Nuprl Lemma : permutation-generators2
∀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[f o (0, 1)]) supposing 1 < n
⇒ (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . (P[f] ⇒ P[f o rot(n)]))
⇒ (∀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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
compose: f o g,
subtype_rel: A ⊆r B,
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,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
nat_plus: ℕ+,
uiff: uiff(P;Q),
subtract: n - m
Lemmas referenced :
permutation-generators,
identity-injection,
int_seg_wf,
inject_wf,
funinv_wf2,
nat_wf,
funinv-unique,
isect_wf,
less_than_wf,
all_wf,
compose-injections,
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,
member-less_than,
set_wf,
equal_wf,
squash_wf,
true_wf,
funinv-compose,
iff_weakening_equal,
flip_inverse,
rotate-injection,
rotate_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
subtype_rel_sets,
subtype_rel_set,
subtype_rel_dep_function,
subtype_rel_self,
subtype_rel_wf,
int_seg_properties,
intformle_wf,
intformeq_wf,
int_formula_prop_le_lemma,
int_formula_prop_eq_lemma,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
fun_exp0_lemma,
primrec-wf2,
inject-compose,
fun_exp_wf,
fun_exp_add_apply1,
subtract-add-cancel,
fun_exp-injection,
rotate-inverse,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
funinv-funinv
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
lambdaEquality,
because_Cache,
functionExtensionality,
applyEquality,
isect_memberFormation,
sqequalRule,
setEquality,
functionEquality,
independent_functionElimination,
cumulativity,
universeEquality,
addLevel,
hyp_replacement,
equalitySymmetry,
levelHypothesis,
equalityTransitivity,
independent_isectElimination,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
applyLambdaEquality,
instantiate,
addEquality,
minusEquality
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[f o (0, 1)]) supposing 1 < n
{}\mRightarrow{} (\mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} . (P[f] {}\mRightarrow{} P[f o rot(n)]))
{}\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_22_14
Last ObjectModification:
2017_02_27-PM-04_45_54
Theory : list_1
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