Nuprl Lemma : glue-comp-agrees
The type (not displayed) of the equality in this lemma
is composition-structure{i:l}(G, psi; T)
This means that in "extent" psi, when ⌜G ⊢ Glue [psi ⊢→ (T;f)] A = T ∈ {G, psi ⊢ _}⌝, the
composition for ⌜Glue [psi ⊢→ (T;f)] A⌝ is the same as the composition for T.
This property of the compostion for Glue is used in construction of the
composition for c𝕌 (the cubiucal universe type).⋅
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[psi:{G ⊢ _:𝔽}]. ∀[T:{G, psi ⊢ _}]. ∀[cT:G, psi +⊢ Compositon(T)].
∀[f:{G, psi ⊢ _:Equiv(T;A)}].
(comp(Glue [psi ⊢→ (T, f)] A) = cT ∈ G, psi ⊢ Compositon(T))
Proof
Definitions occuring in Statement :
glue-comp: comp(Glue [phi ⊢→ (T, f)] A) ,
composition-structure: Gamma ⊢ Compositon(A),
cubical-equiv: Equiv(T;A),
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
all: ∀x:A. B[x],
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
prop: ℙ,
squash: ↓T,
same-cubical-type: Gamma ⊢ A = B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
glue-comp: comp(Glue [phi ⊢→ (T, f)] A) ,
partial-term-0: u[0],
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
compose: f o g,
partial-term-1: u[1],
let: let,
comp_term: comp cA [phi ⊢→ u] a0,
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
interval-0: 0(𝕀),
face-1: 1(𝔽),
csm-ap-term: (t)s,
csm+: tau+,
cube-context-adjoin: X.A,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
pi2: snd(t),
pi1: fst(t),
cc-adjoin-cube: (v;u),
cubical-term-at: u(a),
same-cubical-term: X ⊢ u=v:A,
equiv-fun: equiv-fun(f),
cubical-fst: p.1,
pres-c1: pres-c1(G;phi;f;t;t0;cA),
pres-c2: pres-c2(G;phi;f;t;t0;cT),
cand: A c∧ B
Lemmas referenced :
glue-type_wf,
equiv-fun_wf,
context-subset_wf,
thin-context-subset,
csm-glue-type,
cube-context-adjoin_wf,
interval-type_wf,
cube_set_map_subtype3,
context-subset-is-subset,
sub_cubical_set_self,
cubical_set_cumulativity-i-j,
cube_set_map_cumulativity-i-j,
istype-cubical-term,
csm-ap-type_wf,
csm-id-adjoin_wf-interval-1,
csm-context-subset-subtype2,
csm-ap-term_wf,
csm-context-subset-subtype3,
subset-cubical-term2,
thin-context-subset-adjoin,
csm-id-adjoin_wf,
interval-1_wf,
subset-cubical-term,
interval-0_wf,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term-eqcd,
cubical-term-eqcd,
face-type_wf,
cube_set_map_wf,
cubical_set_wf,
uniform-comp-function-cumulativity,
uniform-comp-function_wf,
cubical-equiv_wf,
composition-structure_wf,
cubical-type_wf,
squash_wf,
true_wf,
glue-type-constraint,
equal_wf,
istype-universe,
subset-cubical-type,
sub_cubical_set_functionality,
csm-face-type,
cc-fst_wf_interval,
context-adjoin-subset1,
face-1-in-context-subset,
csm-face-1,
cubical-fun_wf,
csm-cubical-fun,
subtype_rel_self,
iff_weakening_equal,
unglue-term_wf2,
context-subset-term-subtype,
glue-type-subset,
cubical-fun-subset,
sub_cubical_set_transitivity,
context-subset-swap,
sub_cubical_set_functionality2,
face-1_wf,
context-1-subset,
context-iterated-subset0,
subtype_rel_wf,
glue-type-1',
cubical-type-cumulativity2,
cubical-term_wf,
cubical-app_wf_fun,
csm-cubical-app,
partial-term-0_wf,
csm-id-adjoin-subset,
comp_term_wf,
csm-comp-structure-composition-function,
composition-structure-cumulativity,
context-adjoin-subset4,
face-forall-1,
csm-id_wf,
context-iterated-subset1,
context-iterated-subset2,
csm-ap-id-term,
subtype_rel-equal,
csm-equal2,
csm-comp_wf,
csm+_wf_interval,
I_cube_wf,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
csm-ap_wf,
cc-adjoin-cube_wf,
cubical-term-equal,
cubical-term-at_wf,
csm-id-comp,
case-term-1,
context-subset-adjoin-subtype,
face-forall_wf,
csm-comp-structure_wf,
composition-structure-subset,
pres_wf,
csm-comp-structure-subset,
pres-c1_wf,
composition-function-subset,
pres-c2_wf,
path-type_wf,
equal_functionality_wrt_subtype_rel2,
constrained-cubical-term_wf,
composition-function_wf,
context-adjoin-subset2,
partial-term-1_wf,
sub_cubical_set_wf,
comp_term-subset,
csm-equiv-fun,
path-type-subset,
fiber-comp_wf,
context-subset-map,
csm-cubical-equiv,
cubical-equiv-subset,
face-or_wf,
equiv-term_wf,
cubical-fiber-subset,
cubical-fiber_wf,
equal-wf-T-base,
fiber-member_wf,
glue-term-1',
face-1-or,
fiber-member-fiber-point
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality_alt,
functionExtensionality,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
instantiate,
applyEquality,
independent_isectElimination,
sqequalRule,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
equalityTransitivity,
dependent_functionElimination,
independent_functionElimination,
productElimination,
universeIsType,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeEquality,
cumulativity,
Error :memTop,
applyLambdaEquality,
independent_pairFormation,
productIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)]. \mforall{}[psi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[T:\{G, psi \mvdash{} \_\}].
\mforall{}[cT:G, psi +\mvdash{} Compositon(T)]. \mforall{}[f:\{G, psi \mvdash{} \_:Equiv(T;A)\}].
(comp(Glue [psi \mvdash{}\mrightarrow{} (T, f)] A) = cT)
Date html generated:
2020_05_20-PM-07_03_45
Last ObjectModification:
2020_04_25-AM-11_28_41
Theory : cubical!type!theory
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