Nuprl Lemma : iroot-positive
∀[n,x:ℕ+]. (1 ≤ iroot(n;x))
Proof
Definitions occuring in Statement :
iroot: iroot(n;x),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
nat: ℕ,
guard: {T},
nat_plus: ℕ+,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
int_formula_prop_less_lemma,
intformless_wf,
iff_weakening_equal,
exp-one,
true_wf,
squash_wf,
less_than_wf,
int_subtype_base,
subtype_base_sq,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_plus_properties,
le_wf,
nat_properties,
nat_plus_subtype_nat,
nat_plus_wf,
less_than'_wf,
iroot_wf,
decidable__le,
iroot-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
productElimination,
independent_pairEquality,
lambdaEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
setElimination,
rename,
setEquality,
intEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
instantiate,
cumulativity,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[n,x:\mBbbN{}\msupplus{}]. (1 \mleq{} iroot(n;x))
Date html generated:
2019_06_20-PM-02_34_51
Last ObjectModification:
2019_03_19-AM-10_49_36
Theory : num_thy_1
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