Nuprl Lemma : equiv_comp_wf
∀[H:j⊢]. ∀[A,E:{H ⊢ _}]. ∀[cA:H +⊢ Compositon(A)]. ∀[cE:H +⊢ Compositon(E)].
(equiv_comp(H;A;E;cA;cE) ∈ H +⊢ Compositon(Equiv(A;E)))
Proof
Definitions occuring in Statement :
equiv_comp: equiv_comp(H;A;E;cA;cE),
composition-structure: Gamma ⊢ Compositon(A),
cubical-equiv: Equiv(T;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,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
equiv_comp: equiv_comp(H;A;E;cA;cE),
cubical-equiv: Equiv(T;A),
is-cubical-equiv: IsEquiv(T;A;w),
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical-fun_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cc-fst_wf,
cc-snd_wf,
cubical-term-eqcd,
sigma_comp_wf2,
cubical-pi_wf,
contractible-type_wf,
cubical-fiber_wf,
pi_comp_wf_fun,
pi_comp_wf2,
csm-comp-structure_wf2,
subtype_rel_self,
composition-structure_wf,
contractible_comp_wf,
fiber-comp_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-fun-p,
iff_weakening_equal,
cube_set_map_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,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
natural_numberEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[A,E:\{H \mvdash{} \_\}]. \mforall{}[cA:H +\mvdash{} Compositon(A)]. \mforall{}[cE:H +\mvdash{} Compositon(E)].
(equiv\_comp(H;A;E;cA;cE) \mmember{} H +\mvdash{} Compositon(Equiv(A;E)))
Date html generated:
2020_05_20-PM-07_18_22
Last ObjectModification:
2020_04_27-PM-01_53_47
Theory : cubical!type!theory
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