Test theories

2 files changed, 64 insertions, 9 deletions

diff --git a/scratch.thy b/scratch.thy
deleted file mode 100644
index f0514c3..0000000
--- a/scratch.thy
+++ /dev/null
@@ -1,9 +0,0 @@
-theory scratch
-  imports HoTT
-begin
-
-
-
-
-
-end
\ No newline at end of file
diff --git a/tests/Schematic_subgoal.thy b/tests/Schematic_subgoal.thy
new file mode 100644
index 0000000..5f20eac
--- /dev/null
+++ b/tests/Schematic_subgoal.thy
@@ -0,0 +1,64 @@
+theory Schematic_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 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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