Nuprl Lemma : face-forall-q=0
∀[Gamma:j⊢]. ((Gamma ⊢ ∀ (q=0)) = 0(𝔽) ∈ {Gamma ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
face-zero: (i=0),
face-0: 0(𝔽),
face-type: 𝔽,
cc-snd: q,
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],
face-forall: (∀ phi),
face-0: 0(𝔽),
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
names: names(I),
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
squash: ↓T,
bdd-distributive-lattice: BoundedDistributiveLattice,
and: P ∧ Q,
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,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
cc-adjoin-cube: (v;u),
cc-snd: q,
face-zero: (i=0),
cubical-term-at: u(a),
pi2: snd(t),
DeMorgan-algebra: DeMorganAlgebra
Lemmas referenced :
fl_all-fl0,
new-name_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
istype-int,
strong-subtype-self,
add-name_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
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,
lattice-meet_wf,
lattice-join_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
lattice-0_wf,
subtype_rel_self,
cubical-type-at_wf_face-type,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
eq_int_eq_true,
btrue_wf,
iff_weakening_equal,
not_assert_elim,
btrue_neq_bfalse,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
I_cube_wf,
fset_wf,
cubical-term-equal,
face-type_wf,
face-0_wf,
cubical_set_wf,
fl_all_wf,
istype-nat,
dM-to-FL-opp,
dM-to-FL_wf,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
neg-dM_inc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
equalitySymmetry,
functionExtensionality,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
dependent_functionElimination,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
intEquality,
independent_isectElimination,
natural_numberEquality,
hyp_replacement,
imageElimination,
instantiate,
universeEquality,
productEquality,
cumulativity,
isectEquality,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
Error :memTop,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
independent_functionElimination,
voidElimination,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. ((Gamma \mvdash{} \mforall{} (q=0)) = 0(\mBbbF{}))
Date html generated:
2020_05_20-PM-02_50_35
Last ObjectModification:
2020_04_04-PM-05_04_58
Theory : cubical!type!theory
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