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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