Nuprl Lemma : sym_grp_is_swaps
∀n:ℕ. ∀p:Sym(n). ∃abs:(ℕn × ℕn) List. (p = (Π map(λab.let a,b = ab in txpose_perm(a;b);abs)) ∈ Sym(n))
Proof
Definitions occuring in Statement :
mon_reduce: mon_reduce,
txpose_perm: txpose_perm,
sym_grp: Sym(n),
perm_igrp: perm_igrp(T),
map: map(f;as),
list: T List,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
lambda: λx.A[x],
spread: spread def,
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sym_grp: Sym(n),
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
prop: ℙ,
nat: ℕ,
perm_igrp: perm_igrp(T),
mk_igrp: mk_igrp(T;op;id;inv),
grp_car: |g|,
pi1: fst(t),
top: Top,
mon_reduce: mon_reduce,
grp_id: e,
pi2: snd(t),
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
guard: {T},
lelt: i ≤ j < k,
perm: Perm(T),
sq_type: SQType(T),
squash: ↓T,
compose: f o g,
igrp: IGroup,
imon: IMonoid,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
label: ...$L... t,
infix_ap: x f y,
grp_op: *
Lemmas referenced :
perm_wf,
int_seg_wf,
subtract_wf,
list_wf,
list_subtype_base,
product_subtype_base,
set_subtype_base,
lelt_wf,
int_subtype_base,
istype-int,
less_than_wf,
primrec-wf2,
all_wf,
exists_wf,
equal_wf,
mon_reduce_wf,
perm_igrp_wf,
map_wf,
grp_car_wf,
txpose_perm_wf,
nat_wf,
nil_wf,
map_nil_lemma,
istype-void,
reduce_nil_lemma,
zero_sym_grp,
restrict_perm_using_txpose,
decidable__le,
full-omega-unsat,
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,
cons_wf,
subtype_rel_list,
subtype_rel_product,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
comp_perm_wf,
int_seg_properties,
extend_perm_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
zero-add,
squash_wf,
true_wf,
istype-universe,
extend_perm_over_itcomp,
subtype_rel_self,
iff_weakening_equal,
map_map,
subtract-add-cancel,
imon_wf,
fun_thru_spread,
extend_perm_over_txpose,
map_cons_lemma,
reduce_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
rename,
setElimination,
hypothesisEquality,
sqequalRule,
functionIsType,
productIsType,
productEquality,
equalityIsType3,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
because_Cache,
intEquality,
setIsType,
productElimination,
dependent_pairFormation_alt,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
independent_pairFormation,
independent_pairEquality,
addEquality,
minusEquality,
multiplyEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
imageMemberEquality,
hyp_replacement,
spreadEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}p:Sym(n). \mexists{}abs:(\mBbbN{}n \mtimes{} \mBbbN{}n) List. (p = (\mPi{} map(\mlambda{}ab.let a,b = ab in txpose\_perm(a;b);abs)))
Date html generated:
2019_10_16-PM-01_01_57
Last ObjectModification:
2018_10_08-PM-05_43_30
Theory : list_2
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