Nuprl Lemma : tsqrt-property
∀[n:ℕ]. ((t(tsqrt(n)) ≤ n) ∧ n < t(tsqrt(n) + 1))
Proof
Definitions occuring in Statement :
tsqrt: tsqrt(n),
triangular-num: t(n),
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
tsqrt: tsqrt(n),
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
guard: {T},
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
isqrt_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
nat_wf,
equal_wf,
less_than'_wf,
triangular-num_wf,
tsqrt_wf,
member-less_than,
add_nat_wf,
false_wf,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
value-type-has-value,
set-value-type,
int-value-type,
isqrt-property,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
less_than_wf,
squash_wf,
true_wf,
triangular-num-add1,
iff_weakening_equal,
twice-triangular,
multiply-is-int-iff,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
subtract_wf,
subtract-add-cancel,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
independent_pairEquality,
applyEquality,
axiomEquality,
addEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
instantiate,
cumulativity,
callbyvalueReduce,
equalityElimination,
addLevel,
imageElimination,
imageMemberEquality,
universeEquality,
productEquality
Latex:
\mforall{}[n:\mBbbN{}]. ((t(tsqrt(n)) \mleq{} n) \mwedge{} n < t(tsqrt(n) + 1))
Date html generated:
2019_06_20-PM-02_38_39
Last ObjectModification:
2019_06_12-PM-00_26_57
Theory : num_thy_1
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