Nuprl Lemma : face-forall-implies-0
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. ∀[X:Top]. H ⊢ ((∀ phi) ⇒ (phi)[0(𝕀)])
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
top: Top
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
interval-presheaf: 𝕀,
names: names(I),
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
true: True,
cubical-type-at: A(a),
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,
and: P ∧ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
DeMorgan-algebra: DeMorganAlgebra,
nc-0: (i0),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
empty-fset: {},
nil: [],
dM0: 0,
lattice-0: 0,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
bnot: ¬bb,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
assert: ↑b,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
face-forall: (∀ phi),
cubical-term-at: u(a),
nc-p: (i/z),
csm-ap-term: (t)s
Lemmas referenced :
nc-0_wf,
new-name_wf,
interval-type-at,
I_cube_pair_redex_lemma,
dM_inc_wf,
add-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,
cubical-term-at-morph,
cube-context-adjoin_wf,
interval-type_wf,
face-type_wf,
cubical_set_cumulativity-i-j,
cc-adjoin-cube_wf,
cube-set-restriction_wf,
nc-s_wf,
f-subset-add-name,
face-type-at,
face-type-ap-morph,
cube_set_restriction_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term-at_wf,
I_cube_wf,
fset_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical-type_wf,
istype-cubical-type-at,
subtype_rel_self,
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,
face-forall_wf,
lattice-1_wf,
istype-top,
cubical_set_wf,
cube-set-restriction-comp,
iff_weakening_equal,
cube-set-restriction-id,
s-comp-nc-0,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
dM0_wf,
dM-lift-inc,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
assert_elim,
bnot_wf,
bool_wf,
eq_int_eq_true,
bfalse_wf,
btrue_neq_bfalse,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
btrue_wf,
not_assert_elim,
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,
interval-type-ap-morph,
dM0-sq-empty,
fl_all-implies-instance
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalRule,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
dependent_functionElimination,
because_Cache,
dependent_set_memberEquality_alt,
universeIsType,
intEquality,
independent_isectElimination,
natural_numberEquality,
instantiate,
hyp_replacement,
imageElimination,
universeEquality,
dependent_pairEquality_alt,
imageMemberEquality,
baseClosed,
productEquality,
cumulativity,
isectEquality,
equalityIstype,
productElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
independent_pairFormation,
productIsType,
applyLambdaEquality,
voidElimination,
dependent_pairFormation_alt,
promote_hyp,
approximateComputation,
int_eqEquality
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H.\mBbbI{} \mvdash{} \_:\mBbbF{}\}]. \mforall{}[X:Top]. H \mvdash{} ((\mforall{} phi) {}\mRightarrow{} (phi)[0(\mBbbI{})])
Date html generated:
2020_05_20-PM-03_02_48
Last ObjectModification:
2020_04_04-PM-05_19_16
Theory : cubical!type!theory
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