Nuprl Lemma : rat_term_ind_wf_simple
∀[A:Type]. ∀[v:rat_term()]. ∀[Constant,Var:var:ℤ ⟶ A]. ∀[Add,Subtract,Multiply,Divide:num:rat_term()
⟶ denom:rat_term()
⟶ A
⟶ A
⟶ A]. ∀[Minus:num:rat_term()
⟶ A
⟶ A].
(rat_term_ind(v;
rtermConstant(const)⇒ Constant[const];
rtermVar(var)⇒ Var[var];
rtermAdd(left,right)⇒ rec1,rec2.Add[left;right;rec1;rec2];
rtermSubtract(left,right)⇒ rec3,rec4.Subtract[left;right;rec3;rec4];
rtermMultiply(left,right)⇒ rec5,rec6.Multiply[left;right;rec5;rec6];
rtermDivide(num,denom)⇒ rec7,rec8.Divide[num;denom;rec7;rec8];
rtermMinus(num)⇒ rec9.Minus[num;rec9]) ∈ A)
Proof
Definitions occuring in Statement :
rat_term_ind: rat_term_ind,
rat_term: rat_term(),
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
rat_term_ind_wf,
true_wf,
rat_term_wf,
istype-true,
subtype_rel_dep_function,
istype-int,
istype-universe
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
universeIsType,
functionExtensionality,
applyEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
closedConclusion,
intEquality,
functionEquality,
because_Cache,
setEquality,
independent_isectElimination,
lambdaFormation_alt,
setIsType,
setElimination,
rename,
applyLambdaEquality,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[v:rat\_term()]. \mforall{}[Constant,Var:var:\mBbbZ{} {}\mrightarrow{} A].
\mforall{}[Add,Subtract,Multiply,Divide:num:rat\_term() {}\mrightarrow{} denom:rat\_term() {}\mrightarrow{} A {}\mrightarrow{} A {}\mrightarrow{} A].
\mforall{}[Minus:num:rat\_term() {}\mrightarrow{} A {}\mrightarrow{} A].
(rat\_term\_ind(v;
rtermConstant(const){}\mRightarrow{} Constant[const];
rtermVar(var){}\mRightarrow{} Var[var];
rtermAdd(left,right){}\mRightarrow{} rec1,rec2.Add[left;right;rec1;rec2];
rtermSubtract(left,right){}\mRightarrow{} rec3,rec4.Subtract[left;right;rec3;rec4];
rtermMultiply(left,right){}\mRightarrow{} rec5,rec6.Multiply[left;right;rec5;rec6];
rtermDivide(num,denom){}\mRightarrow{} rec7,rec8.Divide[num;denom;rec7;rec8];
rtermMinus(num){}\mRightarrow{} rec9.Minus[num;rec9]) \mmember{} A)
Date html generated:
2019_10_29-AM-09_31_12
Last ObjectModification:
2019_03_31-PM-05_25_40
Theory : reals
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