Nuprl Lemma : composition-op-uniformity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
∀I,J:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀j:{j:ℕ| ¬j ∈ J} . ∀g:J ⟶ I. ∀rho:Gamma(I+i). ∀phi:𝔽(I).
∀u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}. ∀a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((comp I i rho phi u a0 (i1)(rho) g)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi)) (a0 (i0)(rho) g))
∈ A(g((i1)(rho))))
Proof
Definitions occuring in Statement :
composition-op: Gamma ⊢ CompOp(A),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
subset-trans: subset-trans(I;J;f;x),
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-e': g,i=j,
nc-1: (i1),
nc-0: (i0),
nc-s: s,
add-name: I+i,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
composition-op: Gamma ⊢ CompOp(A),
composition-uniformity: composition-uniformity(Gamma;A;comp),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
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]
Lemmas referenced :
cubical-path-0_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-term_wf,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
cubical-type-cumulativity,
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,
names-hom_wf,
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,
composition-op_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
universeIsType,
instantiate,
extract_by_obid,
isectElimination,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality,
inhabitedIsType,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)].
\mforall{}I,J:fset(\mBbbN{}). \mforall{}i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} . \mforall{}j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} . \mforall{}g:J {}\mrightarrow{} I. \mforall{}rho:Gamma(I+i). \mforall{}phi:\mBbbF{}(I).
\mforall{}u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}. \mforall{}a0:cubical-path-0(Gamma;A;I;i;rho;phi;u).
((comp I i rho phi u a0 (i1)(rho) g)
= (comp J j g,i=j(rho) g(phi) (u)subset-trans(I+i;J+j;g,i=j;s(phi)) (a0 (i0)(rho) g)))
Date html generated:
2020_05_20-PM-03_50_08
Last ObjectModification:
2020_04_09-PM-01_13_54
Theory : cubical!type!theory
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