author | huffman |
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(* Title: HOL/ex/ReflectionEx.thy |
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Author: Amine Chaieb, TU Muenchen |
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*) |
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header {* Examples for generic reflection and reification *} |
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theory ReflectionEx |
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imports "~~/src/HOL/Library/Reflection" |
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begin |
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text{* This theory presents two methods: reify and reflection *} |
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text{* |
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Consider an HOL type 'a, the structure of which is not recongnisable on the theory level. This is the case of bool, arithmetical terms such as int, real etc \dots |
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In order to implement a simplification on terms of type 'a we often need its structure. |
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Traditionnaly such simplifications are written in ML, proofs are synthesized. |
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An other strategy is to declare an HOL-datatype tau and an HOL function (the interpretation) that maps elements of tau to elements of 'a. The functionality of @{text reify} is to compute a term s::tau, which is the representant of t. For this it needs equations for the interpretation. |
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NB: All the interpretations supported by @{text reify} must have the type @{text "'b list \<Rightarrow> tau \<Rightarrow> 'a"}. |
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The method @{text reify} can also be told which subterm of the current subgoal should be reified. The general call for @{text reify} is: @{text "reify eqs (t)"}, where @{text eqs} are the defining equations of the interpretation and @{text "(t)"} is an optional parameter which specifies the subterm to which reification should be applied to. If @{text "(t)"} is abscent, @{text reify} tries to reify the whole subgoal. |
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The method reflection uses @{text reify} and has a very similar signature: @{text "reflection corr_thm eqs (t)"}. Here again @{text eqs} and @{text "(t)"} are as described above and @{text corr_thm} is a thorem proving @{term "I vs (f t) = I vs t"}. We assume that @{text I} is the interpretation and @{text f} is some useful and executable simplification of type @{text "tau \<Rightarrow> tau"}. The method @{text reflection} applies reification and hence the theorem @{term "t = I xs s"} and hence using @{text corr_thm} derives @{term "t = I xs (f s)"}. It then uses normalization by evaluation to prove @{term "f s = s'"} which almost finishes the proof of @{term "t = t'"} where @{term "I xs s' = t'"}. |
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*} |
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text{* Example 1 : Propositional formulae and NNF.*} |
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text{* The type @{text fm} represents simple propositional formulae: *} |
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datatype form = TrueF | FalseF | Less nat nat | |
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And form form | Or form form | Neg form | ExQ form |
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fun interp :: "form \<Rightarrow> ('a::ord) list \<Rightarrow> bool" where |
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"interp TrueF e = True" | |
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"interp FalseF e = False" | |
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"interp (Less i j) e = (e!i < e!j)" | |
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"interp (And f1 f2) e = (interp f1 e & interp f2 e)" | |
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"interp (Or f1 f2) e = (interp f1 e | interp f2 e)" | |
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"interp (Neg f) e = (~ interp f e)" | |
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"interp (ExQ f) e = (EX x. interp f (x#e))" |
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lemmas interp_reify_eqs = interp.simps |
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declare interp_reify_eqs[reify] |
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lemma "EX x. x < y & x < z" |
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apply (reify ) |
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oops |
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datatype fm = And fm fm | Or fm fm | Imp fm fm | Iff fm fm | NOT fm | At nat |
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primrec Ifm :: "fm \<Rightarrow> bool list \<Rightarrow> bool" where |
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"Ifm (At n) vs = vs!n" |
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| "Ifm (And p q) vs = (Ifm p vs \<and> Ifm q vs)" |
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| "Ifm (Or p q) vs = (Ifm p vs \<or> Ifm q vs)" |
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| "Ifm (Imp p q) vs = (Ifm p vs \<longrightarrow> Ifm q vs)" |
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| "Ifm (Iff p q) vs = (Ifm p vs = Ifm q vs)" |
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| "Ifm (NOT p) vs = (\<not> (Ifm p vs))" |
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lemma "Q \<longrightarrow> (D & F & ((~ D) & (~ F)))" |
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apply (reify Ifm.simps) |
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oops |
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text{* Method @{text reify} maps a bool to an fm. For this it needs the |
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semantics of fm, i.e.\ the rewrite rules in @{text Ifm.simps}. *} |
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(* You can also just pick up a subterm to reify \<dots> *) |
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lemma "Q \<longrightarrow> (D & F & ((~ D) & (~ F)))" |
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apply (reify Ifm.simps ("((~ D) & (~ F))")) |
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oops |
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text{* Let's perform NNF. This is a version that tends to generate disjunctions *} |
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primrec fmsize :: "fm \<Rightarrow> nat" where |
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"fmsize (At n) = 1" |
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| "fmsize (NOT p) = 1 + fmsize p" |
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| "fmsize (And p q) = 1 + fmsize p + fmsize q" |
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| "fmsize (Or p q) = 1 + fmsize p + fmsize q" |
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| "fmsize (Imp p q) = 2 + fmsize p + fmsize q" |
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| "fmsize (Iff p q) = 2 + 2* fmsize p + 2* fmsize q" |
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lemma [measure_function]: "is_measure fmsize" .. |
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fun nnf :: "fm \<Rightarrow> fm" |
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where |
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"nnf (At n) = At n" |
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| "nnf (And p q) = And (nnf p) (nnf q)" |
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| "nnf (Or p q) = Or (nnf p) (nnf q)" |
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| "nnf (Imp p q) = Or (nnf (NOT p)) (nnf q)" |
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| "nnf (Iff p q) = Or (And (nnf p) (nnf q)) (And (nnf (NOT p)) (nnf (NOT q)))" |
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| "nnf (NOT (And p q)) = Or (nnf (NOT p)) (nnf (NOT q))" |
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| "nnf (NOT (Or p q)) = And (nnf (NOT p)) (nnf (NOT q))" |
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| "nnf (NOT (Imp p q)) = And (nnf p) (nnf (NOT q))" |
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| "nnf (NOT (Iff p q)) = Or (And (nnf p) (nnf (NOT q))) (And (nnf (NOT p)) (nnf q))" |
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| "nnf (NOT (NOT p)) = nnf p" |
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| "nnf (NOT p) = NOT p" |
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text{* The correctness theorem of nnf: it preserves the semantics of fm *} |
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lemma nnf[reflection]: "Ifm (nnf p) vs = Ifm p vs" |
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by (induct p rule: nnf.induct) auto |
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text{* Now let's perform NNF using our @{term nnf} function defined above. First to the whole subgoal. *} |
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lemma "(\<not> (A = B)) \<and> (B \<longrightarrow> (A \<noteq> (B | C \<and> (B \<longrightarrow> A | D)))) \<longrightarrow> A \<or> B \<and> D" |
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apply (reflection Ifm.simps) |
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oops |
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text{* Now we specify on which subterm it should be applied*} |
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lemma "(\<not> (A = B)) \<and> (B \<longrightarrow> (A \<noteq> (B | C \<and> (B \<longrightarrow> A | D)))) \<longrightarrow> A \<or> B \<and> D" |
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apply (reflection Ifm.simps only: "(B | C \<and> (B \<longrightarrow> A | D))") |
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oops |
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(* Example 2 : Simple arithmetic formulae *) |
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text{* The type @{text num} reflects linear expressions over natural number *} |
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datatype num = C nat | Add num num | Mul nat num | Var nat | CN nat nat num |
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text{* This is just technical to make recursive definitions easier. *} |
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primrec num_size :: "num \<Rightarrow> nat" |
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where |
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"num_size (C c) = 1" |
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| "num_size (Var n) = 1" |
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| "num_size (Add a b) = 1 + num_size a + num_size b" |
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| "num_size (Mul c a) = 1 + num_size a" |
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| "num_size (CN n c a) = 4 + num_size a " |
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text{* The semantics of num *} |
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primrec Inum:: "num \<Rightarrow> nat list \<Rightarrow> nat" |
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where |
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Inum_C : "Inum (C i) vs = i" |
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| Inum_Var: "Inum (Var n) vs = vs!n" |
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| Inum_Add: "Inum (Add s t) vs = Inum s vs + Inum t vs " |
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| Inum_Mul: "Inum (Mul c t) vs = c * Inum t vs " |
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| Inum_CN : "Inum (CN n c t) vs = c*(vs!n) + Inum t vs " |
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text{* Let's reify some nat expressions \dots *} |
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lemma "4 * (2*x + (y::nat)) + f a \<noteq> 0" |
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apply (reify Inum.simps ("4 * (2*x + (y::nat)) + f a")) |
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oops |
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text{* We're in a bad situation!! x, y and f a have been recongnized as a constants, which is correct but does not correspond to our intuition of the constructor C. It should encapsulate constants, i.e. numbers, i.e. numerals.*} |
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text{* So let's leave the @{text "Inum_C"} equation at the end and see what happens \dots*} |
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lemma "4 * (2*x + (y::nat)) \<noteq> 0" |
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apply (reify Inum_Var Inum_Add Inum_Mul Inum_CN Inum_C ("4 * (2*x + (y::nat))")) |
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oops |
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text{* Hmmm let's specialize @{text Inum_C} with numerals.*} |
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lemma Inum_number: "Inum (C (numeral t)) vs = numeral t" by simp |
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lemmas Inum_eqs = Inum_Var Inum_Add Inum_Mul Inum_CN Inum_number |
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text{* Second attempt *} |
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lemma "1 * (2*x + (y::nat)) \<noteq> 0" |
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apply (reify Inum_eqs ("1 * (2*x + (y::nat))")) |
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oops |
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text{* That was fine, so let's try another one \dots *} |
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lemma "1 * (2* x + (y::nat) + 0 + 1) \<noteq> 0" |
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apply (reify Inum_eqs ("1 * (2*x + (y::nat) + 0 + 1)")) |
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oops |
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text{* Oh!! 0 is not a variable \dots\ Oh! 0 is not a @{text "numeral"} \dots\ thing. The same for 1. So let's add those equations too *} |
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lemma Inum_01: "Inum (C 0) vs = 0" "Inum (C 1) vs = 1" "Inum (C(Suc n)) vs = Suc n" |
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by simp+ |
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lemmas Inum_eqs'= Inum_eqs Inum_01 |
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text{* Third attempt: *} |
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lemma "1 * (2*x + (y::nat) + 0 + 1) \<noteq> 0" |
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apply (reify Inum_eqs' ("1 * (2*x + (y::nat) + 0 + 1)")) |
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oops |
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text{* Okay, let's try reflection. Some simplifications on num follow. You can skim until the main theorem @{text linum} *} |
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fun lin_add :: "num \<Rightarrow> num \<Rightarrow> num" |
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where |
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"lin_add (CN n1 c1 r1) (CN n2 c2 r2) = |
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(if n1=n2 then |
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(let c = c1 + c2 |
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in (if c=0 then lin_add r1 r2 else CN n1 c (lin_add r1 r2))) |
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else if n1 \<le> n2 then (CN n1 c1 (lin_add r1 (CN n2 c2 r2))) |
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else (CN n2 c2 (lin_add (CN n1 c1 r1) r2)))" |
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| "lin_add (CN n1 c1 r1) t = CN n1 c1 (lin_add r1 t)" |
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| "lin_add t (CN n2 c2 r2) = CN n2 c2 (lin_add t r2)" |
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| "lin_add (C b1) (C b2) = C (b1+b2)" |
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| "lin_add a b = Add a b" |
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lemma lin_add: "Inum (lin_add t s) bs = Inum (Add t s) bs" |
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apply (induct t s rule: lin_add.induct, simp_all add: Let_def) |
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apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all) |
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by (case_tac "n1 = n2", simp_all add: algebra_simps) |
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fun lin_mul :: "num \<Rightarrow> nat \<Rightarrow> num" |
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where |
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"lin_mul (C j) i = C (i*j)" |
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| "lin_mul (CN n c a) i = (if i=0 then (C 0) else CN n (i*c) (lin_mul a i))" |
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| "lin_mul t i = (Mul i t)" |
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lemma lin_mul: "Inum (lin_mul t i) bs = Inum (Mul i t) bs" |
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by (induct t i rule: lin_mul.induct, auto simp add: algebra_simps) |
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lemma [measure_function]: "is_measure num_size" .. |
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fun linum:: "num \<Rightarrow> num" |
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where |
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"linum (C b) = C b" |
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| "linum (Var n) = CN n 1 (C 0)" |
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| "linum (Add t s) = lin_add (linum t) (linum s)" |
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| "linum (Mul c t) = lin_mul (linum t) c" |
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| "linum (CN n c t) = lin_add (linum (Mul c (Var n))) (linum t)" |
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lemma linum[reflection] : "Inum (linum t) bs = Inum t bs" |
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by (induct t rule: linum.induct, simp_all add: lin_mul lin_add) |
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text{* Now we can use linum to simplify nat terms using reflection *} |
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lemma "(Suc (Suc 1)) * (x + (Suc 1)*y) = 3*x + 6*y" |
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apply (reflection Inum_eqs' only: "(Suc (Suc 1)) * (x + (Suc 1)*y)") |
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oops |
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text{* Let's lift this to formulae and see what happens *} |
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datatype aform = Lt num num | Eq num num | Ge num num | NEq num num | |
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Conj aform aform | Disj aform aform | NEG aform | T | F |
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primrec linaformsize:: "aform \<Rightarrow> nat" |
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where |
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"linaformsize T = 1" |
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| "linaformsize F = 1" |
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| "linaformsize (Lt a b) = 1" |
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| "linaformsize (Ge a b) = 1" |
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| "linaformsize (Eq a b) = 1" |
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| "linaformsize (NEq a b) = 1" |
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| "linaformsize (NEG p) = 2 + linaformsize p" |
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| "linaformsize (Conj p q) = 1 + linaformsize p + linaformsize q" |
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| "linaformsize (Disj p q) = 1 + linaformsize p + linaformsize q" |
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lemma [measure_function]: "is_measure linaformsize" .. |
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primrec is_aform :: "aform => nat list => bool" |
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where |
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"is_aform T vs = True" |
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| "is_aform F vs = False" |
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| "is_aform (Lt a b) vs = (Inum a vs < Inum b vs)" |
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| "is_aform (Eq a b) vs = (Inum a vs = Inum b vs)" |
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| "is_aform (Ge a b) vs = (Inum a vs \<ge> Inum b vs)" |
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| "is_aform (NEq a b) vs = (Inum a vs \<noteq> Inum b vs)" |
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| "is_aform (NEG p) vs = (\<not> (is_aform p vs))" |
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| "is_aform (Conj p q) vs = (is_aform p vs \<and> is_aform q vs)" |
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| "is_aform (Disj p q) vs = (is_aform p vs \<or> is_aform q vs)" |
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text{* Let's reify and do reflection *} |
20319 | 247 |
lemma "(3::nat)*x + t < 0 \<and> (2 * x + y \<noteq> 17)" |
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248 |
apply (reify Inum_eqs' is_aform.simps) |
20319 | 249 |
oops |
250 |
||
20337 | 251 |
text{* Note that reification handles several interpretations at the same time*} |
20319 | 252 |
lemma "(3::nat)*x + t < 0 & x*x + t*x + 3 + 1 = z*t*4*z | x + x + 1 < 0" |
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apply (reflection Inum_eqs' is_aform.simps only:"x + x + 1") |
20319 | 254 |
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255 |
||
20337 | 256 |
text{* For reflection we now define a simple transformation on aform: NNF + linum on atoms *} |
35419 | 257 |
|
258 |
fun linaform:: "aform \<Rightarrow> aform" |
|
259 |
where |
|
20319 | 260 |
"linaform (Lt s t) = Lt (linum s) (linum t)" |
35419 | 261 |
| "linaform (Eq s t) = Eq (linum s) (linum t)" |
262 |
| "linaform (Ge s t) = Ge (linum s) (linum t)" |
|
263 |
| "linaform (NEq s t) = NEq (linum s) (linum t)" |
|
264 |
| "linaform (Conj p q) = Conj (linaform p) (linaform q)" |
|
265 |
| "linaform (Disj p q) = Disj (linaform p) (linaform q)" |
|
266 |
| "linaform (NEG T) = F" |
|
267 |
| "linaform (NEG F) = T" |
|
268 |
| "linaform (NEG (Lt a b)) = Ge a b" |
|
269 |
| "linaform (NEG (Ge a b)) = Lt a b" |
|
270 |
| "linaform (NEG (Eq a b)) = NEq a b" |
|
271 |
| "linaform (NEG (NEq a b)) = Eq a b" |
|
272 |
| "linaform (NEG (NEG p)) = linaform p" |
|
273 |
| "linaform (NEG (Conj p q)) = Disj (linaform (NEG p)) (linaform (NEG q))" |
|
274 |
| "linaform (NEG (Disj p q)) = Conj (linaform (NEG p)) (linaform (NEG q))" |
|
275 |
| "linaform p = p" |
|
20319 | 276 |
|
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lemma linaform: "is_aform (linaform p) vs = is_aform p vs" |
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by (induct p rule: linaform.induct) (auto simp add: linum) |
20319 | 279 |
|
280 |
lemma "(((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0)))) + (Suc (Suc (Suc 0))) * ((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0))))< 0) \<and> (Suc 0 + Suc 0< 0)" |
|
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apply (reflection Inum_eqs' is_aform.simps rules: linaform) |
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282 |
oops |
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283 |
|
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284 |
declare linaform[reflection] |
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285 |
lemma "(((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0)))) + (Suc (Suc (Suc 0))) * ((Suc(Suc (Suc 0)))*((x::nat) + (Suc (Suc 0))))< 0) \<and> (Suc 0 + Suc 0< 0)" |
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apply (reflection Inum_eqs' is_aform.simps) |
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287 |
oops |
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288 |
|
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289 |
text{* We now give an example where Interpretaions have 0 or more than only one envornement of different types and show that automatic reification also deals with binding *} |
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datatype rb = BC bool| BAnd rb rb | BOr rb rb |
35419 | 291 |
primrec Irb :: "rb \<Rightarrow> bool" |
292 |
where |
|
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293 |
"Irb (BC p) = p" |
35419 | 294 |
| "Irb (BAnd s t) = (Irb s \<and> Irb t)" |
295 |
| "Irb (BOr s t) = (Irb s \<or> Irb t)" |
|
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|
296 |
|
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|
297 |
lemma "A \<and> (B \<or> D \<and> B) \<and> A \<and> (B \<or> D \<and> B) \<or> A \<and> (B \<or> D \<and> B) \<or> A \<and> (B \<or> D \<and> B)" |
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|
298 |
apply (reify Irb.simps) |
20319 | 299 |
oops |
300 |
||
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|
301 |
|
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302 |
datatype rint = IC int| IVar nat | IAdd rint rint | IMult rint rint | INeg rint | ISub rint rint |
35419 | 303 |
primrec Irint :: "rint \<Rightarrow> int list \<Rightarrow> int" |
304 |
where |
|
305 |
Irint_Var: "Irint (IVar n) vs = vs!n" |
|
306 |
| Irint_Neg: "Irint (INeg t) vs = - Irint t vs" |
|
307 |
| Irint_Add: "Irint (IAdd s t) vs = Irint s vs + Irint t vs" |
|
308 |
| Irint_Sub: "Irint (ISub s t) vs = Irint s vs - Irint t vs" |
|
309 |
| Irint_Mult: "Irint (IMult s t) vs = Irint s vs * Irint t vs" |
|
310 |
| Irint_C: "Irint (IC i) vs = i" |
|
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311 |
lemma Irint_C0: "Irint (IC 0) vs = 0" |
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|
312 |
by simp |
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|
313 |
lemma Irint_C1: "Irint (IC 1) vs = 1" |
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|
314 |
by simp |
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|
315 |
lemma Irint_Cnumeral: "Irint (IC (numeral x)) vs = numeral x" |
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|
316 |
by simp |
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317 |
lemmas Irint_simps = Irint_Var Irint_Neg Irint_Add Irint_Sub Irint_Mult Irint_C0 Irint_C1 Irint_Cnumeral |
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|
318 |
lemma "(3::int) * x + y*y - 9 + (- z) = 0" |
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|
319 |
apply (reify Irint_simps ("(3::int) * x + y*y - 9 + (- z)")) |
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|
320 |
oops |
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|
321 |
datatype rlist = LVar nat| LEmpty| LCons rint rlist | LAppend rlist rlist |
35419 | 322 |
primrec Irlist :: "rlist \<Rightarrow> int list \<Rightarrow> (int list) list \<Rightarrow> (int list)" |
323 |
where |
|
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|
324 |
"Irlist (LEmpty) is vs = []" |
35419 | 325 |
| "Irlist (LVar n) is vs = vs!n" |
326 |
| "Irlist (LCons i t) is vs = ((Irint i is)#(Irlist t is vs))" |
|
327 |
| "Irlist (LAppend s t) is vs = (Irlist s is vs) @ (Irlist t is vs)" |
|
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|
328 |
lemma "[(1::int)] = []" |
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|
329 |
apply (reify Irlist.simps Irint_simps ("[1]:: int list")) |
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|
330 |
oops |
20374 | 331 |
|
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|
332 |
lemma "([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs = [y*y - z - 9 + (3::int) * x]" |
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|
333 |
apply (reify Irlist.simps Irint_simps ("([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs")) |
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|
334 |
oops |
20374 | 335 |
|
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|
336 |
datatype rnat = NC nat| NVar nat| NSuc rnat | NAdd rnat rnat | NMult rnat rnat | NNeg rnat | NSub rnat rnat | Nlgth rlist |
35419 | 337 |
primrec Irnat :: "rnat \<Rightarrow> int list \<Rightarrow> (int list) list \<Rightarrow> nat list \<Rightarrow> nat" |
338 |
where |
|
339 |
Irnat_Suc: "Irnat (NSuc t) is ls vs = Suc (Irnat t is ls vs)" |
|
340 |
| Irnat_Var: "Irnat (NVar n) is ls vs = vs!n" |
|
341 |
| Irnat_Neg: "Irnat (NNeg t) is ls vs = 0" |
|
342 |
| Irnat_Add: "Irnat (NAdd s t) is ls vs = Irnat s is ls vs + Irnat t is ls vs" |
|
343 |
| Irnat_Sub: "Irnat (NSub s t) is ls vs = Irnat s is ls vs - Irnat t is ls vs" |
|
344 |
| Irnat_Mult: "Irnat (NMult s t) is ls vs = Irnat s is ls vs * Irnat t is ls vs" |
|
345 |
| Irnat_lgth: "Irnat (Nlgth rxs) is ls vs = length (Irlist rxs is ls)" |
|
346 |
| Irnat_C: "Irnat (NC i) is ls vs = i" |
|
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|
347 |
lemma Irnat_C0: "Irnat (NC 0) is ls vs = 0" |
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|
348 |
by simp |
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|
349 |
lemma Irnat_C1: "Irnat (NC 1) is ls vs = 1" |
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|
350 |
by simp |
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|
351 |
lemma Irnat_Cnumeral: "Irnat (NC (numeral x)) is ls vs = numeral x" |
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|
352 |
by simp |
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Reifiaction now deals with Interpretations with an arbtrary number of parameters. It deals with binding. The Atomic cases can be I ... = f (xs!n)
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|
353 |
lemmas Irnat_simps = Irnat_Suc Irnat_Var Irnat_Neg Irnat_Add Irnat_Sub Irnat_Mult Irnat_lgth |
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|
354 |
Irnat_C0 Irnat_C1 Irnat_Cnumeral |
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|
355 |
lemma "(Suc n) * length (([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs) = length xs" |
6857bd9f1a79
Reifiaction now deals with Interpretations with an arbtrary number of parameters. It deals with binding. The Atomic cases can be I ... = f (xs!n)
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changeset
|
356 |
apply (reify Irnat_simps Irlist.simps Irint_simps ("(Suc n) *length (([(3::int) * x + y*y - 9 + (- z)] @ []) @ xs)")) |
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|
357 |
oops |
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|
358 |
datatype rifm = RT | RF | RVar nat |
6857bd9f1a79
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|
359 |
| RNLT rnat rnat | RNILT rnat rint | RNEQ rnat rnat |
6857bd9f1a79
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|
360 |
|RAnd rifm rifm | ROr rifm rifm | RImp rifm rifm| RIff rifm rifm |
6857bd9f1a79
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|
361 |
| RNEX rifm | RIEX rifm| RLEX rifm | RNALL rifm | RIALL rifm| RLALL rifm |
6857bd9f1a79
Reifiaction now deals with Interpretations with an arbtrary number of parameters. It deals with binding. The Atomic cases can be I ... = f (xs!n)
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|
362 |
| RBEX rifm | RBALL rifm |
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|
363 |
|
35419 | 364 |
primrec Irifm :: "rifm \<Rightarrow> bool list \<Rightarrow> int list \<Rightarrow> (int list) list \<Rightarrow> nat list \<Rightarrow> bool" |
365 |
where |
|
366 |
"Irifm RT ps is ls ns = True" |
|
367 |
| "Irifm RF ps is ls ns = False" |
|
368 |
| "Irifm (RVar n) ps is ls ns = ps!n" |
|
369 |
| "Irifm (RNLT s t) ps is ls ns = (Irnat s is ls ns < Irnat t is ls ns)" |
|
370 |
| "Irifm (RNILT s t) ps is ls ns = (int (Irnat s is ls ns) < Irint t is)" |
|
371 |
| "Irifm (RNEQ s t) ps is ls ns = (Irnat s is ls ns = Irnat t is ls ns)" |
|
372 |
| "Irifm (RAnd p q) ps is ls ns = (Irifm p ps is ls ns \<and> Irifm q ps is ls ns)" |
|
373 |
| "Irifm (ROr p q) ps is ls ns = (Irifm p ps is ls ns \<or> Irifm q ps is ls ns)" |
|
374 |
| "Irifm (RImp p q) ps is ls ns = (Irifm p ps is ls ns \<longrightarrow> Irifm q ps is ls ns)" |
|
375 |
| "Irifm (RIff p q) ps is ls ns = (Irifm p ps is ls ns = Irifm q ps is ls ns)" |
|
376 |
| "Irifm (RNEX p) ps is ls ns = (\<exists>x. Irifm p ps is ls (x#ns))" |
|
377 |
| "Irifm (RIEX p) ps is ls ns = (\<exists>x. Irifm p ps (x#is) ls ns)" |
|
378 |
| "Irifm (RLEX p) ps is ls ns = (\<exists>x. Irifm p ps is (x#ls) ns)" |
|
379 |
| "Irifm (RBEX p) ps is ls ns = (\<exists>x. Irifm p (x#ps) is ls ns)" |
|
380 |
| "Irifm (RNALL p) ps is ls ns = (\<forall>x. Irifm p ps is ls (x#ns))" |
|
381 |
| "Irifm (RIALL p) ps is ls ns = (\<forall>x. Irifm p ps (x#is) ls ns)" |
|
382 |
| "Irifm (RLALL p) ps is ls ns = (\<forall>x. Irifm p ps is (x#ls) ns)" |
|
383 |
| "Irifm (RBALL p) ps is ls ns = (\<forall>x. Irifm p (x#ps) is ls ns)" |
|
20374 | 384 |
|
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changeset
|
385 |
lemma " \<forall>x. \<exists>n. ((Suc n) * length (([(3::int) * x + (f t)*y - 9 + (- z)] @ []) @ xs) = length xs) \<and> m < 5*n - length (xs @ [2,3,4,x*z + 8 - y]) \<longrightarrow> (\<exists>p. \<forall>q. p \<and> q \<longrightarrow> r)" |
6857bd9f1a79
Reifiaction now deals with Interpretations with an arbtrary number of parameters. It deals with binding. The Atomic cases can be I ... = f (xs!n)
chaieb
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changeset
|
386 |
apply (reify Irifm.simps Irnat_simps Irlist.simps Irint_simps) |
20374 | 387 |
oops |
388 |
||
22199
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
389 |
|
32960
69916a850301
eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents:
31021
diff
changeset
|
390 |
(* An example for equations containing type variables *) |
22199
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Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
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changeset
|
391 |
datatype prod = Zero | One | Var nat | Mul prod prod |
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Now deals with simples cases where the input equations contain type variables
chaieb
parents:
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changeset
|
392 |
| Pw prod nat | PNM nat nat prod |
35419 | 393 |
primrec Iprod :: " prod \<Rightarrow> ('a::{linordered_idom}) list \<Rightarrow>'a" |
394 |
where |
|
23624
82091387f6d7
The order for parameter for interpretation is now inversted:
chaieb
parents:
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diff
changeset
|
395 |
"Iprod Zero vs = 0" |
35419 | 396 |
| "Iprod One vs = 1" |
397 |
| "Iprod (Var n) vs = vs!n" |
|
398 |
| "Iprod (Mul a b) vs = (Iprod a vs * Iprod b vs)" |
|
399 |
| "Iprod (Pw a n) vs = ((Iprod a vs) ^ n)" |
|
400 |
| "Iprod (PNM n k t) vs = (vs ! n)^k * Iprod t vs" |
|
39246 | 401 |
|
22199
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Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
402 |
datatype sgn = Pos prod | Neg prod | ZeroEq prod | NZeroEq prod | Tr | F |
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
403 |
| Or sgn sgn | And sgn sgn |
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
404 |
|
35419 | 405 |
primrec Isgn :: " sgn \<Rightarrow> ('a::{linordered_idom}) list \<Rightarrow>bool" |
406 |
where |
|
23624
82091387f6d7
The order for parameter for interpretation is now inversted:
chaieb
parents:
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diff
changeset
|
407 |
"Isgn Tr vs = True" |
35419 | 408 |
| "Isgn F vs = False" |
409 |
| "Isgn (ZeroEq t) vs = (Iprod t vs = 0)" |
|
410 |
| "Isgn (NZeroEq t) vs = (Iprod t vs \<noteq> 0)" |
|
411 |
| "Isgn (Pos t) vs = (Iprod t vs > 0)" |
|
412 |
| "Isgn (Neg t) vs = (Iprod t vs < 0)" |
|
413 |
| "Isgn (And p q) vs = (Isgn p vs \<and> Isgn q vs)" |
|
414 |
| "Isgn (Or p q) vs = (Isgn p vs \<or> Isgn q vs)" |
|
22199
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
415 |
|
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
416 |
lemmas eqs = Isgn.simps Iprod.simps |
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
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diff
changeset
|
417 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
32960
diff
changeset
|
418 |
lemma "(x::'a::{linordered_idom})^4 * y * z * y^2 * z^23 > 0" |
22199
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
419 |
apply (reify eqs) |
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
420 |
oops |
b617ddd200eb
Now deals with simples cases where the input equations contain type variables
chaieb
parents:
20564
diff
changeset
|
421 |
|
20319 | 422 |
end |