lambda a,b. 1-a*b lambda x. NAND(x,x) lambda a,b. NAND(NAND(a,b),NAND(a,b)) lambda a,b. NAND(NAND(a,a),NAND(b,b)) lambda a,b. OR(NOT(a),b) lambda a,b. AND(IMPLIES(a,b),IMPLIES(b,a))

ADMISSIBLE(x) := AND_{I in I_set} I(x)

PRESERVE(x,F) := IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x)))

COLLAPSE(x,F) := AND(ADMISSIBLE(x), OR_{I in I_set} NOT(I(F(x))))

UNIVERSAL(F) := AND_{x} IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x)))

COUNTERMODEL(F) := OR_{x} COLLAPSE(x,F)

FIXED_POINT(x,F) := EQUAL(F(x), x)

CLOSED(F,I_set) := AND_{x} IMPLIES(ADMISSIBLE(x), PRESERVE(x,F))

FAILURE(F,I_set) := OR_{x} AND(ADMISSIBLE(x), NOT(PRESERVE(x,F)))

UNIVERSAL(F) ≡ CLOSED(F, I_set) UNIVERSAL(F) := forall x. ADMISSIBLE(x) → ADMISSIBLE(F(x)) CLOSED(F,I_set) := forall x. ADMISSIBLE(x) → PRESERVE(x,F) PRESERVE(x,F) := ADMISSIBLE(x) → ADMISSIBLE(F(x)) Substitute: CLOSED(F,I_set) := forall x. ADMISSIBLE(x) → (ADMISSIBLE(x) → ADMISSIBLE(F(x))) By IMPLIES idempotence: (A → (A → B)) ≡ (A → B) Therefore: CLOSED(F,I_set) ≡ forall x. ADMISSIBLE(x) → ADMISSIBLE(F(x)) ≡ UNIVERSAL(F) EQUAL(UNIVERSAL(F), CLOSED(F,I_set)) = 1 COUNTERMODEL(F) ≡ FAILURE(F, I_set) COUNTERMODEL(F) := exists x. COLLAPSE(x,F) COLLAPSE(x,F) := ADMISSIBLE(x) ∧ exists I. ¬I(F(x)) exists I. ¬I(F(x)) ≡ ¬ADMISSIBLE(F(x)) COLLAPSE(x,F) ≡ ADMISSIBLE(x) ∧ ¬ADMISSIBLE(F(x)) ¬PRESERVE(x,F) ≡ ADMISSIBLE(x) ∧ ¬ADMISSIBLE(F(x)) Therefore COLLAPSE(x,F) ≡ ¬PRESERVE(x,F) COUNTERMODEL(F) := exists x. ¬PRESERVE(x,F) ≡ FAILURE(F,I_set) EQUAL(COUNTERMODEL(F), FAILURE(F,I_set)) = 1 lambda x. forall I in I_set. I(x) lambda x,F. IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x))) lambda x,F. AND(ADMISSIBLE(x), OR_{I in I_set}. NOT(I(F(x)))) lambda F. forall x. IMPLIES(ADMISSIBLE(x), ADMISSIBLE(F(x))) lambda F. exists x. COLLAPSE(x,F) lambda x,F. EQUAL(F(x),x) lambda F,I_set. forall x. IMPLIES(ADMISSIBLE(x), PRESERVE(x,F)) lambda F,I_set. exists x. AND(ADMISSIBLE(x), NOT(PRESERVE(x,F))) UNIVERSAL ≡ CLOSED COUNTERMODEL ≡ FAILURE PROVEN 0xUNIVERSAL_CLOSURE_A1B2C3D4E5F6 2026-01-15T00:00:00Z