internlm/Lean-Github huggingface dataset:
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
h2 : Γ ⊆ Δ
⊢ Δ ⊢ A
Tactic:
induction h generalizing Δ
Proof state after:
case ax
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : A✝ ∈ Γ✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝

case impI
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
a_ih✝ : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ ⇒ B✝

case impE
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝ ⇒ B✝
a✝ : Γ✝ ⊢ A✝
a_ih✝¹ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ ⇒ B✝
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝

case andI
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝
a✝ : Γ✝ ⊢ B✝
a_ih✝¹ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ && B✝

case andE1
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ && B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝

case andE2
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ && B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝

case orI1
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ || B✝

case orI2
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ || B✝

case orE
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ C✝ : Formula
a✝² : Γ✝ ⊢ A✝ || B✝
a✝¹ : insert A✝ Γ✝ ⊢ C✝
a✝ : insert B✝ Γ✝ ⊢ C✝
a_ih✝² : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ || B✝
a_ih✝¹ : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ C✝
a_ih✝ : ∀ {Δ : Set Formula}, insert B✝ Γ✝ ⊆ Δ → Δ ⊢ C✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ C✝

case botE
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
a_ih✝ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ bot
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : A✝ ∈ Γ✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
case ax => apply ax; solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ ⇒ B✝
Tactic:
case impI ih => apply impI; solve_by_elim [insert_subset_insert]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝ ⇒ B✝
a✝ : Γ✝ ⊢ A✝
ih₁ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ ⇒ B✝
ih₂ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝
Tactic:
case impE ih₁ ih₂ => apply impE <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝
a✝ : Γ✝ ⊢ B✝
ih₁ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
ih₂ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ && B✝
Tactic:
case andI ih₁ ih₂ => apply andI <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ && B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
case andE1 ih => apply andE1 <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ && B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝
Tactic:
case andE2 ih => apply andE2 <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ || B✝
Tactic:
case orI1 ih => apply orI1; solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ || B✝
Tactic:
case orI2 ih => apply orI2; solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ C✝ : Formula
a✝² : Γ✝ ⊢ A✝ || B✝
a✝¹ : insert A✝ Γ✝ ⊢ C✝
a✝ : insert B✝ Γ✝ ⊢ C✝
ih₁ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ || B✝
ih₂ : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ C✝
ih₃ : ∀ {Δ : Set Formula}, insert B✝ Γ✝ ⊆ Δ → Δ ⊢ C✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ C✝
Tactic:
case orE ih₁ ih₂ ih₃ => apply orE <;> solve_by_elim [insert_subset_insert]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ bot
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
case botE ih => apply botE; solve_by_elim [insert_subset_insert]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : A✝ ∈ Γ✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
apply ax
Proof state after:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : A✝ ∈ Γ✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ A✝ ∈ Δ
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : A✝ ∈ Γ✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ A✝ ∈ Δ
Tactic:
solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ ⇒ B✝
Tactic:
apply impI
Proof state after:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ insert A✝ Δ ⊢ B✝
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ insert A✝ Δ ⊢ B✝
Tactic:
solve_by_elim [insert_subset_insert]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝ ⇒ B✝
a✝ : Γ✝ ⊢ A✝
ih₁ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ ⇒ B✝
ih₂ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝
Tactic:
apply impE <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝
a✝ : Γ✝ ⊢ B✝
ih₁ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
ih₂ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ && B✝
Tactic:
apply andI <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ && B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
apply andE1 <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ && B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝
Tactic:
apply andE2 <;> solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ || B✝
Tactic:
apply orI1
Proof state after:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝ || B✝
Tactic:
apply orI2
Proof state after:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ B✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ B✝
Tactic:
solve_by_elim
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ B✝ C✝ : Formula
a✝² : Γ✝ ⊢ A✝ || B✝
a✝¹ : insert A✝ Γ✝ ⊢ C✝
a✝ : insert B✝ Γ✝ ⊢ C✝
ih₁ : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ A✝ || B✝
ih₂ : ∀ {Δ : Set Formula}, insert A✝ Γ✝ ⊆ Δ → Δ ⊢ C✝
ih₃ : ∀ {Δ : Set Formula}, insert B✝ Γ✝ ⊆ Δ → Δ ⊢ C✝
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ C✝
Tactic:
apply orE <;> solve_by_elim [insert_subset_insert]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ bot
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ A✝
Tactic:
apply botE
Proof state after:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ bot
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ bot
Theorem: IntuitionisticPropositionalLogic.weakening
Proof state before:
case a
H : Type ?u.33194
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
ih : ∀ {Δ : Set Formula}, Γ✝ ⊆ Δ → Δ ⊢ bot
Δ : Set Formula
h2 : Γ✝ ⊆ Δ
⊢ Δ ⊢ bot
Tactic:
solve_by_elim [insert_subset_insert]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.ProvableFrom.insert
Proof state before:
H : Type ?u.36542
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ Insert.insert B Γ ⊢ A
Tactic:
apply weakening h
Proof state after:
H : Type ?u.36542
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ Γ ⊆ Insert.insert B Γ
Theorem: IntuitionisticPropositionalLogic.ProvableFrom.insert
Proof state before:
H : Type ?u.36542
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ Γ ⊆ Insert.insert B Γ
Tactic:
exact subset_insert B Γ
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.deduction_theorem
Proof state before:
H : Type ?u.36610
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ insert (A ⇒ B) Γ ⊢ B
Tactic:
apply impE (ax $ mem_insert _ _)
Proof state after:
H : Type ?u.36610
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ insert (A ⇒ B) Γ ⊢ A
Theorem: IntuitionisticPropositionalLogic.deduction_theorem
Proof state before:
H : Type ?u.36610
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ insert (A ⇒ B) Γ ⊢ A
Tactic:
exact h.insert
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.Provable.mp
Proof state before:
H : Type ?u.36701
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h1 : Provable (A ⇒ B)
h2 : Γ ⊢ A
⊢ Γ ⊢ B
Tactic:
apply impE _ h2
Proof state after:
H : Type ?u.36701
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h1 : Provable (A ⇒ B)
h2 : Γ ⊢ A
⊢ Γ ⊢ A ⇒ B
Theorem: IntuitionisticPropositionalLogic.Provable.mp
Proof state before:
H : Type ?u.36701
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h1 : Provable (A ⇒ B)
h2 : Γ ⊢ A
⊢ Γ ⊢ A ⇒ B
Tactic:
apply weakening h1
Proof state after:
H : Type ?u.36701
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h1 : Provable (A ⇒ B)
h2 : Γ ⊢ A
⊢ ∅ ⊆ Γ
Theorem: IntuitionisticPropositionalLogic.Provable.mp
Proof state before:
H : Type ?u.36701
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h1 : Provable (A ⇒ B)
h2 : Γ ⊢ A
⊢ ∅ ⊆ Γ
Tactic:
exact empty_subset Γ
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H : Type ?u.36751
inst✝ : HeytingAlgebra H
v : Variable → H
A B : Formula
Γ Δ : Set Formula
h : Γ ⊢ A
⊢ Γ ⊨ A
Tactic:
induction h <;> intros H hH v c hv
Proof state after:
case ax
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ : Formula
a✝ : A✝ ∈ Γ✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v A✝

case impI
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
a_ih✝ : insert A✝ Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ ⇒ B✝)

case impE
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝ ⇒ B✝
a✝ : Γ✝ ⊢ A✝
a_ih✝¹ : Γ✝ ⊨ A✝ ⇒ B✝
a_ih✝ : Γ✝ ⊨ A✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v B✝

case andI
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝
a✝ : Γ✝ ⊢ B✝
a_ih✝¹ : Γ✝ ⊨ A✝
a_ih✝ : Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ && B✝)

case andE1
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
a_ih✝ : Γ✝ ⊨ A✝ && B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v A✝

case andE2
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
a_ih✝ : Γ✝ ⊨ A✝ && B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v B✝

case orI1
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
a_ih✝ : Γ✝ ⊨ A✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ || B✝)

case orI2
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
a_ih✝ : Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ || B✝)

case orE
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ C✝ : Formula
a✝² : Γ✝ ⊢ A✝ || B✝
a✝¹ : insert A✝ Γ✝ ⊢ C✝
a✝ : insert B✝ Γ✝ ⊢ C✝
a_ih✝² : Γ✝ ⊨ A✝ || B✝
a_ih✝¹ : insert A✝ Γ✝ ⊨ C✝
a_ih✝ : insert B✝ Γ✝ ⊨ C✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v C✝

case botE
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
a_ih✝ : Γ✝ ⊨ bot
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v A✝
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : insert A✝ Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ ⇒ B✝)
Tactic:
case impI ih =>
  simp
  apply ih
  simp
  intros B hB
  exact inf_le_of_left_le (hv B hB)
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝ ⇒ B✝
a✝ : Γ✝ ⊢ A✝
ih₁ : Γ✝ ⊨ A✝ ⇒ B✝
ih₂ : Γ✝ ⊨ A✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v B✝
Tactic:
case impE ih₁ ih₂ =>
  specialize ih₁ hv
  simp at ih₁
  rwa [inf_eq_left.mpr (ih₂ hv)] at ih₁
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝¹ : Γ✝ ⊢ A✝
a✝ : Γ✝ ⊢ B✝
ih₁ : Γ✝ ⊨ A✝
ih₂ : Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ && B✝)
Tactic:
case andI ih₁ ih₂ => simp [ih₁ hv, ih₂ hv]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
ih : Γ✝ ⊨ A✝ && B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v A✝
Tactic:
case andE1 ih =>
  specialize ih hv
  simp at ih
  exact ih.1
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝ && B✝
ih : Γ✝ ⊨ A✝ && B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v B✝
Tactic:
case andE2 ih =>
  specialize ih hv
  simp at ih
  exact ih.2
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ A✝
ih : Γ✝ ⊨ A✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ || B✝)
Tactic:
case orI1 ih =>
  simp
  exact le_trans (ih hv) le_sup_left
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : Γ✝ ⊢ B✝
ih : Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ || B✝)
Tactic:
case orI2 ih =>
  simp
  exact le_trans (ih hv) le_sup_right
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A✝ B✝ : Formula
Γ✝ Δ Γ : Set Formula
A B C : Formula
_h1 : Γ ⊢ A || B
_h2 : insert A Γ ⊢ C
_h3 : insert B Γ ⊢ C
ih₁ : Γ ⊨ A || B
ih₂ : insert A Γ ⊨ C
ih₃ : insert B Γ ⊨ C
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ, c ≤ eval v B
⊢ c ≤ eval v C
Tactic:
case orE Γ A B C _h1 _h2 _h3 ih₁ ih₂ ih₃ =>
  specialize ih₁ hv
  have h2v : ∀ D ∈ insert A Γ, c ⊓ eval v A ≤ eval v D := by
    simp; intros D hD; exact inf_le_of_left_le (hv D hD) have h3v : ∀ D ∈ insert B Γ, c ⊓ eval v B ≤ eval v D := by
    simp; intros D hD; exact inf_le_of_left_le (hv D hD) simp at ih₁
  rw [← inf_eq_left.mpr ih₁, inf_sup_left]
  rw [← sup_idem (a := eval v C)]
  exact sup_le_sup (ih₂ h2v) (ih₃ h3v)
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ : Formula
a✝ : Γ✝ ⊢ bot
ih : Γ✝ ⊨ bot
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v A✝
Tactic:
case botE ih =>
  specialize ih hv
  simp at ih
  simp [ih]
Proof state after:
no goals
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : insert A✝ Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ≤ eval v (A✝ ⇒ B✝)
Tactic:
simp
Proof state after:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : insert A✝ Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ⊓ eval v A✝ ≤ eval v B✝
Theorem: IntuitionisticPropositionalLogic.soundness_theorem
Proof state before:
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : insert A✝ Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ c ⊓ eval v A✝ ≤ eval v B✝
Tactic:
apply ih
Proof state after:
case a
H✝ : Type ?u.36751
inst✝ : HeytingAlgebra H✝
v✝ : Variable → H✝
A B : Formula
Γ Δ Γ✝ : Set Formula
A✝ B✝ : Formula
a✝ : insert A✝ Γ✝ ⊢ B✝
ih : insert A✝ Γ✝ ⊨ B✝
H : Type
hH : HeytingAlgebra H
v : Variable → H
c : H
hv : ∀ B ∈ Γ✝, c ≤ eval v B
⊢ ∀ B ∈ insert A✝ Γ✝, c ⊓ eval v A✝ ≤ eval v B