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137 lines
4.1 KiB
Plaintext
137 lines
4.1 KiB
Plaintext
theory HSV_chapter5 imports Main begin
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section \<open>Representing circuits (cf. worksheet Section 5.1)\<close>
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text \<open>Defining a data structure to represent fan-out-free circuits with numbered inputs\<close>
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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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text \<open>A few example circuits\<close>
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definition "circuit1 == AND (INPUT 1) (INPUT 2)"
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definition "circuit2 == OR (NOT circuit1) FALSE"
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definition "circuit3 == NOT (NOT circuit2)"
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definition "circuit4 == AND circuit3 (INPUT 3)"
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section \<open>Simulating circuits (cf. worksheet Section 5.2)\<close>
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text \<open>Simulates a circuit given a valuation for each input wire\<close>
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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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text \<open>A few example valuations\<close>
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definition "\<rho>0 == \<lambda>_. True"
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definition "\<rho>1 == \<rho>0(1 := True, 2 := False, 3 := True)"
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definition "\<rho>2 == \<rho>0(1 := True, 2 := True, 3 := True)"
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text \<open>Trying out the simulator\<close>
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value "simulate circuit1 \<rho>1"
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value "simulate circuit2 \<rho>1"
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value "simulate circuit3 \<rho>1"
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value "simulate circuit4 \<rho>1"
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value "simulate circuit1 \<rho>2"
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value "simulate circuit2 \<rho>2"
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value "simulate circuit3 \<rho>2"
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value "simulate circuit4 \<rho>2"
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section \<open>Structural induction on circuits (cf. worksheet Section 5.3)\<close>
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text \<open>A function that switches each pair of wires entering an OR or AND gate\<close>
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fun mirror where
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"mirror (NOT c) = NOT (mirror c)"
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| "mirror (AND c1 c2) = AND (mirror c2) (mirror c1)"
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| "mirror (OR c1 c2) = OR (mirror c2) (mirror c1)"
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| "mirror TRUE = TRUE"
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| "mirror FALSE = FALSE"
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| "mirror (INPUT i) = INPUT i"
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value "circuit1"
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value "mirror circuit1"
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value "circuit2"
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value "mirror circuit2"
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text \<open>The following non-theorem is easily contradicted.\<close>
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theorem "mirror c = c"
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oops
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text \<open>Proving that mirroring doesn't affect simulation behaviour.\<close>
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theorem mirror_is_sound: "simulate (mirror c) \<rho> = simulate c \<rho>"
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by (induct c, auto)
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section \<open>A simple circuit optimiser (cf. worksheet Section 5.4)\<close>
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text \<open>A function that optimises a circuit by removing pairs of consecutive NOT gates\<close>
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fun opt_NOT where
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"opt_NOT (NOT (NOT c)) = opt_NOT c"
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| "opt_NOT (NOT c) = NOT (opt_NOT c)"
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| "opt_NOT (AND c1 c2) = AND (opt_NOT c1) (opt_NOT c2)"
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| "opt_NOT (OR c1 c2) = OR (opt_NOT c1) (opt_NOT c2)"
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| "opt_NOT TRUE = TRUE"
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| "opt_NOT FALSE = FALSE"
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| "opt_NOT (INPUT i) = INPUT i"
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text \<open>Trying out the optimiser\<close>
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value "circuit1"
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value "opt_NOT circuit1"
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value "circuit2"
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value "opt_NOT circuit2"
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value "circuit3"
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value "opt_NOT circuit3"
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value "circuit4"
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value "opt_NOT circuit4"
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section \<open>Rule induction (cf. worksheet Section 5.5)\<close>
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text \<open>A Fibonacci function that demonstrates complex recursion schemes\<close>
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fun f :: "nat \<Rightarrow> nat" where
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"f (Suc (Suc n)) = f n + f (Suc n)"
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| "f (Suc 0) = 1"
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| "f 0 = 1"
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thm f.induct (* rule induction theorem for f *)
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text \<open>We need to prove a stronger version of the theorem below
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first, in order to make the inductive step work. Just like how
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it often goes with loop invariants in Dafny!\<close>
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lemma helper: "f n \<ge> n \<and> f n \<ge> 1"
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by (rule f.induct[of "\<lambda>n. f n \<ge> n \<and> f n \<ge> 1"], auto)
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text \<open>The nth Fibonacci number is greater than or equal to n\<close>
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theorem "f n \<ge> n"
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using helper by simp
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section \<open>Verifying our optimiser (cf. worksheet Section 5.6)\<close>
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text \<open>The following non-theorem is easily contradicted.\<close>
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theorem "opt_NOT c = c"
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oops
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text \<open>The following theorem says that the optimiser is sound.\<close>
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theorem opt_NOT_is_sound: "simulate (opt_NOT c) \<rho> = simulate c \<rho>"
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by (induct rule:opt_NOT.induct, auto)
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end
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