Nuprl Lemma : composition-structure-equal
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[c1,c2:Gamma ⊢ Compositon(A)].
c1 = c2 ∈ Gamma ⊢ Compositon(A)
supposing ∀H:j⊢. ∀sigma:H.𝕀 j⟶ Gamma. ∀phi:{H ⊢ _:𝔽}. ∀u:{H, phi.𝕀 ⊢ _:(A)sigma}.
∀a0:{H ⊢ _:((A)sigma)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}.
((c1 H sigma phi u a0) = (c2 H sigma phi u a0) ∈ {H ⊢ _:((A)sigma)[1(𝕀)]})
Proof
Definitions occuring in Statement :
composition-structure: Gamma ⊢ Compositon(A),
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,
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
prop: ℙ,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
interval-1: 1(𝕀),
csm-adjoin: (s;u),
csm-ap: (s)x
Lemmas referenced :
cube_set_map_wf,
cube-context-adjoin_wf,
interval-type_wf,
istype-cubical-term,
face-type_wf,
context-subset_wf,
cubical_set_cumulativity-i-j,
thin-context-subset-adjoin,
csm-ap-type_wf,
cubical-type-cumulativity2,
csm-context-subset-subtype3,
constrained-cubical-term_wf,
csm-id-adjoin_wf-interval-0,
csm-ap-term_wf,
csm-id-adjoin_wf-interval-1,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
cubical-term_wf,
uniform-comp-function_wf,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
subset-cubical-term2,
sub_cubical_set_self,
subset-cubical-term,
context-subset-is-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
functionIsType,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
applyEquality,
because_Cache,
equalityIstype,
setElimination,
rename,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
functionExtensionality,
dependent_functionElimination,
lambdaFormation_alt,
independent_functionElimination,
independent_isectElimination,
productElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[c1,c2:Gamma \mvdash{} Compositon(A)].
c1 = c2
supposing \mforall{}H:j\mvdash{}. \mforall{}sigma:H.\mBbbI{} j{}\mrightarrow{} Gamma. \mforall{}phi:\{H \mvdash{} \_:\mBbbF{}\}. \mforall{}u:\{H, phi.\mBbbI{} \mvdash{} \_:(A)sigma\}.
\mforall{}a0:\{H \mvdash{} \_:((A)sigma)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}.
((c1 H sigma phi u a0) = (c2 H sigma phi u a0))
Date html generated:
2020_05_20-PM-04_22_45
Last ObjectModification:
2020_04_17-PM-04_41_42
Theory : cubical!type!theory
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