Nuprl Lemma : fact_add1
∀[n:ℕ]. ((n + 1)! ~ (n + 1) * (n)!)
Proof
Definitions occuring in Statement :
fact: (n)!,
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
subtype_base_sq,
nat_plus_wf,
set_subtype_base,
less_than_wf,
istype-int,
int_subtype_base,
nat_properties,
decidable__lt,
fact_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
fact_unroll_1,
istype-less_than,
add-associates,
add-swap,
add-commutes,
zero-add,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
hypothesisEquality,
setElimination,
rename,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
addEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
applyEquality,
Error :inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
Error :lambdaFormation_alt,
because_Cache,
multiplyEquality,
axiomSqEquality
Latex:
\mforall{}[n:\mBbbN{}]. ((n + 1)! \msim{} (n + 1) * (n)!)
Date html generated:
2019_06_20-PM-02_30_20
Last ObjectModification:
2019_02_01-PM-01_13_24
Theory : num_thy_1
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