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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