Nuprl Lemma : interval-meet-comm
∀[X:j⊢]. ∀[z,y:{X ⊢ _:𝕀}]. (z ∧ y = y ∧ z ∈ {X ⊢ _:𝕀})
Proof
Definitions occuring in Statement :
interval-meet: r ∧ s,
interval-type: 𝕀,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
interval-meet: r ∧ s,
cubical-term-at: u(a),
interval-presheaf: 𝕀,
subtype_rel: A ⊆r B,
guard: {T},
and: P ∧ Q,
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s]
Lemmas referenced :
cubical-term-equal2,
interval-type_wf,
interval-meet_wf,
interval-type-at,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
lattice_properties,
dM_wf,
bdd-distributive-lattice-subtype-lattice,
DeMorgan-algebra-subtype,
subtype_rel_transitivity,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
lattice_wf,
cubical-term-at_wf,
subtype_rel_self,
lattice-point_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
independent_isectElimination,
lambdaFormation_alt,
sqequalRule,
Error :memTop,
universeIsType,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
dependent_functionElimination,
applyEquality,
productElimination,
lambdaEquality_alt,
productEquality,
cumulativity,
because_Cache,
isectEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[z,y:\{X \mvdash{} \_:\mBbbI{}\}]. (z \mwedge{} y = y \mwedge{} z)
Date html generated:
2020_05_20-PM-02_37_48
Last ObjectModification:
2020_04_04-AM-09_56_20
Theory : cubical!type!theory
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