Nuprl Lemma : face-term-distrib3
∀[Gamma:j⊢]. ∀[a,b,c:{Gamma ⊢ _:𝔽}]. Gamma ⊢ (((b ∨ c) ∧ a) ⇐⇒ ((b ∧ a) ∨ (c ∧ a)))
Proof
Definitions occuring in Statement :
face-term-iff: Gamma ⊢ (phi ⇐⇒ psi),
face-or: (a ∨ b),
face-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
face-term-iff: Gamma ⊢ (phi ⇐⇒ psi),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x]
Lemmas referenced :
face-term-iff_wf,
face-and_wf,
face-or_wf,
face-and-com,
iff_weakening_equal,
face-term-distrib1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
hypothesisEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
independent_pairEquality,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[a,b,c:\{Gamma \mvdash{} \_:\mBbbF{}\}]. Gamma \mvdash{} (((b \mvee{} c) \mwedge{} a) \mLeftarrow{}{}\mRightarrow{} ((b \mwedge{} a) \mvee{} (c \mwedge{} a)))
Date html generated:
2020_05_20-PM-02_49_20
Last ObjectModification:
2020_04_04-PM-05_03_09
Theory : cubical!type!theory
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