Nuprl Lemma : sigma_comp_wf2
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X ⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(sigma_comp(cA;cB) ∈ X ⊢ Compositon(Σ A B))
Proof
Definitions occuring in Statement :
sigma_comp: sigma_comp(cA;cB),
composition-structure: Gamma ⊢ Compositon(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,
composition-structure: Gamma ⊢ Compositon(A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
uimplies: b supposing a,
prop: ℙ,
sigma_comp: sigma_comp(cA;cB),
csm+: tau+,
guard: {T},
implies: P ⇒ Q,
respects-equality: respects-equality(S;T),
compose: f o g,
cat-comp: cat-comp(C),
names-hom: I ⟶ J,
type-cat: TypeCat,
pi2: snd(t),
cat-arrow: cat-arrow(C),
quotient: x,y:A//B[x; y],
fset: fset(T),
cube-cat: CubeCat,
spreadn: spread4,
op-cat: op-cat(C),
pi1: fst(t),
cat-ob: cat-ob(C),
nat-trans: nat-trans(C;D;F;G),
psc_map: A ⟶ B,
cube_set_map: A ⟶ B,
true: True,
squash: ↓T,
partial-term-0: u[0],
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
let: let,
csm-ap-term: (t)s,
interval-1: 1(𝕀),
and: P ∧ Q,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
composition-function: composition-function{j:l,i:l}(Gamma;A)
Lemmas referenced :
sigma_comp_wf,
istype-cubical-term,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical-sigma_wf,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
csm-id-adjoin_wf-interval-1,
context-subset-map,
csm-context-subset-subtype3,
context-subset-term-subtype,
interval-0_wf,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term-eqcd,
cube_set_map_wf,
uniform-comp-function_wf,
composition-structure_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
csm+_wf,
cube_set_map_cumulativity-i-j,
subset-cubical-type,
thin-context-subset-adjoin,
sub_cubical_set_functionality,
context-subset-is-subset,
csm-cubical-sigma,
equal_wf,
equal_functionality_wrt_subtype_rel2,
cubical-term-eqcd,
cubical-fst_wf,
cubical-sigma-subset-adjoin,
cubical-snd_wf,
respects-equality-context-subset-term,
context-adjoin-subset1,
cc-fst_wf,
subset-cubical-term2,
partial-term-0_wf,
subtype_rel_self,
true_wf,
squash_wf,
cubical-term_wf,
cubical-sigma-subset,
csm-ap-cubical-fst,
csm-id-adjoin-subset,
fill_term_0,
cc-fst_wf_interval,
csm-comp-structure_wf,
csm-fill_term,
csm-cubical-pair,
csm-cubical-fst,
csm+_wf_interval,
subset-cubical-term,
sub_cubical_set_transitivity,
sub_cubical_set_self,
fill_term_wf,
csm-comp-structure-composition-function,
csm-constrained-cubical-term,
csm-adjoin_wf,
csm-id_wf,
context-adjoin-subset2,
thin-context-subset,
iff_weakening_equal,
istype-universe,
csm-cubical-snd,
context-adjoin-subset4,
csm-comp_term,
comp_term_wf,
composition-function_wf,
csm-comp_wf,
composition-structure-cumulativity,
subtype_rel-equal,
csm-comp-type,
constrained-cubical-term_wf,
cubical-pair_wf,
csm-interval-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
dependent_set_memberEquality_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaFormation_alt,
equalityIstype,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
instantiate,
applyEquality,
because_Cache,
universeIsType,
independent_isectElimination,
inhabitedIsType,
dependent_functionElimination,
hyp_replacement,
applyLambdaEquality,
independent_functionElimination,
lambdaEquality_alt,
cumulativity,
universeEquality,
baseClosed,
imageMemberEquality,
natural_numberEquality,
imageElimination,
rename,
setElimination,
independent_pairFormation,
productElimination,
productIsType,
setEquality,
functionEquality
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{} X \mvdash{} Compositon(\mSigma{} A B))
Date html generated:
2020_05_20-PM-05_00_23
Last ObjectModification:
2020_05_01-PM-11_53_10
Theory : cubical!type!theory
Home
Index