Nuprl Lemma : qexp-nonneg
∀[n:ℕ]. ∀[r:ℚ]. 0 ≤ r ↑ n supposing 0 ≤ r
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
,
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
,
all: ∀x:A. B[x]
,
top: Top
,
and: P ∧ Q
,
prop: ℙ
,
subtype_rel: A ⊆r B
,
squash: ↓T
,
true: True
,
guard: {T}
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
qle: r ≤ s
,
grp_leq: a ≤ b
,
assert: ↑b
,
ifthenelse: if b then t else f fi
,
infix_ap: x f y
,
grp_le: ≤b
,
pi1: fst(t)
,
pi2: snd(t)
,
qadd_grp: <ℚ+>
,
q_le: q_le(r;s)
,
callbyvalueall: callbyvalueall,
evalall: evalall(t)
,
bor: p ∨bq
,
qpositive: qpositive(r)
,
qsub: r - s
,
qadd: r + s
,
qmul: r * s
,
btrue: tt
,
lt_int: i <z j
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
decidable: Dec(P)
,
or: P ∨ Q
,
nat_plus: ℕ+
,
rev_uimplies: rev_uimplies(P;Q)
,
qge: a ≥ b
,
bfalse: ff
,
qeq: qeq(r;s)
,
eq_int: (i =z j)
Lemmas referenced :
qmul_zero_qrng,
qmul_functionality_wrt_qle,
qle_weakening_eq_qorder,
qle_functionality_wrt_implies,
qle_reflexivity,
qmul_wf,
nat_wf,
int-subtype-rationals,
exp_unroll_q,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
le_wf,
false_wf,
iff_weakening_equal,
exp_zero_q,
true_wf,
squash_wf,
rationals_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,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
dependent_set_memberEquality,
unionElimination
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[r:\mBbbQ{}]. 0 \mleq{} r \muparrow{} n supposing 0 \mleq{} r
Date html generated:
2016_05_15-PM-11_09_20
Last ObjectModification:
2016_01_16-PM-09_26_56
Theory : rationals
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