Nuprl Lemma : strict-fun-connected-induction
∀[T:Type]
∀f:T ⟶ T
∀[R:T ⟶ T ⟶ ℙ]
((∀x,y,z:T. (y is f*(z) ⇒ (R[y;z] ∨ (y = z ∈ T)) ⇒ R[x;z]) supposing ((¬(x = y ∈ T)) and (x = (f y) ∈ T)))
⇒ {∀x,y:T. (x = f+(y) ⇒ R[x;y])})
Proof
Definitions occuring in Statement :
strict-fun-connected: y = f+(x),
fun-connected: y is f*(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
prop: ℙ,
uimplies: b supposing a,
not: ¬A,
false: False,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
strict-fun-connected: y = f+(x),
and: P ∧ Q
Lemmas referenced :
fun-connected-induction,
or_wf,
equal_wf,
fun-connected_wf,
not_wf,
all_wf,
isect_wf,
strict-fun-connected_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
cumulativity,
hypothesis,
independent_functionElimination,
inrFormation,
because_Cache,
axiomEquality,
rename,
voidElimination,
inlFormation,
functionEquality,
universeEquality,
independent_isectElimination,
unionElimination,
productElimination,
equalitySymmetry
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T
\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]
((\mforall{}x,y,z:T.
(y is f*(z) {}\mRightarrow{} (R[y;z] \mvee{} (y = z)) {}\mRightarrow{} R[x;z]) supposing ((\mneg{}(x = y)) and (x = (f y))))
{}\mRightarrow{} \{\mforall{}x,y:T. (x = f+(y) {}\mRightarrow{} R[x;y])\})
Date html generated:
2018_05_21-PM-07_45_27
Last ObjectModification:
2017_07_26-PM-05_22_51
Theory : general
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