Nuprl Lemma : exp-increasing
∀[b:{2...}]. ∀[i:ℕ]. ∀j:ℕ. b^i < b^j supposing i < j
Proof
Definitions occuring in Statement :
exp: i^n,
int_upper: {i...},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
prop: ℙ,
int_upper: {i...},
top: Top,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
and: P ∧ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
less_than: a < b,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
le: A ≤ B
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
less_than_wf,
nat_wf,
member-less_than,
exp_wf2,
int_upper_wf,
exp0_lemma,
exp-ge-1,
less_than_transitivity2,
le_weakening,
iff_weakening_equal,
true_wf,
squash_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_upper_properties,
nat_properties,
subtract_wf,
exp_add,
decidable__lt,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
false_wf,
exp_wf_nat_plus,
mul_preserves_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
dependent_set_memberEquality,
productElimination,
universeEquality,
baseClosed,
imageMemberEquality,
imageElimination,
applyEquality,
computeAll,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality
Latex:
\mforall{}[b:\{2...\}]. \mforall{}[i:\mBbbN{}]. \mforall{}j:\mBbbN{}. b\^{}i < b\^{}j supposing i < j
Date html generated:
2017_04_14-AM-09_22_37
Last ObjectModification:
2017_02_24-PM-06_03_27
Theory : int_2
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