Nuprl Lemma : uabeta_aux_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}].
(uabeta_aux(G;A;B;f) ∈ {G.decode(A) ⊢ _:let b = app(equiv-fun((f)p); q) in
let b' = transprt-const(G.decode(A);(CompFun(B))p;b) in
let b'' = transprt-const(G.decode(A);(CompFun(B))p;b') in
(Path_(decode(B))p b'' b)})
Proof
Definitions occuring in Statement :
uabeta_aux: uabeta_aux(G;A;B;f),
universe-comp-fun: CompFun(A),
universe-decode: decode(t),
cubical-universe: c𝕌,
transprt-const: transprt-const(G;cA;a),
csm-comp-structure: (cA)tau,
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
path-type: (Path_A a b),
cubical-app: app(w; u),
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_set: CubicalSet,
let: let,
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,
guard: {T},
uabeta_aux: uabeta_aux(G;A;B;f),
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
let: let,
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),
squash: ↓T,
prop: ℙ,
true: True
Lemmas referenced :
universe-decode_wf,
csm-ap-term-universe,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cc-fst_wf,
transprt-const_wf,
csm-ap-type_wf,
csm-comp-structure_wf2,
universe-comp-fun_wf,
cubical-app_wf_fun,
equiv-fun_wf,
csm-ap-term_wf,
cubical-equiv-p,
cubical-term-eqcd,
cc-snd_wf,
istype-cubical-term,
cubical-equiv_wf,
istype-cubical-universe-term,
cubical_set_wf,
csm-universe-decode,
subtype_rel_self,
composition-structure_wf,
trans-const-path_wf,
squash_wf,
true_wf,
cubical-type_wf,
path-type_wf,
comp_path_wf,
subset-cubical-term2,
sub_cubical_set_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
Error :memTop,
rename,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}].
(uabeta\_aux(G;A;B;f) \mmember{} \{G.decode(A) \mvdash{} \_:let b = app(equiv-fun((f)p); q) in
let b' = transprt-const(G.decode(A);(CompFun(B))p;b) in
let b'' = transprt-const(G.decode(A);(CompFun(B))p;b') in
(Path\_(decode(B))p b'' b)\})
Date html generated:
2020_05_20-PM-07_42_17
Last ObjectModification:
2020_05_01-AM-09_50_23
Theory : cubical!type!theory
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