Nuprl Lemma : universe-comp-op-encode
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[cT:X ⊢ CompOp(T)]. (compOp(encode(T;cT)) = cT ∈ X ⊢ CompOp(T))
Proof
Definitions occuring in Statement :
universe-comp-op: compOp(t),
universe-encode: encode(T;cT),
composition-op: Gamma ⊢ CompOp(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
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,
uimplies: b supposing a,
squash: ↓T,
true: True,
all: ∀x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
universe-encode: encode(T;cT),
universe-comp-op: compOp(t),
context-map: <rho>,
csm-composition: (comp)sigma,
cubical-term-at: u(a),
pi2: snd(t),
functor-arrow: arrow(F),
csm-ap: (s)x,
cube-set-restriction: f(s),
composition-op: Gamma ⊢ CompOp(A),
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u)
Lemmas referenced :
universe-encode_wf,
equal-composition-op2,
universe-comp-op_wf,
subtype_rel-equal,
composition-op_wf,
cubical_set_cumulativity-i-j,
universe-decode_wf,
cubical-type-cumulativity2,
universe-decode-encode,
cubical-path-0_wf,
istype-cubical-term,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_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,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
cubical-type_wf,
cubical_set_wf,
nh-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
equal_wf,
cube-set-restriction-id,
iff_weakening_equal,
subtype_rel_set,
cubical-type-at_wf,
nc-1_wf,
cubical-path-condition'_wf,
istype-cubical-type-at,
squash_wf,
true_wf,
istype-universe,
names-hom_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
sqequalRule,
because_Cache,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
lambdaFormation_alt,
universeIsType,
setElimination,
rename,
dependent_functionElimination,
dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
cumulativity,
universeEquality,
inhabitedIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[cT:X \mvdash{} CompOp(T)]. (compOp(encode(T;cT)) = cT)
Date html generated:
2020_05_20-PM-07_17_34
Last ObjectModification:
2020_04_25-PM-09_43_38
Theory : cubical!type!theory
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