[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 35 from proof shiftzero-max-neg-01 (previous step)

j ≤ 1 + n→i(n - 1), i < 0, i < ~1, # int2bits(i) ≤ n, j ≤ n→i(n), 2 ^ (n→i(n) - j) ≤ >>>(n - 1, (i - 1) / 2, j - 1), 1 < j, 0 < j, ∀ n, i, j0. i→n(j0) < i→n(j) → 0 < j0 ∧ i < 0 ∧ 2 ^ (n→i(n) - j0) ≤ >>>(n, i, j0) ∧ j0 ≤ n→i(n) ∧ # int2bits(i) ≤ n → n = 0 ⊦ n = 1, n = 0, n - 1 = 0, ¬ n ≤ # int2bits((i - 1) / 2) ∧ ¬ >>>(n - 1, (i - 1) / 2, j - 1) < 2 ^ (n→i(n - 1) + 1 - j)

User applied rule cut formula

And obtained the following new goals

Proof step 36:
j ≤ 1 + n→i(n - 1), i < 0, i < ~1, # int2bits(i) ≤ n, j ≤ n→i(n), 2 ^ (n→i(n) - j) ≤ >>>(n - 1, (i - 1) / 2, j - 1), 1 < j, 0 < j, ∀ n, i, j0. i→n(j0) < i→n(j) → 0 < j0 ∧ i < 0 ∧ 2 ^ (n→i(n) - j0) ≤ >>>(n, i, j0) ∧ j0 ≤ n→i(n) ∧ # int2bits(i) ≤ n → n = 0 ⊦ ¬ n ≤ # int2bits((i - 1) / 2), n = 1, n = 0, n - 1 = 0, ¬ n ≤ # int2bits((i - 1) / 2) ∧ ¬ >>>(n - 1, (i - 1) / 2, j - 1) < 2 ^ (n→i(n - 1) + 1 - j)

Rule argument

fma: ¬ n ≤ # int2bits((i - 1) / 2)

Additional information

Used simplifier rules: ⊦ ¬ m ≤ n ↔ n < m;% ⊦ m < n → n ≠ 0;% ⊦ m < n → (n ≤ m ↔ false);% ⊦ m < n → n ≠ 0; (forward)% ⊦ i ≤ j ∧ j < j0 → i < j0; (forward)% ⊦ i + j = j + i;% ⊦ i + j + i0 = (i + j) + i0;% ⊦ i + 0 = i;% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j;%0 ≤ j ⊦ 2 ^ j < 2 ^ j0 ↔ j < j0;% ⊦ i - j < i0 ↔ i < j + i0;% ⊦ i + (j - i) = j;% ⊦ i < j + 1 ↔ i ≤ j;% ⊦ n→i(m) ≤ n→i(n) ↔ m ≤ n;% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1);% ⊦ m ≠ n → (m ≤ n ↔ m < n);%n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n;% ⊦ m + m0 ≤ m ↔ m0 = 0;% ⊦ (m + n0) + n1 = m + n0 + n1;% ⊦ m + n = n + m; Used formulas: ¬ n ≤ # int2bits((i - 1) / 2)