Nuprl Lemma : permutation-sign-flip
∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ]. ∀[u,v:ℕn].
permutation-sign(n;f o (u, v)) = (-permutation-sign(n;f)) ∈ ℤ supposing ¬(u = v ∈ 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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
int_seg: {i..j-},
and: P ∧ Q,
cand: A c∧ B,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
nat: ℕ,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
top: Top,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
uiff: uiff(P;Q),
absval: |i|,
compose: f o g,
flip: (i, j),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
permutation-sign-flip-adjacent,
absval_wf,
subtract_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-void,
int_seg_wf,
inject_wf,
istype-nat,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
ge_wf,
istype-less_than,
subtract-1-ge-0,
istype-le,
decidable__lt,
subtype_base_sq,
equal_wf,
squash_wf,
true_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
permutation-sign_wf,
equal-wf-base,
iff_weakening_equal,
absval_ubound,
false_wf,
le_wf,
decidable__equal_int,
intformeq_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_minus_lemma,
flip_symmetry,
compose_wf,
less_than_wf,
set_wf,
nat_wf,
subtract-add-cancel,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
ifthenelse_wf,
flip_wf,
compose-injections,
flip-injection,
absval_pos,
minus_functionality_wrt_eq,
absval-diff-symmetry
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
productElimination,
imageElimination,
applyEquality,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
addEquality,
because_Cache,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
universeIsType,
voidElimination,
functionIsType,
equalityIstype,
intEquality,
sqequalBase,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
setIsType,
lambdaFormation_alt,
intWeakElimination,
functionIsTypeImplies,
productIsType,
instantiate,
cumulativity,
lambdaEquality,
universeEquality,
dependent_set_memberEquality,
dependent_pairFormation,
isect_memberEquality,
voidEquality,
minusEquality,
setEquality,
baseApply,
closedConclusion,
baseClosed,
imageMemberEquality,
lambdaFormation,
dependent_set_memberEquality_alt,
isect_memberFormation,
productEquality,
functionEquality,
functionExtensionality,
hyp_replacement,
applyLambdaEquality,
equalityElimination,
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} ]. \mforall{}[u,v:\mBbbN{}n].
permutation-sign(n;f o (u, v)) = (-permutation-sign(n;f)) supposing \mneg{}(u = v)
Date html generated:
2020_05_19-PM-10_02_14
Last ObjectModification:
2020_01_04-PM-08_25_03
Theory : num_thy_1
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