url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | case or Δ p q d ih =>
simpa[or_assoc] using ih | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
d : Derivation (p :: q :: Δ)
ih : v ⊧ (p :: q :: Δ).disj
⊢ v ⊧ (p ⋎ q :: Δ).disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | case wk Δ Γ _ ss ih =>
have : ∃ p ∈ Δ, v ⊧ p := by simpa [List.map_disj] using ih
rcases this with ⟨p, hp, hvp⟩
simp [List.map_disj]; exact ⟨p, ss hp, hvp⟩ | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
⊢ v ⊧ List.disj Γ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simp[List.map_disj] | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
a : α
⊢ v ⊧ (Formula.atom a :: Formula.natom a :: Δ).disj | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
a : α
⊢ v.val a ∨ ¬v.val a ∨ ∃ p ∈ Δ, v ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | by_cases v a <;> simp[*] | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
a : α
⊢ v.val a ∨ ¬v.val a ∨ ∃ p ∈ Δ, v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simp[List.map_disj] | α : Type u_1
Δ : Sequent α
v : Valuation α
Δ✝ : List (Formula α)
⊢ v ⊧ (⊤ :: Δ✝).disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | by_cases hv : v ⊧ Δ.disj | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
⊢ v ⊧ (p ⋏ q :: Δ).disj | case pos
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : v ⊧ Δ.disj
⊢ v ⊧ (p ⋏ q :: Δ).disj
case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simp[hv] | case pos
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : v ⊧ Δ.disj
⊢ v ⊧ (p ⋏ q :: Δ).disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | have : v ⊧ p := by simpa[hv] using ihp | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
⊢ v ⊧ (p ⋏ q :: Δ).disj | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
this : v ⊧ p
⊢ v ⊧ (p ⋏ q :: Δ).disj |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | have : v ⊧ q := by simpa[hv] using ihq | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
this : v ⊧ p
⊢ v ⊧ (p ⋏ q :: Δ).disj | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
this✝ : v ⊧ p
this : v ⊧ q
⊢ v ⊧ (p ⋏ q :: Δ).disj |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simp[*] | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
this✝ : v ⊧ p
this : v ⊧ q
⊢ v ⊧ (p ⋏ q :: Δ).disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simpa[hv] using ihp | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
⊢ v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simpa[hv] using ihq | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (q :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihq : v ⊧ (q :: Δ).disj
hv : ¬v ⊧ Δ.disj
this : v ⊧ p
⊢ v ⊧ q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simpa[or_assoc] using ih | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p q : Formula α
d : Derivation (p :: q :: Δ)
ih : v ⊧ (p :: q :: Δ).disj
⊢ v ⊧ (p ⋎ q :: Δ).disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | have : ∃ p ∈ Δ, v ⊧ p := by simpa [List.map_disj] using ih | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
⊢ v ⊧ List.disj Γ | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
this : ∃ p ∈ Δ, v ⊧ p
⊢ v ⊧ List.disj Γ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | rcases this with ⟨p, hp, hvp⟩ | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
this : ∃ p ∈ Δ, v ⊧ p
⊢ v ⊧ List.disj Γ | case intro.intro
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
p : Formula α
hp : p ∈ Δ
hvp : v ⊧ p
⊢ v ⊧ List.disj Γ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simp [List.map_disj] | case intro.intro
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
p : Formula α
hp : p ∈ Δ
hvp : v ⊧ p
⊢ v ⊧ List.disj Γ | case intro.intro
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
p : Formula α
hp : p ∈ Δ
hvp : v ⊧ p
⊢ ∃ p ∈ Γ, v ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | exact ⟨p, ss hp, hvp⟩ | case intro.intro
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
p : Formula α
hp : p ∈ Δ
hvp : v ⊧ p
⊢ ∃ p ∈ Γ, v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simpa [List.map_disj] using ih | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ Γ : Sequent α
a✝ : Derivation Δ
ss : Δ ⊆ Γ
ih : v ⊧ List.disj Δ
⊢ ∃ p ∈ Δ, v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | by_cases hv : v ⊧ Δ.disj | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
⊢ v ⊧ Δ.disj | case pos
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : v ⊧ Δ.disj
⊢ v ⊧ Δ.disj
case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivat... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simp[hv] | case pos
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : v ⊧ Δ.disj
⊢ v ⊧ Δ.disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | have : v ⊧ p := by simpa[hv] using ihp | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
⊢ v ⊧ Δ.disj | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
this : v ⊧ p
⊢ v ⊧ Δ.disj |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | have : ¬v ⊧ p := by simpa[hv] using ihn | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
this : v ⊧ p
⊢ v ⊧ Δ.disj | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
this✝ : v ⊧ p
this : ¬v ⊧ p
⊢ v ⊧ Δ.disj |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | contradiction | case neg
α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
this✝ : v ⊧ p
this : ¬v ⊧ p
⊢ v ⊧ Δ.disj | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simpa[hv] using ihp | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
⊢ v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.Derivation.sound | [13, 1] | [37, 20] | simpa[hv] using ihn | α : Type u_1
Δ✝ : Sequent α
v : Valuation α
Δ : List (Formula α)
p : Formula α
a✝¹ : Derivation (p :: Δ)
a✝ : Derivation (~p :: Δ)
ihp : v ⊧ (p :: Δ).disj
ihn : v ⊧ (~p :: Δ).disj
hv : ¬v ⊧ Δ.disj
this : v ⊧ p
⊢ ¬v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | rintro ⟨Γ, hΓ, d⟩ v hT | α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
⊢ T ⊢! p → T ⊨[Valuation α] p | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
⊢ v ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | by_contra hv | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
⊢ v ⊧ p | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | have : ∀ v : Valuation α, v ⊧ p ∨ ∃ q ∈ Γ, ¬(v ⊧ q) := by
simpa [Semantics.Valid, List.map_disj] using Derivation.sound (Tait.ofConsRight d) | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
⊢ False | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | have : ∃ q ∈ Γ, ¬v ⊧ q := by simpa [hv] using this v | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
⊢ False | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this✝ : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
this : ∃ q ∈ Γ, ¬v ⊧ q
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | rcases this with ⟨q, hqΓ, hq⟩ | case intro.mk
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this✝ : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
this : ∃ q ∈ Γ, ¬v ⊧ q
⊢ False | case intro.mk.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
q : Formula α
hqΓ : q ∈ Γ
hq : ¬v ⊧ q
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | have : v ⊧ q := hT.realize (hΓ q hqΓ) | case intro.mk.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
q : Formula α
hqΓ : q ∈ Γ
hq : ¬v ⊧ q
⊢ False | case intro.mk.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this✝ : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
q : Formula α
hqΓ : q ∈ Γ
hq : ¬v ⊧ q
this : v ⊧ q
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | contradiction | case intro.mk.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this✝ : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
q : Formula α
hqΓ : q ∈ Γ
hq : ¬v ⊧ q
this : v ⊧ q
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | simpa [Semantics.Valid, List.map_disj] using Derivation.sound (Tait.ofConsRight d) | α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
⊢ ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.soundness | [41, 1] | [49, 16] | simpa [hv] using this v | α : Type u_1
Δ : Sequent α
T : Theory α
p : Formula α
Γ : List (Formula α)
hΓ : ∀ p ∈ Γ, p ∈ T
d : Γ ⊢² [p]
v : Valuation α
hT : v ∈ Semantics.models (Valuation α) T
hv : ¬v ⊧ p
this : ∀ (v : Valuation α), v ⊧ p ∨ ∃ q ∈ Γ, ¬v ⊧ q
⊢ ∃ q ∈ Γ, ¬v ⊧ q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | simp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
⊢ ⋃₀ c ∈ {U | Consistent U} | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
⊢ Consistent (⋃₀ c) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | by_contra A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
⊢ Consistent (⋃₀ c) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | rcases System.inconsistent_compact.mp (System.not_consistent_iff_inconsistent.mp A) with ⟨𝓕, h𝓕, fin, 𝓕_consis⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
⊢ False | case intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | rcases Set.subset_mem_chain_of_finite c hnc chain (s := 𝓕) fin h𝓕 with ⟨U, hUc, hsU⟩ | case intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
⊢ False | case intro.intro.intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
U ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | have : Consistent U := hc hUc | case intro.intro.intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
U ... | case intro.intro.intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
U ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | have : ¬Consistent U := (𝓕_consis.of_supset hsU).not_con | case intro.intro.intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
U ... | case intro.intro.intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
U ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | contradiction | case intro.intro.intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
c : Set (Set (Formula α))
hc : c ⊆ {U | Consistent U}
chain : IsChain (fun x x_1 => x ⊆ x_1) c
hnc : c.Nonempty
A : ¬Consistent (⋃₀ c)
𝓕 : Theory α
h𝓕 : 𝓕 ⊆ ⋃₀ c
fin : Collection.Finite 𝓕
𝓕_consis : Inconsistent 𝓕
U ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | rcases this with ⟨Z, con, ss, hZ⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
this : ∃ Z, Consistent Z ∧ T ⊆ Z ∧ ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z
⊢ ∃ Z, Consistent Z ∧ T ⊆ Z ∧ ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z | case intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
Z : Theory α
con : Consistent Z
ss : T ⊆ Z
hZ : ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z
⊢ ∃ Z, Consistent Z ∧ T ⊆ Z ∧ ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | exact ⟨Z, con, ss, by intro U conU ssU; simpa using hZ U conU ssU⟩ | case intro.intro.intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
Z : Theory α
con : Consistent Z
ss : T ⊆ Z
hZ : ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z
⊢ ∃ Z, Consistent Z ∧ T ⊆ Z ∧ ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | intro U conU ssU | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
Z : Theory α
con : Consistent Z
ss : T ⊆ Z
hZ : ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z
⊢ ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
Z : Theory α
con : Consistent Z
ss : T ⊆ Z
hZ : ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z
U : Theory α
conU : Consistent U
ssU : Z ⊆ U
⊢ U = Z |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.exists_maximal_consistent_theory | [60, 1] | [74, 72] | simpa using hZ U conU ssU | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
Z : Theory α
con : Consistent Z
ss : T ⊆ Z
hZ : ∀ (U : Theory α), Consistent U → Z ⊆ U → U = Z
U : Theory α
conU : Consistent U
ssU : Z ⊆ U
⊢ U = Z | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.theory_maximalConsistentTheory_eq | [91, 1] | [94, 85] | simp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
⊢ Consistent (theory (maximalConsistentTheory consisT)) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.theory_maximalConsistentTheory_eq | [91, 1] | [94, 85] | simpa using System.Axiomatized.axm_subset (maximalConsistentTheory consisT) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
⊢ maximalConsistentTheory consisT ⊆ theory (maximalConsistentTheory consisT) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | by_contra A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | have hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT := by simpa [not_or] using A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | have : Consistent (insert p (maximalConsistentTheory consisT)) :=
consistent_insert_iff_not_refutable.mpr
(show ~p ∉ theory (maximalConsistentTheory consisT) from by simpa using hp.2) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this : Consistent (insert p (maximalConsistentTheory consisT))
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | have : insert p (maximalConsistentTheory consisT) ≠ maximalConsistentTheory consisT := by
simp [hp] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this : Consistent (insert p (maximalConsistentTheory consisT))
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this✝ : Consistent (insert p (maximalConsistentTheory consisT))
this : insert p (... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | have : insert p (maximalConsistentTheory consisT) = maximalConsistentTheory consisT :=
maximalConsistentTheory_maximal _ (by assumption) (by simp) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this✝ : Consistent (insert p (maximalConsistentTheory consisT))
this : insert p (... | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this✝¹ : Consistent (insert p (maximalConsistentTheory consisT))
this✝ : insert p... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | contradiction | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this✝¹ : Consistent (insert p (maximalConsistentTheory consisT))
this✝ : insert p... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | simpa [not_or] using A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
⊢ p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | simpa using hp.2 | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
⊢ ~p ∉ theory (maximalConsistentTheory consisT) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | simp [hp] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this : Consistent (insert p (maximalConsistentTheory consisT))
⊢ insert p (maxima... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | assumption | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this✝ : Consistent (insert p (maximalConsistentTheory consisT))
this : insert p (... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_or_neg_mem_maximalConsistentTheory | [96, 1] | [107, 16] | simp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
A : ¬(p ∈ maximalConsistentTheory consisT ∨ ~p ∈ maximalConsistentTheory consisT)
hp : p ∉ maximalConsistentTheory consisT ∧ ~p ∉ maximalConsistentTheory consisT
this✝ : Consistent (insert p (maximalConsistentTheory consisT))
this : insert p (... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_iff | [109, 1] | [111, 116] | have : p ∈ theory (maximalConsistentTheory consisT) := h | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : maximalConsistentTheory consisT ⊢! p
⊢ p ∈ maximalConsistentTheory consisT | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : maximalConsistentTheory consisT ⊢! p
this : p ∈ theory (maximalConsistentTheory consisT)
⊢ p ∈ maximalConsistentTheory consisT |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_iff | [109, 1] | [111, 116] | simpa using this | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : maximalConsistentTheory consisT ⊢! p
this : p ∈ theory (maximalConsistentTheory consisT)
⊢ p ∈ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_consistent' | [113, 1] | [120, 11] | intro h hn | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ p ∈ maximalConsistentTheory consisT → ~p ∉ maximalConsistentTheory consisT | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_consistent' | [113, 1] | [120, 11] | have : Inconsistent (maximalConsistentTheory consisT) :=
System.inconsistent_iff_provable_bot.mpr
(by prover [mem_maximalConsistentTheory_iff.mp h, mem_maximalConsistentTheory_iff.mp hn]) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_consistent' | [113, 1] | [120, 11] | have := this.not_con | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_consistent' | [113, 1] | [120, 11] | simp_all | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_consistent' | [113, 1] | [120, 11] | prover [mem_maximalConsistentTheory_iff.mp h, mem_maximalConsistentTheory_iff.mp hn] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
h : p ∈ maximalConsistentTheory consisT
hn : ~p ∈ maximalConsistentTheory consisT
⊢ maximalConsistentTheory consisT ⊢! ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | by_cases hp : p ∈ maximalConsistentTheory consisT <;> simp[hp] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ p ∉ maximalConsistentTheory consisT ↔ maximalConsistentTheory consisT ⊢! ~p | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ ¬maximalConsistentTheory consisT ⊢! ~p
case neg
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∉ maximalConsistentTheory consisT
⊢ maximalConsistentTheor... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | intro bnp | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ ¬maximalConsistentTheory consisT ⊢! ~p | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | have : Inconsistent (maximalConsistentTheory consisT) :=
System.inconsistent_of_provable (by prover [mem_maximalConsistentTheory_iff.mp hp, bnp]) | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
⊢ False | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | have := this.not_con | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | simp_all | case pos
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | prover [mem_maximalConsistentTheory_iff.mp hp, bnp] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
bnp : maximalConsistentTheory consisT ⊢! ~p
⊢ maximalConsistentTheory consisT ⊢! ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | exact mem_maximalConsistentTheory_iff.mp
(by simpa [hp] using mem_or_neg_mem_maximalConsistentTheory (consisT := consisT) p) | case neg
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∉ maximalConsistentTheory consisT
⊢ maximalConsistentTheory consisT ⊢! ~p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.not_mem_maximalConsistentTheory_iff | [122, 1] | [131, 90] | simpa [hp] using mem_or_neg_mem_maximalConsistentTheory (consisT := consisT) p | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∉ maximalConsistentTheory consisT
⊢ ~p ∈ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_and | [133, 1] | [137, 65] | have : maximalConsistentTheory consisT ⊢! p ⋏ q := mem_maximalConsistentTheory_iff.mp h | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋏ q ∈ maximalConsistentTheory consisT
⊢ p ∈ maximalConsistentTheory consisT ∧ q ∈ maximalConsistentTheory consisT | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋏ q ∈ maximalConsistentTheory consisT
this : maximalConsistentTheory consisT ⊢! p ⋏ q
⊢ p ∈ maximalConsistentTheory consisT ∧ q ∈ maximalConsistentTheory consisT |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_and | [133, 1] | [137, 65] | exact ⟨mem_maximalConsistentTheory_iff.mpr (by prover [this]),
mem_maximalConsistentTheory_iff.mpr (by prover [this])⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋏ q ∈ maximalConsistentTheory consisT
this : maximalConsistentTheory consisT ⊢! p ⋏ q
⊢ p ∈ maximalConsistentTheory consisT ∧ q ∈ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_and | [133, 1] | [137, 65] | prover [this] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋏ q ∈ maximalConsistentTheory consisT
this : maximalConsistentTheory consisT ⊢! p ⋏ q
⊢ maximalConsistentTheory consisT ⊢! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_and | [133, 1] | [137, 65] | prover [this] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋏ q ∈ maximalConsistentTheory consisT
this : maximalConsistentTheory consisT ⊢! p ⋏ q
⊢ maximalConsistentTheory consisT ⊢! q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | by_contra A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
⊢ p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | have b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q := by
simpa [not_or, not_mem_maximalConsistentTheory_iff] using A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | have : Inconsistent (maximalConsistentTheory consisT) :=
System.inconsistent_of_provable (by prover [b.1, b.2, mem_maximalConsistentTheory_iff.mp h]) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this : Inconsistent (maximalConsis... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | have := this.not_con | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this : Inconsistent (maximalConsis... | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this✝ : Inconsistent (maximalConsi... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | simp_all | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
this✝ : Inconsistent (maximalConsi... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | simpa [not_or, not_mem_maximalConsistentTheory_iff] using A | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
⊢ maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.mem_maximalConsistentTheory_or | [139, 1] | [147, 11] | prover [b.1, b.2, mem_maximalConsistentTheory_iff.mp h] | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
h : p ⋎ q ∈ maximalConsistentTheory consisT
A : ¬(p ∈ maximalConsistentTheory consisT ∨ q ∈ maximalConsistentTheory consisT)
b : maximalConsistentTheory consisT ⊢! ~p ∧ maximalConsistentTheory consisT ⊢! ~q
⊢ maximalConsistentTheory consisT ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | intro p hp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
⊢ ∀ ⦃f : Formula α⦄,
f ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ f | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | induction p using Formula.rec' <;> simp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p | case hfalsum
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
⊢ False
case hatom
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.atom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∈ maximalConsistentTheory consisT
case ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | case hatom => simpa | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.atom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∈ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | case hnatom =>
simpa using maximalConsistentTheory_consistent' hp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.natom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∉ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | case hfalsum =>
have : Inconsistent (maximalConsistentTheory consisT) := System.inconsistent_of_provable ⟨System.byAxm hp⟩
have := this.not_con
simp_all | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | case hand p q ihp ihq =>
exact ⟨ihp (mem_maximalConsistentTheory_and hp).1, ihq (mem_maximalConsistentTheory_and hp).2⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q
hp : p ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | simpa | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.atom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∈ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | simpa using maximalConsistentTheory_consistent' hp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.natom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∉ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | have : Inconsistent (maximalConsistentTheory consisT) := System.inconsistent_of_provable ⟨System.byAxm hp⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | have := this.not_con | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
this : Inconsistent (maximalConsistentTheory consisT)
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | simp_all | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
this✝ : Inconsistent (maximalConsistentTheory consisT)
this : ¬Consistent (maximalConsistentTheory consisT)
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | exact ⟨ihp (mem_maximalConsistentTheory_and hp).1, ihq (mem_maximalConsistentTheory_and hp).2⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q
hp : p ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | rcases mem_maximalConsistentTheory_or hp with (hp | hq) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q
hp : p ... | case inl
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | left | case inl
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ ... | case inl.h
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | exact ihp hp | case inl.h
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | right | case inr
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ ... | case inr.h
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ... |
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