Nuprl Lemma : restrict_perm_using_txpose
∀n:{1...}. ∀p:Sym(n).  ∃q:Sym(n - 1). ∃i,j:ℕn. (p = txpose_perm(i;j) O ↑{n - 1}(q) ∈ Sym(n))
Proof
Definitions occuring in Statement : 
extend_perm: ↑{n}(p)
, 
txpose_perm: txpose_perm, 
sym_grp: Sym(n)
, 
comp_perm: comp_perm, 
int_upper: {i...}
, 
int_seg: {i..j-}
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
subtract: n - m
, 
natural_number: $n
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
sym_grp: Sym(n)
, 
uall: ∀[x:A]. B[x]
, 
int_upper: {i...}
, 
exists: ∃x:A. B[x]
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
uimplies: b supposing a
, 
le: A ≤ B
, 
and: P ∧ Q
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
comp_perm: comp_perm, 
mk_perm: mk_perm(f;b)
, 
perm_f: p.f
, 
pi1: fst(t)
, 
txpose_perm: txpose_perm, 
compose: f o g
, 
swap: swap(i;j)
, 
ifthenelse: if b then t else f fi 
, 
eq_int: (i =z j)
, 
subtract: n - m
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
perm: Perm(T)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
bfalse: ff
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
true: True
, 
squash: ↓T
Lemmas referenced : 
perm_wf, 
int_seg_wf, 
int_upper_wf, 
restrict_perm_wf, 
int_seg_subtype_nat, 
false_wf, 
comp_perm_wf, 
txpose_perm_wf, 
subtract_wf, 
subtract-add-cancel, 
int_seg_properties, 
int_upper_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
le_wf, 
subtype_base_sq, 
set_subtype_base, 
int_subtype_base, 
add-associates, 
add-swap, 
add-commutes, 
zero-add, 
perm_f_wf, 
decidable__equal_int, 
intformeq_wf, 
itermSubtract_wf, 
int_formula_prop_eq_lemma, 
int_term_value_subtract_lemma, 
decidable__lt, 
lelt_wf, 
equal_wf, 
int_upper_subtype_nat, 
extend_perm_wf, 
subtype_rel_self, 
exists_wf, 
eq_int_wf, 
bool_wf, 
uiff_transitivity, 
equal-wf-T-base, 
assert_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
iff_transitivity, 
bnot_wf, 
not_wf, 
iff_weakening_uiff, 
eqff_to_assert, 
assert_of_bnot, 
squash_wf, 
true_wf, 
extend_restrict_perm_cancel, 
iff_weakening_equal, 
perm_assoc, 
txpose_perm_order_2, 
perm_ident
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
isectElimination, 
natural_numberEquality, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
dependent_pairFormation, 
equalityTransitivity, 
equalitySymmetry, 
applyEquality, 
independent_isectElimination, 
sqequalRule, 
independent_pairFormation, 
addEquality, 
lambdaEquality, 
because_Cache, 
dependent_set_memberEquality, 
applyLambdaEquality, 
productElimination, 
unionElimination, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll, 
instantiate, 
cumulativity, 
independent_functionElimination, 
equalityElimination, 
baseClosed, 
impliesFunctionality, 
imageElimination, 
universeEquality, 
imageMemberEquality
Latex:
\mforall{}n:\{1...\}.  \mforall{}p:Sym(n).    \mexists{}q:Sym(n  -  1).  \mexists{}i,j:\mBbbN{}n.  (p  =  txpose\_perm(i;j)  O  \muparrow{}\{n  -  1\}(q))
Date html generated:
2017_10_01-AM-09_53_36
Last ObjectModification:
2017_03_03-PM-00_48_33
Theory : perms_1
Home
Index