Nuprl Lemma : int-sq-root
∀x:ℕ. (∃r:ℕ [(((r * r) ≤ x) ∧ x < (r + 1) * (r + 1))])
Proof
Definitions occuring in Statement : 
nat: ℕ
, 
less_than: a < b
, 
le: A ≤ B
, 
all: ∀x:A. B[x]
, 
sq_exists: ∃x:A [B[x]]
, 
and: P ∧ Q
, 
multiply: n * m
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
rev_uimplies: rev_uimplies(P;Q)
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
top: Top
, 
exists: ∃x:A. B[x]
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
or: P ∨ Q
, 
decidable: Dec(P)
, 
ge: i ≥ j 
, 
subtype_rel: A ⊆r B
, 
cand: A c∧ B
, 
le: A ≤ B
, 
false: False
, 
sq_type: SQType(T)
, 
uimplies: b supposing a
, 
not: ¬A
, 
nequal: a ≠ b ∈ T 
, 
int_nzero: ℤ-o
, 
nat_plus: ℕ+
, 
sq_exists: ∃x:A [B[x]]
, 
implies: P 
⇒ Q
, 
so_apply: x[s]
, 
nat: ℕ
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
guard: {T}
, 
uall: ∀[x:A]. B[x]
, 
and: P ∧ Q
, 
true: True
, 
less_than': less_than'(a;b)
, 
squash: ↓T
, 
less_than: a < b
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
Lemmas referenced : 
le_weakening, 
less_than_functionality, 
int_term_value_subtract_lemma, 
itermSubtract_wf, 
subtract_wf, 
mul_preserves_le, 
iff_weakening_equal, 
subtype_rel_self, 
true_wf, 
squash_wf, 
int_term_value_add_lemma, 
int_term_value_mul_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_var_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_and_lemma, 
itermAdd_wf, 
itermMultiply_wf, 
intformeq_wf, 
itermVar_wf, 
intformle_wf, 
intformand_wf, 
decidable__le, 
int-value-type, 
equal_wf, 
set-value-type, 
int_formula_prop_wf, 
int_term_value_constant_lemma, 
int_formula_prop_less_lemma, 
istype-void, 
int_formula_prop_not_lemma, 
itermConstant_wf, 
intformless_wf, 
intformnot_wf, 
full-omega-unsat, 
decidable__lt, 
nat_plus_properties, 
nat_properties, 
nat_plus_subtype_nat, 
rem_bounds_1, 
div_rem_sum, 
false_wf, 
nat_plus_wf, 
nequal_wf, 
istype-int, 
int_subtype_base, 
subtype_base_sq, 
divide_wfa, 
istype-le, 
istype-nat, 
less_than_wf, 
le_wf, 
nat_wf, 
sq_exists_wf, 
istype-less_than, 
div_nat_induction-ext
Rules used in proof : 
promote_hyp, 
universeEquality, 
int_eqEquality, 
imageElimination, 
Error :dependent_set_memberFormation_alt, 
cutEval, 
Error :isect_memberEquality_alt, 
Error :dependent_pairFormation_alt, 
approximateComputation, 
unionElimination, 
applyEquality, 
productElimination, 
lambdaFormation, 
dependent_set_memberEquality, 
dependent_set_memberFormation, 
Error :universeIsType, 
sqequalBase, 
Error :equalityIstype, 
voidElimination, 
equalitySymmetry, 
equalityTransitivity, 
independent_isectElimination, 
intEquality, 
cumulativity, 
instantiate, 
Error :productIsType, 
Error :setIsType, 
Error :lambdaFormation_alt, 
independent_functionElimination, 
Error :inhabitedIsType, 
addEquality, 
because_Cache, 
rename, 
setElimination, 
multiplyEquality, 
productEquality, 
Error :lambdaEquality_alt, 
isectElimination, 
hypothesis, 
baseClosed, 
hypothesisEquality, 
imageMemberEquality, 
independent_pairFormation, 
sqequalRule, 
natural_numberEquality, 
Error :dependent_set_memberEquality_alt, 
thin, 
dependent_functionElimination, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
sqequalHypSubstitution, 
extract_by_obid, 
introduction, 
cut
Latex:
\mforall{}x:\mBbbN{}.  (\mexists{}r:\mBbbN{}  [(((r  *  r)  \mleq{}  x)  \mwedge{}  x  <  (r  +  1)  *  (r  +  1))])
Date html generated:
2019_06_20-PM-02_33_24
Last ObjectModification:
2019_06_19-PM-02_52_33
Theory : num_thy_1
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