Step
*
of Lemma
tsqrt-property
∀[n:ℕ]. ((t(tsqrt(n)) ≤ n) ∧ n < t(tsqrt(n) + 1))
BY
{ TACTIC:((D 0 THENA Auto)
THEN Unfold `tsqrt` 0
THEN (GenConcl ⌜isqrt(2 * n) = r ∈ ℕ⌝⋅ THENA Auto)
THEN (CallByValueReduce 0 THENA Auto)
THEN (InstLemma `isqrt-property` [⌜2 * n⌝]⋅ THENA Auto)
THEN HypSubst' (-2) (-1)) }
1
1. n : ℕ
2. r : ℕ@i
3. isqrt(2 * n) = r ∈ ℕ
4. ((r * r) ≤ (2 * n)) ∧ 2 * n < (r + 1) * (r + 1)
⊢ (t(if (r * r) + r ≤z 2 * n then r else r - 1 fi ) ≤ n) ∧ n < t(if (r * r) + r ≤z 2 * n then r else r - 1 fi + 1)
Latex:
Latex:
\mforall{}[n:\mBbbN{}]. ((t(tsqrt(n)) \mleq{} n) \mwedge{} n < t(tsqrt(n) + 1))
By
Latex:
TACTIC:((D 0 THENA Auto)
THEN Unfold `tsqrt` 0
THEN (GenConcl \mkleeneopen{}isqrt(2 * n) = r\mkleeneclose{}\mcdot{} THENA Auto)
THEN (CallByValueReduce 0 THENA Auto)
THEN (InstLemma `isqrt-property` [\mkleeneopen{}2 * n\mkleeneclose{}]\mcdot{} THENA Auto)
THEN HypSubst' (-2) (-1))
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