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
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