[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 19 from proof uint2bits-add-high (previous step)

0 ≤ i, 0 ≤ 2 ^ i + i, 0 ≤ 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 2 < 2 ^ i + i, 2 ^ (i - 1) + 2 < 2 ^ i + i, 2 ^ (i - 1) < 2 ^ i, 1 < 2 ^ i, 0 < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 < 2 ^ i + 2 ^ i, 2 ^ (i - 1) + 2 < 2 ^ i + 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ false ' + true ' = sgn(sgn(sgn(false ' + (false ' + true ' + mklist(false, i→n(i)))+1).rest).rest)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ bits ≠ [] ∧ ¬ bits.last → bits +1 = bits.butlast + true
Proof step 21:
¬(false ' + true ' + mklist(false, i→n(i)) ≠ [] ∧ ¬ (false ' + true ' + mklist(false, i→n(i))).last), 0 ≤ i, 0 ≤ 2 ^ i + i, 0 ≤ 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 2 < 2 ^ i + i, 2 ^ (i - 1) + 2 < 2 ^ i + i, 2 ^ (i - 1) < 2 ^ i, 1 < 2 ^ i, 0 < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 < 2 ^ i + 2 ^ i, 2 ^ (i - 1) + 2 < 2 ^ i + 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ false ' + true ' = sgn(sgn(sgn(false ' + (false ' + true ' + mklist(false, i→n(i)))+1).rest).rest)
Proof step 26:
false ' + true ' + mklist(false, i→n(i)) ≠ [] ∧ ¬ (false ' + true ' + mklist(false, i→n(i))).last, 0 ≤ i, 0 ≤ 2 ^ i + i, 0 ≤ 2 ^ i, 0 < 2 ^ i, 0 ≤ 2 ^ i + 2 ^ i, 2 < 2 ^ i + i, 2 ^ (i - 1) + 2 < 2 ^ i + i, 2 ^ (i - 1) < 2 ^ i, 1 < 2 ^ i, 0 < i, 2 ^ (i - 1) + 1 < 2 ^ i, 2 < 2 ^ i + 2 ^ i, 2 ^ (i - 1) + 2 < 2 ^ i + 2 ^ i, ∀ i0, j. i→n(i0) < i→n(i) → true → j < 0 ∧ 0 < i0 ∧ 2 ^ (i0 - 1) < abs(j) ∧ abs(j) ≤ 2 ^ i0 → uint2bits(j + 2 ^ i0) = sgn(int2bits(j).rest) ⊦ false ' + true ' = sgn(sgn(sgn(false ' + ((false ' + true ' + mklist(false, i→n(i))).butlast + true)).rest).rest)

Rule argument

seq: ⊦ bits ≠ [] ∧ ¬ bits.last → bits +1 = bits.butlast + true subst: [bits] → [false ' + true ' + mklist(false, i→n(i))] rotate: F spec: bitopsaux inst: name: succ-zero-03

Additional information

Used terms: false ' + true ' + mklist(false, i→n(i))