feat(library,hott): add eq.mpr and eq.mp lemmas

hott/init/logic.hlean

@@ -51,6 +51,12 @@ namespace eq
   definition symm [unfold 4] (H : a = b) : b = a :=
   subst H (refl a)
 
+  theorem mp {a b : Type} : (a = b) → a → b :=
+  eq.rec_on
+
+  theorem mpr {a b : Type} : (a = b) → b → a :=
+  assume H₁ H₂, eq.rec_on (eq.symm H₁) H₂
+
   namespace ops
     postfix ⁻¹ := symm --input with \sy or \-1 or \inv
     infixl ⬝ := trans

library/init/logic.lean

@@ -72,6 +72,12 @@ namespace eq
   theorem substr {P : A → Prop} (H₁ : b = a) : P a → P b :=
   subst (symm H₁)
 
+  theorem mp {a b : Type} : (a = b) → a → b :=
+  eq.rec_on
+
+  theorem mpr {a b : Type} : (a = b) → b → a :=
+  assume H₁ H₂, eq.rec_on (eq.symm H₁) H₂
+
   namespace ops
     notation H `⁻¹` := symm H --input with \sy or \-1 or \inv
     notation H1 ⬝ H2 := trans H1 H2

src/library/constants.cpp

@@ -27,6 +27,8 @@ name const * g_eq_elim_inv_inv = nullptr;
 name const * g_eq_intro = nullptr;
 name const * g_eq_rec = nullptr;
 name const * g_eq_drec = nullptr;
+name const * g_eq_mp = nullptr;
+name const * g_eq_mpr = nullptr;
 name const * g_eq_nrec = nullptr;
 name const * g_eq_rec_eq = nullptr;
 name const * g_eq_refl = nullptr;
@@ -49,6 +51,8 @@ name const * g_iff = nullptr;
 name const * g_iff_refl = nullptr;
 name const * g_iff_symm = nullptr;
 name const * g_iff_trans = nullptr;
+name const * g_iff_mp = nullptr;
+name const * g_iff_mpr = nullptr;
 name const * g_iff_false_intro = nullptr;
 name const * g_iff_true_intro = nullptr;
 name const * g_implies = nullptr;
@@ -187,6 +191,8 @@ void initialize_constants() {
     g_eq_intro = new name{"eq", "intro"};
     g_eq_rec = new name{"eq", "rec"};
     g_eq_drec = new name{"eq", "drec"};
+    g_eq_mp = new name{"eq", "mp"};
+    g_eq_mpr = new name{"eq", "mpr"};
     g_eq_nrec = new name{"eq", "nrec"};
     g_eq_rec_eq = new name{"eq_rec_eq"};
     g_eq_refl = new name{"eq", "refl"};
@@ -209,6 +215,8 @@ void initialize_constants() {
     g_iff_refl = new name{"iff", "refl"};
     g_iff_symm = new name{"iff", "symm"};
     g_iff_trans = new name{"iff", "trans"};
+    g_iff_mp = new name{"iff", "mp"};
+    g_iff_mpr = new name{"iff", "mpr"};
     g_iff_false_intro = new name{"iff_false_intro"};
     g_iff_true_intro = new name{"iff_true_intro"};
     g_implies = new name{"implies"};
@@ -348,6 +356,8 @@ void finalize_constants() {
     delete g_eq_intro;
     delete g_eq_rec;
     delete g_eq_drec;
+    delete g_eq_mp;
+    delete g_eq_mpr;
     delete g_eq_nrec;
     delete g_eq_rec_eq;
     delete g_eq_refl;
@@ -370,6 +380,8 @@ void finalize_constants() {
     delete g_iff_refl;
     delete g_iff_symm;
     delete g_iff_trans;
+    delete g_iff_mp;
+    delete g_iff_mpr;
     delete g_iff_false_intro;
     delete g_iff_true_intro;
     delete g_implies;
@@ -508,6 +520,8 @@ name const & get_eq_elim_inv_inv_name() { return *g_eq_elim_inv_inv; }
 name const & get_eq_intro_name() { return *g_eq_intro; }
 name const & get_eq_rec_name() { return *g_eq_rec; }
 name const & get_eq_drec_name() { return *g_eq_drec; }
+name const & get_eq_mp_name() { return *g_eq_mp; }
+name const & get_eq_mpr_name() { return *g_eq_mpr; }
 name const & get_eq_nrec_name() { return *g_eq_nrec; }
 name const & get_eq_rec_eq_name() { return *g_eq_rec_eq; }
 name const & get_eq_refl_name() { return *g_eq_refl; }
@@ -530,6 +544,8 @@ name const & get_iff_name() { return *g_iff; }
 name const & get_iff_refl_name() { return *g_iff_refl; }
 name const & get_iff_symm_name() { return *g_iff_symm; }
 name const & get_iff_trans_name() { return *g_iff_trans; }
+name const & get_iff_mp_name() { return *g_iff_mp; }
+name const & get_iff_mpr_name() { return *g_iff_mpr; }
 name const & get_iff_false_intro_name() { return *g_iff_false_intro; }
 name const & get_iff_true_intro_name() { return *g_iff_true_intro; }
 name const & get_implies_name() { return *g_implies; }

src/library/constants.h

@@ -29,6 +29,8 @@ name const & get_eq_elim_inv_inv_name();
 name const & get_eq_intro_name();
 name const & get_eq_rec_name();
 name const & get_eq_drec_name();
+name const & get_eq_mp_name();
+name const & get_eq_mpr_name();
 name const & get_eq_nrec_name();
 name const & get_eq_rec_eq_name();
 name const & get_eq_refl_name();
@@ -51,6 +53,8 @@ name const & get_iff_name();
 name const & get_iff_refl_name();
 name const & get_iff_symm_name();
 name const & get_iff_trans_name();
+name const & get_iff_mp_name();
+name const & get_iff_mpr_name();
 name const & get_iff_false_intro_name();
 name const & get_iff_true_intro_name();
 name const & get_implies_name();

src/library/constants.txt

@@ -22,6 +22,8 @@ eq.elim_inv_inv
 eq.intro
 eq.rec
 eq.drec
+eq.mp
+eq.mpr
 eq.nrec
 eq_rec_eq
 eq.refl
@@ -44,6 +46,8 @@ iff
 iff.refl
 iff.symm
 iff.trans
+iff.mp
+iff.mpr
 iff_false_intro
 iff_true_intro
 implies