Nuprl Lemma : primrec-induction-factorial
∀n:ℕ. (∃x:ℤ [((n)! = x ∈ ℤ)])
Proof
Definitions occuring in Statement :
fact: (n)!,
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
sq_exists: ∃x:A [B[x]],
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
not: ¬A,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
primrec-induction,
sq_exists_wf,
equal-wf-T-base,
fact_wf,
nat_plus_wf,
int_subtype_base,
nat_wf,
fact0_redex_lemma,
equal-wf-base,
value-type-has-value,
int-value-type,
fact_unroll_1,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
add-subtract-cancel,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
intEquality,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
independent_functionElimination,
lambdaFormation,
because_Cache,
dependent_set_memberFormation,
natural_numberEquality,
baseClosed,
callbyvalueReduce,
independent_isectElimination,
addEquality,
multiplyEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
productElimination,
dependent_set_memberEquality,
unionElimination
Latex:
\mforall{}n:\mBbbN{}. (\mexists{}x:\mBbbZ{} [((n)! = x)])
Date html generated:
2018_05_21-PM-06_59_42
Last ObjectModification:
2017_07_26-PM-05_02_10
Theory : general
Home
Index