Nuprl Lemma : restrict_perm_wf
∀n:ℕ. ∀p:Sym(n + 1). (((p.f n) = n ∈ ℕn + 1) ⇒ (restrict_perm(p;n) ∈ Sym(n)))
Proof
Definitions occuring in Statement :
restrict_perm: restrict_perm(p;n),
sym_grp: Sym(n),
perm_f: p.f,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
apply: f a,
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
sym_grp: Sym(n),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
perm: Perm(T),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
restrict_perm: restrict_perm(p;n),
ge: i ≥ j ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
guard: {T},
inv_funs: InvFuns(A;B;f;g),
compose: f o g,
tidentity: Id{T},
identity: Id,
sq_type: SQType(T),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
perm_sig: perm_sig(T),
perm_f: p.f,
pi1: fst(t),
perm_b: p.b,
pi2: snd(t),
le: A ≤ B,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m
Lemmas referenced :
int_seg_wf,
perm_f_wf,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
le_wf,
less_than_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
perm_wf,
nat_wf,
nat_properties,
intformand_wf,
int_formula_prop_and_lemma,
perm_properties,
perm_b_wf,
fun_with_inv_is_bij,
int_seg_properties,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
not_wf,
equal_wf,
subtype_base_sq,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
subtype_rel_dep_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
inv_funs_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
equalityIsType4,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
dependent_functionElimination,
because_Cache,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
intEquality,
baseApply,
closedConclusion,
baseClosed,
functionExtensionality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
inhabitedIsType,
equalityIsType1,
instantiate,
cumulativity,
imageElimination,
universeEquality,
imageMemberEquality,
dependent_pairEquality_alt,
minusEquality,
multiplyEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}p:Sym(n + 1). (((p.f n) = n) {}\mRightarrow{} (restrict\_perm(p;n) \mmember{} Sym(n)))
Date html generated:
2019_10_16-PM-01_00_13
Last ObjectModification:
2018_10_08-AM-09_12_46
Theory : perms_1
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