Nuprl Lemma : qexp_preserves_qless
∀[a,b:ℚ]. (∀[n:ℕ]. a ↑ n < b ↑ n supposing 0 < n) supposing (a < b and (0 ≤ a))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
qless: r < s,
rationals: ℚ,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_type: SQType(T)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
less_than_wf,
qless_witness,
qexp_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
nat_wf,
qless_wf,
qle_wf,
rationals_wf,
int-subtype-rationals,
decidable__lt,
squash_wf,
true_wf,
exp_unroll_q,
iff_weakening_equal,
decidable__qless,
qmul_functionality_wrt_qless2,
qexp-nonneg,
qless_transitivity_1_qorder,
qle_weakening_lt_qorder,
qexp-positive-iff,
assert_wf,
isEven_wf,
equal-wf-base,
int_subtype_base,
equal-wf-T-base,
qless_complement_qorder,
qle_antisymmetry,
equal_wf,
qmul_zero_qrng,
subtype_base_sq,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
qmul_one_qrng,
qmul_wf,
exp_zero_q
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
imageElimination,
productElimination,
unionElimination,
dependent_set_memberEquality,
applyEquality,
imageMemberEquality,
baseClosed,
universeEquality,
inrFormation,
inlFormation,
productEquality,
hyp_replacement,
applyLambdaEquality,
instantiate,
cumulativity
Latex:
\mforall{}[a,b:\mBbbQ{}]. (\mforall{}[n:\mBbbN{}]. a \muparrow{} n < b \muparrow{} n supposing 0 < n) supposing (a < b and (0 \mleq{} a))
Date html generated:
2018_05_22-AM-00_00_59
Last ObjectModification:
2017_07_26-PM-06_49_47
Theory : rationals
Home
Index