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Proof step 18 from proof coef-le-2pot (previous step)

2 ^ (2 * j) * 2 = Σ countFun(0, 2 * j + 1, λ j0. binom_coef(2 * j + 1, j0)), binom_coef(2 * j + 1, j) = binom_coef(2 * j + 1, j + 1), 0 ≤ j, 0 < j, 2 ^ (2 * j) < binom_coef(2 * j + 1, j + 1) ⊦ ∃ i1, j1. 0 ≤ i1 ∧ i1 ≤ 2 * j + 1 ∧ binom_coef(2 * j + 1, i1) = binom_coef(2 * j + 1, j + 1) ∧ 0 ≤ j1 ∧ j1 ≤ 2 * j + 1 ∧ binom_coef(2 * j + 1, j1) = binom_coef(2 * j + 1, j + 1) ∧ i1 ≠ j1

User applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: j, j + 1 abort: F discard: T computed: T

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%le-add-02: ⊦ i + j0 ≤ i0 ∧ j0 ≥ 0 → i ≤ i0; from specification int-abs%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%le-add: ⊦ i + j ≤ i0 + j ↔ i ≤ i0; from specification int-abs%le-same-01: ⊦ 0 < i → (i ≤ i * j ↔ 1 ≤ j); from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%eq: ⊦ i = i + j ↔ j = 0; from specification int-basic2 Used terms: j, j + 1