Nuprl Lemma : implies-face-forall-holds
∀H:j⊢. ∀phi:{H.𝕀 ⊢ _:𝔽}. (H.𝕀 ⊢ (1(𝔽) ⇒ phi) ⇒ H ⊢ (1(𝔽) ⇒ (∀ phi)))
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
face-1: 1(𝔽),
face-type: 𝔽,
interval-type: 𝕀,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
face-forall: (∀ phi),
cubical-term-at: u(a),
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
interval-presheaf: 𝕀,
names: names(I),
nat: ℕ,
face-1: 1(𝔽),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
face-type_wf,
face-1_wf,
subtype_rel_self,
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,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
face-term-implies_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
cubical-term_wf,
add-name_wf,
new-name_wf,
cc-adjoin-cube_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
trivial-member-add-name1,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
fl_all-1,
iff_weakening_equal,
squash_wf,
true_wf,
istype-universe,
fl_all_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
hypothesis,
equalityIstype,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
independent_isectElimination,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
Error :memTop,
dependent_set_memberEquality_alt,
intEquality,
natural_numberEquality,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
universeEquality
Latex:
\mforall{}H:j\mvdash{}. \mforall{}phi:\{H.\mBbbI{} \mvdash{} \_:\mBbbF{}\}. (H.\mBbbI{} \mvdash{} (1(\mBbbF{}) {}\mRightarrow{} phi) {}\mRightarrow{} H \mvdash{} (1(\mBbbF{}) {}\mRightarrow{} (\mforall{} phi)))
Date html generated:
2020_05_20-PM-03_07_01
Last ObjectModification:
2020_04_04-PM-05_23_55
Theory : cubical!type!theory
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