Nuprl Lemma : sign-inverse
∀[n:ℕ]. ∀[f:{p:ℕn ⟶ ℕn| Inj(ℕn;ℕn;p)} ]. (permutation-sign(n;inv(f)) = permutation-sign(n;f) ∈ {s:ℤ| |s| = 1 ∈ ℤ} )
Proof
Definitions occuring in Statement :
permutation-sign: permutation-sign(n;f),
funinv: inv(f),
inject: Inj(A;B;f),
absval: |i|,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
sq_stable: SqStable(P),
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
and: P ∧ Q,
lelt: i ≤ j < k,
squash: ↓T,
int_seg: {i..j-},
guard: {T},
label: ...$L... t,
compose: f o g,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
absval: |i|,
sq_type: SQType(T),
uiff: uiff(P;Q),
less_than': less_than'(a;b),
le: A ≤ B
Lemmas referenced :
nat_wf,
set_wf,
int_seg_wf,
inject_wf,
funinv_wf2,
permutation-sign-compose,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
sq_stable__equal,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
int_seg_properties,
permutation-sign-id,
permutation-sign_wf,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
funinv-property,
iff_weakening_equal,
set_subtype_base,
int_subtype_base,
int_term_value_mul_lemma,
itermMultiply_wf,
subtype_base_sq,
absval_pos,
le_wf,
false_wf,
absval_cases
Rules used in proof :
lambdaEquality,
sqequalRule,
functionEquality,
applyEquality,
functionExtensionality,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality,
rename,
setElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
imageElimination,
productElimination,
baseClosed,
imageMemberEquality,
equalityTransitivity,
intEquality,
setEquality,
applyLambdaEquality,
hyp_replacement,
equalitySymmetry,
universeEquality,
lambdaFormation,
closedConclusion,
baseApply,
promote_hyp,
minusEquality,
cumulativity,
instantiate
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\{p:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;p)\} ]. (permutation-sign(n;inv(f)) = permutation-sign(n;f))
Date html generated:
2018_05_21-PM-00_59_16
Last ObjectModification:
2017_12_11-AM-10_07_02
Theory : num_thy_1
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