Nuprl Lemma : expfact_wf
∀[m:ℕ+]. ∀[k:ℕ]. ∀[n:ℕ+]. ∀b:{b:ℕ| n * k^b < (b)!} . ((m ≤ b) ⇒ (expfact(m;k;n * k^m;(m)!) ∈ {b:ℕ+| (n * k^b) ≤ (b)!} \000C))
Proof
Definitions occuring in Statement :
expfact: expfact(n;x;p;b),
fact: (n)!,
exp: i^n,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat_plus: ℕ+,
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
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,
expfact: expfact(n;x;p;b),
decidable: Dec(P),
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
le: A ≤ B,
subtract: n - m,
less_than: a < b,
has-value: (a)↓,
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
le-add-cancel,
add_functionality_wrt_le,
add-commutes,
add-swap,
condition-implies-le,
less-iff-le,
not-le-2,
set_subtype_base,
zero-add,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-associates,
minus-one-mul,
minus-minus,
minus-add,
trivial-int-eq1,
equal_wf,
fact_unroll_1,
add-subtract-cancel,
exp_step,
mul-swap,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
int-value-type,
value-type-has-value,
decidable__lt,
false_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
nat_plus_subtype_nat,
add-zero,
minus-zero,
assert_of_lt_int,
bnot_of_le_int,
assert_functionality_wrt_uiff,
eqff_to_assert,
assert_of_le_int,
eqtt_to_assert,
uiff_transitivity,
bnot_wf,
lt_int_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
nat_plus_properties,
sq_stable__less_than,
le_int_wf,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
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,
nat_plus_wf,
fact_wf,
exp_wf2,
less_than_wf,
nat_wf,
set_wf,
le_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
multiplyEquality,
applyEquality,
setEquality,
because_Cache,
isect_memberFormation,
introduction,
lambdaFormation,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityElimination,
productElimination,
equalityEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
callbyvalueReduce,
addEquality,
instantiate,
minusEquality,
cumulativity
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}].
\mforall{}b:\{b:\mBbbN{}| n * k\^{}b < (b)!\} . ((m \mleq{} b) {}\mRightarrow{} (expfact(m;k;n * k\^{}m;(m)!) \mmember{} \{b:\mBbbN{}\msupplus{}| (n * k\^{}b) \mleq{} (b)!\} ))
Date html generated:
2016_05_15-PM-04_06_59
Last ObjectModification:
2016_01_16-AM-11_04_42
Theory : general
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