Nuprl Lemma : mk_ctt-type-mng_wf
∀[X:⊢''']. ∀[lvl:ℕ4]. ∀[T:{X ⊢lvl _}]. ∀[cT:if (lvl =z 0) then composition-structure{i''':l, i:l}(X; T)
if (lvl =z 1) then composition-structure{i''':l, i':l}(X; T)
if (lvl =z 2) then composition-structure{i''':l, i'':l}(X; T)
else composition-structure{i''':l, i''':l}(X; T)
fi ].
(cttType(levl= lvl
type= T
comp= cT) ∈ cttType(X))
Proof
Definitions occuring in Statement :
mk_ctt-type-mng: mk_ctt-type-mng,
ctt-type-meaning: cttType(X),
ctt-level-type: {X ⊢lvl _},
composition-structure: Gamma ⊢ Compositon(A),
cubical_set: CubicalSet,
int_seg: {i..j-},
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
subtype_rel: A ⊆r B,
ctt-level-type: {X ⊢lvl _},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
exists: ∃x:A. B[x],
bnot: ¬bb,
assert: ↑b,
false: False,
nequal: a ≠ b ∈ T ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
mk_ctt-type-mng: mk_ctt-type-mng,
ctt-type-meaning: cttType(X),
istype: istype(T),
cubical-type: {X ⊢ _},
nat: ℕ,
less_than: a < b,
squash: ↓T
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
composition-structure_wf,
subtype_rel-equal,
ifthenelse_wf,
eq_int_wf,
cubical-type_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformnot_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
int_seg_subtype_special,
int_seg_cases,
ctt-level-type-subtype,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
subtype_rel_self,
ctt-level-type_wf,
int_seg_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
productElimination,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
hypothesisEquality,
sqequalRule,
universeIsType,
applyEquality,
universeEquality,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
productIsType,
dependent_pairEquality_alt,
imageElimination
Latex:
\mforall{}[X:\mvdash{}''']. \mforall{}[lvl:\mBbbN{}4]. \mforall{}[T:\{X \mvdash{}lvl \_\}]. \mforall{}[cT:if (lvl =\msubz{} 0)
then composition-structure\{i''':l, i:l\}(X; T)
if (lvl =\msubz{} 1)
then composition-structure\{i''':l, i':l\}(X; T)
if (lvl =\msubz{} 2)
then composition-structure\{i''':l, i'':l\}(X; T)
else composition-structure\{i''':l, i''':l\}(X; T)
fi ].
(cttType(levl= lvl
type= T
comp= cT) \mmember{} cttType(X))
Date html generated:
2020_05_20-PM-07_59_01
Last ObjectModification:
2020_05_05-PM-00_16_35
Theory : cubical!type!theory
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