Nuprl Lemma : pi_comp_wf2
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[cA:X +⊢ Compositon(A)]. ∀[cB:X.A +⊢ Compositon(B)].
(pi_comp(X;A;cA;cB) ∈ X ⊢ Compositon(ΠA B))
Proof
Definitions occuring in Statement :
pi_comp: pi_comp(X;A;cA;cB),
composition-structure: Gamma ⊢ Compositon(A),
cubical-pi: ΠA 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],
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
prop: ℙ,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
csm+: tau+,
cubical-type: {X ⊢ _},
cc-snd: q,
csm-ap-type: (AF)s,
interval-1: 1(𝕀),
cc-fst: p,
interval-type: 𝕀,
csm-comp: G o F,
csm-ap: (s)x,
csm-adjoin: (s;u),
compose: f o g,
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
pi_comp: pi_comp(X;A;cA;cB),
csm-comp-structure: (cA)tau,
constant-cubical-type: (X),
squash: ↓T,
true: True,
pi2: snd(t),
pi1: fst(t),
let: let,
interval-0: 0(𝕀),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
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),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
csm-ap-term: (t)s,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cubical-pi: ΠA B,
composition-function: composition-function{j:l,i:l}(Gamma;A)
Lemmas referenced :
pi_comp_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical-pi_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
context-subset_wf,
csm-context-subset-subtype3,
istype-cubical-term,
face-type_wf,
cube_set_map_wf,
uniform-comp-function_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
csm-cubical-pi,
csm-id-adjoin_wf-interval-1,
csm+_wf,
cube_set_map_cumulativity-i-j,
csm-id-adjoin_wf,
interval-1_wf,
csm-adjoin_wf,
csm-comp_wf,
cc-fst_wf,
cc-snd_wf,
cubical-term-eqcd,
subset-cubical-type,
thin-context-subset-adjoin,
sub_cubical_set_functionality,
context-subset-is-subset,
interval-0_wf,
equal_wf,
equal_functionality_wrt_subtype_rel2,
revfill-1,
csm+_wf_interval,
csm-comp-structure_wf,
subtype_rel-equal,
csm-interval-type,
revfill_wf,
cubical-term_wf,
squash_wf,
true_wf,
subtype_rel_self,
istype-universe,
cubical-type-cumulativity,
cubical-app_wf,
cubical-lambda_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-context-subset-subtype2,
context-adjoin-subset,
cubical-pi-subset-adjoin2,
csm-comp-type,
subset-cubical-term,
cubical-pi-p,
csm-cubical-app,
thin-context-subset,
cubical-pi-context-subset,
cubical-pi-subset,
iff_weakening_equal,
csm-face-type,
context-adjoin-subset0,
comp_term_wf,
csm-cubical-lambda,
csm-comp_term,
csm-revfill,
context-subset-map,
cube_set_map_subtype2,
cubical-app_wf-csm,
cube_set_map_subtype3,
csm-constrained-cubical-term,
context-subset-term-subtype
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,
universeIsType,
instantiate,
applyEquality,
because_Cache,
sqequalRule,
inhabitedIsType,
dependent_functionElimination,
applyLambdaEquality,
setElimination,
rename,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
cumulativity,
universeEquality,
hyp_replacement,
independent_functionElimination,
imageElimination,
Error :memTop,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
sqequalBase,
independent_pairFormation,
productIsType,
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)].
(pi\_comp(X;A;cA;cB) \mmember{} X \mvdash{} Compositon(\mPi{}A B))
Date html generated:
2020_05_20-PM-05_07_48
Last ObjectModification:
2020_04_20-PM-02_31_25
Theory : cubical!type!theory
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