Nuprl Lemma : divides-fact
∀m:ℕ. ∀c:{2..m + 1-}. (c | (m)!)
Proof
Definitions occuring in Statement :
fact: (n)!,
divides: b | a,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
fact: (n)!,
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
nat_plus: ℕ+,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
primrec-unroll,
all_wf,
subtract_wf,
divides_wf,
fact_wf,
subtract-add-cancel,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_plus_wf,
nat_wf,
lt_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_int_wf,
bnot_wf,
divides_product,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
decidable__lt,
lelt_wf,
divides_weakening,
assoced_nelim,
int_seg_subtype_nat,
false_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
addEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
because_Cache,
dependent_set_memberEquality,
unionElimination,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
inrFormation,
inlFormation
Latex:
\mforall{}m:\mBbbN{}. \mforall{}c:\{2..m + 1\msupminus{}\}. (c | (m)!)
Date html generated:
2018_05_21-PM-07_59_20
Last ObjectModification:
2018_05_19-PM-04_53_38
Theory : general
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