Nuprl Lemma : no-halt-decider

`¬(∃h:partial(ℤ) ⟶ 𝔹. (h 0 = tt ∧ h ⊥ = ff))`

Proof

Definitions occuring in Statement :  partial: `partial(T)` bottom: `⊥` bfalse: `ff` btrue: `tt` bool: `𝔹` exists: `∃x:A. B[x]` not: `¬A` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` false: `False` prop: `ℙ` exists: `∃x:A. B[x]` and: `P ∧ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` sq_type: `SQType(T)` guard: `{T}` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Lemmas referenced :  bottom_diverge base-member-partial int-value-type has-value_wf_base not-is-exception-bottom fixpoint-induction-bottom partial_wf int-mono ifthenelse_wf exists_wf bool_wf equal-wf-T-base inclusion-partial subtype_base_sq bool_subtype_base btrue_neq_bfalse equal_wf
Rules used in proof :  cut introduction extract_by_obid sqequalHypSubstitution sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isectElimination thin intEquality independent_isectElimination hypothesis baseClosed isect_memberFormation independent_functionElimination voidElimination sqequalRule axiomEquality equalityTransitivity equalitySymmetry lambdaFormation productElimination because_Cache lambdaEquality hypothesisEquality applyEquality functionExtensionality natural_numberEquality functionEquality productEquality unionElimination equalityElimination addLevel instantiate cumulativity dependent_functionElimination levelHypothesis

Latex:
\mneg{}(\mexists{}h:partial(\mBbbZ{})  {}\mrightarrow{}  \mBbbB{}.  (h  0  =  tt  \mwedge{}  h  \mbot{}  =  ff))

Date html generated: 2017_04_14-AM-07_40_55
Last ObjectModification: 2017_02_27-PM-03_12_40

Theory : partial_1

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