Nuprl Lemma : sigma_comp_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X ⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(sigma_comp(cA;cB) ∈ composition-function{j:l,i:l}(X;Σ A B))
Proof
Definitions occuring in Statement :
sigma_comp: sigma_comp(cA;cB),
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sigma_comp: sigma_comp(cA;cB),
composition-function: composition-function{j:l,i:l}(Gamma;A),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
csm+: tau+,
prop: ℙ,
guard: {T},
implies: P ⇒ Q,
csm-id-adjoin: [u],
csm-id: 1(X),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
true: True,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
respects-equality: respects-equality(S;T),
partial-term-0: u[0],
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
let: let,
so_lambda: λ2x.t[x],
so_apply: x[s],
composition-structure: Gamma ⊢ Compositon(A),
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
csm-ap: (s)x,
interval-0: 0(𝕀),
csm-ap-term: (t)s,
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
and: P ∧ Q,
cubical-term: {X ⊢ _:A},
interval-1: 1(𝕀),
cubical-sigma: Σ A B,
cc-adjoin-cube: (v;u),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
context-subset: Gamma, phi,
I_cube: A(I),
bdd-distributive-lattice: BoundedDistributiveLattice,
cubical-type-at: A(a),
face-type: 𝔽,
functor-ob: ob(F),
face-presheaf: 𝔽,
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
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm+_wf,
cube_set_map_cumulativity-i-j,
subset-cubical-type,
context-subset_wf,
thin-context-subset-adjoin,
csm-context-subset-subtype3,
sub_cubical_set_functionality,
context-subset-is-subset,
csm-cubical-sigma,
csm-id-adjoin_wf,
interval-0_wf,
equal_wf,
cubical-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-sigma_wf,
interval-1_wf,
csm-id-adjoin_wf-interval-1,
equal_functionality_wrt_subtype_rel2,
cubical-term-eqcd,
cubical-fst_wf,
cubical-sigma-subset-adjoin,
cubical-snd_wf,
constrained-cubical-term_wf,
csm-ap-term_wf,
istype-cubical-term,
face-type_wf,
cube_set_map_wf,
composition-structure_wf,
cubical_set_wf,
cubical-term_wf,
squash_wf,
true_wf,
subtype_rel_self,
csm-context-subset-subtype2,
partial-term-0_wf,
subset-cubical-term2,
csm-face-type,
cc-fst_wf,
context-adjoin-subset1,
respects-equality-context-subset-term,
cubical-sigma-subset,
csm-ap-cubical-fst,
csm-id-adjoin-subset,
fill_term_0,
cc-fst_wf_interval,
csm-comp-structure_wf,
fill_term_wf,
csm-comp-structure-composition-function,
comp_term_wf,
respects-equality-set-trivial2,
context-subset-term-subtype,
composition-function-cumulativity,
cube_set_map_subtype3,
sub_cubical_set_self,
csm-adjoin_wf,
csm-id_wf,
subset-cubical-term,
context-adjoin-subset2,
thin-context-subset,
csm-cubical-snd,
context-adjoin-subset4,
respects-equality-set,
fset_wf,
nat_wf,
I_cube_wf,
cubical-type-at_wf,
names-hom_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
subset-I_cube,
subtype-respects-equality,
subtype_rel_set,
all_wf,
subtype_rel_dep_function,
subtype_rel-equal,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
cubical-pair_wf,
cubical-pair-eta,
istype-universe,
cubical-type-cumulativity,
iff_weakening_equal,
csm-ap-cubical-pair,
ob_pair_lemma,
lattice-point_wf,
face_lattice_wf,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
face-term-at-restriction-eq-1,
I_cube_pair_redex_lemma,
cubical-term-at_wf,
lattice-1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaEquality_alt,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
independent_isectElimination,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
equalityTransitivity,
independent_functionElimination,
universeIsType,
Error :memTop,
inhabitedIsType,
setElimination,
rename,
imageElimination,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
equalityIstype,
lambdaFormation_alt,
productElimination,
independent_pairFormation,
productIsType,
functionEquality,
functionIsType,
universeEquality,
productEquality,
isectEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[cA:X \mvdash{} Compositon(A)]. \mforall{}[cB:X.A +\mvdash{} Compositon(B)].
(sigma\_comp(cA;cB) \mmember{} composition-function\{j:l,i:l\}(X;\mSigma{} A B))
Date html generated:
2020_05_20-PM-04_59_16
Last ObjectModification:
2020_04_20-AM-10_03_33
Theory : cubical!type!theory
Home
Index