Nuprl Lemma : face-term-and-implies
∀[Gamma:j⊢]. ∀[phi,psi,phi',psi':{Gamma ⊢ _:𝔽}].
(Gamma ⊢ ((phi ∧ psi) ⇒ (phi' ∧ psi'))) supposing (Gamma ⊢ (phi ⇒ phi') and Gamma ⊢ (psi ⇒ psi'))
Proof
Definitions occuring in Statement :
face-term-implies: Gamma ⊢ (phi ⇒ psi),
face-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
cubical-type-at: A(a),
pi1: fst(t),
face-type: 𝔽,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
guard: {T}
Lemmas referenced :
face-and-eq-1,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-term-at_wf,
face-type_wf,
face-and_wf,
subtype_rel_self,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
face-term-implies_wf,
cubical-term_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation_alt,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
productElimination,
independent_functionElimination,
independent_pairFormation,
equalityIstype,
universeIsType,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi,phi',psi':\{Gamma \mvdash{} \_:\mBbbF{}\}].
(Gamma \mvdash{} ((phi \mwedge{} psi) {}\mRightarrow{} (phi' \mwedge{} psi'))) supposing
(Gamma \mvdash{} (phi {}\mRightarrow{} phi') and
Gamma \mvdash{} (psi {}\mRightarrow{} psi'))
Date html generated:
2020_05_20-PM-02_47_26
Last ObjectModification:
2020_04_04-PM-05_01_31
Theory : cubical!type!theory
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