Nuprl Lemma : path-term-equal
∀X:j⊢. ∀psi:{X ⊢ _:𝔽}.
∀[r:{X ⊢ _:𝕀}]
∀T:{X ⊢ _}. ∀a,b:{X ⊢ _:T}. ∀w:{X, psi ⊢ _:(Path_T a b)}.
∀[z:{X ⊢ _:T}]
path-term(psi;w;a;b;r) = z ∈ {X, (psi ∨ ((r=0) ∨ (r=1))) ⊢ _:T}
supposing (w @ r = z ∈ {X, psi ⊢ _:T}) ∧ (a = z ∈ {X, (r=0) ⊢ _:T}) ∧ (b = z ∈ {X, (r=1) ⊢ _:T})
Proof
Definitions occuring in Statement :
path-term: path-term(phi;w;a;b;r),
cubical-path-app: pth @ r,
path-type: (Path_A a b),
context-subset: Gamma, phi,
face-zero: (i=0),
face-one: (i=1),
face-or: (a ∨ b),
face-type: 𝔽,
interval-type: 𝕀,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
and: P ∧ Q,
guard: {T},
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
top: Top,
implies: P ⇒ Q,
true: True,
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
same-cubical-term: X ⊢ u=v:A,
path-term: path-term(phi;w;a;b;r)
Lemmas referenced :
cubical-path-app_wf,
context-subset_wf,
thin-context-subset,
context-subset-term-subtype,
subset-cubical-term2,
sub_cubical_set_self,
path-type_wf,
subset-cubical-term,
context-subset-is-subset,
path-type-subset,
interval-type_wf,
cubical-term_wf,
face-zero_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
face-one_wf,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
istype-void,
context-subset-term-0,
face-0_wf,
sub_cubical_set_transitivity,
face-and_wf,
context-iterated-subset,
subset-cubical-type,
context-iterated-subset0,
context-subset-swap,
sub_cubical_set_functionality2,
sub_cubical_set_wf,
squash_wf,
true_wf,
face-one-and-zero,
iff_weakening_equal,
case-term-same2,
face-or_wf,
case-term_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
productIsType,
equalityIstype,
inhabitedIsType,
universeIsType,
instantiate,
dependent_set_memberEquality_alt,
productElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}X:j\mvdash{}. \mforall{}psi:\{X \mvdash{} \_:\mBbbF{}\}.
\mforall{}[r:\{X \mvdash{} \_:\mBbbI{}\}]
\mforall{}T:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:T\}. \mforall{}w:\{X, psi \mvdash{} \_:(Path\_T a b)\}.
\mforall{}[z:\{X \mvdash{} \_:T\}]. path-term(psi;w;a;b;r) = z supposing (w @ r = z) \mwedge{} (a = z) \mwedge{} (b = z)
Date html generated:
2020_05_20-PM-05_09_45
Last ObjectModification:
2020_04_10-AM-11_41_34
Theory : cubical!type!theory
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