Nuprl Lemma : qexp-greater-one
∀e:{e:ℚ| 0 < e} . ∀r:{r:ℚ| 1 + e < r} . ∀n:ℕ. 1 + (n * e) < r ↑ n supposing 1 ≤ n
Proof
Definitions occuring in Statement :
qexp: r ↑ n,
qless: r < s,
qmul: r * s,
qadd: r + s,
rationals: ℚ,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
decidable: Dec(P),
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
uiff: uiff(P;Q),
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
qge: a ≥ b,
qgt: a > b,
nat_plus: ℕ+,
qsub: r - s
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,
qadd_wf,
qmul_wf,
qexp_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
sq_stable_from_decidable,
qless_wf,
int-subtype-rationals,
decidable__qless,
nat_wf,
set_wf,
rationals_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
squash_wf,
true_wf,
qexp1,
iff_weakening_equal,
qmul_ident,
qexp_preserves_qless,
false_wf,
qless-int,
qadd_preserves_qless,
qless_transitivity,
qle_weakening_lt_qorder,
qadd_comm_q,
mon_ident_q,
qexp2,
qmul_over_plus_qrng,
qmul_one_qrng,
mon_assoc_q,
q_distrib,
qmul-positive,
qless_functionality_wrt_implies_1,
qle_weakening_eq_qorder,
qadd_ac_1_q,
qinverse_q,
qadd_inv_assoc_q,
qadd_assoc,
intformeq_wf,
int_formula_prop_eq_lemma,
qmul_functionality_wrt_qless2,
qexp-nonneg,
qless_transitivity_2_qorder,
exp_unroll_q,
qsub-sub,
equal_wf,
qmul_over_minus_qrng,
qmul_comm_qrng,
qmul_assoc_qrng,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation,
productElimination,
unionElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
universeEquality,
minusEquality,
inlFormation,
productEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}e:\{e:\mBbbQ{}| 0 < e\} . \mforall{}r:\{r:\mBbbQ{}| 1 + e < r\} . \mforall{}n:\mBbbN{}. 1 + (n * e) < r \muparrow{} n supposing 1 \mleq{} n
Date html generated:
2018_05_22-AM-00_01_56
Last ObjectModification:
2017_07_26-PM-06_50_28
Theory : rationals
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