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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