Nuprl Lemma : permutation-generators
∀n:ℕ
∀[P:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ⟶ ℙ]
(P[λx.x]
⇒ ∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . (P[f] ⇒ P[(0, 1) o f]) supposing 1 < n
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . (P[f] ⇒ P[rot(n) o f]))
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . P[f]))
Proof
Definitions occuring in Statement :
flip: (i, j),
rotate: rot(n),
inject: Inj(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
guard: {T},
squash: ↓T,
compose: f o g,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
compose-flips: compose-flips(flips),
label: ...$L... t
Lemmas referenced :
set_wf,
int_seg_wf,
inject_wf,
all_wf,
compose-injections,
rotate-injection,
rotate_wf,
isect_wf,
less_than_wf,
flip-injection,
false_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
lelt_wf,
flip_wf,
identity-injection,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
intformeq_wf,
intformle_wf,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
singleton_int_seg,
decidable__le,
itermAdd_wf,
int_term_value_add_lemma,
zero-add,
flip-generators,
list_induction,
bool_wf,
reduce_wf,
compose_wf,
eqtt_to_assert,
int_upper_subtype_nat,
le_wf,
nequal-le-implies,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
int_upper_properties,
list_wf,
reduce_nil_lemma,
reduce_cons_lemma,
inject-compose,
squash_wf,
true_wf,
comp_assoc,
iff_weakening_equal,
sq_stable__inject,
cycle-as-flips,
no_repeats_wf,
cycle-injection,
map_wf,
map_nil_lemma,
map_cons_lemma,
cycle-decomp,
cycle_wf,
l_all_wf,
l_member_wf,
length_wf,
l_all_wf_nil,
and_wf,
nil_wf,
l_all_cons,
injection-if-compose-injection,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
lambdaEquality,
functionExtensionality,
applyEquality,
setEquality,
dependent_set_memberEquality,
universeEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
cumulativity,
instantiate,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
addLevel,
hyp_replacement,
levelHypothesis,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
equalityElimination,
hypothesis_subsumption,
promote_hyp,
comment,
productEquality,
spreadEquality,
independent_pairEquality
Latex:
\mforall{}n:\mBbbN{}
\mforall{}[P:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} {}\mrightarrow{} \mBbbP{}]
(P[\mlambda{}x.x]
{}\mRightarrow{} \mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . (P[f] {}\mRightarrow{} P[(0, 1) o f]) supposing 1 < n
{}\mRightarrow{} (\mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . (P[f] {}\mRightarrow{} P[rot(n) o f]))
{}\mRightarrow{} (\mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . P[f]))
Date html generated:
2017_04_17-AM-08_21_47
Last ObjectModification:
2017_02_27-PM-04_47_11
Theory : list_1
Home
Index