@@ -209,63 +209,48 @@ object Kernel2Code extends lisa.Main {
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}
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val s_2 = have(∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ⊢ (Q (f(x)) ==> Q (f(f(x))))) subproof {
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val s_2_0 = have(∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ⊢ (Q (f(x)) ==> Q (f(f(x))))) by InstFunSchema (Map (x -> f(x)))(s_1)
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+ val s_2_1 = have(∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ⊢ (Q (f(x)) ==> Q (f(f(x))))) by InstFunSchema (Map (x -> f(x)))(s_1)
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}
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val s_3 = have(∀ (x, (Q (x) ==> Q (f(x)))) ⊢ (Q (x) ==> Q (f(f(x))))) subproof {
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val s_3_0 = have(() ⊢ (∀ (x, (Q (x) ==> Q (f(x)))) ==> (Q (x) ==> Q (f(x))))) by Restate .from(s_1)
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val s_3_1 = have(() ⊢ (∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ==> (Q (f(x)) ==> Q (f(f(x)))))) by Restate .from(s_2)
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val s_3_2 = have((∀ (x, (Q (x) ==> Q (f(x)))), (∀ (x, (Q (x) ==> Q (f(x)))) ==> (Q (x) ==> Q (f(x)))), (∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ==> (Q (f(x)) ==> Q (f(f(x)))))) ⊢ (Q (x) ==> Q (f(f(x))))) subproof {
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val s_3_2_0 = have(
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- Q (f(x)) ⊢ ¬ ((((( ¬ ((( ∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊤ ) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((( ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (⊤ )))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x) ) ∧ ¬ (Q (f(f(x))))))
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+ Q (f(x)) ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊤ ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (⊤ ))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Restate
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val s_3_2_1 = thenHave(
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- Q (f(x)) ⊢ ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (Q (f(f(x)))))
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- )
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- ) by RightSubstIff (
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+ Q (f(x)) ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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+ ) by RightSubstIff .withParametersSimple(
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List (((Q (f(x))), (⊤ ))),
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lambda(
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Seq (MaRvIn_1 ),
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- ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ MaRvIn_1 ) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (MaRvIn_1 )))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (
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- Q (f(f(x)))
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- ))
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- )
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+ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ MaRvIn_1 ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (MaRvIn_1 ))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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)
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)
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val s_3_2_2 = have(
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- ¬ (Q (f(x))) ⊢ ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊥ ) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (⊥ )))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (Q (f(f(x)))))
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- )
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+ ¬ (Q (f(x))) ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊥ ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (⊥ ))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Restate
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val s_3_2_3 = thenHave(
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¬ (Q (f(x))) ⊢ ¬ (
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- (((( ¬ ((( ∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((( ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x) ) ∧ ¬ (Q (f(f(x)))))
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+ (¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x)))))
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)
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- ) by RightSubstIff (
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+ ) by RightSubstIff .withParametersSimple (
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List (((Q (f(x))), (⊥ ))),
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lambda(
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Seq (MaRvIn_1 ),
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- ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ MaRvIn_1 ) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (MaRvIn_1 )))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (
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- Q (f(f(x)))
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- ))
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- )
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+ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ MaRvIn_1 ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (MaRvIn_1 ))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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)
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)
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val s_3_2_4 = thenHave(
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() ⊢ (Q (f(x)), ¬ (
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- (((( ¬ ((( ∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((( ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x) ) ∧ ¬ (Q (f(f(x)))))
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+ (¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x)))))
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))
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) by Restate
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val s_3_2_5 = have(
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- () ⊢ ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (Q (f(f(x)))))
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- )
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+ () ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Cut (s_3_2_4, s_3_2_1)
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val s_3_2_6 = thenHave(
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- () ⊢ ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (Q (f(f(x)))))
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- )
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+ () ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Restate
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val s_3_2_7 =
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thenHave((∀ (x, (Q (x) ==> Q (f(x)))), (∀ (x, (Q (x) ==> Q (f(x)))) ==> (Q (x) ==> Q (f(x)))), (∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ==> (Q (f(x)) ==> Q (f(f(x)))))) ⊢ (Q (x) ==> Q (f(f(x))))) by Restate
@@ -282,25 +267,21 @@ object Kernel2Code extends lisa.Main {
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val s_3 = have(() ⊢ (∀ (x, (Q (x) ==> Q (f(x)))) ==> (Q (x) ==> Q (f(x))))) by Restate .from(s_1)
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val s_4 = have(() ⊢ (∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ==> (Q (f(x)) ==> Q (f(f(x)))))) by Restate .from(s_2)
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val s_5 = have(
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- Q (f(x)) ⊢ ¬ ((((( ¬ ((( ∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊤ ) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((( ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (⊤ )))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x) ) ∧ ¬ (Q (f(f(x))))))
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+ Q (f(x)) ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊤ ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (⊤ ))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Restate
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val s_6 = thenHave(
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- Q (f(x)) ⊢ ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (Q (f(f(x)))))
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- )
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+ Q (f(x)) ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Restate
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val s_7 = have(
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- ¬ (Q (f(x))) ⊢ ¬ ((((( ¬ ((( ∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊥ ) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((( ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (⊥ )))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x) ) ∧ ¬ (Q (f(f(x))))))
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+ ¬ (Q (f(x))) ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ ⊥ ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (⊥ ))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Restate
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val s_8 = thenHave(
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() ⊢ (Q (f(x)), ¬ (
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- (((( ¬ ((( ∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((( ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x) ) ∧ ¬ (Q (f(f(x)))))
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+ (¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x)))))
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))
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) by Restate
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val s_9 = have(
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- () ⊢ ¬ (
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- ((((¬ (((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x))) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ (((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x)) ∧ ¬ (Q (f(x)))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x))))))) ∧ Q (x)) ∧ ¬ (Q (f(f(x)))))
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- )
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+ () ⊢ ¬ ((¬ ((∀ (x_1, ¬ ((Q (x_1) ∧ ¬ (Q (f(x_1)))))) ∧ Q (f(x)) ∧ ¬ (Q (f(f(x)))))) ∧ ¬ ((∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(x))))) ∧ ∀ (x, ¬ ((Q (x) ∧ ¬ (Q (f(x)))))) ∧ Q (x) ∧ ¬ (Q (f(f(x))))))
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) by Cut (s_8, s_6)
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val s_10 = thenHave((∀ (x, (Q (x) ==> Q (f(x)))), (∀ (x, (Q (x) ==> Q (f(x)))) ==> (Q (x) ==> Q (f(x)))), (∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ==> (Q (f(x)) ==> Q (f(f(x)))))) ⊢ (Q (x) ==> Q (f(f(x))))) by Restate
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val s_11 = have((∀ (x, (Q (x) ==> Q (f(x)))), (∀ (x_1, (Q (x_1) ==> Q (f(x_1)))) ==> (Q (f(x)) ==> Q (f(f(x)))))) ⊢ (Q (x) ==> Q (f(f(x))))) by Cut (s_3, s_10)
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