Nuprl Lemma : csm-equiv_comp
∀[H,K:j⊢]. ∀[tau:K j⟶ H]. ∀[A,E:{H ⊢ _}]. ∀[cA:H +⊢ Compositon(A)]. ∀[cE:H +⊢ Compositon(E)].
((equiv_comp(H;A;E;cA;cE))tau = equiv_comp(K;(A)tau;(E)tau;(cA)tau;(cE)tau) ∈ K ⊢ Compositon(Equiv((A)tau;(E)tau)))
Proof
Definitions occuring in Statement :
equiv_comp: equiv_comp(H;A;E;cA;cE),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
cubical-equiv: Equiv(T;A),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
prop: ℙ,
squash: ↓T,
and: P ∧ Q,
true: True,
uimplies: b supposing a,
equiv_comp: equiv_comp(H;A;E;cA;cE),
cubical-equiv: Equiv(T;A),
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
compose: f o g,
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cubical-type: {X ⊢ _},
cc-fst: p,
csm-ap-type: (AF)s,
csm+: tau+,
cc-snd: q,
csm-adjoin: (s;u),
csm-ap: (s)x,
pi1: fst(t),
csm-ap-term: (t)s,
pi2: snd(t),
is-cubical-equiv: IsEquiv(T;A;w)
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical-fun_wf,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cc-fst_wf,
cc-snd_wf,
cubical-fun-p,
composition-structure_wf,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term-eqcd,
csm-equiv_comp-sq,
sigma_comp_wf2,
is-cubical-equiv_wf,
pi_comp_wf_fun,
csm-comp-structure_wf,
cube_set_map_cumulativity-i-j,
fiber-comp_wf,
csm-comp-structure_wf2,
csm-cubical-fiber,
subtype_rel_self,
iff_weakening_equal,
csm+_wf,
subtype_rel-equal,
csm-cubical-fun,
contractible_comp_wf,
pi_comp_wf2,
contractible-type_wf,
cubical-pi_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
dependent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
hyp_replacement,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
universeEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
Error :memTop,
lambdaFormation_alt,
independent_functionElimination
Latex:
\mforall{}[H,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} H]. \mforall{}[A,E:\{H \mvdash{} \_\}]. \mforall{}[cA:H +\mvdash{} Compositon(A)]. \mforall{}[cE:H +\mvdash{} Compositon(E)].
((equiv\_comp(H;A;E;cA;cE))tau = equiv\_comp(K;(A)tau;(E)tau;(cA)tau;(cE)tau))
Date html generated:
2020_05_20-PM-07_20_03
Last ObjectModification:
2020_04_27-PM-03_46_41
Theory : cubical!type!theory
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