Nuprl Lemma : permutation-sign-flip-adjacent
∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. ∀[u:ℕn - 1].
(permutation-sign(n;f o (u, u + 1)) = (-permutation-sign(n;f)) ∈ ℤ)
Proof
Definitions occuring in Statement :
permutation-sign: permutation-sign(n;f),
flip: (i, j),
inject: Inj(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
minus: -n,
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,
compose: f o g,
permutation-sign: permutation-sign(n;f),
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_type: SQType(T),
uimplies: b supposing a,
true: True,
false: False,
assert: ↑b,
bnot: ¬bb,
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
squash: ↓T,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
lelt: i ≤ j < k,
ge: i ≥ j ,
not: ¬A,
nequal: a ≠ b ∈ T ,
flip: (i, j),
decidable: Dec(P),
subtract: n - m,
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sign: sign(x),
lt_int: i <z j,
le_int: i ≤z j,
inject: Inj(A;B;f)
Lemmas referenced :
nat_wf,
inject_wf,
set_wf,
subtract_wf,
int_seg_wf,
int_subtype_base,
subtype_base_sq,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_wf,
eq_int_wf,
true_wf,
squash_wf,
int-prod_wf,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
le_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
decidable__le,
int-prod-unroll-hi,
add-subtract-cancel,
lelt_wf,
int_formula_prop_less_lemma,
int_term_value_subtract_lemma,
intformless_wf,
itermSubtract_wf,
add-member-int_seg2,
sign_wf,
false_wf,
int_seg_subtype_nat,
decidable__lt,
int_term_value_mul_lemma,
itermMultiply_wf,
decidable__equal_int,
mul_assoc,
iff_weakening_equal,
assert_of_le_int,
le_int_wf,
mul-one,
ifthenelse_wf,
btrue_wf,
eq_int_eq_true,
int_seg_subtype,
flip_wf,
int-prod-isolate,
bfalse_wf,
equal-wf-base,
eq_int_eq_false,
minus_functionality_wrt_eq,
minus-one-mul
Rules used in proof :
applyEquality,
functionExtensionality,
lambdaEquality,
functionEquality,
because_Cache,
axiomEquality,
isect_memberEquality,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
dependent_functionElimination,
independent_isectElimination,
intEquality,
cumulativity,
instantiate,
baseClosed,
imageMemberEquality,
voidElimination,
promote_hyp,
dependent_pairFormation,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation,
imageElimination,
independent_pairFormation,
voidEquality,
int_eqEquality,
approximateComputation,
addEquality,
dependent_set_memberEquality,
levelHypothesis,
equalityUniverse,
universeEquality,
multiplyEquality,
applyLambdaEquality,
closedConclusion,
baseApply,
minusEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} ]. \mforall{}[u:\mBbbN{}n - 1].
(permutation-sign(n;f o (u, u + 1)) = (-permutation-sign(n;f)))
Date html generated:
2018_05_21-PM-00_58_42
Last ObjectModification:
2017_12_11-AM-00_05_06
Theory : num_thy_1
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