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theory Subgoal
imports "../HoTT"
begin
text "
Proof of \<open>rpathcomp_type\<close> (see EqualProps.thy) in apply-style.
Subgoaling can be used to fix variables and apply the elimination rules.
"
lemma rpathcomp_type:
assumes "A: U(i)"
shows "rpathcomp: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)"
unfolding rpathcomp_def
apply standard
subgoal premises 1 for x \<comment> \<open>\<open>subgoal\<close> is the proof script version of \<open>fix-assume-show\<close>.\<close>
apply standard
subgoal premises 2 for y
apply standard
subgoal premises 3 for p
apply (rule Equal_elim[where ?x=x and ?y=y and ?A=A])
\<comment> \<open>specifying \<open>?A=A\<close> is crucial here to prevent the next \<open>subgoal\<close> from binding a schematic ?A which should be instantiated to \<open>A\<close>.\<close>
prefer 4
apply standard
apply (rule Prod_intro)
subgoal premises 4 for u z q
apply (rule Equal_elim[where ?x=u and ?y=z])
apply (simple lems: assms 4)
done
apply (simple lems: assms 1 2 3)
done
apply (simple lems: assms 1 2)
done
apply fact
done
apply fact
done
text "
\<open>subgoal\<close> converts schematic variables to fixed free variables, making it unsuitable for use in \<open>schematic_goal\<close> proofs.
This is the same thing as being unable to start a ``sub schematic-goal'' inside an ongoing proof.
This is a problem for syntheses which need to use induction (elimination rules), as these often have to be applied to fixed variables, while keeping any schematic variables intact.
"
schematic_goal rpathcomp_synthesis:
assumes "A: U(i)"
shows "?a: \<Prod>x:A. \<Prod>y:A. x =\<^sub>A y \<rightarrow> (\<Prod>z:A. y =\<^sub>A z \<rightarrow> x =\<^sub>A z)"
text "
Try (and fail) to synthesize the constant for path composition, following the proof of \<open>rpathcomp_type\<close> below.
"
apply (rule Prod_intro)
apply (rule Prod_intro)
apply (rule Prod_intro)
subgoal 123 for x y p
apply (rule Equal_elim[where ?x=x and ?y=y and ?A=A])
oops
end
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