diff --git a/ahmed/day1.agda b/ahmed/day1.agda
index 60712c7..d6b00ed 100644
--- a/ahmed/day1.agda
+++ b/ahmed/day1.agda
@@ -50,7 +50,7 @@ subst (x ∙ x₁) v = (subst x v) ∙ (subst x₁ v)
 data value-rel : type → term → Set where 
   v-`true : value-rel bool `true
   v-`false : value-rel bool `false
-  v-`λ[_]_ : ∀ {τ e} → value-rel τ (`λ[ τ ] e)
+  v-`λ[_]_ : ∀ {τ₁ τ₂ e} → value-rel (τ₁ -→ τ₂) (`λ[ τ₁ ] e)
 
 data good-subst : ctx → sub → Set where
   nil : good-subst nil nil
@@ -60,14 +60,20 @@ data good-subst : ctx → sub → Set where
     → good-subst (cons ctx τ) γ
 
 data step : term → term → Set where
+  step-if-e : ∀ {e e' e₁ e₂} → step e e' → step (`if e then e₁ else e₂) (`if e' then e₁ else e₂)
   step-if-1 : ∀ {e₁ e₂} → step (`if `true then e₁ else e₂) e₁
   step-if-2 : ∀ {e₁ e₂} → step (`if `false then e₁ else e₂) e₂
-  step-`λ : ∀ {τ e v} → step ((`λ[ τ ] e) ∙ v) (subst e v)
+  step-`λ-1 : ∀ {e₁ e₁' e₂} → step e₁ e₁' → step (e₁ ∙ e₂) (e₁' ∙ e₂)
+  step-`λ-2 : ∀ {τ e₁ e₂ e₂'} → step e₂ e₂' → step ((`λ[ τ ] e₁) ∙ e₂) ((`λ[ τ ] e₁) ∙ e₂')
+  step-`λ-β : ∀ {τ e v} → step ((`λ[ τ ] e) ∙ v) (subst e v)
 
 data steps : ℕ → term → term → Set where
   zero : ∀ {e} → steps zero e e
   suc : ∀ {e e' e''} → (n : ℕ) → step e e' → steps n e' e'' → steps (suc n) e e''
 
+safe : (τ : type) → (e : term) → Set
+safe τ e = ∀ {n} → (e' : term) → steps n e e' → (value-rel τ e') ⊎ (∃ λ e'' → step e' e'')
+
 data _⊢_∶_ : ctx → term → type → Set where
   type-`true : ∀ {ctx} → ctx ⊢ `true ∶ bool
   type-`false : ∀ {ctx} → ctx ⊢ `false ∶ bool
@@ -79,9 +85,9 @@ data _⊢_∶_ : ctx → term → type → Set where
   type-`x : ∀ {ctx x}
     → (p : Is-just (lookup ctx x))
     → ctx ⊢ (` x) ∶ (to-witness p)
-  type-`λ : ∀ {ctx τ τ₂ e}
-    → (cons ctx τ) ⊢ e ∶ τ₂
-    → ctx ⊢ (`λ[ τ ] e) ∶ (τ -→ τ₂)
+  type-`λ : ∀ {ctx τ₁ τ₂ e}
+    → (cons ctx τ₁) ⊢ e ∶ τ₂
+    → ctx ⊢ (`λ[ τ₁ ] e) ∶ (τ₁ -→ τ₂)
   type-∙ : ∀ {ctx τ₁ τ₂ e₁ e₂}
     → ctx ⊢ e₁ ∶ (τ₁ -→ τ₂)
     → ctx ⊢ e₂ ∶ τ₂
@@ -90,19 +96,81 @@ data _⊢_∶_ : ctx → term → type → Set where
 irreducible : term → Set
 irreducible e = ¬ (∃ λ e' → step e e')
 
-data term-relation : type → term → Set where
+data term-rel : type → term → Set where
   e-term : ∀ {τ e}
     → (∀ {n} → (e' : term) → steps n e e' → irreducible e' → value-rel τ e')
-    → term-relation τ e
+    → term-rel τ e
 
-type-sound : ∀ {Γ e τ} → Γ ⊢ e ∶ τ → Set
-type-sound {Γ} {e} {τ} s = ∀ {n} → (e' : term) → steps n e e' → value-rel τ e' ⊎ ∃ λ e'' → step e' e''
+type-soundness : ∀ {Γ e τ} → Γ ⊢ e ∶ τ → Set
+type-soundness {Γ} {e} {τ} s = ∀ {n} → (e' : term) → steps n e e' → (value-rel τ e') ⊎ (∃ λ e'' → step e' e'')
+
+type-sound : (e : term) → (τ : type) → (p : nil ⊢ e ∶ τ) → type-soundness p
+type-sound .`true .bool type-`true .`true zero = inj₁ v-`true
+type-sound .`false .bool type-`false .`false zero = inj₁ v-`false
+type-sound .(`if _ then _ else _) τ (type-`ifthenelse p p₁ p₂) e' steps = {!   !}
+type-sound .(`λ[ _ ] _) .(_ -→ _) (type-`λ p) e' steps = {!   !}
+type-sound .(_ ∙ _) τ (type-∙ p p₁) e' steps = {!   !}
 
 _⊨_∶_ : (Γ : ctx) → (e : term) → (τ : type) → Set
-_⊨_∶_ Γ e τ = (γ : sub) → (good-subst Γ γ) → term-relation τ e
-
-fundamental : ∀ {Γ e τ} → (well-typed : Γ ⊢ e ∶ τ) → type-sound well-typed → Γ ⊨ e ∶ τ
-fundamental {Γ} {e} {τ} well-typed type-sound γ good-sub = e-term f
+_⊨_∶_ Γ e τ = (γ : sub) → (good-subst Γ γ) → term-rel τ e
+  
+fundamental : ∀ {Γ e τ} → (well-typed : Γ ⊢ e ∶ τ) → type-soundness well-typed → Γ ⊨ e ∶ τ
+fundamental {Γ} {e} {τ} well-typed tsound γ good-sub = e-term f
   where
-    f : {n : ℕ} (e' : term) → steps n e e' → irreducible e' → value-rel τ e'
-    f e' steps irred = [ id , (λ exists → ⊥-elim (irred exists)) ] (type-sound e' steps)
\ No newline at end of file
+    f : {n : ℕ} (e' : term) → steps n e e' → irreducible e' → value-rel τ e'   
+    f e' steps irred = [ id , (λ exists → ⊥-elim (irred exists)) ] (tsound e' steps)
+
+module semantic-type-soundness where
+  part1 : (e : term) → (τ : type) → nil ⊢ e ∶ τ → term-rel τ e
+  part1 (` x) _ (type-`x ())
+  part1 `true bool p = e-term f
+    where
+      f : {n : ℕ} (e' : term) → steps n `true e' → irreducible e' → value-rel bool e'
+      f {.zero} .`true zero irred = v-`true
+  part1 `false bool p = e-term f
+    where
+      f : {n : ℕ} (e' : term) → steps n `false e' → irreducible e' → value-rel bool e'
+      f {zero} .`false zero irred = v-`false
+  part1 (`λ[ τ₁ ] e) (.τ₁ -→ τ₂) (type-`λ p) = e-term f
+    where
+      f : {n : ℕ} (e' : term) → steps n (`λ[ τ₁ ] e) e' → irreducible e' → value-rel (τ₁ -→ τ₂) e'
+      f .(`λ[ τ₁ ] e) zero irred = v-`λ[_]_
+  part1 (`if e then e₁ else e₂) τ (type-`ifthenelse p p₁ p₂) = e-term f
+    where
+      v : ∀ {n} → (e' : term) → steps n e e' → value-rel bool e' → ∃ (step (`if e then e₁ else e₂))
+      v .`true zero v-`true = e₁ , step-if-1
+      v .`false zero v-`false = e₂ , step-if-2
+      v e' (suc {_} {e''} n step steps) val =
+        (`if e'' then e₁ else e₂) , step-if-e step
+
+      f : {n : ℕ} (e' : term) → steps n (`if e then e₁ else e₂) e' → irreducible e' → value-rel τ e'
+      f .(`if e then e₁ else e₂) zero irred with part1 e bool p
+      f .(`if e then e₁ else e₂) zero irred | e-term f' =
+        let
+          wtf = v e zero {!   !}
+          wtf3 = {! irred ?  !}
+          -- wtf2 = v e wtf
+          -- wtf3 = irred wtf2
+        -- in ⊥-elim wtf3
+        in {!   !}
+      f e' (suc n (step-if-e step) steps) irred = {!   !}
+      f e' (suc n step-if-1 steps) irred with part1 e₁ τ p₁
+      f e' (suc n step-if-1 steps) irred | e-term f' = f' e' steps irred
+      f e' (suc n step-if-2 steps) irred with part1 e₂ τ p₂
+      f e' (suc n step-if-2 steps) irred | e-term f' = f' e' steps irred
+  part1 (e₁ ∙ e₂) τ₂ (type-∙ {_} {τ₁} p₁ p₂) = e-term f  
+    where
+      f : {n : ℕ} (e' : term) → steps n (e₁ ∙ e₂) e' → irreducible e' → value-rel τ₂ e'
+      f .(e₁ ∙ e₂) zero irred = {!   !}
+      f e' (suc n (step-`λ-1 step) steps) irred with part1 e₁ (τ₁ -→ τ₂) p₁
+      f e' (suc _ (step-`λ-1 step) steps) irred | e-term f' = {!    !}
+      f e' (suc n (step-`λ-2 step) steps) irred = {!   !}
+      f e' (suc n step-`λ-β steps) irred = {!   !}
+
+  part2 : (e : term) → (τ : type) → term-rel τ e → safe τ e 
+  part2 e τ (e-term p) .e zero =
+    inj₁ (p e zero λ q → {!   !})  
+    where
+      v : ⊥
+      v = {!   !}     
+  part2 e τ (e-term p) e' (suc {f} {g} n step steps) = inj₂ ({!   !} , {! step  !}) 
\ No newline at end of file
diff --git a/holtzen/notes.typ b/holtzen/notes.typ
index 9ce465d..aba275d 100644
--- a/holtzen/notes.typ
+++ b/holtzen/notes.typ
@@ -195,4 +195,20 @@ ret z$
 
 You can draw a search tree of probabilities. Add up the probabilities to get the probability that a program returns a specific value.
 
-You can share $z$ since it doesn't depend directly on $x$. This builds a *binary decision diagram*.
\ No newline at end of file
+You can share $z$ since it doesn't depend directly on $x$. This builds a *binary decision diagram*.
+
+=== Knowledge compilation
+
+https://en.wikipedia.org/wiki/Knowledge_compilation
+
+Relationship between hardness of propositional reasoning tasks and its syntax of the formula
+
+SAT for DNF is easy. What kinds of structure enables efficient reasoning?
+
+==== Succinctness
+
+$cal(L)_1$ is more succinct than $cal(L)_2$ if it's efficient (polynomial-time) to translate (in a semantics-preserving way) programs written in $cal(L)_2$ to programs written in $cal(L)_1$
+
+==== Canonicity of BDDs
+
+There is only 1 structural BDD for any particular formula.
\ No newline at end of file