Nuprl Lemma : exp-nondecreasing
∀[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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
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,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
int_upper: {i...},
prop: ℙ,
guard: {T},
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T)
Lemmas referenced :
decidable__lt,
less_than'_wf,
exp_wf2,
le_wf,
nat_wf,
int_upper_wf,
exp-increasing,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
itermConstant_wf,
int_term_value_constant_lemma
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,
unionElimination,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
hypothesisEquality,
isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
natural_numberEquality,
independent_isectElimination,
dependent_set_memberEquality,
imageElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
independent_pairFormation,
instantiate,
cumulativity
Latex:
\mforall{}[b:\{2...\}]. \mforall{}[i:\mBbbN{}]. \mforall{}j:\mBbbN{}. b\^{}i \mleq{} b\^{}j supposing i \mleq{} j
Date html generated:
2017_09_29-PM-05_57_39
Last ObjectModification:
2017_07_03-PM-05_35_05
Theory : int_2
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