Nuprl Lemma : csm-equiv-term
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G ⊢ _}]. ∀[f:{G ⊢ _:Equiv(T;A)}]. ∀[t:{G, phi ⊢ _:T}]. ∀[a:{G ⊢ _:A}].
∀[c:{G, phi ⊢ _:(Path_A a app(equiv-fun(f); t))}]. ∀[cF:G ⊢ Compositon(Fiber(equiv-fun(f);a))]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((equiv f [phi ⊢→ (t, c)] a)s
= equiv (f)s [(phi)s ⊢→ ((t)s, (c)s)] (a)s
∈ {H ⊢ _:Fiber(equiv-fun((f)s);(a)s)[(phi)s |⟶ fiber-point((t)s;(c)s)]})
Proof
Definitions occuring in Statement :
equiv-term: equiv f [phi ⊢→ (t, c)] a,
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
fiber-point: fiber-point(t;c),
cubical-fiber: Fiber(w;a),
path-type: (Path_A a b),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-app: app(w; u),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _: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,
uimplies: b supposing a,
equiv-term: equiv f [phi ⊢→ (t, c)] a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
let: let,
guard: {T},
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
cubical-path-app: pth @ r,
cubicalpath-app: pth @ r,
squash: ↓T,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
csm+: tau+,
csm-comp: G o F,
csm-ap-term: (t)s,
csm-ap: (s)x,
csm-adjoin: (s;u),
pi1: fst(t),
compose: f o g,
same-cubical-type: Gamma ⊢ A = B,
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
cubical-type: {X ⊢ _},
cubical-fiber: Fiber(w;a),
cubical-sigma: Σ A B,
cc-adjoin-cube: (v;u),
comp_trm: comp_trm,
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
pi2: snd(t),
csm-path-ap-q: csm-path-ap-q(H;G;s;t),
interval-0: 0(𝕀),
respects-equality: respects-equality(S;T)
Lemmas referenced :
cubical-app_wf_fun,
context-subset_wf,
thin-context-subset,
cubical-fun-subset,
equiv-fun_wf,
subset-cubical-term,
context-subset-is-subset,
cubical-fun_wf,
equiv-contr_wf,
cubical-fiber_wf,
fiber-subset,
cubical-term-eqcd,
fiber-point_wf,
context-subset-term-subtype,
csm-comp_term,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf_interval,
csm-comp-structure_wf,
contr-center_wf,
contr-path_wf,
contractible-type-subset,
contractible-type_wf,
cube_set_map_wf,
composition-structure_wf,
istype-cubical-term,
path-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-equiv_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
cubical-path-app_wf,
csm-ap-term_wf,
csm-path-type,
cc-snd_wf,
cubical-path-app-0,
cubical-path-ap-id-adjoin,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-ap-id-type,
subset-cubical-type,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm-id_wf,
csm+_wf_interval,
csm-id-adjoin_wf-interval-1,
csm-face-type,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
context-subset-map,
csm-ap-term-wf-subset,
face-term-implies-same,
csm-equiv-fun,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
csm-cubical-fiber,
comp_trm_wf,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
thin-context-subset-adjoin,
composition-function_wf,
csm+_wf,
subtype_rel-equal,
csm-interval-type,
csm-equiv-contr,
csm-fiber-point,
csm-contr-path,
csm-cubical-path-app,
csm-path-ap-q_wf,
cube_set_map_cumulativity-i-j,
context-adjoin-subset3,
csm-contr-center,
interval-0_wf,
respects-equality-context-subset-term,
cubical-path-app-1,
subset-cubical-term2,
sub_cubical_set_self,
csm-cubical-fun,
csm-cubical-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
Error :memTop,
applyEquality,
independent_isectElimination,
inhabitedIsType,
lambdaFormation_alt,
rename,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
hyp_replacement,
instantiate,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality_alt,
setEquality,
applyLambdaEquality,
setElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(T;A)\}]. \mforall{}[t:\{G, phi \mvdash{} \_:T\}].
\mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[c:\{G, phi \mvdash{} \_:(Path\_A a app(equiv-fun(f); t))\}].
\mforall{}[cF:G \mvdash{} Compositon(Fiber(equiv-fun(f);a))]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((equiv f [phi \mvdash{}\mrightarrow{} (t, c)] a)s = equiv (f)s [(phi)s \mvdash{}\mrightarrow{} ((t)s, (c)s)] (a)s)
Date html generated:
2020_05_20-PM-05_38_00
Last ObjectModification:
2020_04_18-PM-11_52_57
Theory : cubical!type!theory
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