Nuprl Lemma : composition-op-nc-e
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)].
∀I:fset(ℕ). ∀i,j:{j:ℕ| ¬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) = (comp I j e(i;j)(rho) phi (u)subset-trans(I+i;I+j;e(i;j);s(phi)) a0) ∈ A((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-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: e(i;j),
nc-1: (i1),
nc-s: s,
add-name: I+i,
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],
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],
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
composition-op: Gamma ⊢ CompOp(A),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-type: {X ⊢ _},
subset-iota: iota,
csm-comp: G o F,
csm-ap-type: (AF)s,
compose: f o g,
csm-ap: (s)x,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
sq_stable: SqStable(P),
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
cubical-term-at: u(a),
subset-trans: subset-trans(I;J;f;x),
csm-ap-term: (t)s,
bdd-distributive-lattice: BoundedDistributiveLattice,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
context-map: <rho>,
functor-arrow: arrow(F),
cube-set-restriction: f(s)
Lemmas referenced :
composition-op-uniformity,
nh-id_wf,
cubical-path-0_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
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,
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-type-at_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
nc-1_wf,
cube-set-restriction-comp,
nc-e_wf,
subtype_rel_self,
iff_weakening_equal,
nc-e-comp-nc-1,
nc-e'-1,
cubical-subset-term-trans,
subtype_rel-equal,
cube-set-restriction-id,
cubical-type-ap-morph-id,
names-hom_wf,
cube_set_map_wf,
nc-0_wf,
nc-e-comp-nc-0,
sq_stable__cubical-path-condition,
cubical-subset-I_cube-member,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
subtype_rel_weakening,
ext-eq_weakening,
nh-comp-assoc,
nh-comp_wf,
cubical-term-at_wf,
cubical-subset-I_cube,
name-morph-satisfies_wf,
nh-id-right,
uiff_transitivity2,
name-morph-satisfies-comp,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
s-comp-nc-0,
csm-ap-type-at,
cubical-path-condition_wf,
subtype_rel_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
dependent_functionElimination,
universeIsType,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
setElimination,
rename,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
inhabitedIsType,
setIsType,
functionIsType,
intEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
hyp_replacement,
equalityIstype,
productIsType,
applyLambdaEquality,
productEquality,
cumulativity,
isectEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)].
\mforall{}I:fset(\mBbbN{}). \mforall{}i,j:\{j:\mBbbN{}| \mneg{}j \mmember{} 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) = (comp I j e(i;j)(rho) phi (u)subset-trans(I+i;I+j;e(i;j);s(phi)) a0))
Date html generated:
2020_05_20-PM-03_50_34
Last ObjectModification:
2020_04_09-PM-01_57_10
Theory : cubical!type!theory
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