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adding isabelle coursework
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John Wickerson
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isabelle/2021/HSV_tasks_2021.thy
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179
isabelle/2021/HSV_tasks_2021.thy
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theory HSV_tasks_2021 imports Complex_Main begin
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section \<open>Task 1: Factorising circuits.\<close>
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(* Datatype for representing simple circuits. *)
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datatype "circuit" =
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NOT "circuit"
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| AND "circuit" "circuit"
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| OR "circuit" "circuit"
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| TRUE
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| FALSE
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| INPUT "int"
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(* Simulates a circuit given a valuation for each input wire. *)
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fun simulate where
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"simulate (AND c1 c2) \<rho> = ((simulate c1 \<rho>) \<and> (simulate c2 \<rho>))"
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| "simulate (OR c1 c2) \<rho> = ((simulate c1 \<rho>) \<or> (simulate c2 \<rho>))"
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| "simulate (NOT c) \<rho> = (\<not> (simulate c \<rho>))"
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| "simulate TRUE \<rho> = True"
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| "simulate FALSE \<rho> = False"
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| "simulate (INPUT i) \<rho> = \<rho> i"
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(* Equivalence between circuits. *)
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fun circuits_equiv (infix "\<sim>" 50) (* the "50" indicates the operator precedence *) where
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"c1 \<sim> c2 = (\<forall>\<rho>. simulate c1 \<rho> = simulate c2 \<rho>)"
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(* An optimisation that exploits the following Boolean identities:
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`(a | b) & (a | c) = a | (b & c)`
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`(a | b) & (c | a) = a | (b & c)`
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`(a | b) & (b | c) = b | (a & c)`
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`(a | b) & (c | b) = b | (a & c)`
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*)
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fun factorise where
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"factorise (NOT c) = NOT (factorise c)"
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| "factorise (AND (OR c1 c2) (OR c3 c4)) = (
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let c1' = factorise c1; c2' = factorise c2; c3' = factorise c3; c4' = factorise c4 in
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if c1' = c3' then OR c1' (AND c2' c4')
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else if c1' = c4' then OR c1' (AND c2' c3')
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else if c2' = c3' then OR c2' (AND c1' c4')
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else if c2' = c4' then OR c2' (AND c1' c3')
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else AND (OR c1' c2') (OR c3' c4'))"
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| "factorise (AND c1 c2) = AND (factorise c1) (factorise c2)"
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| "factorise (OR c1 c2) = OR (factorise c1) (factorise c2)"
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| "factorise TRUE = TRUE"
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| "factorise FALSE = FALSE"
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| "factorise (INPUT i) = INPUT i"
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lemma (* test case *)
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"factorise (AND TRUE TRUE) = AND TRUE TRUE"
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by eval
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lemma (* test case *)
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"factorise (AND (OR (INPUT 1) FALSE) (OR TRUE (INPUT 1))) =
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OR (INPUT 1) (AND FALSE TRUE)"
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by eval
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lemma (* test case *)
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"factorise (NOT (AND (OR FALSE (INPUT 2)) (OR TRUE (INPUT 2)))) =
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NOT (OR (INPUT 2) (AND FALSE TRUE))"
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by eval
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theorem factorise_is_sound: "factorise c \<sim> c"
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sorry
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fun factorise2 where "factorise2 c = c" (* dummy definition *)
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lemma (* test case *)
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"factorise2 (OR (AND (INPUT 1) (INPUT 2)) (AND TRUE (INPUT 1))) =
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AND (INPUT 1) (OR (INPUT 2) TRUE)"
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sorry
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theorem factorise2_is_sound: "factorise2 c \<sim> c"
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sorry
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section \<open>Task 2: A theorem about divisibility.\<close>
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(* NB: Without the "::int" annotation, Isabelle will try to prove a
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slightly more general theorem where "a" and "b" can be either ints
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or nats. That more general theorem is a little harder to prove. *)
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theorem plus_dvd_odd_power:
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"(a::int) + b dvd a ^ (2 * n + 1) + b ^ (2 * n + 1)"
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sorry
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section \<open>Task 3: Proving that the shift-and-add-3 algorithm is correct.\<close>
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subsection \<open>Binary and its conversion to nat\<close>
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type_synonym bit = "bool"
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abbreviation B0 where "B0 == False"
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abbreviation B1 where "B1 == True"
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fun binary_to_nat :: "bit list \<Rightarrow> nat"
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where
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"binary_to_nat [] = 0"
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| "binary_to_nat (b # bs) = (if b then 2 ^ length bs else 0) + binary_to_nat bs"
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lemma (* test case *) "binary_to_nat [B0, B1, B0, B1] = 5" by eval
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lemma (* test case *) "binary_to_nat [B0, B0, B1, B0, B1] = 5" by eval
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lemma (* test case *) "binary_to_nat [B1] = 1" by eval
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lemma (* test case *) "binary_to_nat [B0] = 0" by eval
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subsection \<open>BCD and its conversion to nat\<close>
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type_synonym nibble = "bit * bit * bit * bit"
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fun nibble_to_nat :: "nibble \<Rightarrow> nat"
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where
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"nibble_to_nat (B0,B0,B0,B0) = 0"
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| "nibble_to_nat (B0,B0,B0,B1) = 1"
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| "nibble_to_nat (B0,B0,B1,B0) = 2"
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| "nibble_to_nat (B0,B0,B1,B1) = 3"
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| "nibble_to_nat (B0,B1,B0,B0) = 4"
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| "nibble_to_nat (B0,B1,B0,B1) = 5"
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| "nibble_to_nat (B0,B1,B1,B0) = 6"
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| "nibble_to_nat (B0,B1,B1,B1) = 7"
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| "nibble_to_nat (B1,B0,B0,B0) = 8"
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| "nibble_to_nat (B1,B0,B0,B1) = 9"
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| "nibble_to_nat (B1,B0,B1,B0) = 10"
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| "nibble_to_nat (B1,B0,B1,B1) = 11"
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| "nibble_to_nat (B1,B1,B0,B0) = 12"
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| "nibble_to_nat (B1,B1,B0,B1) = 13"
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| "nibble_to_nat (B1,B1,B1,B0) = 14"
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| "nibble_to_nat (B1,B1,B1,B1) = 15"
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fun bcd_to_nat :: "nibble list \<Rightarrow> nat"
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where
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"bcd_to_nat [] = 0"
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| "bcd_to_nat (n # ns) = bcd_to_nat ns + nibble_to_nat n * 10 ^ length ns"
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lemma (* test case *) "bcd_to_nat [(B0,B1,B1,B0)] = 6" by eval
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lemma (* test case *) "bcd_to_nat [(B0,B1,B1,B0),(B1,B0,B0,B1)] = 69" by eval
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lemma (* test case *) "bcd_to_nat [(B0,B0,B0,B0),(B1,B0,B0,B1)] = 9" by eval
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lemma (* test case *) "bcd_to_nat [(B0,B0,B1,B1),(B0,B0,B0,B0)] = 30" by eval
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subsection \<open>Converting binary to BCD\<close>
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fun binary_to_bcd :: "bit list \<Rightarrow> nibble list"
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where
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"binary_to_bcd bs = []" (* dummy definition *)
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lemma (* test case *)
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"binary_to_bcd [B1,B0,B1,B0,B1,B0,B1] = [(B1,B0,B0,B0), (B0,B1,B0,B1)]"
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sorry
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subsection \<open>Checking that nibbles correspond to valid BCD digits\<close>
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fun valid_nibble :: "nibble \<Rightarrow> bool"
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where
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"valid_nibble (B0,B0,B0,B0) = True"
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| "valid_nibble (B0,B0,B0,B1) = True"
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| "valid_nibble (B0,B0,B1,B0) = True"
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| "valid_nibble (B0,B0,B1,B1) = True"
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| "valid_nibble (B0,B1,B0,B0) = True"
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| "valid_nibble (B0,B1,B0,B1) = True"
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| "valid_nibble (B0,B1,B1,B0) = True"
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| "valid_nibble (B0,B1,B1,B1) = True"
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| "valid_nibble (B1,B0,B0,B0) = True"
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| "valid_nibble (B1,B0,B0,B1) = True"
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| "valid_nibble (B1,B0,B1,B0) = False"
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| "valid_nibble (B1,B0,B1,B1) = False"
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| "valid_nibble (B1,B1,B0,B0) = False"
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| "valid_nibble (B1,B1,B0,B1) = False"
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| "valid_nibble (B1,B1,B1,B0) = False"
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| "valid_nibble (B1,B1,B1,B1) = False"
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theorem binary_to_bcd_valid:
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"list_all valid_nibble (binary_to_bcd bs)"
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sorry
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subsection \<open>Proof that the binary_to_bcd translation is correct.\<close>
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theorem binary_to_bcd_correct:
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"bcd_to_nat (binary_to_bcd bs) = binary_to_nat bs"
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sorry
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end
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BIN
isabelle/2021/isabelle_exercises_2021.pdf
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isabelle/2021/isabelle_exercises_2021.pdf
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