Nuprl Lemma : composition-in-subset
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G ⊢ Compositon(A)]. ∀[H1,H2:j⊢].
∀[sigma:H1.𝕀 j⟶ G]. ∀[phi:{H1 ⊢ _:𝔽}]. ∀[u:{H1, phi.𝕀 ⊢ _:(A)sigma}].
∀[a0:{H1 ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}].
((cA H1 sigma phi u a0) = (cA H2 sigma phi u a0) ∈ {H2 ⊢ _:((A)sigma)[1(𝕀)]})
supposing sub_cubical_set{j:l}(H2; H1)
Proof
Definitions occuring in Statement :
composition-structure: Gamma ⊢ Compositon(A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
sub_cubical_set: Y ⊆ X,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
composition-structure: Gamma ⊢ Compositon(A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
csm-id-adjoin: [u],
csm-id: 1(X),
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-1: 1(𝕀),
csm-adjoin: (s;u),
csm-ap: (s)x,
prop: ℙ,
true: True,
composition-function: composition-function{j:l,i:l}(Gamma;A),
implies: P ⇒ Q,
csm+: tau+,
csm-comp: G o F,
cube-context-adjoin: X.A,
interval-type: 𝕀,
compose: f o g,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
pi2: snd(t),
pi1: fst(t),
cc-adjoin-cube: (v;u),
and: P ∧ Q,
interval-0: 0(𝕀),
csm-ap-term: (t)s
Lemmas referenced :
csm-id_wf,
cube_set_map_subtype3,
sub_cubical_set_self,
constrained-cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
context-subset_wf,
thin-context-subset-adjoin,
istype-cubical-term,
csm-context-subset-subtype3,
face-type_wf,
cube_set_map_wf,
sub_cubical_set_wf,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-id-adjoin_wf,
interval-1_wf,
sub_cubical_set-cumulativity1,
cubical-term_wf,
csm-ap-id-term,
subset-cubical-term2,
csm-id-adjoin_wf-interval-1,
csm-equal,
sub_cubical_set_functionality,
csm-comp_wf,
csm+_wf_interval,
cube-set-map-subtype,
I_cube_wf,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
csm-ap_wf,
cc-adjoin-cube_wf,
subset-cubical-term,
sub_cubical_set_transitivity,
context-subset-is-subset,
sub_cubical_set_functionality2,
interval-0_wf,
csm-context-subset-subtype2,
subset-cubical-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
applyEquality,
because_Cache,
independent_isectElimination,
sqequalRule,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
instantiate,
inhabitedIsType,
productElimination,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
equalityTransitivity,
universeEquality,
natural_numberEquality,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
functionExtensionality,
Error :memTop,
independent_pairFormation,
dependent_set_memberEquality_alt
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[cA:G \mvdash{} Compositon(A)]. \mforall{}[H1,H2:j\mvdash{}].
\mforall{}[sigma:H1.\mBbbI{} j{}\mrightarrow{} G]. \mforall{}[phi:\{H1 \mvdash{} \_:\mBbbF{}\}]. \mforall{}[u:\{H1, phi.\mBbbI{} \mvdash{} \_:(A)sigma\}].
\mforall{}[a0:\{H1 \mvdash{} \_:((A)sigma)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}].
((cA H1 sigma phi u a0) = (cA H2 sigma phi u a0))
supposing sub\_cubical\_set\{j:l\}(H2; H1)
Date html generated:
2020_05_20-PM-04_23_42
Last ObjectModification:
2020_04_17-PM-04_43_43
Theory : cubical!type!theory
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