### Nuprl Lemma : int_formula-ext

`int_formula() ≡ lbl:Atom × if lbl =a "less" then left:int_term() × int_term()`
`                           if lbl =a "le" then left:int_term() × int_term()`
`                           if lbl =a "eq" then left:int_term() × int_term()`
`                           if lbl =a "and" then left:int_formula() × int_formula()`
`                           if lbl =a "or" then left:int_formula() × int_formula()`
`                           if lbl =a "implies" then left:int_formula() × int_formula()`
`                           if lbl =a "not" then int_formula()`
`                           else Void`
`                           fi `

Proof

Definitions occuring in Statement :  int_formula: `int_formula()` int_term: `int_term()` ifthenelse: `if b then t else f fi ` eq_atom: `x =a y` ext-eq: `A ≡ B` product: `x:A × B[x]` token: `"\$token"` atom: `Atom` void: `Void`
Definitions unfolded in proof :  ext-eq: `A ≡ B` and: `P ∧ Q` subtype_rel: `A ⊆r B` member: `t ∈ T` int_formula: `int_formula()` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` uimplies: `b supposing a` ifthenelse: `if b then t else f fi ` sq_type: `SQType(T)` guard: `{T}` eq_atom: `x =a y` int_formulaco_size: `int_formulaco_size(p)` has-value: `(a)↓` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` bnot: `¬bb` assert: `↑b` false: `False` pi1: `fst(t)` pi2: `snd(t)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_formula_size: `int_formula_size(p)` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A`
Lemmas referenced :  int_formulaco-ext eq_atom_wf bool_wf eqtt_to_assert assert_of_eq_atom subtype_base_sq atom_subtype_base int_term_wf eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_atom int_subtype_base int_formulaco_size_wf subtype_partial_sqtype_base nat_wf set_subtype_base le_wf base_wf value-type-has-value int-value-type has-value_wf-partial set-value-type int_formula_wf int_formulaco_wf add-nat false_wf int_formula_size_wf nat_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity independent_pairFormation lambdaEquality sqequalHypSubstitution setElimination thin rename cut introduction extract_by_obid hypothesis promote_hyp productElimination hypothesis_subsumption hypothesisEquality applyEquality sqequalRule dependent_pairEquality isectElimination tokenEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry independent_isectElimination because_Cache instantiate cumulativity atomEquality dependent_functionElimination independent_functionElimination dependent_pairFormation voidElimination dependent_set_memberEquality natural_numberEquality intEquality baseApply closedConclusion baseClosed callbyvalueAdd productEquality voidEquality sqleReflexivity

Latex:
int\_formula()  \mequiv{}  lbl:Atom  \mtimes{}  if  lbl  =a  "less"  then  left:int\_term()  \mtimes{}  int\_term()
if  lbl  =a  "le"  then  left:int\_term()  \mtimes{}  int\_term()
if  lbl  =a  "eq"  then  left:int\_term()  \mtimes{}  int\_term()
if  lbl  =a  "and"  then  left:int\_formula()  \mtimes{}  int\_formula()
if  lbl  =a  "or"  then  left:int\_formula()  \mtimes{}  int\_formula()
if  lbl  =a  "implies"  then  left:int\_formula()  \mtimes{}  int\_formula()
if  lbl  =a  "not"  then  int\_formula()
else  Void
fi

Date html generated: 2017_04_14-AM-08_59_53
Last ObjectModification: 2017_02_27-PM-03_42_32

Theory : omega

Home Index