Nuprl Lemma : fact-as-genfact
∀[n:ℕ]. ((n)! ~ genfact(n;1;m.m))
Proof
Definitions occuring in Statement :
fact: (n)!,
genfact: genfact(n;b;m.f[m]),
nat: ℕ,
uall: ∀[x:A]. B[x],
natural_number: $n,
sqequal: s ~ t
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,
fact: (n)!,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
genfact: genfact(n;b;m.f[m]),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
subtract-1-ge-0,
istype-nat,
int_subtype_base,
nat_plus_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
genfact_wf,
subtract_wf,
fact_unroll,
genfact-step,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomSqEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
because_Cache,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
unionElimination,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
intEquality,
multiplyEquality,
sqequalIntensionalEquality
Latex:
\mforall{}[n:\mBbbN{}]. ((n)! \msim{} genfact(n;1;m.m))
Date html generated:
2019_06_20-PM-02_30_06
Last ObjectModification:
2019_02_08-PM-00_47_40
Theory : num_thy_1
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