Nuprl Lemma : nh-comp-cancel
∀I,J,K:fset(ℕ). ∀f:I ⟶ J. ∀g,h:J ⟶ K. ∀x:names(K).
(g ⋅ f x) = (h ⋅ f x) ∈ Point(dM(I)) supposing (g x) = (h x) ∈ Point(dM(J))
Proof
Definitions occuring in Statement :
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
dM: dM(I),
names: names(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
member: t ∈ T,
uall: ∀[x:A]. B[x],
deq: EqDecider(T),
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
bool: 𝔹,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
assert: ↑b,
rev_implies: P ⇐ Q,
names-hom: I ⟶ J,
subtype_rel: A ⊆r B,
dM: dM(I),
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
free-dma-lift_wf,
names_wf,
names-deq_wf,
free-DeMorgan-algebra_wf,
free-dml-deq_wf,
subtype_rel_self,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
names-hom_wf,
fset_wf,
nat_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
because_Cache,
functionExtensionality,
applyEquality,
inhabitedIsType,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
instantiate,
lambdaEquality_alt,
productEquality,
independent_isectElimination,
cumulativity,
setElimination,
rename
Latex:
\mforall{}I,J,K:fset(\mBbbN{}). \mforall{}f:I {}\mrightarrow{} J. \mforall{}g,h:J {}\mrightarrow{} K. \mforall{}x:names(K). (g \mcdot{} f x) = (h \mcdot{} f x) supposing (g x) = (h x)
Date html generated:
2019_11_04-PM-05_31_18
Last ObjectModification:
2018_11_08-AM-11_03_43
Theory : cubical!type!theory
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