Nuprl Lemma : equipollent-factorial
∀n:ℕ. ℕn →⟶ ℕn ~ ℕ(n)!
Proof
Definitions occuring in Statement :
injection: A →⟶ B,
fact: (n)!,
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
prop: ℙ,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
sq_type: SQType(T)
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
fact0_redex_lemma,
intformand_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_term_value_mul_lemma,
combinations-formula,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
ext-eq_weakening,
equipollent_weakening_ext-eq,
count-combinations,
injections-combinations,
equipollent_transitivity,
equipollent_functionality_wrt_equipollent,
nat_wf,
injection_wf,
int_seg_wf,
combinations_wf_int,
fact_wf,
nat_plus_wf,
combination_wf,
equipollent-nsub,
combinations_wf,
nat_plus_subtype_nat
Rules used in proof :
independent_functionElimination,
productElimination,
dependent_functionElimination,
because_Cache,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_isectElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
independent_pairFormation,
equalitySymmetry,
equalityTransitivity,
instantiate
Latex:
\mforall{}n:\mBbbN{}. \mBbbN{}n \mrightarrow{}{}\mrightarrow{} \mBbbN{}n \msim{} \mBbbN{}(n)!
Date html generated:
2018_05_21-PM-08_20_53
Last ObjectModification:
2017_12_07-PM-06_15_21
Theory : general
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