Nuprl Lemma : permr_upto_split
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
(EquivRel(T;x,y.R[x;y])
⇒ (∀as,bs:T List. (as ≡ bs upto x,y.R[x;y] ⇐⇒ ∃cs:T List. ((as ≡(T) cs) ∧ cs = bs upto {x,y.R[x;y]}))))
Proof
Definitions occuring in Statement :
lequiv: as = bs upto {x,y.R[x; y]},
permr_upto: as ≡ bs upto x,y.R[x; y] ,
permr: as ≡(T) bs,
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
permr_upto: as ≡ bs upto x,y.R[x; y] ,
cand: A c∧ B,
tlambda: λx:T. b[x],
sym_grp: Sym(n),
perm: Perm(T),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
ge: i ≥ j ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
false: False,
nat: ℕ,
less_than: a < b,
squash: ↓T,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
permr: as ≡(T) bs,
le: A ≤ B,
true: True,
lequiv: as = bs upto {x,y.R[x; y]},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T)
Lemmas referenced :
permr_upto_wf,
istype-universe,
permr_wf,
lequiv_wf,
list_wf,
equiv_rel_wf,
listify_wf,
select_wf,
perm_f_wf,
int_seg_wf,
non_neg_length,
int_seg_properties,
decidable__le,
le_wf,
less_than_wf,
length_wf,
length_wf_nat,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
listify_length,
le_weakening2,
decidable__equal_int,
intformeq_wf,
intformor_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_formula_prop_or_lemma,
int_term_value_subtract_lemma,
equal_wf,
squash_wf,
true_wf,
select_listify_id,
subtype_rel_self,
iff_weakening_equal,
subtype_base_sq,
nat_wf,
set_subtype_base,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
isectElimination,
hypothesis,
productIsType,
because_Cache,
functionIsType,
universeEquality,
productElimination,
dependent_pairFormation_alt,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
independent_isectElimination,
dependent_set_memberEquality_alt,
unionElimination,
applyLambdaEquality,
imageElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
imageMemberEquality,
baseClosed,
instantiate,
equalityIsType1,
cumulativity,
intEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}as,bs:T List.
(as \mequiv{} bs upto x,y.R[x;y] \mLeftarrow{}{}\mRightarrow{} \mexists{}cs:T List. ((as \mequiv{}(T) cs) \mwedge{} cs = bs upto \{x,y.R[x;y]\}))))
Date html generated:
2019_10_16-PM-01_01_34
Last ObjectModification:
2018_10_08-PM-05_38_47
Theory : perms_2
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