Nuprl Lemma : permutation-sign-compose
∀[n:ℕ]. ∀[f,g:{p:ℕn ⟶ ℕn| Inj(ℕn;ℕn;p)} ].
(permutation-sign(n;f o g) = (permutation-sign(n;f) * permutation-sign(n;g)) ∈ {s:ℤ| |s| = 1 ∈ ℤ} )
Proof
Definitions occuring in Statement :
permutation-sign: permutation-sign(n;f),
inject: Inj(A;B;f),
compose: f o g,
absval: |i|,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
multiply: n * m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
inject: Inj(A;B;f),
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
flip: (i, j),
surject: Surj(A;B;f),
compose: f o g,
so_apply: x[s],
less_than: a < b,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
false: False,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
sq_type: SQType(T),
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
absval-minus,
int_term_value_minus_lemma,
itermMinus_wf,
inject-compose,
permutation-sign-flip,
decidable__equal_int_seg,
int_seg_properties,
nat_properties,
intformand_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
le_wf,
less_than_wf,
bfalse_wf,
lelt_wf,
eq_int_eq_true,
btrue_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
bool_wf,
bool_subtype_base,
flip_wf,
injection-is-surjection,
permutation-sign-id,
one-mul,
nat_wf,
istype-nat,
istype-less_than,
absval_wf,
set_subtype_base,
compose_wf,
permutation-sign_wf,
equal-wf-base,
inject_wf,
int_seg_wf,
permutation-generators3,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
iff_weakening_equal,
subtype_rel_self,
absval_mul,
istype-universe,
true_wf,
squash_wf,
equal_wf
Rules used in proof :
Error :equalityIsType4,
int_eqEquality,
independent_pairFormation,
Error :productIsType,
equalityElimination,
Error :equalityIsType1,
promote_hyp,
functionExtensionality,
applyLambdaEquality,
hyp_replacement,
Error :functionExtensionality_alt,
Error :functionIsTypeImplies,
axiomEquality,
Error :functionIsType,
setEquality,
functionEquality,
Error :setIsType,
sqequalBase,
closedConclusion,
baseApply,
Error :equalityIstype,
voidElimination,
Error :isect_memberEquality_alt,
Error :dependent_pairFormation_alt,
approximateComputation,
unionElimination,
dependent_functionElimination,
cumulativity,
independent_functionElimination,
productElimination,
independent_isectElimination,
because_Cache,
baseClosed,
imageMemberEquality,
sqequalRule,
natural_numberEquality,
intEquality,
universeEquality,
instantiate,
Error :inhabitedIsType,
Error :universeIsType,
equalitySymmetry,
hypothesis,
equalityTransitivity,
isectElimination,
extract_by_obid,
introduction,
imageElimination,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
applyEquality,
hypothesisEquality,
multiplyEquality,
Error :dependent_set_memberEquality_alt,
rename,
setElimination,
Error :lambdaFormation_alt,
thin,
cut,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f,g:\{p:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;p)\} ].
(permutation-sign(n;f o g) = (permutation-sign(n;f) * permutation-sign(n;g)))
Date html generated:
2019_06_20-PM-02_26_55
Last ObjectModification:
2019_06_19-PM-00_17_42
Theory : num_thy_1
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