I_set = {I1, I2, I3} F(x,y) = (x+1, y+x)

ADMISSIBLE(x,y) = (x>=0) AND (y>=0) AND (x+y<=10)

66 states in Z x Z

I1(F(x,y)) = x+1 >= 0 ← satisfied whenever x >= -1 I2(F(x,y)) = y+x >= 0 ← satisfied when x,y >= 0 I3(F(x,y)) = (x+1)+(y+x) <= 10 ≡ 2x+y <= 9

Given x>=0: I1 always satisfied. Given x,y>=0: I2 always satisfied. Binding constraint: 2x+y <= 9. PRESERVE(x,y,F) = IMPLIES(ADMISSIBLE(x,y), 2x+y <= 9)

COLLAPSE(x,y,F) = ADMISSIBLE(x,y) AND (2x+y > 9)

C1: y+x < 0 — impossible when x,y >= 0 Enumeration below is illustrative, not exhaustive. See exact predicate. M(F) = exists (x,y) in Z^2: x>=0 AND y>=0 AND x+y<=10 AND 2x+y>9 (5,0) — F(5,0)=(6,5), 6+5=11 > 10 violates I3 (0,10) also qualifies: 2(0)+10=10 > 9 x+1 = x → 1 = 0 (CONTRADICTION) y+x = y → x = 0 Fix(F) = empty set 66 IMPLIES(x>=0 AND y>=0 AND x+y<=10, 2x+y<=9) x>=0 AND y>=0 AND x+y<=10 AND 2x+y>9 false — F is not closed under I_set true — exact predicate above; minimal witness (5,0) empty false (equivalent to UNIVERSAL = false) true (equivalent to COUNTERMODEL = true) UNIVERSAL ≡ CLOSED: both FALSE COUNTERMODEL ≡ FAILURE: both TRUE INSTANTIATED_AND_PROVEN 0xINST_F_xy_3F8A7B2E9C4D 2026-01-15T00:00:01Z