url stringclasses 147
values | commit stringclasses 147
values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M |
|---|---|---|---|---|---|---|---|---|
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.maximalConsistentTheory_satisfiable | [149, 1] | [165, 27] | exact ihq hq | case inr.h
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p q : Formula α
ihp : p ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
ihq : q ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.satisfiable_of_consistent | [167, 1] | [169, 82] | simp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT✝ consisT : Consistent T
⊢ T ⊆ maximalConsistentTheory consisT | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | suffices Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T) by
contrapose
intro hp hs
have : Semantics.Satisfiable (Valuation α) (insert (~p) T) :=
this (consistent_insert_iff_not_refutable.mpr $ by simpa)
rcases this with ⟨v, hv⟩
have : v ⊧* T := Semantics.RealizeSet.of_subs... | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ T ⊨[Valuation α] p → T ⊢! p | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | intro consis | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
⊢ Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
consis : Consistent (insert (~p) T)
⊢ Semantics.Satisfiable (Valuation α) (insert (~p) T) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | exact satisfiable_of_consistent consis | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
consis : Consistent (insert (~p) T)
⊢ Semantics.Satisfiable (Valuation α) (insert (~p) T) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | contrapose | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
⊢ T ⊨[Valuation α] p → T ⊢! p | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
⊢ ¬T ⊢! p → ¬T ⊨[Valuation α] p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | intro hp hs | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
⊢ ¬T ⊢! p → ¬T ⊨[Valuation α] p | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | have : Semantics.Satisfiable (Valuation α) (insert (~p) T) :=
this (consistent_insert_iff_not_refutable.mpr $ by simpa) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
⊢ False | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
this : Semantics.Satisfiable (Valuation α) (insert (~p) T)
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | rcases this with ⟨v, hv⟩ | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
this : Semantics.Satisfiable (Valuation α) (insert (~p) T)
⊢ False | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | have : v ⊧* T := Semantics.RealizeSet.of_subset hv (by simp) | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
⊢ False | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this : v ⊧* T
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | have : v ⊧ p := hs this | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this : v ⊧* T
⊢ False | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝¹ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this✝ : v ⊧* T
this : v ⊧ p
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | have : ¬v ⊧ p := by simpa using hv.realize (Set.mem_insert (~p) T) | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝¹ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this✝ : v ⊧* T
this : v ⊧ p
⊢ False | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝² : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this✝¹ : v ⊧* T
this✝ : v ⊧ p
this : ¬v ⊧ p
⊢ False |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | contradiction | case intro
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝² : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this✝¹ : v ⊧* T
this✝ : v ⊧ p
this : ¬v ⊧ p
⊢ False | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | simpa | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
⊢ T ⊬! ~~p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | simp | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
⊢ T ⊆ insert (~p) T | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Classical/Basic/Completeness.lean | LO.Propositional.Classical.completeness! | [171, 1] | [183, 41] | simpa using hv.realize (Set.mem_insert (~p) T) | α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
this✝¹ : Consistent (insert (~p) T) → Semantics.Satisfiable (Valuation α) (insert (~p) T)
hp : ¬T ⊢! p
hs : T ⊨[Valuation α] p
v : Valuation α
hv : v ⊧* insert (~p) T
this✝ : v ⊧* T
this : v ⊧ p
⊢ ¬v ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.Necessitation.nec! | [21, 1] | [21, 78] | rintro ⟨hp⟩ | F : Type u_1
inst✝⁵ : StandardModalLogicalConnective F
inst✝⁴ : DecidableEq F
S : Type u_2
inst✝³ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝² : Classical 𝓢
inst✝¹ : System.NegationEquiv 𝓢
inst✝ : Necessitation 𝓢
⊢ 𝓢 ⊢! p → 𝓢 ⊢! □p | case intro
F : Type u_1
inst✝⁵ : StandardModalLogicalConnective F
inst✝⁴ : DecidableEq F
S : Type u_2
inst✝³ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝² : Classical 𝓢
inst✝¹ : System.NegationEquiv 𝓢
inst✝ : Necessitation 𝓢
hp : 𝓢 ⊢ p
⊢ 𝓢 ⊢! □p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.Necessitation.nec! | [21, 1] | [21, 78] | exact ⟨nec hp⟩ | case intro
F : Type u_1
inst✝⁵ : StandardModalLogicalConnective F
inst✝⁴ : DecidableEq F
S : Type u_2
inst✝³ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝² : Classical 𝓢
inst✝¹ : System.NegationEquiv 𝓢
inst✝ : Necessitation 𝓢
hp : 𝓢 ⊢ p
⊢ 𝓢 ⊢! □p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.Necessitation.multinec! | [29, 1] | [29, 92] | rintro ⟨hp⟩ | F : Type u_1
inst✝⁵ : StandardModalLogicalConnective F
inst✝⁴ : DecidableEq F
S : Type u_2
inst✝³ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝² : Classical 𝓢
inst✝¹ : System.NegationEquiv 𝓢
inst✝ : Necessitation 𝓢
n : ℕ
⊢ 𝓢 ⊢! p → 𝓢 ⊢! □^[n]p | case intro
F : Type u_1
inst✝⁵ : StandardModalLogicalConnective F
inst✝⁴ : DecidableEq F
S : Type u_2
inst✝³ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝² : Classical 𝓢
inst✝¹ : System.NegationEquiv 𝓢
inst✝ : Necessitation 𝓢
n : ℕ
hp : 𝓢 ⊢ p
⊢ 𝓢 ⊢! □^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.Necessitation.multinec! | [29, 1] | [29, 92] | exact ⟨multinec hp⟩ | case intro
F : Type u_1
inst✝⁵ : StandardModalLogicalConnective F
inst✝⁴ : DecidableEq F
S : Type u_2
inst✝³ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝² : Classical 𝓢
inst✝¹ : System.NegationEquiv 𝓢
inst✝ : Necessitation 𝓢
n : ℕ
hp : 𝓢 ⊢ p
⊢ 𝓢 ⊢! □^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | constructor | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n]p ↔ 𝓢 ⊢! ~(◇^[n](~p)) | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n]p → 𝓢 ⊢! ~(◇^[n](~p))
case mpr
F : Type u... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | . intro h; exact (conj₁'! multiboxDuality!) ⨀ h; | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n]p → 𝓢 ⊢! ~(◇^[n](~p))
case mpr
F : Type u... | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(◇^[n](~p)) → 𝓢 ⊢! □^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | . intro h; exact (conj₂'! multiboxDuality!) ⨀ h; | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(◇^[n](~p)) → 𝓢 ⊢! □^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | intro h | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n]p → 𝓢 ⊢! ~(◇^[n](~p)) | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! □^[n]p
⊢ 𝓢 ⊢! ~(◇^[n](~p)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | exact (conj₁'! multiboxDuality!) ⨀ h | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! □^[n]p
⊢ 𝓢 ⊢! ~(◇^[n](~p)) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | intro h | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(◇^[n](~p)) → 𝓢 ⊢! □^[n]p | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ~(◇^[n](~p))
⊢ 𝓢 ⊢! □^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxDuality'! | [101, 1] | [104, 51] | exact (conj₂'! multiboxDuality!) ⨀ h | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ~(◇^[n](~p))
⊢ 𝓢 ⊢! □^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | constructor | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n]p ↔ 𝓢 ⊢! ~(□^[n](~p)) | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n]p → 𝓢 ⊢! ~(□^[n](~p))
case mpr
F : Type u... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | . intro h; exact (conj₁'! multidiaDuality!) ⨀ h; | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n]p → 𝓢 ⊢! ~(□^[n](~p))
case mpr
F : Type u... | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~p)) → 𝓢 ⊢! ◇^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | . intro h; exact (conj₂'! multidiaDuality!) ⨀ h; | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~p)) → 𝓢 ⊢! ◇^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | intro h | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n]p → 𝓢 ⊢! ~(□^[n](~p)) | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ◇^[n]p
⊢ 𝓢 ⊢! ~(□^[n](~p)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | exact (conj₁'! multidiaDuality!) ⨀ h | case mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ◇^[n]p
⊢ 𝓢 ⊢! ~(□^[n](~p)) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | intro h | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~p)) → 𝓢 ⊢! ◇^[n]p | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ~(□^[n](~p))
⊢ 𝓢 ⊢! ◇^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multidiaDuality'! | [122, 1] | [125, 51] | exact (conj₂'! multidiaDuality!) ⨀ h | case mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ~(□^[n](~p))
⊢ 𝓢 ⊢! ◇^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | induction Γ using List.induction_with_singleton with
| hnil => simp;
| hsingle => simp;
| hcons p Γ h ih =>
simp_all;
constructor;
. intro h;
have := distribute_multibox_and'! h;
constructor;
. exact conj₁'! this;
. exact ih.mp (conj₂'! this);
. rintro ⟨h₁, h₂⟩;
exact collect_multibox_and'! ... | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n]Γ.conj' ↔ ∀ p ∈ Γ, 𝓢 ⊢! □^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | simp | case hnil
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n][].conj' ↔ ∀ p ∈ [], 𝓢 ⊢! □^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | simp | case hsingle
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
a✝ : F
⊢ 𝓢 ⊢! □^[n][a✝].conj' ↔ ∀ p ∈ [a✝], 𝓢 ⊢! ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | simp_all | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : Γ ≠ []
ih : 𝓢 ⊢! □^[n]Γ.conj'... | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.conj... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | constructor | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.conj... | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.c... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | . intro h;
have := distribute_multibox_and'! h;
constructor;
. exact conj₁'! this;
. exact ih.mp (conj₂'! this); | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.c... | case hcons.mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | . rintro ⟨h₁, h₂⟩;
exact collect_multibox_and'! $ conj₃'! h₁ (ih.mpr h₂); | case hcons.mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | intro h | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.c... | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | have := distribute_multibox_and'! h | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | constructor | case hcons.mp
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... | case hcons.mp.left
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | . exact conj₁'! this; | case hcons.mp.left
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^... | case hcons.mp.right
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | . exact ih.mp (conj₂'! this); | case hcons.mp.right
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | exact conj₁'! this | case hcons.mp.left
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □^... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | exact ih.mp (conj₂'! this) | case hcons.mp.right
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h✝ : ¬Γ = []
ih : 𝓢 ⊢! □... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | rintro ⟨h₁, h₂⟩ | case hcons.mpr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □^[n]Γ.... | case hcons.mpr.intro
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxConj'_iff! | [172, 1] | [185, 61] | exact collect_multibox_and'! $ conj₃'! h₁ (ih.mpr h₂) | case hcons.mpr.intro
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! □... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxconj'_of_conj'multibox! | [188, 1] | [191, 57] | apply multiboxConj'_iff!.mpr | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
d : 𝓢 ⊢! (□'^[n]Γ).conj'
⊢ 𝓢 ⊢! □^[n]Γ.conj' | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
d : 𝓢 ⊢! (□'^[n]Γ).conj'
⊢ ∀ p ∈ Γ, 𝓢 ⊢! □^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxconj'_of_conj'multibox! | [188, 1] | [191, 57] | intro p hp | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
d : 𝓢 ⊢! (□'^[n]Γ).conj'
⊢ ∀ p ∈ Γ, 𝓢 ⊢! □^[n]p | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
d : 𝓢 ⊢! (□'^[n]Γ).conj'
p : F
hp : p ∈ Γ
⊢ 𝓢 ⊢! □^[n]p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxconj'_of_conj'multibox! | [188, 1] | [191, 57] | exact iff_provable_list_conj.mp d (□^[n]p) (by aesop) | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
d : 𝓢 ⊢! (□'^[n]Γ).conj'
p : F
hp : p ∈ Γ
⊢ 𝓢 ⊢! □^[n]p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multiboxconj'_of_conj'multibox! | [188, 1] | [191, 57] | aesop | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
d : 𝓢 ⊢! (□'^[n]Γ).conj'
p : F
hp : p ∈ Γ
⊢ □^[n]p ∈ □'^[n]Γ | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multibox_cons_conj'! | [193, 1] | [196, 12] | apply conj'conj'_subset | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! (□'^[n](p :: Γ)).conj' ⟶ (□'^[n]Γ).conj' | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ ∀ p_1 ∈ □'^[n]Γ, p_1 ∈ □'^[n](p :: Γ) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.multibox_cons_conj'! | [193, 1] | [196, 12] | simp_all | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ ∀ p_1 ∈ □'^[n]Γ, p_1 ∈ □'^[n](p :: Γ) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | induction Γ using List.induction_with_singleton with
| hnil => simpa using dhyp! multiboxverum!;
| hsingle => simp;
| hcons p Γ h ih =>
simp_all;
exact imp_trans! (implyRightAnd! (generalConj'! (by simp)) (imp_trans! (by simp) ih)) collect_multibox_and!; | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! (□'^[n]Γ).conj' ⟶ □^[n]Γ.conj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | simpa using dhyp! multiboxverum! | case hnil
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! (□'^[n][]).conj' ⟶ □^[n][].conj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | simp | case hsingle
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
a✝ : F
⊢ 𝓢 ⊢! (□'^[n][a✝]).conj' ⟶ □^[n][a✝].conj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | simp_all | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : Γ ≠ []
ih : 𝓢 ⊢! (□'^[n]Γ).co... | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! (□'^[n]Γ).c... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | exact imp_trans! (implyRightAnd! (generalConj'! (by simp)) (imp_trans! (by simp) ih)) collect_multibox_and! | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! (□'^[n]Γ).c... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | simp | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! (□'^[n]Γ).conj' ⟶ □^[n... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.collect_multibox_conj'! | [198, 1] | [205, 113] | simp | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! (□'^[n]Γ).conj' ⟶ □^[n... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | suffices h : 𝓢 ⊢! ~(□^[n](~(p ⋏ q))) ⟶ ~(□^[n](~p)) ⋏ ~(□^[n](~q)) by
exact imp_trans! (imp_trans! (conj₁'! multidiaDuality!) h) $ andReplace! (conj₂'! multidiaDuality!) (conj₂'! multidiaDuality!); | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n](p ⋏ q) ⟶ ◇^[n]p ⋏ ◇^[n]q | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~(p ⋏ q))) ⟶ ~(□^[n](~p)) ⋏ ~(□^[n](~q)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | apply FiniteContext.deduct'! | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~(p ⋏ q))) ⟶ ~(□^[n](~p)) ⋏ ~(□^[n](~q)) | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ [~(□^[n](~(p ⋏ q)))] ⊢[𝓢]! ~(□^[n](~p)) ⋏ ~(□^[n](~q)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | apply demorgan₃'! | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ [~(□^[n](~(p ⋏ q)))] ⊢[𝓢]! ~(□^[n](~p)) ⋏ ~(□^[n](~q)) | case h.b
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ { ctx := [~(□^[n](~(p ⋏ q)))] } ⊢! ~(□^[n](~p) ⋎ □^[n... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | apply FiniteContext.deductInv'! | case h.b
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ { ctx := [~(□^[n](~(p ⋏ q)))] } ⊢! ~(□^[n](~p) ⋎ □^[n... | case h.b.h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~(p ⋏ q))) ⟶ ~(□^[n](~p) ⋎ □^[n](~q)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | apply contra₀'! | case h.b.h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ~(□^[n](~(p ⋏ q))) ⟶ ~(□^[n](~p) ⋎ □^[n](~q)) | case h.b.h.b
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n](~p) ⋎ □^[n](~q) ⟶ □^[n](~(p ⋏ q)) |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | apply imp_trans! collect_multibox_or! (imply_multibox_distribute'! demorgan₁!) | case h.b.h.b
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! □^[n](~p) ⋎ □^[n](~q) ⟶ □^[n](~(p ⋏ q)) | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.distribute_multidia_and! | [238, 1] | [245, 81] | exact imp_trans! (imp_trans! (conj₁'! multidiaDuality!) h) $ andReplace! (conj₂'! multidiaDuality!) (conj₂'! multidiaDuality!) | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
h : 𝓢 ⊢! ~(□^[n](~(p ⋏ q))) ⟶ ~(□^[n](~p)) ⋏ ~(□^[n](~q))
⊢ 𝓢 ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | induction Γ using List.induction_with_singleton with
| hnil => simpa using dhyp! verum!;
| hsingle p => simp;
| hcons p Γ h ih =>
simp_all;
exact imp_trans! distribute_multidia_and! $ by
apply deduct'!;
apply iff_provable_list_conj.mpr;
intro q hq;
simp at hq;
cases hq with
| inl => subst_va... | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ (◇'^[n]Γ).conj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | simpa using dhyp! verum! | case hnil
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
⊢ 𝓢 ⊢! ◇^[n][].conj' ⟶ (◇'^[n][]).conj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | simp | case hsingle
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
⊢ 𝓢 ⊢! ◇^[n][p].conj' ⟶ (◇'^[n][p]).conj' | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | simp_all | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : Γ ≠ []
ih : 𝓢 ⊢! ◇^[n]Γ.conj'... | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | exact imp_trans! distribute_multidia_and! $ by
apply deduct'!;
apply iff_provable_list_conj.mpr;
intro q hq;
simp at hq;
cases hq with
| inl => subst_vars; exact conj₁'! id!;
| inr hq =>
obtain ⟨r, hr₁, hr₂⟩ := hq;
exact (iff_provable_list_conj.mp $ (of'! ih) ⨀ (conj₂'! $ id!)) q (by aesop); | case hcons
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | apply deduct'! | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ (◇'^[n]... | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | apply iff_provable_list_conj.mpr | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ ... | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | intro q hq | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ ... | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | simp at hq | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶... | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | cases hq with
| inl => subst_vars; exact conj₁'! id!;
| inr hq =>
obtain ⟨r, hr₁, hr₂⟩ := hq;
exact (iff_provable_list_conj.mp $ (of'! ih) ⨀ (conj₂'! $ id!)) q (by aesop); | case h
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | subst_vars | case h.inl
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.con... | case h.inl
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | exact conj₁'! id! | case h.inl
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | obtain ⟨r, hr₁, hr₂⟩ := hq | case h.inr
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.con... | case h.inr.intro.intro
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r✝ : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | exact (iff_provable_list_conj.mp $ (of'! ih) ⨀ (conj₂'! $ id!)) q (by aesop) | case h.inr.intro.intro
F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r✝ : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.iff_conj'multidia_multidiaconj'! | [251, 1] | [266, 86] | aesop | F : Type u_1
inst✝⁷ : StandardModalLogicalConnective F
inst✝⁶ : DecidableEq F
S : Type u_2
inst✝⁵ : System F S
p✝ q✝ r✝ : F
Γ✝ Δ : List F
𝓢 : S
inst✝⁴ : Classical 𝓢
inst✝³ : System.NegationEquiv 𝓢
inst✝² : Necessitation 𝓢
inst✝¹ inst✝ : HasAxiomK 𝓢
n : ℕ
p : F
Γ : List F
h : ¬Γ = []
ih : 𝓢 ⊢! ◇^[n]Γ.conj' ⟶ (◇'^[... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.LoebRule.loeb! | [435, 1] | [435, 92] | rintro ⟨hp⟩ | F : Type u_1
inst✝⁸ : StandardModalLogicalConnective F
inst✝⁷ : DecidableEq F
S : Type u_2
inst✝⁶ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁵ : Classical 𝓢
inst✝⁴ : System.NegationEquiv 𝓢
inst✝³ : Necessitation 𝓢
inst✝² inst✝¹ : HasAxiomK 𝓢
inst✝ : LoebRule 𝓢
⊢ 𝓢 ⊢! □p ⟶ p → 𝓢 ⊢! p | case intro
F : Type u_1
inst✝⁸ : StandardModalLogicalConnective F
inst✝⁷ : DecidableEq F
S : Type u_2
inst✝⁶ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁵ : Classical 𝓢
inst✝⁴ : System.NegationEquiv 𝓢
inst✝³ : Necessitation 𝓢
inst✝² inst✝¹ : HasAxiomK 𝓢
inst✝ : LoebRule 𝓢
hp : 𝓢 ⊢ □p ⟶ p
⊢ 𝓢 ⊢! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.LoebRule.loeb! | [435, 1] | [435, 92] | exact ⟨loeb hp⟩ | case intro
F : Type u_1
inst✝⁸ : StandardModalLogicalConnective F
inst✝⁷ : DecidableEq F
S : Type u_2
inst✝⁶ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁵ : Classical 𝓢
inst✝⁴ : System.NegationEquiv 𝓢
inst✝³ : Necessitation 𝓢
inst✝² inst✝¹ : HasAxiomK 𝓢
inst✝ : LoebRule 𝓢
hp : 𝓢 ⊢ □p ⟶ p
⊢ 𝓢 ⊢! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.HenkinRule.henkin! | [438, 1] | [438, 100] | rintro ⟨hp⟩ | F : Type u_1
inst✝⁸ : StandardModalLogicalConnective F
inst✝⁷ : DecidableEq F
S : Type u_2
inst✝⁶ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁵ : Classical 𝓢
inst✝⁴ : System.NegationEquiv 𝓢
inst✝³ : Necessitation 𝓢
inst✝² inst✝¹ : HasAxiomK 𝓢
inst✝ : HenkinRule 𝓢
⊢ 𝓢 ⊢! □p ⟷ p → 𝓢 ⊢! p | case intro
F : Type u_1
inst✝⁸ : StandardModalLogicalConnective F
inst✝⁷ : DecidableEq F
S : Type u_2
inst✝⁶ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁵ : Classical 𝓢
inst✝⁴ : System.NegationEquiv 𝓢
inst✝³ : Necessitation 𝓢
inst✝² inst✝¹ : HasAxiomK 𝓢
inst✝ : HenkinRule 𝓢
hp : 𝓢 ⊢ □p ⟷ p
⊢ 𝓢 ⊢! p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Modal/Standard/HilbertStyle.lean | LO.System.HenkinRule.henkin! | [438, 1] | [438, 100] | exact ⟨henkin hp⟩ | case intro
F : Type u_1
inst✝⁸ : StandardModalLogicalConnective F
inst✝⁷ : DecidableEq F
S : Type u_2
inst✝⁶ : System F S
p q r : F
Γ Δ : List F
𝓢 : S
inst✝⁵ : Classical 𝓢
inst✝⁴ : System.NegationEquiv 𝓢
inst✝³ : Necessitation 𝓢
inst✝² inst✝¹ : HasAxiomK 𝓢
inst✝ : HenkinRule 𝓢
hp : 𝓢 ⊢ □p ⟷ p
⊢ 𝓢 ⊢! p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_provably_subtheory | [9, 1] | [12, 54] | intro p hp | L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
⊢ ∀ ⦃f : Sentence L⦄, f ∈ T → inst✝.toStruc ⊧ f | L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
p : Sentence L
hp : p ∈ T
⊢ inst✝.toStruc ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_provably_subtheory | [9, 1] | [12, 54] | have : U ⊢ p := System.Subtheory.prf (System.byAxm hp) | L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
p : Sentence L
hp : p ∈ T
⊢ inst✝.toStruc ⊧ p | L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
p : Sentence L
hp : p ∈ T
this : U ⊢ p
⊢ inst✝.toStruc ⊧ p |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_provably_subtheory | [9, 1] | [12, 54] | exact consequence_iff'.{u, w}.mp (sound! ⟨this⟩) M | L : Language
M : Type w
inst✝¹ : Nonempty M
inst✝ : Structure L M
T U V : Theory L
x✝ : T ≼ U
h : M ⊧ₘ* U
p : Sentence L
hp : p ∈ T
this : U ⊢ p
⊢ inst✝.toStruc ⊧ p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_add_left | [16, 1] | [16, 112] | simp [Theory.add_def] | L : Language
M : Type w
inst✝² : Nonempty M
inst✝¹ : Structure L M
T U V : Theory L
inst✝ : M ⊧ₘ* T + U
⊢ T ⊆ T + U | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/FirstOrder/Completeness/Lemmata.lean | LO.FirstOrder.ModelsTheory.of_add_right | [18, 1] | [18, 113] | simp [Theory.add_def] | L : Language
M : Type w
inst✝² : Nonempty M
inst✝¹ : Structure L M
T U V : Theory L
inst✝ : M ⊧ₘ* T + U
⊢ U ⊆ T + U | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/DP.lean | LO.Propositional.Superintuitionistic.Kripke.satisfies_left_on_IntDPCounterexampleModel | [70, 1] | [88, 79] | induction p using rec' generalizing w with
| himp p q ihp ihq =>
constructor;
. simp only [Satisfies.imp_def];
intro h X hWX hp;
obtain ⟨x, hx, ex⟩ : ∃ x, w ≺ x ∧ (Sum.inr $ Sum.inl x) = X := by
replace hWX : (IntDPCounterexampleModel M₁ M₂ w₁ w₂).Frame.Rel _ X := hWX;
simp [IntDPCounterexampleF... | α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p q : Formula α
M₁ M₂ : Model α
w w₁ : M₁.World
w₂ : M₂.World
⊢ Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/DP.lean | LO.Propositional.Superintuitionistic.Kripke.satisfies_left_on_IntDPCounterexampleModel | [70, 1] | [88, 79] | constructor | case himp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁ w ... | case himp.mp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/DP.lean | LO.Propositional.Superintuitionistic.Kripke.satisfies_left_on_IntDPCounterexampleModel | [70, 1] | [88, 79] | . simp only [Satisfies.imp_def];
intro h X hWX hp;
obtain ⟨x, hx, ex⟩ : ∃ x, w ≺ x ∧ (Sum.inr $ Sum.inl x) = X := by
replace hWX : (IntDPCounterexampleModel M₁ M₂ w₁ w₂).Frame.Rel _ X := hWX;
simp [IntDPCounterexampleFrame] at hWX;
split at hWX;
all_goals simp_all;
subst ex;
exact ihq.mp $ h hx ... | case himp.mp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁... | case himp.mpr
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/DP.lean | LO.Propositional.Superintuitionistic.Kripke.satisfies_left_on_IntDPCounterexampleModel | [70, 1] | [88, 79] | . simp only [Satisfies.imp_def];
intro h v hv hp;
exact ihq.mpr $ h (by simpa [IntDPCounterexampleModel]) $ ihp.mp hp; | case himp.mpr
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M... | no goals |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/DP.lean | LO.Propositional.Superintuitionistic.Kripke.satisfies_left_on_IntDPCounterexampleModel | [70, 1] | [88, 79] | simp only [Satisfies.imp_def] | case himp.mp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁... | case himp.mp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁... |
https://github.com/iehality/lean4-logic.git | 9cee05ba7c48d586f7e488ef44f6445dea8102f8 | Logic/Propositional/Superintuitionistic/Kripke/DP.lean | LO.Propositional.Superintuitionistic.Kripke.satisfies_left_on_IntDPCounterexampleModel | [70, 1] | [88, 79] | intro h X hWX hp | case himp.mp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁... | case himp.mp
α : Type u_1
inst✝² : Inhabited α
inst✝¹ : DecidableEq α
inst✝ : Encodable α
p✝ q✝ : Formula α
M₁ M₂ : Model α
w₁ : M₁.World
w₂ : M₂.World
p q : Formula α
ihp : ∀ {w : M₁.World}, Satisfies M₁ w p ↔ Satisfies (IntDPCounterexampleModel M₁ M₂ w₁ w₂) (Sum.inr (Sum.inl w)) p
ihq : ∀ {w : M₁.World}, Satisfies M₁... |
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