Nuprl Lemma : sym_grp_is_swaps_a
∀n:{1...}. ∀p:Sym(n).
∃abs:{ab:ℕn × ℕn| fst(ab) < snd(ab)} 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_upper: {i...},
int_seg: {i..j-},
less_than: a < b,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
set: {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],
int_upper: {i...},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
pi1: fst(t),
int_seg: {i..j-},
pi2: snd(t),
so_lambda: λ2x.t[x],
igrp: IGroup,
imon: IMonoid,
perm_igrp: perm_igrp(T),
mk_igrp: mk_igrp(T;op;id;inv),
grp_car: |g|,
perm: Perm(T),
so_apply: x[s],
top: Top,
mon_reduce: mon_reduce,
grp_id: e,
infix_ap: x f y,
grp_op: *,
gt: i > j,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
istype: istype(T)
Lemmas referenced :
perm_wf,
int_seg_wf,
int_upper_wf,
sym_grp_is_swaps,
upper_subtype_nat,
istype-false,
exists_wf,
list_wf,
less_than_wf,
equal_wf,
mon_reduce_wf,
perm_igrp_wf,
map_wf,
grp_car_wf,
txpose_perm_wf,
subtype_rel_self,
list_induction,
map_nil_lemma,
istype-void,
reduce_nil_lemma,
map_cons_lemma,
reduce_cons_lemma,
list_subtype_base,
product_subtype_base,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
le_wf,
nil_wf,
id_perm_wf,
int_seg_properties,
int_upper_properties,
decidable__or,
or_wf,
equal-wf-base,
decidable__lt,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformor_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_or_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
cons_wf,
pi1_wf,
subtype_rel_product,
pi2_wf,
comp_perm_wf,
squash_wf,
true_wf,
istype-universe,
txpose_perm_id,
iff_weakening_equal,
perm_ident,
txpose_perm_sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
independent_isectElimination,
sqequalRule,
independent_pairFormation,
productElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
setEquality,
productEquality,
because_Cache,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
productIsType,
setIsType,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
dependent_pairFormation_alt,
unionElimination,
approximateComputation,
int_eqEquality,
dependent_set_memberEquality_alt,
independent_pairEquality,
imageElimination,
universeEquality,
imageMemberEquality,
instantiate
Latex:
\mforall{}n:\{1...\}. \mforall{}p:Sym(n).
\mexists{}abs:\{ab:\mBbbN{}n \mtimes{} \mBbbN{}n| fst(ab) < snd(ab)\} List. (p = (\mPi{} map(\mlambda{}ab.let a,b = ab in txpose\_perm(a;b);abs))\000C)
Date html generated:
2019_10_16-PM-01_02_01
Last ObjectModification:
2018_10_08-PM-05_41_03
Theory : list_2
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