Nuprl Lemma : csm-presw
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 ⊢ Compositon(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((presw(G;phi;f;t;t0;cT))s+ = presw(H;(phi)s;(f)s+;(t)s+;(t0)s;(cT)s+) ∈ {H.𝕀 ⊢ _:(A)s+})
Proof
Definitions occuring in Statement :
presw: presw(G;phi;f;t;t0;cT),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm+: tau+,
csm-id-adjoin: [u],
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 ⊢ _},
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,
presw: presw(G;phi;f;t;t0;cT),
subtype_rel: A ⊆r B,
csm+: tau+,
csm-comp: G o F,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
true: True,
csm-comp-structure: (cA)tau,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
csm-cubical-app,
csm-ap-term_wf,
cube-context-adjoin_wf,
csm-ap-type_wf,
interval-type_wf,
cubical-fun_wf,
csm+_wf,
cube_set_map_cumulativity-i-j,
subtype_rel-equal,
csm-interval-type,
cubical_set_cumulativity-i-j,
csm+_wf_interval,
cubical-term-eqcd,
csm-cubical-fun,
cubical-app_wf_fun,
istype-cubical-term,
cube_set_map_wf,
composition-structure_wf,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
partial-term-0_wf,
context-subset_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
thin-context-subset,
cubical-type_wf,
cubical_set_wf,
csm-pres-v
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
instantiate,
hypothesisEquality,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_functionElimination,
inhabitedIsType,
lambdaFormation_alt,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityIstype,
independent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
applyLambdaEquality,
setElimination,
rename
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,T:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[f:\{G.\mBbbI{} \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[t:\{G.\mBbbI{}, (phi)p \mvdash{} \_:T\}].
\mforall{}[t0:\{G \mvdash{} \_:(T)[0(\mBbbI{})][phi |{}\mrightarrow{} t[0]]\}]. \mforall{}[cT:G.\mBbbI{} \mvdash{} Compositon(T)]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((presw(G;phi;f;t;t0;cT))s+ = presw(H;(phi)s;(f)s+;(t)s+;(t0)s;(cT)s+))
Date html generated:
2020_05_20-PM-05_27_28
Last ObjectModification:
2020_04_21-PM-01_28_37
Theory : cubical!type!theory
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