Nuprl Lemma : csm-transprt-fun
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 +⊢ Compositon(A)]. ∀[H:j⊢]. ∀[s:H j⟶ Gamma].
((transprt-fun(Gamma;A;cA))s = transprt-fun(H;(A)s+;(cA)s+) ∈ {H ⊢ _:(((A)s+)[0(𝕀)] ⟶ ((A)s+)[1(𝕀)])})
Proof
Definitions occuring in Statement :
transprt-fun: transprt-fun(Gamma;A;cA),
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm+: tau+,
csm-id-adjoin: [u],
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,
subtype_rel: A ⊆r B,
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
interval-type: 𝕀,
csm+: tau+,
csm-ap-type: (AF)s,
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
interval-1: 1(𝕀),
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),
csm-comp-structure: (cA)tau,
transprt-fun: transprt-fun(Gamma;A;cA),
cubical-lam: cubical-lam(X;b),
csm-ap-term: (t)s
Lemmas referenced :
csm+_wf,
interval-type_wf,
cube_set_map_cumulativity-i-j,
csm-interval-type,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
csm-cubical-fun,
csm-ap-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
csm-id-adjoin_wf-interval-1,
cubical-fun_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-term-eqcd,
csm-ap-term_wf,
transprt-fun_wf,
cubical-type-cumulativity2,
composition-structure_wf,
csm-comp-structure_wf,
cubical-fun-as-cubical-pi,
csm-cubical-lambda,
cc-fst_wf,
cube_set_map_wf,
cubical_set_wf,
transprt_wf,
csm+_wf_interval,
subtype_rel-equal,
cc-snd_wf,
cubical-lam_wf,
istype-cubical-term,
csm-transprt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
sqequalRule,
Error :memTop,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
inhabitedIsType,
setElimination,
rename,
cumulativity,
hyp_replacement,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} +\mvdash{} Compositon(A)]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} Gamma].
((transprt-fun(Gamma;A;cA))s = transprt-fun(H;(A)s+;(cA)s+))
Date html generated:
2020_05_20-PM-04_38_49
Last ObjectModification:
2020_04_18-PM-02_24_11
Theory : cubical!type!theory
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