Nuprl Lemma : fact-non-decreasing
∀[m,n:ℕ]. ((n ≤ m) ⇒ ((n)! ≤ (m)!))
Proof
Definitions occuring in Statement :
fact: (n)!,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
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,
le: A ≤ B,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
subtract: n - m,
less_than': less_than'(a;b),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
nat_plus: ℕ+
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,
less_than'_wf,
fact_wf,
le_wf,
nat_wf,
fact_unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
false_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
nat_plus_wf,
fact_unroll_1,
nat_plus_properties,
mul_preserves_le,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma
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,
productElimination,
independent_pairEquality,
applyEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality,
applyLambdaEquality,
multiplyEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[m,n:\mBbbN{}]. ((n \mleq{} m) {}\mRightarrow{} ((n)! \mleq{} (m)!))
Date html generated:
2018_05_21-PM-01_01_25
Last ObjectModification:
2018_05_19-AM-06_39_19
Theory : num_thy_1
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