Nuprl Lemma : permr_upto_transitivity
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
(EquivRel(T;x,y.R[x;y])
⇒ (∀as,bs,cs:T List. (as ≡ bs upto x,y.R[x;y] ⇒ bs ≡ cs upto x,y.R[x;y] ⇒ as ≡ cs upto x,y.R[x;y] )))
Proof
Definitions occuring in Statement :
permr_upto: as ≡ bs upto x,y.R[x; y] ,
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
permr_upto: as ≡ bs upto x,y.R[x; y] ,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
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,
cand: A c∧ B,
exists: ∃x:A. B[x],
sym_grp: Sym(n),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
true: True,
perm: Perm(T),
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,
not: ¬A,
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),
compose: f o g
Lemmas referenced :
permr_upto_wf,
istype-universe,
list_wf,
equiv_rel_wf,
comp_perm_wf,
int_seg_wf,
length_wf,
subtype_rel-equal,
perm_wf,
squash_wf,
true_wf,
istype-int,
select_wf,
perm_f_wf,
non_neg_length,
int_seg_properties,
decidable__le,
le_wf,
less_than_wf,
length_wf_nat,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
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,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
universeIsType,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
isectElimination,
hypothesis,
functionIsType,
universeEquality,
equalityTransitivity,
independent_pairFormation,
rename,
dependent_pairFormation_alt,
natural_numberEquality,
independent_isectElimination,
imageElimination,
equalitySymmetry,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
productIsType,
equalityIsType1,
applyLambdaEquality,
setElimination,
because_Cache,
unionElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
isect_memberEquality_alt,
voidElimination
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}as,bs,cs:T List.
(as \mequiv{} bs upto x,y.R[x;y] {}\mRightarrow{} bs \mequiv{} cs upto x,y.R[x;y] {}\mRightarrow{} as \mequiv{} cs upto x,y.R[x;y] )))
Date html generated:
2019_10_16-PM-01_01_16
Last ObjectModification:
2018_10_08-PM-00_29_14
Theory : perms_2
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