Nuprl Lemma : case-term-same2
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}]. ∀[v:{Gamma, psi ⊢ _:A}].
∀[w:{Gamma ⊢ _:A}].
(Gamma, (phi ∨ psi) ⊢ (u ∨ v)=w:A) supposing (Gamma, phi ⊢ u=w:A and Gamma, psi ⊢ v=w:A)
Proof
Definitions occuring in Statement :
case-term: (u ∨ v),
same-cubical-term: X ⊢ u=v:A,
context-subset: Gamma, phi,
face-or: (a ∨ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
same-cubical-term: X ⊢ u=v:A,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
case-term: (u ∨ v),
cubical-term-at: u(a),
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
context-subset: Gamma, phi,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: 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,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(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,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q
Lemmas referenced :
I_cube_wf,
context-subset_wf,
face-or_wf,
fset_wf,
nat_wf,
cubical-term-equal,
thin-context-subset,
context-subset-term-subtype,
subset-cubical-term,
context-subset-is-subset,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
face-type_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
face-or-eq-1,
fl-eq_wf,
cubical-term-at_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,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
equalitySymmetry,
cut,
functionExtensionality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
applyEquality,
equalityTransitivity,
independent_isectElimination,
equalityIstype,
universeIsType,
instantiate,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
productElimination,
independent_functionElimination,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
dependent_pairFormation_alt,
promote_hyp,
voidElimination,
dependent_set_memberEquality_alt,
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[u:\{Gamma, phi \mvdash{} \_:A\}].
\mforall{}[v:\{Gamma, psi \mvdash{} \_:A\}]. \mforall{}[w:\{Gamma \mvdash{} \_:A\}].
(Gamma, (phi \mvee{} psi) \mvdash{} (u \mvee{} v)=w:A) supposing (Gamma, phi \mvdash{} u=w:A and Gamma, psi \mvdash{} v=w:A)
Date html generated:
2020_05_20-PM-03_10_53
Last ObjectModification:
2020_04_06-PM-00_53_53
Theory : cubical!type!theory
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