Nuprl Lemma : case-term-equal-right'
∀[Gamma:j⊢]. ∀[phi,psi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma, (phi ∨ psi) ⊢ _}]. ∀[u:{Gamma, phi ⊢ _:A}].
∀[v,x:{Gamma, psi ⊢ _:A}].
(Gamma, psi ⊢ (u ∨ v)=x:A) supposing ((x = v ∈ {Gamma, psi ⊢ _:A}) and Gamma, (phi ∧ psi) ⊢ u=v: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-and: (a ∧ b),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
same-cubical-term: X ⊢ u=v:A,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
guard: {T}
Lemmas referenced :
case-term-equal-right,
context-subset-subtype,
face-or_wf,
face-and_wf,
face-or-eq-1,
face-and-eq-1,
cubical-term-at_wf,
face-type_wf,
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,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
same-cubical-term_wf,
cubical-type-cumulativity2,
context-subset_wf,
subset-cubical-term2,
face-term-implies-subset,
face-term-and-implies1,
context-subset-subtype-or,
face-term-and-implies2,
context-subset-subtype-or2,
cubical-term_wf,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
inhabitedIsType,
universeIsType,
applyEquality,
because_Cache,
lambdaFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
inlFormation_alt,
sqequalRule,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma, (phi \mvee{} psi) \mvdash{} \_\}]. \mforall{}[u:\{Gamma, phi \mvdash{} \_:A\}].
\mforall{}[v,x:\{Gamma, psi \mvdash{} \_:A\}].
(Gamma, psi \mvdash{} (u \mvee{} v)=x:A) supposing ((x = v) and Gamma, (phi \mwedge{} psi) \mvdash{} u=v:A)
Date html generated:
2020_05_20-PM-03_12_06
Last ObjectModification:
2020_04_07-PM-00_59_03
Theory : cubical!type!theory
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