Nuprl Lemma : csm-pres-c1
∀[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]]}]. ∀[cA:G.𝕀 ⊢ Compositon(A)]. ∀[H:j⊢]. ∀[s:H j⟶ G].
((pres-c1(G;phi;f;t;t0;cA))s
= pres-c1(H;(phi)s;(f)s+;(t)s+;(t0)s;(cA)s+)
∈ {H ⊢ _:((A)s+)[1(𝕀)][(phi)s |⟶ app((f)s+; (t)s+)[1]]})
Proof
Definitions occuring in Statement :
pres-c1: pres-c1(G;phi;f;t;t0;cA),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
partial-term-1: u[1],
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-app: app(w; u),
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,
composition-structure: Gamma ⊢ Compositon(A),
squash: ↓T,
subtype_rel: A ⊆r B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
uimplies: b supposing a,
csm+: tau+,
csm-comp: G o F,
cubical-type: {X ⊢ _},
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
interval-type: 𝕀,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
pi2: snd(t),
compose: f o g,
pi1: fst(t),
prop: ℙ,
true: True,
partial-term-1: u[1],
csm-ap-term: (t)s,
pres-c1: pres-c1(G;phi;f;t;t0;cA),
all: ∀x:A. B[x],
implies: P ⇒ Q,
partial-term-0: u[0],
interval-0: 0(𝕀)
Lemmas referenced :
pres-c1_wf,
context-subset-term-subtype,
cube-context-adjoin_wf,
interval-type_wf,
cubical-fun_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
cubical-app_wf_fun,
context-subset_wf,
thin-context-subset,
cubical-fun-subset,
subset-cubical-term,
context-subset-is-subset,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm+_wf_interval,
cube_set_map_cumulativity-i-j,
csm-id-adjoin_wf-interval-1,
cube_set_map_wf,
composition-structure_wf,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
partial-term-0_wf,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
context-subset-map,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
cubical-term-eqcd,
csm-cubical-app,
csm-comp_term,
context-adjoin-subset4,
csm-cubical-fun,
interval-0_wf,
csm+_wf,
subtype_rel-equal,
cubical-term_wf,
csm-interval-type
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
instantiate,
Error :memTop,
because_Cache,
dependent_set_memberEquality_alt,
equalityIstype,
inhabitedIsType,
independent_isectElimination,
universeIsType,
productElimination,
hyp_replacement,
lambdaEquality_alt,
universeEquality,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
cumulativity,
lambdaFormation_alt
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{}[cA:G.\mBbbI{} \mvdash{} Compositon(A)]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} G].
((pres-c1(G;phi;f;t;t0;cA))s = pres-c1(H;(phi)s;(f)s+;(t)s+;(t0)s;(cA)s+))
Date html generated:
2020_05_20-PM-05_25_55
Last ObjectModification:
2020_04_21-PM-02_58_28
Theory : cubical!type!theory
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