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Diffstat (limited to 'tests/Subgoal.thy')
| -rw-r--r-- | tests/Subgoal.thy | 65 |
1 files changed, 65 insertions, 0 deletions
diff --git a/tests/Subgoal.thy b/tests/Subgoal.thy new file mode 100644 index 0000000..2226f08 --- /dev/null +++ b/tests/Subgoal.thy @@ -0,0 +1,65 @@ +theory Subgoal + imports "../HoTT" +begin + +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) +prefer 2 + apply (rule Prod_intro) + prefer 2 + apply (rule Prod_intro) + prefer 2 + subgoal 123 for x y p + apply (rule Equal_elim[where ?x=x and ?y=y and ?A=A]) + oops + + +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 +prefer 2 + subgoal premises 1 for x \<comment> \<open>\<open>subgoal\<close> is the proof script version of \<open>fix-assume-show\<close>.\<close> + apply standard + prefer 2 + subgoal premises 2 for y + apply standard + prefer 2 + 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 2 + apply standard + prefer 2 + apply (rule Prod_intro) + prefer 2 + 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 + + +end
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