Nuprl Lemma : pi_term_ind_wf

∀[A:Type]. ∀[R:A ⟶ pi_term() ⟶ ℙ]. ∀[v:pi_term()]. ∀[zero:{x:A| R[x;pizero()]} ]. ∀[comm:pre:pi_prefix()
                                                                                          ⟶ body:pi_term()
                                                                                          ⟶ {x:A| R[x;body]} 
                                                                                          ⟶ {x:A| 
                                                                                              R[x;picomm(pre;body)]} ].
∀[option:left:pi_term()
         ⟶ right:pi_term()
         ⟶ {x:A| R[x;left]} 
         ⟶ {x:A| R[x;right]} 
         ⟶ {x:A| R[x;pioption(left;right)]} ]. ∀[par:left:pi_term()
                                                     ⟶ right:pi_term()
                                                     ⟶ {x:A| R[x;left]} 
                                                     ⟶ {x:A| R[x;right]} 
                                                     ⟶ {x:A| R[x;pipar(left;right)]} ]. ∀[rep:body:pi_term()
                                                                                              ⟶ {x:A| R[x;body]} 
                                                                                              ⟶ {x:A| 
                                                                                                  R[x;pirep(body)]} ].
∀[new:name:Name ⟶ body:pi_term() ⟶ {x:A| R[x;body]}  ⟶ {x:A| R[x;pinew(name;body)]} ].
  (... ∈ {x:A| R[x;v]} )


Proof



Definitions occuring in Statement :  pi_term_ind: pi_term_ind(v;zero;pre,body,rec1....;left,right,rec2,rec3....;left,right,rec4,rec5....;body,rec6....;name,body,rec7....),  pinew: pinew(name;body),  pirep: pirep(body),  pipar: pipar(left;right),  pioption: pioption(left;right),  picomm: picomm(pre;body),  pizero: pizero(),  pi_term: pi_term(),  pi_prefix: pi_prefix(),  name: Name,  uall: ∀[x:A]. B[x],  prop: ℙ,  so_apply: x[s1;s2;s3;s4],  so_apply: x[s1;s2;s3],  so_apply: x[s1;s2],  member: t ∈ T,  set: {x:A| B[x]} ,  function: x:A ⟶ B[x],  universe: Type
Definitions unfolded in proof :  uall: ∀[x:A]. B[x],  member: t ∈ T,  pi_term_ind: pi_term_ind(v;zero;pre,body,rec1....;left,right,rec2,rec3....;left,right,rec4,rec5....;body,rec6....;name,body,rec7....),  so_apply: x[s1;s2;s3],  so_apply: x[s1;s2],  so_apply: x[s1;s2;s3;s4],  pi_term-definition,  pi_term-induction,  uniform-comp-nat-induction,  pi_term-ext,  eq_atom: x =a y,  bool_cases_sqequal,  eqff_to_assert,  any: any x,  btrue: tt,  bfalse: ff,  it: ⋅,  top: Top,  all: ∀x:A. B[x],  implies: P ⇒ Q,  has-value: (a)↓,  so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),  so_lambda: λ2x.t[x],  so_apply: x[s],  uimplies: b supposing a,  strict4: strict4(F),  and: P ∧ Q,  prop: ℙ,  guard: {T},  or: P ∨ Q,  squash: ↓T,  subtype_rel: A ⊆r B
Latex:
\mforall{}[A:Type].  \mforall{}[R:A  {}\mrightarrow{}  pi\_term()  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[v:pi\_term()].  \mforall{}[zero:\{x:A|  R[x;pizero()]\}  ].
\mforall{}[comm:pre:pi\_prefix()  {}\mrightarrow{}  body:pi\_term()  {}\mrightarrow{}  \{x:A|  R[x;body]\}    {}\mrightarrow{}  \{x:A|  R[x;picomm(pre;body)]\}  ].
\mforall{}[option:left:pi\_term()
                  {}\mrightarrow{}  right:pi\_term()
                  {}\mrightarrow{}  \{x:A|  R[x;left]\} 
                  {}\mrightarrow{}  \{x:A|  R[x;right]\} 
                  {}\mrightarrow{}  \{x:A|  R[x;pioption(left;right)]\}  ].  \mforall{}[par:left:pi\_term()
                                                                                                          {}\mrightarrow{}  right:pi\_term()
                                                                                                          {}\mrightarrow{}  \{x:A|  R[x;left]\} 
                                                                                                          {}\mrightarrow{}  \{x:A|  R[x;right]\} 
                                                                                                          {}\mrightarrow{}  \{x:A|  R[x;pipar(left;right)]\}  ].
\mforall{}[rep:body:pi\_term()  {}\mrightarrow{}  \{x:A|  R[x;body]\}    {}\mrightarrow{}  \{x:A|  R[x;pirep(body)]\}  ].
\mforall{}[new:name:Name  {}\mrightarrow{}  body:pi\_term()  {}\mrightarrow{}  \{x:A|  R[x;body]\}    {}\mrightarrow{}  \{x:A|  R[x;pinew(name;body)]\}  ].
    (...  \mmember{}  \{x:A|  R[x;v]\}  )


Date html generated: 2016_05_17-AM-11_22_45
Last ObjectModification: 2016_01_18-AM-07_51_02

Theory : event-logic-applications


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