Nuprl Lemma : sigma-path-fst
∀X:j⊢. ∀A:{X ⊢ _}. ∀B:{X.A ⊢ _}. ∀x,y:{X ⊢ _:Σ A B}. ({X ⊢ _:(Path_Σ A B x y)} ⇒ {X ⊢ _:(Path_A x.1 y.1)})
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
cubical-fst: p.1,
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
and: P ∧ Q,
cubical-type: {X ⊢ _},
cc-snd: q,
cc-fst: p,
csm-ap-type: (AF)s,
interval-type: 𝕀,
csm-comp: G o F,
csm-ap: (s)x,
constant-cubical-type: (X),
compose: f o g,
uimplies: b supposing a,
cubical-path-app: pth @ r,
true: True,
same-cubical-term: X ⊢ u=v:A,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
cubical-term_wf,
path-type_wf,
cubical-sigma_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type_wf,
cubical_set_wf,
path-eta_wf,
path-type-subtype,
cubical-sigma-p,
interval-type_wf,
term-to-path-wf,
cubical-fst_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-adjoin_wf,
csm-comp_wf,
cc-snd_wf,
csm-cubical-fst,
path-eta-1,
path-eta-0,
same-cubical-term_wf,
squash_wf,
true_wf,
cubical-path-app-1,
iff_weakening_equal,
cubical-path-app-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
inhabitedIsType,
because_Cache,
rename,
dependent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality_alt,
independent_pairFormation,
equalityTransitivity,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
productElimination,
lambdaEquality_alt,
hyp_replacement,
independent_isectElimination,
Error :memTop,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}X:j\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}B:\{X.A \mvdash{} \_\}. \mforall{}x,y:\{X \mvdash{} \_:\mSigma{} A B\}.
(\{X \mvdash{} \_:(Path\_\mSigma{} A B x y)\} {}\mRightarrow{} \{X \mvdash{} \_:(Path\_A x.1 y.1)\})
Date html generated:
2020_05_20-PM-03_25_06
Last ObjectModification:
2020_04_07-PM-04_41_22
Theory : cubical!type!theory
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