Nuprl Lemma : extend_restrict_perm_cancel
∀n:{1...}. ∀p:Sym(n). (((p.f (n - 1)) = (n - 1) ∈ ℕn) ⇒ (↑{n - 1}(restrict_perm(p;n - 1)) = p ∈ Sym(n)))
Proof
Definitions occuring in Statement :
restrict_perm: restrict_perm(p;n),
extend_perm: ↑{n}(p),
sym_grp: Sym(n),
perm_f: p.f,
int_upper: {i...},
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
subtract: n - m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
int_upper: {i...},
sym_grp: Sym(n),
perm: Perm(T),
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
restrict_perm: restrict_perm(p;n),
extend_perm: ↑{n}(p),
extend_permf: extend_permf(pf;n),
squash: ↓T,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
inv_funs: InvFuns(A;B;f;g),
compose: f o g,
tidentity: Id{T},
identity: Id,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
int_seg_wf,
perm_f_wf,
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
perm_wf,
int_upper_wf,
inv_funs_wf,
perm_b_wf,
perm_properties,
squash_wf,
true_wf,
perm_sig_wf,
mk_perm_eta_rw,
mk_perm_wf,
eq_int_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
iff_weakening_equal,
equal-wf-T-base,
assert_wf,
int_subtype_base,
int_seg_properties,
decidable__equal_int_seg,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
set_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
applyEquality,
dependent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
hypothesisEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
computeAll,
equalitySymmetry,
imageElimination,
equalityTransitivity,
universeEquality,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
imageMemberEquality,
baseClosed,
functionEquality,
equalityUniverse,
levelHypothesis,
functionExtensionality,
applyLambdaEquality,
impliesFunctionality
Latex:
\mforall{}n:\{1...\}. \mforall{}p:Sym(n). (((p.f (n - 1)) = (n - 1)) {}\mRightarrow{} (\muparrow{}\{n - 1\}(restrict\_perm(p;n - 1)) = p))
Date html generated:
2017_10_01-AM-09_53_34
Last ObjectModification:
2017_03_03-PM-00_48_13
Theory : perms_1
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