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/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | have : ¬(rel r v ∈ ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂)] ∧ nrel r v ∈ ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂)]) := by
rw[not_and_or]
have : ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂) + 1] ≺[Code.axL r v] ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂)] := chain_spec' nwf _ _
generalize ⛓️[(encode $ Code.a... | case intro.intro.intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r... | case intro.intro.intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | contradiction | case intro.intro.intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | have h : rel r v ∈ ⛓️ ∧ nrel r v ∈ ⛓️ := by simpa[not_or] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h✝ : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h✝ : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | simpa[not_or] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | exact ⟨chain_monotone nwf (le_trans (by simp) (Nat.right_le_pair _ _)) hs₁,
chain_monotone nwf (le_trans (by simp) (Nat.right_le_pair _ _)) hs₂⟩ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | simp | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | simp | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | rw[not_and_or] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | have : ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂) + 1] ≺[Code.axL r v] ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂)] := chain_spec' nwf _ _ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | generalize ⛓️[(encode $ Code.axL r v).pair (max s₁ s₂) + 1] = Δ' at this | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | rcases this | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ nrel r v ∉ ⛓️... | case axLRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | assumption | case axLRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
k : ℕ
r : L.Rel k
v : Fin k → SyntacticTerm L
h : ¬(rel r v ∉ ⛓️ ∨ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | have : ∃ s, p ⋏ q ∈ ⛓️[s] := by simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
⊢ p ∈ ⛓️ ∨ q ∈ ⛓️ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
this : ∃ s, p ⋏ q ∈ ⛓️[s]
⊢ p ∈ ⛓️ ∨ q ∈... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | rcases this with ⟨s, hs⟩ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
this : ∃ s, p ⋏ q ∈ ⛓️[s]
⊢ p ∈ ⛓️ ∨ q ∈... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
⊢ p ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | have : ⛓️[(encode $ Code.and p q).pair s + 1] ≺[Code.and p q] ⛓️[(encode $ Code.and p q).pair s] := chain_spec' nwf _ _ | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
⊢ p ... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
this... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | generalize hΔ : ⛓️[(encode $ Code.and p q).pair s + 1] = Δ at this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
this... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
Δ : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | rcases this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
Δ : ... | case intro.and₁
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | case and₁ =>
exact Or.inl (Set.mem_iUnion.mpr ⟨(encode $ Code.and p q).pair s + 1, by simp[hΔ]⟩) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
a✝ : p ⋏ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | case and₂ =>
exact Or.inr (Set.mem_iUnion.mpr ⟨(encode $ Code.and p q).pair s + 1, by simp[hΔ]⟩) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
a✝ : p ⋏ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | case andRefl =>
have : p ⋏ q ∈ ⛓️[(encode $ Code.and p q).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs
contradiction | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
hΔ : ⛓️[(encode... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
⊢ ∃ s, p ⋏ q ∈ ⛓️[s] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | exact Or.inl (Set.mem_iUnion.mpr ⟨(encode $ Code.and p q).pair s + 1, by simp[hΔ]⟩) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
a✝ : p ⋏ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
a✝ : p ⋏ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | exact Or.inr (Set.mem_iUnion.mpr ⟨(encode $ Code.and p q).pair s + 1, by simp[hΔ]⟩) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
a✝ : p ⋏ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
a✝ : p ⋏ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | have : p ⋏ q ∈ ⛓️[(encode $ Code.and p q).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
hΔ : ⛓️[(encode... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
hΔ : ⛓️[(encode... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_and | [211, 1] | [223, 18] | contradiction | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
s : ℕ
hs : p ⋏ q ∈ ⛓️[s]
hΔ : ⛓️[(encode... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | have : ∃ s, p ⋎ q ∈ ⛓️[s] := by simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
⊢ p ∈ ⛓️ ∧ q ∈ ⛓️ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
this : ∃ s, p ⋎ q ∈ ⛓️[s]
⊢ p ∈ ⛓️ ∧ q ∈... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | rcases this with ⟨s, hs⟩ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
this : ∃ s, p ⋎ q ∈ ⛓️[s]
⊢ p ∈ ⛓️ ∧ q ∈... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
⊢ p ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | have : ⛓️[(encode $ Code.or p q).pair s + 1] ≺[Code.or p q] ⛓️[(encode $ Code.or p q).pair s] := chain_spec' nwf _ _ | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
⊢ p ... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
this... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | generalize hΔ : ⛓️[(encode $ Code.or p q).pair s + 1] = Δ at this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
this... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
Δ : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | rcases this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
Δ : ... | case intro.or
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
a... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | { exact ⟨Set.mem_iUnion.mpr ⟨(encode $ Code.or p q).pair s + 1, by simp[hΔ]⟩,
Set.mem_iUnion.mpr ⟨(encode $ Code.or p q).pair s + 1, by simp[hΔ]⟩⟩ } | case intro.or
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
a... | case intro.orRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | { have : p ⋎ q ∈ ⛓️[(encode $ Code.or p q).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs
contradiction } | case intro.orRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
⊢ ∃ s, p ⋎ q ∈ ⛓️[s] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | exact ⟨Set.mem_iUnion.mpr ⟨(encode $ Code.or p q).pair s + 1, by simp[hΔ]⟩,
Set.mem_iUnion.mpr ⟨(encode $ Code.or p q).pair s + 1, by simp[hΔ]⟩⟩ | case intro.or
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
a... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
a✝ : p ⋎ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[s]
a✝ : p ⋎ q ∈ ⛓️... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | have : p ⋎ q ∈ ⛓️[(encode $ Code.or p q).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs | case intro.orRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[... | case intro.orRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_or | [225, 1] | [234, 20] | contradiction | case intro.orRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
s : ℕ
hs : p ⋎ q ∈ ⛓️[... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | have : ∃ s, ∀' p ∈ ⛓️[s] := by simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
⊢ ∃ t, [→ t].hom p ∈ ⛓️ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
this : ∃ s, ∀' p ∈ ⛓️[s]
⊢ ∃ t, [→ t]... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | rcases this with ⟨s, hs⟩ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
this : ∃ s, ∀' p ∈ ⛓️[s]
⊢ ∃ t, [→ t]... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
⊢ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | have : ⛓️[(encode $ Code.all p).pair s + 1] ≺[Code.all p] ⛓️[(encode $ Code.all p).pair s] := chain_spec' nwf _ _ | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
⊢ ... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
th... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | generalize hΔ : ⛓️[(encode $ Code.all p).pair s + 1] = Δ at this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
th... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
Δ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | rcases this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
Δ ... | case intro.all
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | { exact ⟨&(newVar ⛓️[(encode $ Code.all p).pair s]), Set.mem_iUnion.mpr ⟨(encode $ Code.all p).pair s + 1, by simp[hΔ]⟩⟩ } | case intro.all
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s... | case intro.allRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | { have : ∀' p ∈ ⛓️[(encode $ Code.all p).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs
contradiction } | case intro.allRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
⊢ ∃ s, ∀' p ∈ ⛓️[s] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | exact ⟨&(newVar ⛓️[(encode $ Code.all p).pair s]), Set.mem_iUnion.mpr ⟨(encode $ Code.all p).pair s + 1, by simp[hΔ]⟩⟩ | case intro.all
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ⛓️[s]
a✝ : ∀' p ∈ ⛓... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | have : ∀' p ∈ ⛓️[(encode $ Code.all p).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs | case intro.allRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ... | case intro.allRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_all | [236, 1] | [244, 20] | contradiction | case intro.allRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∀' p ∈ ⛓️
s : ℕ
hs : ∀' p ∈ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | have : ∃ s, ∃' p ∈ ⛓️[s] := by simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
⊢ [→ t].hom p ∈ ⛓️ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
this : ∃ s, ∃' p ∈... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | rcases this with ⟨s, hs⟩ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
this : ∃ s, ∃' p ∈... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
h... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | have : ⛓️[(encode $ Code.ex p t).pair s + 1] ≺[Code.ex p t] ⛓️[(encode $ Code.ex p t).pair s] := chain_spec' nwf _ _ | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
h... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
h... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | generalize hΔ : ⛓️[(encode $ Code.ex p t).pair s + 1] = Δ at this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
h... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
h... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | rcases this | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
h... | case intro.ex
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | { exact Set.mem_iUnion.mpr ⟨(encode $ Code.ex p t).pair s + 1, by simp[hΔ]⟩ } | case intro.ex
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ... | case intro.exRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | { have : ∃' p ∈ ⛓️[(encode $ Code.ex p t).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs
contradiction } | case intro.exRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | simpa[chainSet] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
⊢ ∃ s, ∃' p ∈ ⛓️[s... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | exact Set.mem_iUnion.mpr ⟨(encode $ Code.ex p t).pair s + 1, by simp[hΔ]⟩ | case intro.ex
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
s : ℕ
hs : ∃' p ∈ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | have : ∃' p ∈ ⛓️[(encode $ Code.ex p t).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) hs | case intro.exRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
... | case intro.exRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_ex | [246, 1] | [254, 20] | contradiction | case intro.exRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L 1
h : ∃' p ∈ ⛓️
t : Semiterm L ℕ 0
... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | have : ⛓️[(encode $ Code.id σ).pair 0 + 1] ≺[Code.id σ] ⛓️[(encode $ Code.id σ).pair 0] := chain_spec' nwf _ _ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
⊢ ~Rew.emb.hom σ ∈ ⛓️ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
this : ⛓️[(encode (Code.id σ)).pair 0 + 1] ≺[Code.id σ]... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | generalize hΔ : ⛓️[(encode $ Code.id σ).pair 0 + 1] = Δ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
this : ⛓️[(encode (Code.id σ)).pair 0 + 1] ≺[Code.id σ]... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
this : ⛓️[(encode (Code.id σ)).pair 0 + 1] ≺[Code.id σ]... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | rw[hΔ] at this | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
this : ⛓️[(encode (Code.id σ)).pair 0 + 1] ≺[Code.id σ]... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
Δ : Sequent L
this : Δ ≺[Code.id σ] ⛓️[(encode (Code.id... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | rcases this | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
Δ : Sequent L
this : Δ ≺[Code.id σ] ⛓️[(encode (Code.id... | case id
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h hσ✝ : σ ∈ T
hΔ : ⛓️[(encode (Code.id σ)).pair 0 + 1] = ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | { exact Set.mem_iUnion.mpr ⟨(encode $ Code.id σ).pair 0 + 1, by simp[hΔ]⟩ } | case id
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h hσ✝ : σ ∈ T
hΔ : ⛓️[(encode (Code.id σ)).pair 0 + 1] = ... | case idRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
hσ✝ : σ ∉ T
hΔ : ⛓️[(encode (Code.id σ)).pa... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | { contradiction } | case idRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
hσ✝ : σ ∉ T
hΔ : ⛓️[(encode (Code.id σ)).pa... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | exact Set.mem_iUnion.mpr ⟨(encode $ Code.id σ).pair 0 + 1, by simp[hΔ]⟩ | case id
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h hσ✝ : σ ∈ T
hΔ : ⛓️[(encode (Code.id σ)).pair 0 + 1] = ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | simp[hΔ] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h hσ✝ : σ ∈ T
hΔ : ⛓️[(encode (Code.id σ)).pair 0 + 1] = ~Rew.emb... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_id | [256, 1] | [261, 20] | contradiction | case idRefl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
σ : Sentence L
h : σ ∈ T
hσ✝ : σ ∉ T
hΔ : ⛓️[(encode (Code.id σ)).pa... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.Model.val | [274, 1] | [276, 60] | induction t <;> simp[*, Semiterm.val_func, Rew.func] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
n : ℕ
e : Fin n → SyntacticTerm L
ε : ℕ → Model T Γ
t : SyntacticSemiterm L n
⊢ ... | case func
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
n : ℕ
e : Fin n → SyntacticTerm L
ε : ℕ → Model T Γ
arity✝ : ℕ
a✝¹ : L... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.Model.val | [274, 1] | [276, 60] | rfl | case func
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
n : ℕ
e : Fin n → SyntacticTerm L
ε : ℕ → Model T Γ
arity✝ : ℕ
a✝¹ : L... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | by_contra | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
h : ⊤ ∈ ⛓️
⊢ ¬(Evalf (Model.structure T Γ) Semiterm.fvar) ⊤ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
h : ⊤ ∈ ⛓️
a✝ : (Evalf (Model.structure T Γ) Semiterm.fvar) ⊤
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | exact chainSet_verum nwf h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
h : ⊤ ∈ ⛓️
a✝ : (Evalf (Model.structure T Γ) Semiterm.fvar) ⊤
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simp | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
x✝ : ⊥ ∈ ⛓️
⊢ ¬(Evalf (Model.structure T Γ) Semiterm.fvar) ⊥ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | rcases chainSet_axL nwf r v with (hr | hr) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v ∈ ⛓️
⊢ ¬... | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | { contradiction } | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v... | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | { simpa[eval_rel] using hr } | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | contradiction | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simpa[eval_rel] using hr | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : rel r v... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simpa[eval_nrel] using h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
arity✝ : ℕ
r : L.Rel arity✝
v : Fin arity✝ → Semiterm L ℕ 0
h : nrel r v ∈ ⛓️
⊢ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simp | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
⊢ ¬(Evalf (Model.structure T Γ) Semiterm... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
⊢ (Evalf (Model.structure T Γ) Semiterm.... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | intro _ _ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
⊢ (Evalf (Model.structure T Γ) Semiterm.... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure T Γ) Semit... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : p ∈ ⛓️ ∨ q ∈ ⛓️ := chainSet_and nwf h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure T Γ) Semit... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure T Γ) Semit... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | rcases this with (h | h) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure T Γ) Semit... | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : ¬Evalf (Model.structure T Γ) Semiterm.fvar p := semanticMainLemma_val p h | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | contradiction | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : ¬Evalf (Model.structure T Γ) Semiterm.fvar q := semanticMainLemma_val q h | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | contradiction | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋏ q ∈ ⛓️
a✝¹ : (Evalf (Model.structure ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have hpq : p ∈ ⛓️ ∧ q ∈ ⛓️ := chainSet_or nwf h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
⊢ ¬(Evalf (Model.structure T Γ) Semiterm... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
⊢ ¬(Evalf (Model.s... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | simp only [LogicalConnective.HomClass.map_or, LogicalConnective.Prop.or_eq] | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
⊢ ¬(Evalf (Model.s... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
⊢ ¬((Evalf (Model.... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | rintro (h | h) | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
⊢ ¬((Evalf (Model.... | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
h : (Eva... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | exact semanticMainLemma_val p hpq.1 h | case inl
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
h : (Eva... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | exact semanticMainLemma_val q hpq.2 h | case inr
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p q : SyntacticFormula L
h✝ : p ⋎ q ∈ ⛓️
hpq : p ∈ ⛓️ ∧ q ∈ ⛓️
h : (Eva... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | have : ∃ u, [→ u].hom p ∈ ⛓️ := chainSet_all nwf h | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
⊢ ¬(Evalf (Model.structure T Γ)... | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
this : ∃ u, [→ u].hom p ∈ ⛓️
⊢ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.semanticMainLemma_val | [281, 1] | [312, 37] | rcases this with ⟨u, hu⟩ | L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
this : ∃ u, [→ u].hom p ∈ ⛓️
⊢ ... | case intro
L : Language
inst✝³ : (k : ℕ) → DecidableEq (L.Func k)
inst✝² : (k : ℕ) → DecidableEq (L.Rel k)
inst✝¹ : (k : ℕ) → Encodable (L.Func k)
inst✝ : (k : ℕ) → Encodable (L.Rel k)
T : Theory L
Γ : Sequent L
nwf : ¬WellFounded (SearchTree.Lt T Γ)
p : SyntacticSemiformula L (0 + 1)
h : ∀' p ∈ ⛓️
u : Semiterm L ℕ 0
h... |
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