Nuprl Lemma : fact-increasing
∀[m:ℕ]. ∀[n:ℕ+]. (n < m ⇒ (n)! < (m)!)
Proof
Definitions occuring in Statement :
fact: (n)!,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q
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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
squash: ↓T,
fact: (n)!,
primrec: primrec(n;b;c),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
member-less_than,
fact_wf,
nat_plus_properties,
nat_plus_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
nat_plus_subtype_nat,
nat_wf,
fact_unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__equal_int,
int_subtype_base,
fact0_redex_lemma,
fact_unroll_1,
intformeq_wf,
int_formula_prop_eq_lemma,
mul_preserves_lt,
decidable__lt,
intformimplies_wf,
int_formual_prop_imp_lemma,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-commutes,
le-add-cancel2,
condition-implies-le,
minus-add,
add-swap,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
le-add-cancel
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,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
applyEquality,
because_Cache,
unionElimination,
dependent_set_memberEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
instantiate,
cumulativity,
imageElimination,
imageMemberEquality,
baseClosed,
multiplyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
addEquality,
minusEquality
Latex:
\mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (n < m {}\mRightarrow{} (n)! < (m)!)
Date html generated:
2018_05_21-PM-01_01_13
Last ObjectModification:
2018_05_19-AM-06_39_24
Theory : num_thy_1
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