Nuprl Lemma : face-forall-decomp
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. H.𝕀 ⊢ (phi ⇐⇒ (((∀ phi))p ∨ ((phi ∧ (q=0)) ∨ (phi ∧ (q=1)))))
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
face-term-iff: Gamma ⊢ (phi ⇐⇒ psi),
face-zero: (i=0),
face-one: (i=1),
face-or: (a ∨ b),
face-and: (a ∧ b),
face-type: 𝔽,
interval-type: 𝕀,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
face-term-iff: Gamma ⊢ (phi ⇐⇒ psi),
and: P ∧ Q,
subtype_rel: A ⊆r B,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x],
implies: P ⇒ Q,
cube-context-adjoin: X.A,
interval-presheaf: 𝕀,
names: names(I),
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
bdd-distributive-lattice: BoundedDistributiveLattice,
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,
cc-adjoin-cube: (v;u),
pi2: snd(t),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
DeMorgan-algebra: DeMorganAlgebra,
nc-p: (i/z),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
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-zero: (i=0),
cc-snd: q,
cubical-term-at: u(a),
face-one: (i=1),
face-forall: (∀ phi),
cc-fst: p,
csm-ap: (s)x,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
interval-type: 𝕀,
ge: i ≥ j ,
decidable: Dec(P),
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
csm-ap-type: (AF)s,
respects-equality: respects-equality(S;T)
Lemmas referenced :
face-forall-implies,
cubical-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
face-type_wf,
cubical_set_wf,
cc-fst_wf,
I_cube_pair_redex_lemma,
interval-type-at,
nc-p_wf,
new-name_wf,
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,
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,
subtype_rel_self,
lattice-1_wf,
I_cube_wf,
fset_wf,
cubical-term-at-morph,
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,
squash_wf,
true_wf,
istype-universe,
cubical-type-cumulativity2,
cubical-type_wf,
istype-cubical-type-at,
cube-set-restriction-comp,
iff_weakening_equal,
cube-set-restriction-id,
s-comp-nc-p,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_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,
face-or-at,
face-and-at,
csm-ap-term-at,
fl_all_wf,
dM-to-FL_wf,
dm-neg_wf,
names_wf,
names-deq_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
fl-morph_wf,
cubical-type-at_wf,
nat_properties,
decidable__le,
intformand_wf,
intformle_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
istype-le,
face_lattice-induction,
sq_stable__all,
sq_stable__equal,
lattice-0_wf,
fl_all-1,
lattice-join-1,
bdd-distributive-lattice-subtype-bdd-lattice,
fl0_wf,
fl1_wf,
fl-morph-0,
face-lattice-0-not-1,
iff_transitivity,
fl-morph-join,
face_lattice-1-join-irreducible,
fl_all-join,
fl-morph-meet,
lattice-meet-eq-1,
fl_all-meet,
fl-morph-fl0,
assert_wf,
not_wf,
equal-wf-base,
set_subtype_base,
istype-nat,
int_subtype_base,
istype-assert,
istype-void,
bool_cases,
iff_weakening_uiff,
assert_of_bnot,
fl_all-fl0,
neg-dM_inc,
not-added-name,
dM-to-FL-opp,
fl-morph-fl1,
fl_all-id,
dM-to-FL-inc,
lattice-1-meet,
face-or_wf,
csm-ap-term_wf,
csm-face-type,
face-forall_wf,
face-and_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
respects-equality_weakening,
face-or-eq-1,
face-and-eq-1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
universeIsType,
instantiate,
applyEquality,
sqequalRule,
because_Cache,
lambdaFormation_alt,
dependent_functionElimination,
Error :memTop,
productElimination,
rename,
lambdaEquality_alt,
setElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
intEquality,
independent_isectElimination,
natural_numberEquality,
equalityIstype,
productEquality,
cumulativity,
isectEquality,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
dependent_pairEquality_alt,
independent_functionElimination,
unionElimination,
equalityElimination,
productIsType,
applyLambdaEquality,
voidElimination,
dependent_pairFormation_alt,
promote_hyp,
approximateComputation,
int_eqEquality,
functionEquality,
functionIsType,
unionEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
sqequalBase
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H.\mBbbI{} \mvdash{} \_:\mBbbF{}\}]. H.\mBbbI{} \mvdash{} (phi \mLeftarrow{}{}\mRightarrow{} (((\mforall{} phi))p \mvee{} ((phi \mwedge{} (q=0)) \mvee{} (phi \mwedge{} (q=1)))))
Date html generated:
2020_05_20-PM-03_03_42
Last ObjectModification:
2020_04_06-AM-10_57_11
Theory : cubical!type!theory
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