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/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | by_cases hx₃ : x = w₃ <;> simp_all [Satisfies, hx₃] | case neg
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
x : F.World
hx : w₁ ≺ x
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w' (atom default)
hx₂ : ¬x = w₂
⊢ S... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | existsi w₂ | case intro.intro.intro.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ ∃ x, w₁ ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } x (atom default) | case intro.intro.intro.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ w₁ ≺ w₂ ∧ ¬Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w₂ (atom default) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_transitive | [15, 1] | [28, 35] | simpa [Satisfies] | case intro.intro.intro.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
w₁ w₂ : F.World
r₁₂ : F.Rel w₁ w₂
w₃ : F.World
r₂₃ : F.Rel w₂ w₃
nr₁₃ : ¬F.Rel w₁ w₃
⊢ w₁ ≺ w₂ ∧ ¬Satisfies { Frame := F, Valuation := fun w' x => w' ≠ w₂ ∧ w' ≠ w₃ } w₂ (atom default) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | contrapose | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ F ⊧* 𝗟 → ConverseWellFounded F.Rel | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ ¬ConverseWellFounded F.Rel → ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | intro hCF | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ ¬ConverseWellFounded F.Rel → ¬F ⊧* 𝗟 | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
⊢ ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | obtain ⟨X, hX₁, hX₂⟩ := by simpa using ConverseWellFounded.iff_has_max.not.mp hCF; | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
⊢ ¬F ⊧* 𝗟 | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ¬F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | simp only [Semantics.RealizeSet.setOf_iff, ValidOnFrame.models_iff, ValidOnFrame,
ValidOnModel.iff_models, ValidOnModel, Satisfies.iff_models, forall_exists_index,
forall_apply_eq_imp_iff, Satisfies.imp_def, Satisfies.box_def, not_forall, exists_prop] | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ¬F ⊧* 𝗟 | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∃ x x_1 x_2,
(∀ (w' : F.World),
x_2 ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuatio... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | existsi (atom default), (λ w _ => w ∉ X), hX₁.some | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∃ x x_1 x_2,
(∀ (w' : F.World),
x_2 ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuatio... | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ (∀ (w' : F.World),
hX₁.some ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w x => ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | constructor | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ (∀ (w' : F.World),
hX₁.some ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w x => ... | case intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∀ (w' : F.World),
hX₁.some ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w x => ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | . intro x _;
by_cases hxs : x ∈ X
. obtain ⟨y, hy₁, hy₂⟩ := hX₂ x hxs;
intro h;
exact h x (by simp_all only [Satisfies]);
. aesop; | case intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∀ (w' : F.World),
hX₁.some ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w x => ... | case intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∃ x, hX₁.some ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fun w x => w ∉ X } x (atom default) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | . obtain ⟨w', hw'₁, hw'₂⟩ := hX₂ hX₁.some (by apply Set.Nonempty.some_mem);
existsi w';
constructor;
. simpa using hw'₂;
. simpa [Satisfies]; | case intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∃ x, hX₁.some ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fun w x => w ∉ X } x (atom default) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | simpa using ConverseWellFounded.iff_has_max.not.mp hCF | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
⊢ ?m.20147 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | intro x _ | case intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∀ (w' : F.World),
hX₁.some ≺ w' →
(∀ (w'_1 : F.World), w' ≺ w'_1 → Satisfies { Frame := F, Valuation := fun w x => ... | case intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom de... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | by_cases hxs : x ∈ X | case intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom de... | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom def... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | . obtain ⟨y, hy₁, hy₂⟩ := hX₂ x hxs;
intro h;
exact h x (by simp_all only [Satisfies]); | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom def... | case neg
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∉ X
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom def... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | . aesop; | case neg
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∉ X
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom def... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | obtain ⟨y, hy₁, hy₂⟩ := hX₂ x hxs | case pos
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom def... | case pos.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
y : F.World
hy₁ : y ∈ X
hy₂ : F.Rel x y
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Fram... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | intro h | case pos.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
y : F.World
hy₁ : y ∈ X
hy₂ : F.Rel x y
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Fram... | case pos.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
y : F.World
hy₁ : y ∈ X
hy₂ : F.Rel x y
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Fra... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | exact h x (by simp_all only [Satisfies]) | case pos.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
y : F.World
hy₁ : y ∈ X
hy₂ : F.Rel x y
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Fra... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | simp_all only [Satisfies] | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∈ X
y : F.World
hy₁ : y ∈ X
hy₂ : F.Rel x y
h : ∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation :=... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | aesop | case neg
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
x : F.World
a✝ : hX₁.some ≺ x
hxs : x ∉ X
⊢ (∀ (w' : F.World), x ≺ w' → Satisfies { Frame := F, Valuation := fun w x => w ∉ X } w' (atom def... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | obtain ⟨w', hw'₁, hw'₂⟩ := hX₂ hX₁.some (by apply Set.Nonempty.some_mem) | case intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ ∃ x, hX₁.some ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fun w x => w ∉ X } x (atom default) | case intro.intro.right.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ ∃ x, hX₁.some ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fu... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | existsi w' | case intro.intro.right.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ ∃ x, hX₁.some ≺ x ∧ ¬Satisfies { Frame := F, Valuation := fu... | case intro.intro.right.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ hX₁.some ≺ w' ∧ ¬Satisfies { Frame := F, Valuation := fun w ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | constructor | case intro.intro.right.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ hX₁.some ≺ w' ∧ ¬Satisfies { Frame := F, Valuation := fun w ... | case intro.intro.right.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ hX₁.some ≺ w'
case intro.intro.right.intro.intro.right... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | . simpa using hw'₂; | case intro.intro.right.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ hX₁.some ≺ w'
case intro.intro.right.intro.intro.right... | case intro.intro.right.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ ¬Satisfies { Frame := F, Valuation := fun w x => w ∉ X... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | . simpa [Satisfies]; | case intro.intro.right.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ ¬Satisfies { Frame := F, Valuation := fun w x => w ∉ X... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | apply Set.Nonempty.some_mem | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
⊢ hX₁.some ∈ X | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | simpa using hw'₂ | case intro.intro.right.intro.intro.left
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ hX₁.some ≺ w' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.implies_cwf | [30, 1] | [49, 25] | simpa [Satisfies] | case intro.intro.right.intro.intro.right
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hCF : ¬ConverseWellFounded F.Rel
X : Set F.World
hX₁ : X.Nonempty
hX₂ : ∀ x ∈ X, ∃ x_1 ∈ X, F.Rel x x_1
w' : F.World
hw'₁ : w' ∈ X
hw'₂ : F.Rel hX₁.some w'
⊢ ¬Satisfies { Frame := F, Valuation := fun w x => w ∉ X... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | rintro ⟨hTrans, hWF⟩ | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
⊢ Transitive F.Rel ∧ ConverseWellFounded F.Rel → F ⊧* 𝗟 | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
⊢ F ⊧* 𝗟 |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simp [AxiomSet.L, Axioms.L] | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
⊢ F ⊧* 𝗟 | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
⊢ ∀ (a : Formula α), ValidOnFrame F (□(□a ⟶ a) ⟶ □a) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | intro p V w | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
⊢ ∀ (a : Formula α), ValidOnFrame F (□(□a ⟶ a) ⟶ □a) | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
⊢ w ⊧ □(□p ⟶ p) ⟶ □p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simp only [Satisfies.iff_models, Satisfies.imp_def] | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
⊢ w ⊧ □(□p ⟶ p) ⟶ □p | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
⊢ Satisfies { Frame := F, Valuation := V } w (□(□p ⟶ p)) → Satisfies { Frame := F, Valuation := V } w (□p... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | contrapose | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
⊢ Satisfies { Frame := F, Valuation := V } w (□(□p ⟶ p)) → Satisfies { Frame := F, Valuation := V } w (□p... | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
⊢ ¬Satisfies { Frame := F, Valuation := V } w (□p) → ¬Satisfies { Frame := F, Valuation := V } w (□(□p ⟶ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | intro h | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
⊢ ¬Satisfies { Frame := F, Valuation := V } w (□p) → ¬Satisfies { Frame := F, Valuation := V } w (□(□p ⟶ ... | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
⊢ ¬Satisfies { Frame := F, Valuation := V } w (□(□p ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | obtain ⟨z, rwz, hz⟩ := by simpa using h; | case intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
⊢ ¬Satisfies { Frame := F, Valuation := V } w (□(□p ... | case intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | obtain ⟨xm, ⟨hxm₁, hxm₂⟩⟩ := hWF.has_min ({ x | (F.Rel w x) ∧ ¬(Satisfies ⟨F, V⟩ x p) }) (by existsi z; simp_all) | case intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies ... | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simp [Satisfies.box_def] | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | existsi xm | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | have : Satisfies ⟨F, V⟩ xm (□p) := by
by_contra hC;
obtain ⟨y, hy₁, hy₂⟩ := by simpa using hC;
have : ¬(xm ≺ y) := hxm₂ y ⟨(hTrans (by simp_all) hy₁), hy₂⟩;
contradiction; | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simp_all | case intro.intro.intro.intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz :... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simpa using h | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
⊢ ?m.44125 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | existsi z | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simp_all | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | by_contra hC | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | obtain ⟨y, hy₁, hy₂⟩ := by simpa using hC; | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Fram... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | have : ¬(xm ≺ y) := hxm₂ y ⟨(hTrans (by simp_all) hy₁), hy₂⟩ | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Fram... | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Fram... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | contradiction | case intro.intro
W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Fram... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simpa using hC | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/Kripke/GL/Definability.lean | LO.Modal.Standard.AxiomSet.L.definability.impliedby | [51, 1] | [67, 12] | simp_all | W α : Type u
inst✝¹ : Inhabited W
inst✝ : Inhabited α
F : Frame' α
hTrans : Transitive F.Rel
hWF : ConverseWellFounded F.Rel
p : Formula α
V : Valuation F.World α
w : { Frame := F, Valuation := V }.World
h : ¬Satisfies { Frame := F, Valuation := V } w (□p)
z : F.World
rwz : w ≺ z
hz : ¬Satisfies { Frame := F, Valuation... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.Redux.antimonotone | [43, 1] | [44, 72] | cases h <;> simp[List.subset_cons_of_subset _ (List.subset_cons _ _)] | 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
c : Code L
Δ₂ Δ₁ : Sequent L
h : Δ₂ ≺[c] Δ₁
⊢ Δ₁ ⊆ Δ₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.antimonotone | [46, 1] | [47, 87] | cases 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
s : ℕ
Δ₂ Δ₁ : Sequent L
h : Δ₂ ≺⟨s⟩ Δ₁
⊢ Δ₁ ⊆ Δ₂ | case redux
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
s : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
a✝¹ : decode s.unpair.1 = some c✝
a✝ : Δ₂ ≺[c✝] Δ₁
⊢ Δ₁ ⊆ Δ₂
case refl
... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.antimonotone | [46, 1] | [47, 87] | { exact Redux.antimonotone (by assumption) } | case redux
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
s : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
a✝¹ : decode s.unpair.1 = some c✝
a✝ : Δ₂ ≺[c✝] Δ₁
⊢ Δ₁ ⊆ Δ₂
case refl
... | case refl
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
s : ℕ
Δ₂ : Sequent L
a✝ : decode s.unpair.1 = none
⊢ Δ₂ ⊆ Δ₂ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.antimonotone | [46, 1] | [47, 87] | { exact List.Subset.refl Δ₂ } | case refl
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
s : ℕ
Δ₂ : Sequent L
a✝ : decode s.unpair.1 = none
⊢ Δ₂ ⊆ Δ₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.antimonotone | [46, 1] | [47, 87] | exact Redux.antimonotone (by assumption) | case redux
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
s : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
a✝¹ : decode s.unpair.1 = some c✝
a✝ : Δ₂ ≺[c✝] Δ₁
⊢ Δ₁ ⊆ Δ₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.antimonotone | [46, 1] | [47, 87] | assumption | 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
s : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
a✝¹ : decode s.unpair.1 = some c✝
a✝ : Δ₂ ≺[c✝] Δ₁
⊢ Redux ?m.13639 ?m.13640 Δ₂ Δ₁ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.antimonotone | [46, 1] | [47, 87] | exact List.Subset.refl Δ₂ | case refl
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
s : ℕ
Δ₂ : Sequent L
a✝ : decode s.unpair.1 = none
⊢ Δ₂ ⊆ Δ₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.toRedux | [49, 1] | [50, 79] | rcases h with (⟨h, r⟩ | ⟨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
c : Code L
i : ℕ
Δ₂ Δ₁ : Sequent L
h : Δ₂ ≺⟨(encode c).pair i⟩ Δ₁
⊢ Δ₂ ≺[c] Δ₁ | case redux
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
c : Code L
i : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
h : decode ((encode c).pair i).unpair.1 = some c✝
r : Δ₂ ≺[c✝... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.toRedux | [49, 1] | [50, 79] | { simp at h; simpa[h] using r } | case redux
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
c : Code L
i : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
h : decode ((encode c).pair i).unpair.1 = some c✝
r : Δ₂ ≺[c✝... | case refl
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
c : Code L
i : ℕ
Δ₂ : Sequent L
h : decode ((encode c).pair i).unpair.1 = none
⊢ Δ₂ ≺[c] Δ₂ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.toRedux | [49, 1] | [50, 79] | { simp at h } | case refl
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
c : Code L
i : ℕ
Δ₂ : Sequent L
h : decode ((encode c).pair i).unpair.1 = none
⊢ Δ₂ ≺[c] Δ₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.toRedux | [49, 1] | [50, 79] | simp at h | case redux
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
c : Code L
i : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
h : decode ((encode c).pair i).unpair.1 = some c✝
r : Δ₂ ≺[c✝... | case redux
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
c : Code L
i : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
r : Δ₂ ≺[c✝] Δ₁
h : c = c✝
⊢ Δ₂ ≺[c] Δ₁ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.toRedux | [49, 1] | [50, 79] | simpa[h] using r | case redux
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
c : Code L
i : ℕ
Δ₂ Δ₁ : Sequent L
c✝ : Code L
r : Δ₂ ≺[c✝] Δ₁
h : c = c✝
⊢ Δ₂ ≺[c] Δ₁ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.ReduxNat.toRedux | [49, 1] | [50, 79] | simp at h | case refl
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
c : Code L
i : ℕ
Δ₂ : Sequent L
h : decode ((encode c).pair i).unpair.1 = none
⊢ Δ₂ ≺[c] Δ₂ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.SearchTree.rank_of_lt | [67, 1] | [68, 16] | cases 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
τ₁ τ₂ : SearchTree T Γ
h : Lt T Γ τ₂ τ₁
⊢ τ₂.rank = τ₁.rank + 1 | 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
s✝ : ℕ
Δ₂✝ Δ₁✝ : Sequent L
a✝ : SearchTreeAux T Γ s✝ Δ₁✝
r✝ : Δ₂✝ ≺⟨s✝⟩ Δ₁✝
⊢ rank ⟨s✝ + 1, ⟨Δ₂✝, a✝.succ r✝... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.SearchTree.rank_of_lt | [67, 1] | [68, 16] | simp | 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
s✝ : ℕ
Δ₂✝ Δ₁✝ : Sequent L
a✝ : SearchTreeAux T Γ s✝ Δ₁✝
r✝ : Δ₂✝ ≺⟨s✝⟩ Δ₁✝
⊢ rank ⟨s✝ + 1, ⟨Δ₂✝, a✝.succ r✝... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.SearchTree.seq_of_lt | [70, 1] | [71, 39] | cases 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
τ₁ τ₂ : SearchTree T Γ
h : Lt T Γ τ₂ τ₁
⊢ τ₂.seq ≺⟨τ₁.rank⟩ τ₁.seq | 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
s✝ : ℕ
Δ₂✝ Δ₁✝ : Sequent L
a✝ : SearchTreeAux T Γ s✝ Δ₁✝
r✝ : Δ₂✝ ≺⟨s✝⟩ Δ₁✝
⊢ seq ⟨s✝ + 1, ⟨Δ₂✝, a✝.succ r✝⟩... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.SearchTree.seq_of_lt | [70, 1] | [71, 39] | simp[rank, seq] | 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
s✝ : ℕ
Δ₂✝ Δ₁✝ : Sequent L
a✝ : SearchTreeAux T Γ s✝ Δ₁✝
r✝ : Δ₂✝ ≺⟨s✝⟩ Δ₁✝
⊢ seq ⟨s✝ + 1, ⟨Δ₂✝, a✝.succ r✝⟩... | 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
s✝ : ℕ
Δ₂✝ Δ₁✝ : Sequent L
a✝ : SearchTreeAux T Γ s✝ Δ₁✝
r✝ : Δ₂✝ ≺⟨s✝⟩ Δ₁✝
⊢ Δ₂✝ ≺⟨s✝⟩ Δ₁✝ |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.SearchTree.seq_of_lt | [70, 1] | [71, 39] | assumption | 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
s✝ : ℕ
Δ₂✝ Δ₁✝ : Sequent L
a✝ : SearchTreeAux T Γ s✝ Δ₁✝
r✝ : Δ₂✝ ≺⟨s✝⟩ Δ₁✝
⊢ Δ₂✝ ≺⟨s✝⟩ Δ₁✝ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | intro A | 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)
⊢ ¬Acc (SearchTree.Lt 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)
A : Acc (SearchTree.Lt T Γ) ⊤
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | have : WellFounded (SearchTree.Lt T Γ) := ⟨by
rintro ⟨s, Δ, a⟩
induction a
case zero => exact A
case succ s Δ₁ Δ₂ a r ih => exact ih.inv (SearchTree.Lt.intro a r)⟩ | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
⊢ False | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
this : WellFounded (SearchTree.Lt T Γ)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
this : WellFounded (SearchTree.Lt T Γ)
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | rintro ⟨s, Δ, a⟩ | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
⊢ ∀ (a : SearchTree T Γ), Acc (SearchTree.Lt T Γ) ... | case mk.mk
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)
A : Acc (SearchTree.Lt T Γ) ⊤
s : ℕ
Δ : Sequent L
a : SearchTreeAux T... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | induction a | case mk.mk
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)
A : Acc (SearchTree.Lt T Γ) ⊤
s : ℕ
Δ : Sequent L
a : SearchTreeAux T... | case mk.mk.zero
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)
A : Acc (SearchTree.Lt T Γ) ⊤
s : ℕ
Δ : Sequent L
⊢ Acc (SearchT... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | case zero => exact A | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
s : ℕ
Δ : Sequent L
⊢ Acc (SearchTree.Lt T Γ) ⟨0, ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | case succ s Δ₁ Δ₂ a r ih => exact ih.inv (SearchTree.Lt.intro a r) | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
s✝ : ℕ
Δ : Sequent L
s : ℕ
Δ₁ Δ₂ : Sequent L
a : S... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | exact A | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
s : ℕ
Δ : Sequent L
⊢ Acc (SearchTree.Lt T Γ) ⟨0, ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.top_inaccessible | [157, 1] | [164, 16] | exact ih.inv (SearchTree.Lt.intro a r) | 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)
A : Acc (SearchTree.Lt T Γ) ⊤
s✝ : ℕ
Δ : Sequent L
s : ℕ
Δ₁ Δ₂ : Sequent L
a : S... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainU_val_fst_eq | [169, 1] | [172, 62] | induction' s with s ih <;> simp[SearchTree.rank] | 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)
s : ℕ
⊢ (chainU T Γ s).rank = s | case zero
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)
⊢ (chainU T Γ 0).fst = 0
case succ
L : Language
inst✝³ : (k : ℕ) → De... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainU_val_fst_eq | [169, 1] | [172, 62] | exact rfl | case zero
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)
⊢ (chainU T Γ 0).fst = 0 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainU_val_fst_eq | [169, 1] | [172, 62] | simpa[ih] using SearchTree.rank_of_lt (chainU_spec nwf s) | case succ
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)
s : ℕ
ih : (chainU T Γ s).rank = s
⊢ (chainU T Γ (s + 1)).fst = s + 1 | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_spec | [174, 1] | [175, 83] | simpa[chainU_val_fst_eq nwf s] using SearchTree.seq_of_lt (chainU_spec nwf 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)
s : ℕ
⊢ ⛓️[s + 1] ≺⟨s⟩ ⛓️[s] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_monotone | [177, 1] | [182, 102] | suffices ∀ d, ⛓️[s] ⊆ ⛓️[s + d] by
simpa[Nat.add_sub_of_le h] using this (u - 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)
s u : ℕ
h : s ≤ u
⊢ ⛓️[s] ⊆ ⛓️[u] | 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)
s u : ℕ
h : s ≤ u
⊢ ∀ (d : ℕ), ⛓️[s] ⊆ ⛓️[s + d] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_monotone | [177, 1] | [182, 102] | intro d | 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)
s u : ℕ
h : s ≤ u
⊢ ∀ (d : ℕ), ⛓️[s] ⊆ ⛓️[s + d] | 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)
s u : ℕ
h : s ≤ u
d : ℕ
⊢ ⛓️[s] ⊆ ⛓️[s + d] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_monotone | [177, 1] | [182, 102] | induction' d with d ih | 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)
s u : ℕ
h : s ≤ u
d : ℕ
⊢ ⛓️[s] ⊆ ⛓️[s + d] | case zero
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)
s u : ℕ
h : s ≤ u
⊢ ⛓️[s] ⊆ ⛓️[s + 0]
case succ
L : Language
inst✝³ :... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_monotone | [177, 1] | [182, 102] | simpa[Nat.add_sub_of_le h] using this (u - 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)
s u : ℕ
h : s ≤ u
this : ∀ (d : ℕ), ⛓️[s] ⊆ ⛓️[s + d]
⊢ ⛓️[s] ⊆ ⛓️[u] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_monotone | [177, 1] | [182, 102] | simp | case zero
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)
s u : ℕ
h : s ≤ u
⊢ ⛓️[s] ⊆ ⛓️[s + 0] | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chain_monotone | [177, 1] | [182, 102] | simpa only [Nat.add_succ] using subset_trans ih <| ReduxNat.antimonotone (chain_spec nwf (s + d)) | case succ
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)
s u : ℕ
h : s ≤ u
d : ℕ
ih : ⛓️[s] ⊆ ⛓️[s + d]
⊢ ⛓️[s] ⊆ ⛓️[s + (d + 1... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | simp[chainSet] | 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)
⊢ ⊤ ∉ ⛓️ | 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 : ℕ), ⊤ ∉ ⛓️[x] |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | intro s 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)
⊢ ∀ (x : ℕ), ⊤ ∉ ⛓️[x] | 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | have : ⊤ ∈ ⛓️[(encode (Code.verum : Code L)).pair s] := chain_monotone nwf (Nat.right_le_pair _ _) 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
⊢ False | 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | have : ¬⊤ ∈ ⛓️[(encode (Code.verum : Code L)).pair s] := by
have : ⛓️[(encode Code.verum).pair s + 1] ≺[Code.verum] ⛓️[(encode Code.verum).pair s] := chain_spec' nwf _ _
generalize ⛓️[(encode (Code.verum : Code L)).pair s + 1] = Δ' at this
rcases this; assumption | 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
⊢ False | 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this✝ : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
this : ⊤ ∉ ⛓️[(en... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this✝ : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
this : ⊤ ∉ ⛓️[(en... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | have : ⛓️[(encode Code.verum).pair s + 1] ≺[Code.verum] ⛓️[(encode Code.verum).pair s] := 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
⊢ ⊤ ∉ ⛓️[(encode C... | 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this✝ : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
this : ⛓️[(encode... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | generalize ⛓️[(encode (Code.verum : Code L)).pair s + 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this✝ : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
this : ⛓️[(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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this✝ : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
Δ' : Sequent L
th... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this✝ : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
Δ' : Sequent L
th... | case verumRefl
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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
a✝ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_verum | [186, 1] | [193, 16] | assumption | case verumRefl
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)
s : ℕ
h : ⊤ ∈ ⛓️[s]
this : ⊤ ∈ ⛓️[(encode Code.verum).pair s]
a✝ ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/SearchTree.lean | LO.FirstOrder.Completeness.chainSet_axL | [195, 1] | [209, 16] | by_contra 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
⊢ 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 : (∃ s₁, rel r v ∈ ⛓️[s₁]) ∧ (∃ s₂, nrel r v ∈ ⛓️[s₂]) := by
have h : rel r v ∈ ⛓️ ∧ nrel r v ∈ ⛓️ := by simpa[not_or] using h
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 ∉ ⛓️... | 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 with ⟨⟨s₁, hs₁⟩, ⟨s₂, 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 ∉ ⛓️... | 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] | 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
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⟩ | 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... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.