Nuprl Lemma : qexp_preserves_qle
∀[a,b:ℚ]. (∀[n:ℕ]. (a ↑ n ≤ b ↑ n)) supposing ((a ≤ b) and (0 ≤ a))
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qle: r ≤ s,
rationals: ℚ,
nat: ℕ,
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,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b
Lemmas referenced :
qle_weakening_eq_qorder,
qmul_functionality_wrt_qle,
qle_functionality_wrt_implies,
qexp-nonneg,
qmul_wf,
exp_unroll_q,
rationals_wf,
int-subtype-rationals,
nat_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
le_wf,
qle_reflexivity,
iff_weakening_equal,
exp_zero_q,
true_wf,
squash_wf,
qle_wf,
qexp_wf,
qle_witness,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_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,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
dependent_set_memberEquality,
unionElimination
Latex:
\mforall{}[a,b:\mBbbQ{}]. (\mforall{}[n:\mBbbN{}]. (a \muparrow{} n \mleq{} b \muparrow{} n)) supposing ((a \mleq{} b) and (0 \mleq{} a))
Date html generated:
2016_05_15-PM-11_09_49
Last ObjectModification:
2016_01_16-PM-09_25_09
Theory : rationals
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