Nuprl Lemma : case-endpoints-0
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a:{G ⊢ _:A}]. ∀[b:Top]. ([0(𝕀)=0 ⊢→ a; 0(𝕀)=1 ⊢→ b] = a ∈ {G ⊢ _:A})
Proof
Definitions occuring in Statement :
case-endpoints: [r=0 ⊢→ a; r=1 ⊢→ b],
interval-0: 0(𝕀),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
top: Top,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
case-endpoints: [r=0 ⊢→ a; r=1 ⊢→ b],
case-term: (u ∨ v),
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
cubical-term: {X ⊢ _:A},
ifthenelse: if b then t else f fi ,
btrue: tt,
cubical-term-at: u(a),
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],
eq_atom: x =a y,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
interval-0: 0(𝕀),
face-zero: (i=0),
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
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),
empty-fset: {},
nil: [],
opposite-lattice: opposite-lattice(L),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
dM1: 1
Lemmas referenced :
subtype_base_sq,
bool_wf,
bool_subtype_base,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
istype-top,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
fl-eq_wf,
cubical-term-at_wf,
face-type_wf,
face-zero_wf,
interval-0_wf,
subtype_rel_self,
lattice-point_wf,
face_lattice_wf,
lattice-1_wf,
eqtt_to_assert,
assert-fl-eq,
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,
btrue_wf,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
iff_weakening_uiff,
assert_wf,
dM-to-FL-dM1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalitySymmetry,
cut,
functionExtensionality,
sqequalRule,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
dependent_functionElimination,
equalityTransitivity,
independent_functionElimination,
applyEquality,
setElimination,
rename,
hypothesisEquality,
universeIsType,
because_Cache,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
lambdaEquality_alt,
productEquality,
isectEquality,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
voidElimination
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[a:\{G \mvdash{} \_:A\}]. \mforall{}[b:Top]. ([0(\mBbbI{})=0 \mvdash{}\mrightarrow{} a; 0(\mBbbI{})=1 \mvdash{}\mrightarrow{} b] = a)
Date html generated:
2020_05_20-PM-04_15_32
Last ObjectModification:
2020_04_10-AM-04_46_05
Theory : cubical!type!theory
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