Nuprl Lemma : csm-comp-structure-id
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ Compositon(A)]. ∀[tau:Gamma j⟶ Gamma].
(cA)tau = cA ∈ Gamma ⊢ Compositon(A) supposing tau = 1(Gamma) ∈ Gamma j⟶ Gamma
Proof
Definitions occuring in Statement :
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
cubical-type: {X ⊢ _},
csm-id: 1(X),
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
and: P ∧ Q,
member: t ∈ T,
squash: ↓T,
true: True,
csm-comp-structure: (cA)tau,
interval-type: 𝕀,
csm-comp: G o F,
compose: f o g,
subtype_rel: A ⊆r B,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
csm-id-adjoin: [u],
csm-id: 1(X)
Lemmas referenced :
composition-structure_wf,
csm-comp-structure_wf,
csm-id_wf,
cube_set_map_wf,
cubical-type_wf,
cubical_set_wf,
csm-ap-id-type,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-ap-type_wf,
subtype_rel_self,
iff_weakening_equal,
constrained-cubical-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
csm-ap-term_wf,
context-subset_wf,
thin-context-subset-adjoin,
cubical-term_wf,
csm-context-subset-subtype3,
face-type_wf,
uniform-comp-function_wf,
csm-id-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
dependent_set_memberEquality_alt,
hypothesis,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productIsType,
equalityIstype,
inhabitedIsType,
hypothesisEquality,
applyLambdaEquality,
thin,
applyEquality,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
setElimination,
rename,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeIsType,
instantiate,
universeEquality,
independent_isectElimination,
independent_functionElimination,
functionExtensionality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} Compositon(A)]. \mforall{}[tau:Gamma j{}\mrightarrow{} Gamma].
(cA)tau = cA supposing tau = 1(Gamma)
Date html generated:
2020_05_20-PM-04_36_04
Last ObjectModification:
2020_04_17-AM-01_04_21
Theory : cubical!type!theory
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