### Nuprl Lemma : formula-ext

`formula() ≡ lbl:Atom × if lbl =a "var" then Atom`
`                       if lbl =a "not" then formula()`
`                       if lbl =a "and" then left:formula() × formula()`
`                       if lbl =a "or" then left:formula() × formula()`
`                       if lbl =a "imp" then left:formula() × formula()`
`                       else Void`
`                       fi `

Proof

Definitions occuring in Statement :  formula: `formula()` 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` formula: `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` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` bnot: `¬bb` assert: `↑b` false: `False` formulaco_size: `formulaco_size(p)` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` has-value: `(a)↓` formula_size: `formula_size(p)` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A`
Lemmas referenced :  formulaco-ext eq_atom_wf bool_wf eqtt_to_assert assert_of_eq_atom subtype_base_sq atom_subtype_base eqff_to_assert equal_wf bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_atom int_subtype_base 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 formula_wf ifthenelse_wf formulaco_wf add-nat false_wf 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 universeEquality productEquality voidEquality sqleReflexivity

Latex:
formula()  \mequiv{}  lbl:Atom  \mtimes{}  if  lbl  =a  "var"  then  Atom
if  lbl  =a  "not"  then  formula()
if  lbl  =a  "and"  then  left:formula()  \mtimes{}  formula()
if  lbl  =a  "or"  then  left:formula()  \mtimes{}  formula()
if  lbl  =a  "imp"  then  left:formula()  \mtimes{}  formula()
else  Void
fi

Date html generated: 2018_05_21-PM-08_47_50
Last ObjectModification: 2017_07_26-PM-06_10_48

Theory : general

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