Nuprl Lemma : exp-ge-1
∀[b:{2...}]. ∀[j:ℕ+]. 1 < b^j
Proof
Definitions occuring in Statement :
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat_plus: ℕ+,
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
guard: {T},
int_upper: {i...},
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,
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
squash: ↓T,
sq_type: SQType(T),
subtract: n - m,
less_than: a < b
Lemmas referenced :
nat_plus_properties,
less_than_wf,
exp_wf2,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
primrec-wf-nat-plus,
nat_plus_wf,
member-less_than,
nat_plus_subtype_nat,
int_upper_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__lt,
false_wf,
exp_wf_nat_plus,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
equal_wf,
squash_wf,
true_wf,
exp1,
iff_weakening_equal,
exp_step,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
add-zero,
add-subtract-cancel,
mul_preserves_lt,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
natural_numberEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
because_Cache,
applyEquality,
instantiate,
cumulativity,
productElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
addEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[b:\{2...\}]. \mforall{}[j:\mBbbN{}\msupplus{}]. 1 < b\^{}j
Date html generated:
2017_04_14-AM-09_22_31
Last ObjectModification:
2017_02_27-PM-03_58_00
Theory : int_2
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