Nuprl Lemma : cube-+
∀[I:fset(ℕ)]. ∀[i:ℕ]. (cube+(I;i) o cube-(I;i) = 1(formal-cube(I+i)) ∈ formal-cube(I+i) j⟶ formal-cube(I+i))
Proof
Definitions occuring in Statement :
cube-: cube-(I;i),
cube+: cube+(I;i),
interval-type: 𝕀,
cube-context-adjoin: X.A,
csm-id: 1(X),
csm-comp: G o F,
cube_set_map: A ⟶ B,
formal-cube: formal-cube(I),
add-name: I+i,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
formal-cube: formal-cube(I),
all: ∀x:A. B[x],
csm-id: 1(X),
cube+: cube+(I;i),
cube-: cube-(I;i),
csm-comp: G o F,
compose: f o g,
names-hom: I ⟶ J,
names: names(I),
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
sq_stable: SqStable(P),
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
label: ...$L... t,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
istype-nat,
fset_wf,
nat_wf,
formal-cube_wf1,
add-name_wf,
csm-comp_wf,
cube-context-adjoin_wf,
interval-type_wf,
cube-_wf,
cube+_wf,
csm-id_wf,
cube-set-map-subtype,
I_cube_pair_redex_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
sq_stable__fset-member,
int-deq_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
fset-member_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
iff_weakening_equal,
trivial-member-add-name1,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
names_wf,
I_cube_wf,
csm-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_set_memberEquality_alt,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
independent_pairFormation,
voidElimination,
equalityIstype,
promote_hyp,
cumulativity
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. (cube+(I;i) o cube-(I;i) = 1(formal-cube(I+i)))
Date html generated:
2020_05_20-PM-02_39_03
Last ObjectModification:
2020_04_04-PM-04_37_33
Theory : cubical!type!theory
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