Nuprl Lemma : isqrrrt
∀n:ℕ. (∃r:ℕ [(((r * r) ≤ n) ∧ n < (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),
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,
sq_stable: SqStable(P),
sq_type: SQType(T),
uimplies: b supposing a,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
false: False,
le: A ≤ B,
sq_exists: ∃x:A [B[x]],
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
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,
add_functionality_wrt_le,
less_than_functionality,
add-commutes,
add-swap,
mul-swap,
mul-commutes,
add-associates,
mul-associates,
mul-distributes-right,
mul-distributes,
remainder_wfa,
mul_preserves_le,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermMultiply_wf,
itermAdd_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,
member-less_than,
sq_stable__less_than,
sq_stable__le,
sq_stable__and,
nat_plus_wf,
nequal_wf,
istype-int,
int_subtype_base,
subtype_base_sq,
divide_wfa,
istype-le,
istype-nat,
false_wf,
less_than_wf,
le_wf,
nat_wf,
sq_exists_wf,
istype-less_than,
div_nat_induction-ext
Rules used in proof :
int_eqEquality,
Error :dependent_set_memberFormation_alt,
cutEval,
Error :dependent_pairFormation_alt,
approximateComputation,
unionElimination,
applyEquality,
imageElimination,
Error :functionIsTypeImplies,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
closedConclusion,
Error :isect_memberEquality_alt,
productElimination,
Error :universeIsType,
sqequalBase,
Error :equalityIstype,
voidElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
intEquality,
cumulativity,
instantiate,
Error :productIsType,
Error :setIsType,
Error :lambdaFormation_alt,
lambdaFormation,
dependent_set_memberEquality,
dependent_set_memberFormation,
addEquality,
because_Cache,
rename,
setElimination,
multiplyEquality,
productEquality,
lambdaEquality,
independent_functionElimination,
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{}n:\mBbbN{}. (\mexists{}r:\mBbbN{} [(((r * r) \mleq{} n) \mwedge{} n < (r + 1) * (r + 1))])
Date html generated:
2019_06_20-PM-02_35_15
Last ObjectModification:
2019_06_12-PM-00_56_24
Theory : num_thy_1
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