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-rw-r--r--spartan/core/Spartan.thy (renamed from spartan/theories/Spartan.thy)18
-rw-r--r--spartan/core/congruence.ML (renamed from spartan/lib/congruence.ML)0
-rw-r--r--spartan/core/elimination.ML (renamed from spartan/lib/elimination.ML)0
-rw-r--r--spartan/core/eqsubst.ML (renamed from spartan/lib/eqsubst.ML)0
-rw-r--r--spartan/core/equality.ML (renamed from spartan/lib/equality.ML)0
-rw-r--r--spartan/core/focus.ML (renamed from spartan/lib/focus.ML)0
-rw-r--r--spartan/core/goals.ML (renamed from spartan/lib/goals.ML)0
-rw-r--r--spartan/core/implicits.ML (renamed from spartan/lib/implicits.ML)0
-rw-r--r--spartan/core/lib.ML (renamed from spartan/lib/lib.ML)0
-rw-r--r--spartan/core/rewrite.ML (renamed from spartan/lib/rewrite.ML)0
-rw-r--r--spartan/core/tactics.ML (renamed from spartan/lib/tactics.ML)0
-rw-r--r--spartan/data/List.thy6
-rw-r--r--spartan/theories/Equivalence.thy416
-rw-r--r--spartan/theories/Identity.thy464
14 files changed, 15 insertions, 889 deletions
diff --git a/spartan/theories/Spartan.thy b/spartan/core/Spartan.thy
index d235041..b4f7772 100644
--- a/spartan/theories/Spartan.thy
+++ b/spartan/core/Spartan.thy
@@ -142,20 +142,20 @@ section \<open>Proof commands\<close>
named_theorems typechk
-ML_file \<open>../lib/lib.ML\<close>
-ML_file \<open>../lib/goals.ML\<close>
-ML_file \<open>../lib/focus.ML\<close>
+ML_file \<open>lib.ML\<close>
+ML_file \<open>goals.ML\<close>
+ML_file \<open>focus.ML\<close>
section \<open>Congruence automation\<close>
(*Potential to be retired*)
-ML_file \<open>../lib/congruence.ML\<close>
+ML_file \<open>congruence.ML\<close>
section \<open>Methods\<close>
-ML_file \<open>../lib/elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close>
+ML_file \<open>elimination.ML\<close> \<comment> \<open>declares the [elims] attribute\<close>
named_theorems intros and comps
lemmas
@@ -165,7 +165,7 @@ lemmas
[comps] = beta Sig_comp and
[cong] = Pi_cong lam_cong Sig_cong
-ML_file \<open>../lib/tactics.ML\<close>
+ML_file \<open>tactics.ML\<close>
method_setup assumptions =
\<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (
@@ -227,7 +227,7 @@ ML_file "~~/src/Tools/misc_legacy.ML"
ML_file "~~/src/Tools/IsaPlanner/isand.ML"
ML_file "~~/src/Tools/IsaPlanner/rw_inst.ML"
ML_file "~~/src/Tools/IsaPlanner/zipper.ML"
-ML_file "../lib/eqsubst.ML"
+ML_file "eqsubst.ML"
\<comment> \<open>\<open>rewrite\<close> method\<close>
consts rewrite_HOLE :: "'a::{}" ("\<hole>")
@@ -254,7 +254,7 @@ lemma imp_cong_eq:
done
ML_file \<open>~~/src/HOL/Library/cconv.ML\<close>
-ML_file \<open>../lib/rewrite.ML\<close>
+ML_file \<open>rewrite.ML\<close>
\<comment> \<open>\<open>reduce\<close> computes terms via judgmental equalities\<close>
setup \<open>map_theory_simpset (fn ctxt => ctxt addSolver (mk_solver "" typechk_tac))\<close>
@@ -273,7 +273,7 @@ consts
iarg :: \<open>'a\<close> ("?")
hole :: \<open>'b\<close> ("{}")
-ML_file \<open>../lib/implicits.ML\<close>
+ML_file \<open>implicits.ML\<close>
attribute_setup implicit = \<open>Scan.succeed Implicits.implicit_defs_attr\<close>
diff --git a/spartan/lib/congruence.ML b/spartan/core/congruence.ML
index bb7348c..bb7348c 100644
--- a/spartan/lib/congruence.ML
+++ b/spartan/core/congruence.ML
diff --git a/spartan/lib/elimination.ML b/spartan/core/elimination.ML
index 617f83e..617f83e 100644
--- a/spartan/lib/elimination.ML
+++ b/spartan/core/elimination.ML
diff --git a/spartan/lib/eqsubst.ML b/spartan/core/eqsubst.ML
index ea6f098..ea6f098 100644
--- a/spartan/lib/eqsubst.ML
+++ b/spartan/core/eqsubst.ML
diff --git a/spartan/lib/equality.ML b/spartan/core/equality.ML
index 023147b..023147b 100644
--- a/spartan/lib/equality.ML
+++ b/spartan/core/equality.ML
diff --git a/spartan/lib/focus.ML b/spartan/core/focus.ML
index 1d8de78..1d8de78 100644
--- a/spartan/lib/focus.ML
+++ b/spartan/core/focus.ML
diff --git a/spartan/lib/goals.ML b/spartan/core/goals.ML
index 9f394f0..9f394f0 100644
--- a/spartan/lib/goals.ML
+++ b/spartan/core/goals.ML
diff --git a/spartan/lib/implicits.ML b/spartan/core/implicits.ML
index 04fc825..04fc825 100644
--- a/spartan/lib/implicits.ML
+++ b/spartan/core/implicits.ML
diff --git a/spartan/lib/lib.ML b/spartan/core/lib.ML
index 615f601..615f601 100644
--- a/spartan/lib/lib.ML
+++ b/spartan/core/lib.ML
diff --git a/spartan/lib/rewrite.ML b/spartan/core/rewrite.ML
index f9c5d8e..f9c5d8e 100644
--- a/spartan/lib/rewrite.ML
+++ b/spartan/core/rewrite.ML
diff --git a/spartan/lib/tactics.ML b/spartan/core/tactics.ML
index f23bfee..f23bfee 100644
--- a/spartan/lib/tactics.ML
+++ b/spartan/core/tactics.ML
diff --git a/spartan/data/List.thy b/spartan/data/List.thy
new file mode 100644
index 0000000..71a879b
--- /dev/null
+++ b/spartan/data/List.thy
@@ -0,0 +1,6 @@
+theory List
+imports Spartan
+
+begin
+
+end
diff --git a/spartan/theories/Equivalence.thy b/spartan/theories/Equivalence.thy
deleted file mode 100644
index 2975738..0000000
--- a/spartan/theories/Equivalence.thy
+++ /dev/null
@@ -1,416 +0,0 @@
-theory Equivalence
-imports Identity
-
-begin
-
-section \<open>Homotopy\<close>
-
-definition "homotopy A B f g \<equiv> \<Prod>x: A. f `x =\<^bsub>B x\<^esub> g `x"
-
-definition homotopy_i (infix "~" 100)
- where [implicit]: "f ~ g \<equiv> homotopy ? ? f g"
-
-translations "f ~ g" \<leftharpoondown> "CONST homotopy A B f g"
-
-Lemma homotopy_type [typechk]:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> B x: U i"
- "f: \<Prod>x: A. B x" "g: \<Prod>x: A. B x"
- shows "f ~ g: U i"
- unfolding homotopy_def by typechk
-
-Lemma (derive) homotopy_refl [refl]:
- assumes
- "A: U i"
- "f: \<Prod>x: A. B x"
- shows "f ~ f"
- unfolding homotopy_def by intros
-
-Lemma (derive) hsym:
- assumes
- "f: \<Prod>x: A. B x"
- "g: \<Prod>x: A. B x"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> B x: U i"
- shows "H: f ~ g \<Longrightarrow> g ~ f"
- unfolding homotopy_def
- apply intros
- apply (rule pathinv)
- \<guillemotright> by (elim H)
- \<guillemotright> by typechk
- done
-
-Lemma (derive) htrans:
- assumes
- "f: \<Prod>x: A. B x"
- "g: \<Prod>x: A. B x"
- "h: \<Prod>x: A. B x"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> B x: U i"
- shows "\<lbrakk>H1: f ~ g; H2: g ~ h\<rbrakk> \<Longrightarrow> f ~ h"
- unfolding homotopy_def
- apply intro
- \<guillemotright> vars x
- apply (rule pathcomp[where ?y="g x"])
- \<^item> by (elim H1)
- \<^item> by (elim H2)
- done
- \<guillemotright> by typechk
- done
-
-text \<open>For calculations:\<close>
-
-lemmas
- homotopy_sym [sym] = hsym[rotated 4] and
- homotopy_trans [trans] = htrans[rotated 5]
-
-Lemma (derive) commute_homotopy:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- "f: A \<rightarrow> B" "g: A \<rightarrow> B"
- "H: homotopy A (\<lambda>_. B) f g"
- shows "(H x) \<bullet> g[p] = f[p] \<bullet> (H y)"
- \<comment> \<open>Need this assumption unfolded for typechecking:\<close>
- supply assms(8)[unfolded homotopy_def]
- apply (eq p)
- focus vars x
- apply reduce
- \<comment> \<open>Here it would really be nice to have automation for transport and
- propositional equality.\<close>
- apply (rule use_transport[where ?y="H x \<bullet> refl (g x)"])
- \<guillemotright> by (rule pathcomp_right_refl)
- \<guillemotright> by (rule pathinv) (rule pathcomp_left_refl)
- \<guillemotright> by typechk
- done
- done
-
-Corollary (derive) commute_homotopy':
- assumes
- "A: U i"
- "x: A"
- "f: A \<rightarrow> A"
- "H: homotopy A (\<lambda>_. A) f (id A)"
- shows "H (f x) = f [H x]"
-oops
-
-Lemma homotopy_id_left [typechk]:
- assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
- shows "homotopy_refl A f: (id B) \<circ> f ~ f"
- unfolding homotopy_refl_def homotopy_def by reduce
-
-Lemma homotopy_id_right [typechk]:
- assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
- shows "homotopy_refl A f: f \<circ> (id A) ~ f"
- unfolding homotopy_refl_def homotopy_def by reduce
-
-Lemma homotopy_funcomp_left:
- assumes
- "H: homotopy B C g g'"
- "f: A \<rightarrow> B"
- "g: \<Prod>x: B. C x"
- "g': \<Prod>x: B. C x"
- "A: U i" "B: U i"
- "\<And>x. x: B \<Longrightarrow> C x: U i"
- shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g' \<circ>\<^bsub>A\<^esub> f)"
- unfolding homotopy_def
- apply (intro; reduce)
- apply (insert \<open>H: _\<close>[unfolded homotopy_def])
- apply (elim H)
- done
-
-Lemma homotopy_funcomp_right:
- assumes
- "H: homotopy A (\<lambda>_. B) f f'"
- "f: A \<rightarrow> B"
- "f': A \<rightarrow> B"
- "g: B \<rightarrow> C"
- "A: U i" "B: U i" "C: U i"
- shows "homotopy A {} (g \<circ>\<^bsub>A\<^esub> f) (g \<circ>\<^bsub>A\<^esub> f')"
- unfolding homotopy_def
- apply (intro; reduce)
- apply (insert \<open>H: _\<close>[unfolded homotopy_def])
- apply (dest PiE, assumption)
- apply (rule ap, assumption)
- done
-
-
-section \<open>Quasi-inverse and bi-invertibility\<close>
-
-subsection \<open>Quasi-inverses\<close>
-
-definition "qinv A B f \<equiv> \<Sum>g: B \<rightarrow> A.
- homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A) \<times> homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B)"
-
-lemma qinv_type [typechk]:
- assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
- shows "qinv A B f: U i"
- unfolding qinv_def by typechk
-
-definition qinv_i ("qinv")
- where [implicit]: "qinv f \<equiv> Equivalence.qinv ? ? f"
-
-translations "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
-
-Lemma (derive) id_qinv:
- assumes "A: U i"
- shows "qinv (id A)"
- unfolding qinv_def
- apply intro defer
- apply intro defer
- apply (rule homotopy_id_right)
- apply (rule homotopy_id_left)
- done
-
-Lemma (derive) quasiinv_qinv:
- assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
- shows "prf: qinv f \<Longrightarrow> qinv (fst prf)"
- unfolding qinv_def
- apply intro
- \<guillemotright> by (rule \<open>f:_\<close>)
- \<guillemotright> apply (elim "prf")
- focus vars g HH
- apply intro
- \<^item> by reduce (snd HH)
- \<^item> by reduce (fst HH)
- done
- done
- done
-
-Lemma (derive) funcomp_qinv:
- assumes
- "A: U i" "B: U i" "C: U i"
- "f: A \<rightarrow> B" "g: B \<rightarrow> C"
- shows "qinv f \<rightarrow> qinv g \<rightarrow> qinv (g \<circ> f)"
- apply (intros, unfold qinv_def, elims)
- focus
- prems prms
- vars _ _ finv _ ginv _ HfA HfB HgB HgC
-
- apply intro
- apply (rule funcompI[where ?f=ginv and ?g=finv])
-
- text \<open>Now a whole bunch of rewrites and we're done.\<close>
-oops
-
-subsection \<open>Bi-invertible maps\<close>
-
-definition "biinv A B f \<equiv>
- (\<Sum>g: B \<rightarrow> A. homotopy A (\<lambda>_. A) (g \<circ>\<^bsub>A\<^esub> f) (id A))
- \<times> (\<Sum>g: B \<rightarrow> A. homotopy B (\<lambda>_. B) (f \<circ>\<^bsub>B\<^esub> g) (id B))"
-
-lemma biinv_type [typechk]:
- assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
- shows "biinv A B f: U i"
- unfolding biinv_def by typechk
-
-definition biinv_i ("biinv")
- where [implicit]: "biinv f \<equiv> Equivalence.biinv ? ? f"
-
-translations "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
-
-Lemma (derive) qinv_imp_biinv:
- assumes
- "A: U i" "B: U i"
- "f: A \<rightarrow> B"
- shows "?prf: qinv f \<rightarrow> biinv f"
- apply intros
- unfolding qinv_def biinv_def
- by (rule Sig_dist_exp)
-
-text \<open>
- Show that bi-invertible maps are quasi-inverses, as a demonstration of how to
- work in this system.
-\<close>
-
-Lemma (derive) biinv_imp_qinv:
- assumes "A: U i" "B: U i" "f: A \<rightarrow> B"
- shows "biinv f \<rightarrow> qinv f"
-
- text \<open>Split the hypothesis \<^term>\<open>biinv f\<close> into its components:\<close>
- apply intro
- unfolding biinv_def
- apply elims
-
- text \<open>Name the components:\<close>
- focus prems vars _ _ _ g H1 h H2
- thm \<open>g:_\<close> \<open>h:_\<close> \<open>H1:_\<close> \<open>H2:_\<close>
-
- text \<open>
- \<^term>\<open>g\<close> is a quasi-inverse to \<^term>\<open>f\<close>, so the proof will be a triple
- \<^term>\<open><g, <?H1, ?H2>>\<close>.
- \<close>
- unfolding qinv_def
- apply intro
- \<guillemotright> by (rule \<open>g: _\<close>)
- \<guillemotright> apply intro
- text \<open>The first part \<^prop>\<open>?H1: g \<circ> f ~ id A\<close> is given by \<^term>\<open>H1\<close>.\<close>
- apply (rule \<open>H1: _\<close>)
-
- text \<open>
- It remains to prove \<^prop>\<open>?H2: f \<circ> g ~ id B\<close>. First show that \<open>g ~ h\<close>,
- then the result follows from @{thm \<open>H2: f \<circ> h ~ id B\<close>}. Here a proof
- block is used to calculate "forward".
- \<close>
- proof -
- have "g ~ g \<circ> (id B)" by reduce refl
- also have "g \<circ> (id B) ~ g \<circ> f \<circ> h"
- by (rule homotopy_funcomp_right) (rule \<open>H2:_\<close>[symmetric])
- also have "g \<circ> f \<circ> h ~ (id A) \<circ> h"
- by (subst funcomp_assoc[symmetric])
- (rule homotopy_funcomp_left, rule \<open>H1:_\<close>)
- also have "(id A) \<circ> h ~ h" by reduce refl
- finally have "g ~ h" by this
-
- then have "f \<circ> g ~ f \<circ> h" by (rule homotopy_funcomp_right)
-
- with \<open>H2:_\<close>
- show "f \<circ> g ~ id B"
- by (rule homotopy_trans) (assumption+, typechk)
- qed
- done
- done
-
-Lemma (derive) id_biinv:
- "A: U i \<Longrightarrow> biinv (id A)"
- by (rule qinv_imp_biinv) (rule id_qinv)
-
-Lemma (derive) funcomp_biinv:
- assumes
- "A: U i" "B: U i" "C: U i"
- "f: A \<rightarrow> B" "g: B \<rightarrow> C"
- shows "biinv f \<rightarrow> biinv g \<rightarrow> biinv (g \<circ> f)"
- apply intros
- focus vars biinv\<^sub>f biinv\<^sub>g
-
- text \<open>Follows from \<open>funcomp_qinv\<close>.\<close>
-oops
-
-
-section \<open>Equivalence\<close>
-
-text \<open>
- Following the HoTT book, we first define equivalence in terms of
- bi-invertibility.
-\<close>
-
-definition equivalence (infix "\<simeq>" 110)
- where "A \<simeq> B \<equiv> \<Sum>f: A \<rightarrow> B. Equivalence.biinv A B f"
-
-lemma equivalence_type [typechk]:
- assumes "A: U i" "B: U i"
- shows "A \<simeq> B: U i"
- unfolding equivalence_def by typechk
-
-Lemma (derive) equivalence_refl:
- assumes "A: U i"
- shows "A \<simeq> A"
- unfolding equivalence_def
- apply intro defer
- apply (rule qinv_imp_biinv) defer
- apply (rule id_qinv)
- done
-
-text \<open>
- The following could perhaps be easier with transport (but then I think we need
- univalence?)...
-\<close>
-
-Lemma (derive) equivalence_symmetric:
- assumes "A: U i" "B: U i"
- shows "A \<simeq> B \<rightarrow> B \<simeq> A"
- apply intros
- unfolding equivalence_def
- apply elim
- \<guillemotright> vars _ f "prf"
- apply (dest (4) biinv_imp_qinv)
- apply intro
- \<^item> unfolding qinv_def by (rule fst[of "(biinv_imp_qinv A B f) prf"])
- \<^item> by (rule qinv_imp_biinv) (rule quasiinv_qinv)
- done
- done
-
-Lemma (derive) equivalence_transitive:
- assumes "A: U i" "B: U i" "C: U i"
- shows "A \<simeq> B \<rightarrow> B \<simeq> C \<rightarrow> A \<simeq> C"
- apply intros
- unfolding equivalence_def
-
- text \<open>Use \<open>funcomp_biinv\<close>.\<close>
-oops
-
-text \<open>
- Equal types are equivalent. We give two proofs: the first by induction, and
- the second by following the HoTT book and showing that transport is an
- equivalence.
-\<close>
-
-Lemma
- assumes
- "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
- shows "A \<simeq> B"
- by (eq p) (rule equivalence_refl)
-
-text \<open>
- The following proof is wordy because (1) the typechecker doesn't rewrite, and
- (2) we don't yet have universe automation.
-\<close>
-
-Lemma (derive) id_imp_equiv:
- assumes
- "A: U i" "B: U i" "p: A =\<^bsub>U i\<^esub> B"
- shows "<trans (id (U i)) p, ?isequiv>: A \<simeq> B"
- unfolding equivalence_def
- apply intros defer
-
- \<comment> \<open>Switch off auto-typechecking, which messes with universe levels\<close>
- supply [[auto_typechk=false]]
-
- \<guillemotright> apply (eq p, typechk)
- \<^item> prems vars A B
- apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
- apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
- apply (rule transport, rule U_in_U)
- apply (rule lift_universe_codomain, rule U_in_U)
- apply (typechk, rule U_in_U)
- done
- \<^item> prems vars A
- apply (subst transport_comp)
- \<^enum> by (rule U_in_U)
- \<^enum> by reduce (rule lift_universe)
- \<^enum> by reduce (rule id_biinv)
- done
- done
-
- \<guillemotright> \<comment> \<open>Similar proof as in the first subgoal above\<close>
- apply (rewrite at A in "A \<rightarrow> B" id_comp[symmetric])
- apply (rewrite at B in "_ \<rightarrow> B" id_comp[symmetric])
- apply (rule transport, rule U_in_U)
- apply (rule lift_universe_codomain, rule U_in_U)
- apply (typechk, rule U_in_U)
- done
- done
-
-(*Uncomment this to see all implicits from here on.
-no_translations
- "f x" \<leftharpoondown> "f `x"
- "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y"
- "g \<circ> f" \<leftharpoondown> "g \<circ>\<^bsub>A\<^esub> f"
- "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p"
- "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
- "fst" \<leftharpoondown> "CONST Spartan.fst A B"
- "snd" \<leftharpoondown> "CONST Spartan.snd A B"
- "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
- "trans P p" \<leftharpoondown> "CONST transport A P x y p"
- "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
- "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
- "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
- "f ~ g" \<leftharpoondown> "CONST homotopy A B f g"
- "qinv f" \<leftharpoondown> "CONST Equivalence.qinv A B f"
- "biinv f" \<leftharpoondown> "CONST Equivalence.biinv A B f"
-*)
-
-
-end
diff --git a/spartan/theories/Identity.thy b/spartan/theories/Identity.thy
deleted file mode 100644
index 3a982f6..0000000
--- a/spartan/theories/Identity.thy
+++ /dev/null
@@ -1,464 +0,0 @@
-chapter \<open>The identity type\<close>
-
-text \<open>
- The identity type, the higher groupoid structure of types,
- and type families as fibrations.
-\<close>
-
-theory Identity
-imports Spartan
-
-begin
-
-axiomatization
- Id :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> and
- refl :: \<open>o \<Rightarrow> o\<close> and
- IdInd :: \<open>o \<Rightarrow> (o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o) \<Rightarrow> (o \<Rightarrow> o) \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close>
-
-syntax "_Id" :: \<open>o \<Rightarrow> o \<Rightarrow> o \<Rightarrow> o\<close> ("(2_ =\<^bsub>_\<^esub>/ _)" [111, 0, 111] 110)
-
-translations "a =\<^bsub>A\<^esub> b" \<rightleftharpoons> "CONST Id A a b"
-
-axiomatization where
- IdF: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^bsub>A\<^esub> b: U i" and
-
- IdI: "a: A \<Longrightarrow> refl a: a =\<^bsub>A\<^esub> a" and
-
- IdE: "\<lbrakk>
- p: a =\<^bsub>A\<^esub> b;
- a: A;
- b: A;
- \<And>x y p. \<lbrakk>p: x =\<^bsub>A\<^esub> y; x: A; y: A\<rbrakk> \<Longrightarrow> C x y p: U i;
- \<And>x. x: A \<Longrightarrow> f x: C x x (refl x)
- \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a b p: C a b p" and
-
- Id_comp: "\<lbrakk>
- a: A;
- \<And>x y p. \<lbrakk>x: A; y: A; p: x =\<^bsub>A\<^esub> y\<rbrakk> \<Longrightarrow> C x y p: U i;
- \<And>x. x: A \<Longrightarrow> f x: C x x (refl x)
- \<rbrakk> \<Longrightarrow> IdInd A (\<lambda>x y p. C x y p) f a a (refl a) \<equiv> f a"
-
-lemmas
- [intros] = IdF IdI and
- [elims "?p" "?a" "?b"] = IdE and
- [comps] = Id_comp and
- [refl] = IdI
-
-
-section \<open>Path induction\<close>
-
-method_setup eq = \<open>
-Args.term >> (fn tm => fn ctxt => CONTEXT_METHOD (K (
- CONTEXT_SUBGOAL (fn (goal, i) =>
- let
- val facts = Proof_Context.facts_of ctxt
- val prems = Logic.strip_assums_hyp goal
- val template = \<^const>\<open>has_type\<close>
- $ tm
- $ (\<^const>\<open>Id\<close> $ Var (("*?A", 0), \<^typ>\<open>o\<close>)
- $ Var (("*?x", 0), \<^typ>\<open>o\<close>)
- $ Var (("*?y", 0), \<^typ>\<open>o\<close>))
- val types =
- map (Thm.prop_of o #1) (Facts.could_unify facts template)
- @ filter (fn prem => Term.could_unify (template, prem)) prems
- |> map Lib.type_of_typing
- in case types of
- (\<^const>\<open>Id\<close> $ _ $ x $ y)::_ =>
- elim_context_tac [tm, x, y] ctxt i
- | _ => Context_Tactic.CONTEXT_TACTIC no_tac
- end) 1)))
-\<close>
-
-
-section \<open>Symmetry and transitivity\<close>
-
-Lemma (derive) pathinv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "y =\<^bsub>A\<^esub> x"
- by (eq p) intro
-
-lemma pathinv_comp [comps]:
- assumes "x: A" "A: U i"
- shows "pathinv A x x (refl x) \<equiv> refl x"
- unfolding pathinv_def by reduce
-
-Lemma (derive) pathcomp:
- assumes
- "A: U i" "x: A" "y: A" "z: A"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
- shows
- "x =\<^bsub>A\<^esub> z"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply intro
- done
- done
-
-lemma pathcomp_comp [comps]:
- assumes "a: A" "A: U i"
- shows "pathcomp A a a a (refl a) (refl a) \<equiv> refl a"
- unfolding pathcomp_def by reduce
-
-
-section \<open>Notation\<close>
-
-definition Id_i (infix "=" 110)
- where [implicit]: "Id_i x y \<equiv> x =\<^bsub>?\<^esub> y"
-
-definition pathinv_i ("_\<inverse>" [1000])
- where [implicit]: "pathinv_i p \<equiv> pathinv ? ? ? p"
-
-definition pathcomp_i (infixl "\<bullet>" 121)
- where [implicit]: "pathcomp_i p q \<equiv> pathcomp ? ? ? ? p q"
-
-translations
- "x = y" \<leftharpoondown> "x =\<^bsub>A\<^esub> y"
- "p\<inverse>" \<leftharpoondown> "CONST pathinv A x y p"
- "p \<bullet> q" \<leftharpoondown> "CONST pathcomp A x y z p q"
-
-
-section \<open>Calculational reasoning\<close>
-
-consts rhs :: \<open>'a\<close> ("''''")
-
-ML \<open>
-local fun last_rhs ctxt =
- let
- val this_name = Name_Space.full_name (Proof_Context.naming_of ctxt)
- (Binding.name Auto_Bind.thisN)
- val this = #thms (the (Proof_Context.lookup_fact ctxt this_name))
- handle Option => []
- val rhs = case map Thm.prop_of this of
- [\<^const>\<open>has_type\<close> $ _ $ (\<^const>\<open>Id\<close> $ _ $ _ $ y)] => y
- | _ => Term.dummy
- in
- map_aterms (fn t => case t of Const (\<^const_name>\<open>rhs\<close>, _) => rhs | _ => t)
- end
-in val _ = Context.>>
- (Syntax_Phases.term_check 5 "" (fn ctxt => map (last_rhs ctxt)))
-end
-\<close>
-
-lemmas
- [sym] = pathinv[rotated 3] and
- [trans] = pathcomp[rotated 4]
-
-
-section \<open>Basic propositional equalities\<close>
-
-Lemma (derive) pathcomp_left_refl:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "(refl x) \<bullet> p = p"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_right_refl:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p \<bullet> (refl y) = p"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_left_inv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p\<inverse> \<bullet> p = refl y"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_right_inv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p \<bullet> p\<inverse> = refl x"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathinv_pathinv:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "p\<inverse>\<inverse> = p"
- apply (eq p)
- apply (reduce; intros)
- done
-
-Lemma (derive) pathcomp_assoc:
- assumes
- "A: U i" "x: A" "y: A" "z: A" "w: A"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z" "r: z =\<^bsub>A\<^esub> w"
- shows "p \<bullet> (q \<bullet> r) = p \<bullet> q \<bullet> r"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply (reduce; intros)
- done
- done
- done
-
-
-section \<open>Functoriality of functions\<close>
-
-Lemma (derive) ap:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y"
- shows "f x = f y"
- apply (eq p)
- apply intro
- done
-
-definition ap_i ("_[_]" [1000, 0])
- where [implicit]: "ap_i f p \<equiv> ap ? ? ? ? f p"
-
-translations "f[p]" \<leftharpoondown> "CONST ap A B x y f p"
-
-Lemma ap_refl [comps]:
- assumes "f: A \<rightarrow> B" "x: A" "A: U i" "B: U i"
- shows "f[refl x] \<equiv> refl (f x)"
- unfolding ap_def by reduce
-
-Lemma (derive) ap_pathcomp:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A" "z: A"
- "f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
- shows
- "f[p \<bullet> q] = f[p] \<bullet> f[q]"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply (reduce; intro)
- done
- done
-
-Lemma (derive) ap_pathinv:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "f: A \<rightarrow> B"
- "p: x =\<^bsub>A\<^esub> y"
- shows "f[p\<inverse>] = f[p]\<inverse>"
- by (eq p) (reduce; intro)
-
-text \<open>The next two proofs currently use some low-level rewriting.\<close>
-
-Lemma (derive) ap_funcomp:
- assumes
- "A: U i" "B: U i" "C: U i"
- "x: A" "y: A"
- "f: A \<rightarrow> B" "g: B \<rightarrow> C"
- "p: x =\<^bsub>A\<^esub> y"
- shows "(g \<circ> f)[p] = g[f[p]]"
- apply (eq p)
- apply (simp only: funcomp_apply_comp[symmetric])
- apply (reduce; intro)
- done
-
-Lemma (derive) ap_id:
- assumes "A: U i" "x: A" "y: A" "p: x =\<^bsub>A\<^esub> y"
- shows "(id A)[p] = p"
- apply (eq p)
- apply (rewrite at "\<hole> = _" id_comp[symmetric])
- apply (rewrite at "_ = \<hole>" id_comp[symmetric])
- apply (reduce; intros)
- done
-
-
-section \<open>Transport\<close>
-
-Lemma (derive) transport:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "P x \<rightarrow> P y"
- by (eq p) intro
-
-definition transport_i ("trans")
- where [implicit]: "trans P p \<equiv> transport ? P ? ? p"
-
-translations "trans P p" \<leftharpoondown> "CONST transport A P x y p"
-
-Lemma transport_comp [comps]:
- assumes
- "a: A"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- shows "trans P (refl a) \<equiv> id (P a)"
- unfolding transport_def by reduce
-
-\<comment> \<open>TODO: Build transport automation!\<close>
-
-Lemma use_transport:
- assumes
- "p: y =\<^bsub>A\<^esub> x"
- "u: P x"
- "x: A" "y: A"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- shows "trans P p\<inverse> u: P y"
- by typechk
-
-Lemma (derive) transport_left_inv:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "(trans P p\<inverse>) \<circ> (trans P p) = id (P x)"
- by (eq p) (reduce; intro)
-
-Lemma (derive) transport_right_inv:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "(trans P p) \<circ> (trans P p\<inverse>) = id (P y)"
- by (eq p) (reduce; intros)
-
-Lemma (derive) transport_pathcomp:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A" "z: A"
- "u: P x"
- "p: x =\<^bsub>A\<^esub> y" "q: y =\<^bsub>A\<^esub> z"
- shows "trans P q (trans P p u) = trans P (p \<bullet> q) u"
- apply (eq p)
- focus prems vars x p
- apply (eq p)
- apply (reduce; intros)
- done
- done
-
-Lemma (derive) transport_compose_typefam:
- assumes
- "A: U i" "B: U i"
- "\<And>x. x: B \<Longrightarrow> P x: U i"
- "f: A \<rightarrow> B"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- "u: P (f x)"
- shows "trans (\<lambda>x. P (f x)) p u = trans P f[p] u"
- by (eq p) (reduce; intros)
-
-Lemma (derive) transport_function_family:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "\<And>x. x: A \<Longrightarrow> Q x: U i"
- "f: \<Prod>x: A. P x \<rightarrow> Q x"
- "x: A" "y: A"
- "u: P x"
- "p: x =\<^bsub>A\<^esub> y"
- shows "trans Q p ((f x) u) = (f y) (trans P p u)"
- by (eq p) (reduce; intros)
-
-Lemma (derive) transport_const:
- assumes
- "A: U i" "B: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "\<Prod>b: B. trans (\<lambda>_. B) p b = b"
- by (intro, eq p) (reduce; intro)
-
-definition transport_const_i ("trans'_const")
- where [implicit]: "trans_const B p \<equiv> transport_const ? B ? ? p"
-
-translations "trans_const B p" \<leftharpoondown> "CONST transport_const A B x y p"
-
-Lemma transport_const_comp [comps]:
- assumes
- "x: A" "b: B"
- "A: U i" "B: U i"
- shows "trans_const B (refl x) b\<equiv> refl b"
- unfolding transport_const_def by reduce
-
-Lemma (derive) pathlift:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- "u: P x"
- shows "<x, u> = <y, trans P p u>"
- by (eq p) (reduce; intros)
-
-definition pathlift_i ("lift")
- where [implicit]: "lift P p u \<equiv> pathlift ? P ? ? p u"
-
-translations "lift P p u" \<leftharpoondown> "CONST pathlift A P x y p u"
-
-Lemma pathlift_comp [comps]:
- assumes
- "u: P x"
- "x: A"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "A: U i"
- shows "lift P (refl x) u \<equiv> refl <x, u>"
- unfolding pathlift_def by reduce
-
-Lemma (derive) pathlift_fst:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "x: A" "y: A"
- "u: P x"
- "p: x =\<^bsub>A\<^esub> y"
- shows "fst[lift P p u] = p"
- apply (eq p)
- text \<open>Some rewriting needed here:\<close>
- \<guillemotright> vars x y
- (*Here an automatic reordering tactic would be helpful*)
- apply (rewrite at x in "x = y" fst_comp[symmetric])
- prefer 4
- apply (rewrite at y in "_ = y" fst_comp[symmetric])
- done
- \<guillemotright> by reduce intro
- done
-
-
-section \<open>Dependent paths\<close>
-
-Lemma (derive) apd:
- assumes
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- "f: \<Prod>x: A. P x"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "trans P p (f x) = f y"
- by (eq p) (reduce; intros; typechk)
-
-definition apd_i ("apd")
- where [implicit]: "apd f p \<equiv> Identity.apd ? ? f ? ? p"
-
-translations "apd f p" \<leftharpoondown> "CONST Identity.apd A P f x y p"
-
-Lemma dependent_map_comp [comps]:
- assumes
- "f: \<Prod>x: A. P x"
- "x: A"
- "A: U i"
- "\<And>x. x: A \<Longrightarrow> P x: U i"
- shows "apd f (refl x) \<equiv> refl (f x)"
- unfolding apd_def by reduce
-
-Lemma (derive) apd_ap:
- assumes
- "A: U i" "B: U i"
- "f: A \<rightarrow> B"
- "x: A" "y: A"
- "p: x =\<^bsub>A\<^esub> y"
- shows "apd f p = trans_const B p (f x) \<bullet> f[p]"
- by (eq p) (reduce; intro)
-
-
-end