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
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