Nuprl Lemma : permr_upto_weakening
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. (EquivRel(T;x,y.R[x;y]) ⇒ (∀as,bs:T List. ((as ≡(T) bs) ⇒ as ≡ bs upto x,y.R[x;y] )))
Proof
Definitions occuring in Statement :
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],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
permr: as ≡(T) bs,
permr_upto: as ≡ bs upto x,y.R[x; y] ,
cand: A c∧ B,
exists: ∃x:A. 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,
and: P ∧ Q,
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,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
permr_wf,
list_wf,
equiv_rel_wf,
istype-universe,
int_seg_wf,
length_wf,
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,
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,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
isectElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
functionIsType,
universeEquality,
productElimination,
independent_pairFormation,
dependent_pairFormation_alt,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
productIsType,
unionElimination,
applyLambdaEquality,
imageElimination,
approximateComputation,
independent_functionElimination,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
instantiate
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{}(T) bs) {}\mRightarrow{} as \mequiv{} bs upto x,y.R[x;y] )))
Date html generated:
2019_10_16-PM-01_01_19
Last ObjectModification:
2018_10_08-PM-00_47_55
Theory : perms_2
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