Nuprl Lemma : rat_term-definition
∀[A:Type]. ∀[R:A ⟶ rat_term() ⟶ ℙ].
((∀const:ℤ. {x:A| R[x;"const"]} )
⇒ (∀var:ℤ. {x:A| R[x;rtermVar(var)]} )
⇒ (∀left,right:rat_term(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "+" right]} ))
⇒ (∀left,right:rat_term(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "-" right]} ))
⇒ (∀left,right:rat_term(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "*" right]} ))
⇒ (∀num,denom:rat_term(). ({x:A| R[x;num]} ⇒ {x:A| R[x;denom]} ⇒ {x:A| R[x;num "/" denom]} ))
⇒ (∀num:rat_term(). ({x:A| R[x;num]} ⇒ {x:A| R[x;rtermMinus(num)]} ))
⇒ {∀v:rat_term(). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
rtermMinus: rtermMinus(num),
rtermDivide: num "/" denom,
rtermMultiply: left "*" right,
rtermSubtract: left "-" right,
rtermAdd: left "+" right,
rtermVar: rtermVar(var),
rtermConstant: "const",
rat_term: rat_term(),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
all: ∀x:A. B[x]
Lemmas referenced :
rat_term-induction,
set_wf,
rat_term_wf,
subtype_rel_self,
rtermMinus_wf,
rtermDivide_wf,
rtermMultiply_wf,
rtermSubtract_wf,
rtermAdd_wf,
istype-int,
rtermVar_wf,
rtermConstant_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
hypothesisEquality,
applyEquality,
because_Cache,
universeIsType,
independent_functionElimination,
functionIsType,
setIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} rat\_term() {}\mrightarrow{} \mBbbP{}].
((\mforall{}const:\mBbbZ{}. \{x:A| R[x;"const"]\} )
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. \{x:A| R[x;rtermVar(var)]\} )
{}\mRightarrow{} (\mforall{}left,right:rat\_term().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;left "+" right]\} ))
{}\mRightarrow{} (\mforall{}left,right:rat\_term().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;left "-" right]\} ))
{}\mRightarrow{} (\mforall{}left,right:rat\_term().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;left "*" right]\} ))
{}\mRightarrow{} (\mforall{}num,denom:rat\_term(). (\{x:A| R[x;num]\} {}\mRightarrow{} \{x:A| R[x;denom]\} {}\mRightarrow{} \{x:A| R[x;num "/" denom]\} )\000C)
{}\mRightarrow{} (\mforall{}num:rat\_term(). (\{x:A| R[x;num]\} {}\mRightarrow{} \{x:A| R[x;rtermMinus(num)]\} ))
{}\mRightarrow{} \{\mforall{}v:rat\_term(). \{x:A| R[x;v]\} \})
Date html generated:
2019_10_29-AM-09_30_53
Last ObjectModification:
2019_03_31-PM-05_21_41
Theory : reals
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