Nuprl Lemma : cube+-
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ].
(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-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
false: False,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
formal-cube: formal-cube(I),
cube-context-adjoin: X.A,
csm-id: 1(X),
cube-: cube-(I;i),
cube+: cube+(I;i),
csm-comp: G o F,
compose: f o g,
sq_type: SQType(T),
guard: {T},
ifthenelse: if b then t else f fi ,
btrue: tt,
interval-presheaf: 𝕀,
names-hom: I ⟶ J,
names: names(I),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
istype-void,
fset_wf,
cube-context-adjoin_wf,
formal-cube_wf1,
interval-type_wf,
csm-comp_wf,
add-name_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
cube+_wf,
cube-_wf,
csm-id_wf,
cube-set-map-subtype,
I_cube_pair_redex_lemma,
interval-type-at,
I_cube_wf,
csm-equal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eq_int_eq_true_intro,
btrue_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
names_wf,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
setIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalRule,
functionIsType,
universeIsType,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
lambdaEquality_alt,
natural_numberEquality,
hypothesisEquality,
instantiate,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
functionExtensionality,
productElimination,
cumulativity,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality_alt,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
equalityIstype,
promote_hyp
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. (cube-(I;i) o cube+(I;i) = 1(formal-cube(I).\mBbbI{}))
Date html generated:
2020_05_20-PM-02_38_53
Last ObjectModification:
2020_04_04-PM-07_14_23
Theory : cubical!type!theory
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