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Step * of 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]} )
BY
{ ProveDatatypeIndWf TERMOF{pi_term-definition:o, 1:l, i:l}⋅ }

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Latex:

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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]\}  )


By

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Latex:
ProveDatatypeIndWf  TERMOF\{pi\_term-definition:o,  1:l,  i:l\}\mcdot{}

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