Nuprl Lemma : int_formula-definition
∀[A:Type]. ∀[R:A ⟶ int_formula() ⟶ ℙ].
((∀left,right:int_term(). {x:A| R[x;(left "<" right)]} )
⇒ (∀left,right:int_term(). {x:A| R[x;left "≤" right]} )
⇒ (∀left,right:int_term(). {x:A| R[x;left "=" right]} )
⇒ (∀left,right:int_formula(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "∧" right]} ))
⇒ (∀left,right:int_formula(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "or" right]} ))
⇒ (∀left,right:int_formula(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;left "=>" right]} ))
⇒ (∀form:int_formula(). ({x:A| R[x;form]} ⇒ {x:A| R[x;"¬"form]} ))
⇒ {∀v:int_formula(). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
intformnot: "¬"form,
intformimplies: left "=>" right,
intformor: left "or" right,
intformand: left "∧" right,
intformeq: left "=" right,
intformle: left "≤" right,
intformless: (left "<" right),
int_formula: int_formula(),
int_term: int_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],
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
Lemmas referenced :
int_formula-induction,
set_wf,
int_formula_wf,
all_wf,
intformnot_wf,
intformimplies_wf,
intformor_wf,
intformand_wf,
int_term_wf,
intformeq_wf,
intformle_wf,
intformless_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
because_Cache,
independent_functionElimination,
functionEquality,
universeEquality,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} int\_formula() {}\mrightarrow{} \mBbbP{}].
((\mforall{}left,right:int\_term(). \{x:A| R[x;(left "<" right)]\} )
{}\mRightarrow{} (\mforall{}left,right:int\_term(). \{x:A| R[x;left "\mleq{}" right]\} )
{}\mRightarrow{} (\mforall{}left,right:int\_term(). \{x:A| R[x;left "=" right]\} )
{}\mRightarrow{} (\mforall{}left,right:int\_formula().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;left "\mwedge{}" right]\} ))
{}\mRightarrow{} (\mforall{}left,right:int\_formula().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;left "or" right]\} ))
{}\mRightarrow{} (\mforall{}left,right:int\_formula().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;left "=>" right]\} ))
{}\mRightarrow{} (\mforall{}form:int\_formula(). (\{x:A| R[x;form]\} {}\mRightarrow{} \{x:A| R[x;"\mneg{}"form]\} ))
{}\mRightarrow{} \{\mforall{}v:int\_formula(). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_14-AM-07_07_02
Last ObjectModification:
2015_12_26-PM-01_09_03
Theory : omega
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