Nuprl Lemma : composition-op-1
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[u:{I+i,s(1) ⊢ _:(A)<rho> o iota}]. ∀[a:cubical-path-0(Gamma;A;I;i;rho;1;u)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((cA I i rho 1 u a (i1)(rho) f) = u((i1) ⋅ f) ∈ A(f((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),
cubical-term-at: u(a),
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-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
face_lattice: face_lattice(I),
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-1: (i1),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T,
lattice-1: 1
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-op: Gamma ⊢ CompOp(A),
subtype_rel: A ⊆r B,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
all: ∀x:A. B[x],
implies: P ⇒ Q,
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
nat: ℕ,
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,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
lattice-1_wf,
face_lattice_wf,
subtype_rel_self,
I_cube_wf,
face-presheaf_wf2,
names-hom_wf,
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,
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,
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,
composition-op_wf,
cubical-type_wf,
cubical_set_wf,
cubical-subset-I_cube,
name-morph-1-satisfies,
name-morph-satisfies_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
extract_by_obid,
isectElimination,
because_Cache,
sqequalRule,
instantiate,
inhabitedIsType,
lambdaFormation_alt,
dependent_functionElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
lambdaEquality_alt,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ].
\mforall{}[rho:Gamma(I+i)]. \mforall{}[u:\{I+i,s(1) \mvdash{} \_:(A)<rho> o iota\}]. \mforall{}[a:cubical-path-0(Gamma;A;I;i;rho;1;u)].
\mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
((cA I i rho 1 u a (i1)(rho) f) = u((i1) \mcdot{} f))
Date html generated:
2020_05_20-PM-03_52_19
Last ObjectModification:
2020_04_09-PM-01_32_52
Theory : cubical!type!theory
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