
h1 : ↑F.primeSet ⊆ Δ_U
h2 : Δ_U' = evalPrimeFfToNot V '' Δ_U
h3 : F' = evalPrimeFfToNot V F
⊢ IsDeduct Δ_U' F' | F F' : Formula
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑F.primeSet ⊆ Δ_U
h3 : F' = evalPrimeFfToNot V F
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) F' |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | subst h3 | F F' : Formula
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑F.primeSet ⊆ Δ_U
h3 : F' = evalPrimeFfToNot V F
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) F' | F : Formula
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑F.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V F) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | induction F | F : Formula
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑F.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V F) | case pred_const_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : ↑(pred_const_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ a✝¹ a✝))
case pred_var_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : PredName
a✝ : List VarName
h1 : ↑(pred_var_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ a✝¹ a✝))
case eq_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ a✝ : VarName
h1 : ↑(eq_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ a✝¹ a✝))
case true_
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V true_)
case false_
Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑false_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_)
case not_
Δ_U : Set Formula
V : VarBoolAssignment
a✝ : Formula
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑a✝.not_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝.not_)
case imp_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : ↑a✝¹.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝¹)
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(a✝¹.imp_ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (a✝¹.imp_ a✝))
case and_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : ↑a✝¹.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝¹)
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(a✝¹.and_ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (a✝¹.and_ a✝))
case or_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : ↑a✝¹.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝¹)
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(a✝¹.or_ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (a✝¹.or_ a✝))
case iff_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ a✝ : Formula
a_ih✝¹ : ↑a✝¹.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝¹)
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(a✝¹.iff_ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (a✝¹.iff_ a✝))
case forall_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(forall_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ a✝¹ a✝))
case exists_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝))
case def_
Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : DefName
a✝ : List VarName
h1 : ↑(def_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ a✝¹ a✝)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case pred_const_ X xs =>
let F := pred_const_ X xs
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case pred_var_ X xs =>
let F := pred_var_ X xs
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case eq_ x y =>
let F := eq_ x y
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto | Δ_U : Set Formula
V : VarBoolAssignment
x y : VarName
h1 : ↑(eq_ x y).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (eq_ x y)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case true_ =>
apply IsDeduct.axiom_
apply IsAxiom.prop_true_ | Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑true_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V true_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case false_ =>
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
simp
sorry | Δ_U : Set Formula
V : VarBoolAssignment
h1 : ↑false_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V false_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case not_ phi phi_ih =>
simp only [Formula.primeSet] at h1
simp only [evalPrimeFfToNot] at phi_ih
simp only [evalPrimeFfToNot]
simp only [evalPrime]
simp
split_ifs
case _ c1 =>
simp only [c1] at phi_ih
simp at phi_ih
apply IsDeduct.mp_ phi
apply proof_imp_deduct
apply T_14_6
exact phi_ih h1
case _ c1 =>
simp only [c1] at phi_ih
simp at phi_ih
exact phi_ih h1 | Δ_U : Set Formula
V : VarBoolAssignment
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑phi.not_.primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi.not_) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case forall_ x phi phi_ih =>
let F := forall_ x phi
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto | Δ_U : Set Formula
V : VarBoolAssignment
x : VarName
phi : Formula
phi_ih : ↑phi.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V phi)
h1 : ↑(forall_ x phi).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (forall_ x phi)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case def_ X xs =>
let F := def_ X xs
simp only [Formula.primeSet] at h1
simp at h1
simp only [evalPrimeFfToNot]
simp only [Formula.evalPrime]
apply IsDeduct.assume_
simp
apply Exists.intro F
tauto | Δ_U : Set Formula
V : VarBoolAssignment
X : DefName
xs : List VarName
h1 : ↑(def_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (def_ X xs)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | case and_ | or_ | iff_ | exists_ =>
sorry | Δ_U : Set Formula
V : VarBoolAssignment
a✝¹ : VarName
a✝ : Formula
a_ih✝ : ↑a✝.primeSet ⊆ Δ_U → IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V a✝)
h1 : ↑(exists_ a✝¹ a✝).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (exists_ a✝¹ a✝)) | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | let F := pred_const_ X xs | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp only [Formula.primeSet] at h1 | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_const_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_const_ X xs} ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp at h1 | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_const_ X xs} ⊆ Δ_U
F : Formula := pred_const_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp only [evalPrimeFfToNot] | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_const_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_const_ X xs) then pred_const_ X xs else (pred_const_ X xs).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp only [Formula.evalPrime] | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_const_ X xs) then pred_const_ X xs else (pred_const_ X xs).not_) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | apply IsDeduct.assume_ | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ (if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ (if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | apply Exists.intro F | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ ∃ x ∈ Δ_U,
(if evalPrime V x then x else x.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_ | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_ |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | tauto | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_const_ X xs
h1 : pred_const_ X xs ∈ Δ_U
⊢ F ∈ Δ_U ∧
(if evalPrime V F then F else F.not_) =
if V (pred_const_ X xs) = true then pred_const_ X xs else (pred_const_ X xs).not_ | no goals |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | let F := pred_var_ X xs | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp only [Formula.primeSet] at h1 | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑(pred_var_ X xs).primeSet ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_var_ X xs} ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp at h1 | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
h1 : ↑{pred_var_ X xs} ⊆ Δ_U
F : Formula := pred_var_ X xs
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp only [evalPrimeFfToNot] | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct (evalPrimeFfToNot V '' Δ_U) (evalPrimeFfToNot V (pred_var_ X xs)) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_var_ X xs) then pred_var_ X xs else (pred_var_ X xs).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | simp only [Formula.evalPrime] | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if evalPrime V (pred_var_ X xs) then pred_var_ X xs else (pred_var_ X xs).not_) | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_) |
https://github.com/pthomas505/FOL.git | 097a4abea51b641d144539b9a0f7516f3b9d818c | FOL/NV/Margaris/Prop.lean | FOL.NV.L_15_7 | [773, 1] | [916, 10] | apply IsDeduct.assume_ | Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ IsDeduct ((fun a => if evalPrime V a then a else a.not_) '' Δ_U)
(if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_) | case a
Δ_U : Set Formula
V : VarBoolAssignment
X : PredName
xs : List VarName
F : Formula := pred_var_ X xs
h1 : pred_var_ X xs ∈ Δ_U
⊢ (if V (pred_var_ X xs) = true then pred_var_ X xs else (pred_var_ X xs).not_) ∈
(fun a => if evalPrime V a then a else a.not_) '' Δ_U |
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