Nuprl Lemma : bijection-preserves-contractible
∀X:j⊢. ∀A,B:{X ⊢ _}. ∀f:{X ⊢ _:(A ⟶ B)}. ∀g:{X ⊢ _:(B ⟶ A)}. ∀cA:X +⊢ Compositon(A).
∀b:{X.A ⊢ _:(Path_(A)p app((g)p; app((f)p; q)) q)}. ∀c:{X ⊢ _:Contractible(B)}.
{X ⊢ _:Contractible(A)}
Proof
Definitions occuring in Statement :
composition-structure: Gamma ⊢ Compositon(A),
contractible-type: Contractible(A),
path-type: (Path_A a b),
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
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),
uimplies: b supposing a
Lemmas referenced :
bij-contr_wf,
cubical-type-cumulativity2,
subtype_rel_self,
composition-structure_wf,
istype-cubical-term,
contractible-type_wf,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
csm-cubical-fun,
cube-context-adjoin_wf,
cc-fst_wf,
cubical-term-eqcd,
csm-ap-term_wf,
path-type_wf,
csm-ap-type_wf,
cubical-app_wf_fun,
cc-snd_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
isectElimination,
universeIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement
Latex:
\mforall{}X:j\mvdash{}. \mforall{}A,B:\{X \mvdash{} \_\}. \mforall{}f:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}. \mforall{}g:\{X \mvdash{} \_:(B {}\mrightarrow{} A)\}. \mforall{}cA:X +\mvdash{} Compositon(A).
\mforall{}b:\{X.A \mvdash{} \_:(Path\_(A)p app((g)p; app((f)p; q)) q)\}. \mforall{}c:\{X \mvdash{} \_:Contractible(B)\}.
\{X \mvdash{} \_:Contractible(A)\}
Date html generated:
2020_05_20-PM-05_25_04
Last ObjectModification:
2020_04_20-PM-02_36_30
Theory : cubical!type!theory
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