Nuprl Lemma : cont-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],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
sq_stable__equal,
le_wf,
false_wf,
primrec-induction-factorial,
nat_wf,
nat_plus_wf,
fact_wf,
equal_wf,
sq_exists_wf,
cont-induction-ext
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
intEquality,
hypothesisEquality,
hypothesis,
applyEquality,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
addEquality,
equalityTransitivity,
equalitySymmetry,
introduction,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache
Latex:
\mforall{}n:\mBbbN{}. (\mexists{}x:\{\mBbbZ{}| ((n)! = x)\})
Date html generated:
2016_05_15-PM-04_09_29
Last ObjectModification:
2016_01_16-AM-11_05_12
Theory : general
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