Nuprl Lemma : assert_of_eq_list
∀s:DSet. ∀as,bs:|s| List. (↑(as =b bs) ⇐⇒ as = bs ∈ (|s| List))
Proof
Definitions occuring in Statement :
eq_list: as =b bs,
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
dset: DSet,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
eq_list: as =b bs,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
cons: [a / b],
bfalse: ff,
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
subtype_rel: A ⊆r B,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
band: p ∧b q,
bnot: ¬bb,
cand: A c∧ B
Lemmas referenced :
list_wf,
set_car_wf,
dset_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
assert_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
list_ind_nil_lemma,
null_nil_lemma,
nil_wf,
true_wf,
product_subtype_list,
null_cons_lemma,
btrue_wf,
null_wf,
bfalse_wf,
btrue_neq_bfalse,
cons_wf,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
eq_list_wf,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
list_ind_cons_lemma,
assert_wf,
set_eq_wf,
eqtt_to_assert,
assert_of_dset_eq,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
equal_wf,
iff_transitivity,
assert_of_band,
reduce_hd_cons_lemma,
hd_wf,
squash_wf,
length_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf,
nat_wf,
reduce_tl_cons_lemma,
tl_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
productElimination,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
because_Cache,
unionElimination,
equalityIsType4,
baseClosed,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
productIsType,
equalityIsType3,
equalityIsType1,
equalityIsType2,
instantiate,
imageElimination,
baseApply,
closedConclusion,
applyEquality,
intEquality,
equalityElimination,
cumulativity,
productEquality,
universeEquality,
imageMemberEquality
Latex:
\mforall{}s:DSet. \mforall{}as,bs:|s| List. (\muparrow{}(as =\msubb{} bs) \mLeftarrow{}{}\mRightarrow{} as = bs)
Date html generated:
2019_10_16-PM-01_01_40
Last ObjectModification:
2018_10_08-PM-00_07_58
Theory : list_2
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