Nuprl Lemma : tsqrt-unique
∀[n,x:ℕ]. (((t(x) ≤ n) ∧ n < t(x + 1)) ⇒ (x = tsqrt(n) ∈ ℤ))
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,
implies: P ⇒ Q,
and: P ∧ Q,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
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,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T}
Lemmas referenced :
tsqrt-property,
le_wf,
triangular-num_wf,
less_than_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_wf,
decidable__lt,
tsqrt_wf,
triangular-num-le,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
add_nat_wf,
false_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
productEquality,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
independent_functionElimination
Latex:
\mforall{}[n,x:\mBbbN{}]. (((t(x) \mleq{} n) \mwedge{} n < t(x + 1)) {}\mRightarrow{} (x = tsqrt(n)))
Date html generated:
2019_06_20-PM-02_38_45
Last ObjectModification:
2019_06_12-PM-00_27_02
Theory : num_thy_1
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