author | kleing |
Sat, 01 Mar 2003 16:45:51 +0100 | |
changeset 13840 | 399c8103a98f |
parent 12450 | 1162b280700a |
child 16417 | 9bc16273c2d4 |
permissions | -rw-r--r-- |
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(* Title: HOL/ex/Intuitionistic.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1991 University of Cambridge |
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Higher-Order Logic: Intuitionistic predicate calculus problems |
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Taken from FOL/ex/int.ML |
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*) |
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theory Intuitionistic = Main: |
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(*Metatheorem (for PROPOSITIONAL formulae...): |
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P is classically provable iff ~~P is intuitionistically provable. |
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Therefore ~P is classically provable iff it is intuitionistically provable. |
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Proof: Let Q be the conjuction of the propositions A|~A, one for each atom A |
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in P. Now ~~Q is intuitionistically provable because ~~(A|~A) is and because |
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~~ distributes over &. If P is provable classically, then clearly Q-->P is |
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provable intuitionistically, so ~~(Q-->P) is also provable intuitionistically. |
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The latter is intuitionistically equivalent to ~~Q-->~~P, hence to ~~P, since |
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~~Q is intuitionistically provable. Finally, if P is a negation then ~~P is |
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intuitionstically equivalent to P. [Andy Pitts] *) |
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lemma "(~~(P&Q)) = ((~~P) & (~~Q))" |
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by rules |
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lemma "~~ ((~P --> Q) --> (~P --> ~Q) --> P)" |
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by rules |
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(* ~~ does NOT distribute over | *) |
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lemma "(~~(P-->Q)) = (~~P --> ~~Q)" |
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by rules |
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lemma "(~~~P) = (~P)" |
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by rules |
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lemma "~~((P --> Q | R) --> (P-->Q) | (P-->R))" |
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by rules |
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lemma "(P=Q) = (Q=P)" |
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by rules |
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lemma "((P --> (Q | (Q-->R))) --> R) --> R" |
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by rules |
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lemma "(((G-->A) --> J) --> D --> E) --> (((H-->B)-->I)-->C-->J) |
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--> (A-->H) --> F --> G --> (((C-->B)-->I)-->D)-->(A-->C) |
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--> (((F-->A)-->B) --> I) --> E" |
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by rules |
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(* Lemmas for the propositional double-negation translation *) |
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lemma "P --> ~~P" |
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by rules |
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lemma "~~(~~P --> P)" |
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by rules |
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lemma "~~P & ~~(P --> Q) --> ~~Q" |
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by rules |
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(* de Bruijn formulae *) |
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(*de Bruijn formula with three predicates*) |
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lemma "((P=Q) --> P&Q&R) & |
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((Q=R) --> P&Q&R) & |
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((R=P) --> P&Q&R) --> P&Q&R" |
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by rules |
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(*de Bruijn formula with five predicates*) |
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lemma "((P=Q) --> P&Q&R&S&T) & |
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((Q=R) --> P&Q&R&S&T) & |
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((R=S) --> P&Q&R&S&T) & |
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((S=T) --> P&Q&R&S&T) & |
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((T=P) --> P&Q&R&S&T) --> P&Q&R&S&T" |
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by rules |
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(*** Problems from Sahlin, Franzen and Haridi, |
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An Intuitionistic Predicate Logic Theorem Prover. |
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J. Logic and Comp. 2 (5), October 1992, 619-656. |
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***) |
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(*Problem 1.1*) |
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lemma "(ALL x. EX y. ALL z. p(x) & q(y) & r(z)) = |
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(ALL z. EX y. ALL x. p(x) & q(y) & r(z))" |
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by (rules del: allE elim 2: allE') |
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(*Problem 3.1*) |
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lemma "~ (EX x. ALL y. p y x = (~ p x x))" |
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by rules |
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(* Intuitionistic FOL: propositional problems based on Pelletier. *) |
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(* Problem ~~1 *) |
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lemma "~~((P-->Q) = (~Q --> ~P))" |
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by rules |
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(* Problem ~~2 *) |
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lemma "~~(~~P = P)" |
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by rules |
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(* Problem 3 *) |
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lemma "~(P-->Q) --> (Q-->P)" |
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by rules |
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(* Problem ~~4 *) |
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lemma "~~((~P-->Q) = (~Q --> P))" |
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by rules |
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(* Problem ~~5 *) |
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lemma "~~((P|Q-->P|R) --> P|(Q-->R))" |
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by rules |
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(* Problem ~~6 *) |
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lemma "~~(P | ~P)" |
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by rules |
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(* Problem ~~7 *) |
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lemma "~~(P | ~~~P)" |
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by rules |
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(* Problem ~~8. Peirce's law *) |
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lemma "~~(((P-->Q) --> P) --> P)" |
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by rules |
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(* Problem 9 *) |
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lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)" |
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by rules |
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(* Problem 10 *) |
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lemma "(Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P=Q)" |
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by rules |
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(* 11. Proved in each direction (incorrectly, says Pelletier!!) *) |
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lemma "P=P" |
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143 |
by rules |
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144 |
|
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145 |
(* Problem ~~12. Dijkstra's law *) |
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146 |
lemma "~~(((P = Q) = R) = (P = (Q = R)))" |
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147 |
by rules |
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148 |
|
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149 |
lemma "((P = Q) = R) --> ~~(P = (Q = R))" |
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150 |
by rules |
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151 |
|
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152 |
(* Problem 13. Distributive law *) |
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153 |
lemma "(P | (Q & R)) = ((P | Q) & (P | R))" |
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154 |
by rules |
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155 |
|
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156 |
(* Problem ~~14 *) |
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157 |
lemma "~~((P = Q) = ((Q | ~P) & (~Q|P)))" |
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158 |
by rules |
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159 |
|
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160 |
(* Problem ~~15 *) |
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161 |
lemma "~~((P --> Q) = (~P | Q))" |
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162 |
by rules |
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163 |
|
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164 |
(* Problem ~~16 *) |
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165 |
lemma "~~((P-->Q) | (Q-->P))" |
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166 |
by rules |
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|
167 |
|
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168 |
(* Problem ~~17 *) |
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169 |
lemma "~~(((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S)))" |
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170 |
oops |
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171 |
|
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172 |
(*Dijkstra's "Golden Rule"*) |
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173 |
lemma "(P&Q) = (P = (Q = (P|Q)))" |
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174 |
by rules |
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175 |
|
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176 |
|
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177 |
(****Examples with quantifiers****) |
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178 |
|
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179 |
(* The converse is classical in the following implications... *) |
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180 |
|
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181 |
lemma "(EX x. P(x)-->Q) --> (ALL x. P(x)) --> Q" |
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182 |
by rules |
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|
183 |
|
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184 |
lemma "((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)" |
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185 |
by rules |
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186 |
|
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187 |
lemma "((ALL x. ~P(x))-->Q) --> ~ (ALL x. ~ (P(x)|Q))" |
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188 |
by rules |
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|
189 |
|
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190 |
lemma "(ALL x. P(x)) | Q --> (ALL x. P(x) | Q)" |
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191 |
by rules |
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|
192 |
|
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193 |
lemma "(EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))" |
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194 |
by rules |
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|
195 |
|
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|
196 |
|
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197 |
(* Hard examples with quantifiers *) |
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|
198 |
|
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199 |
(*The ones that have not been proved are not known to be valid! |
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200 |
Some will require quantifier duplication -- not currently available*) |
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|
201 |
|
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202 |
(* Problem ~~19 *) |
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203 |
lemma "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" |
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204 |
by rules |
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|
205 |
|
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206 |
(* Problem 20 *) |
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207 |
lemma "(ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w))) |
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208 |
--> (EX x y. P(x) & Q(y)) --> (EX z. R(z))" |
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209 |
by rules |
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|
210 |
|
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211 |
(* Problem 21 *) |
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212 |
lemma "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P=Q(x))" |
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213 |
by rules |
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|
214 |
|
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215 |
(* Problem 22 *) |
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216 |
lemma "(ALL x. P = Q(x)) --> (P = (ALL x. Q(x)))" |
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217 |
by rules |
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|
218 |
|
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219 |
(* Problem ~~23 *) |
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220 |
lemma "~~ ((ALL x. P | Q(x)) = (P | (ALL x. Q(x))))" |
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221 |
by rules |
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|
222 |
|
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223 |
(* Problem 25 *) |
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224 |
lemma "(EX x. P(x)) & |
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225 |
(ALL x. L(x) --> ~ (M(x) & R(x))) & |
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226 |
(ALL x. P(x) --> (M(x) & L(x))) & |
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227 |
((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x))) |
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228 |
--> (EX x. Q(x)&P(x))" |
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229 |
by rules |
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|
230 |
|
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|
231 |
(* Problem 27 *) |
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232 |
lemma "(EX x. P(x) & ~Q(x)) & |
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233 |
(ALL x. P(x) --> R(x)) & |
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234 |
(ALL x. M(x) & L(x) --> P(x)) & |
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235 |
((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x))) |
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|
236 |
--> (ALL x. M(x) --> ~L(x))" |
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|
237 |
by rules |
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|
238 |
|
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|
239 |
(* Problem ~~28. AMENDED *) |
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|
240 |
lemma "(ALL x. P(x) --> (ALL x. Q(x))) & |
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241 |
(~~(ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) & |
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242 |
(~~(EX x. S(x)) --> (ALL x. L(x) --> M(x))) |
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243 |
--> (ALL x. P(x) & L(x) --> M(x))" |
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244 |
by rules |
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|
245 |
|
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246 |
(* Problem 29. Essentially the same as Principia Mathematica *11.71 *) |
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247 |
lemma "(((EX x. P(x)) & (EX y. Q(y))) --> |
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248 |
(((ALL x. (P(x) --> R(x))) & (ALL y. (Q(y) --> S(y)))) = |
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249 |
(ALL x y. ((P(x) & Q(y)) --> (R(x) & S(y))))))" |
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250 |
by rules |
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|
251 |
|
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|
252 |
(* Problem ~~30 *) |
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253 |
lemma "(ALL x. (P(x) | Q(x)) --> ~ R(x)) & |
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254 |
(ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x)) |
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255 |
--> (ALL x. ~~S(x))" |
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256 |
by rules |
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|
257 |
|
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|
258 |
(* Problem 31 *) |
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259 |
lemma "~(EX x. P(x) & (Q(x) | R(x))) & |
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260 |
(EX x. L(x) & P(x)) & |
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|
261 |
(ALL x. ~ R(x) --> M(x)) |
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|
262 |
--> (EX x. L(x) & M(x))" |
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|
263 |
by rules |
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|
264 |
|
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|
265 |
(* Problem 32 *) |
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266 |
lemma "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) & |
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267 |
(ALL x. S(x) & R(x) --> L(x)) & |
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|
268 |
(ALL x. M(x) --> R(x)) |
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269 |
--> (ALL x. P(x) & M(x) --> L(x))" |
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|
270 |
by rules |
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|
271 |
|
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|
272 |
(* Problem ~~33 *) |
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|
273 |
lemma "(ALL x. ~~(P(a) & (P(x)-->P(b))-->P(c))) = |
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274 |
(ALL x. ~~((~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c))))" |
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275 |
oops |
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|
276 |
|
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277 |
(* Problem 36 *) |
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|
278 |
lemma |
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|
279 |
"(ALL x. EX y. J x y) & |
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|
280 |
(ALL x. EX y. G x y) & |
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281 |
(ALL x y. J x y | G x y --> (ALL z. J y z | G y z --> H x z)) |
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--> (ALL x. EX y. H x y)" |
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283 |
by rules |
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284 |
|
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285 |
(* Problem 39 *) |
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lemma "~ (EX x. ALL y. F y x = (~F y y))" |
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287 |
by rules |
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288 |
|
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289 |
(* Problem 40. AMENDED *) |
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290 |
lemma "(EX y. ALL x. F x y = F x x) --> |
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291 |
~(ALL x. EX y. ALL z. F z y = (~ F z x))" |
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292 |
by rules |
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293 |
|
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294 |
(* Problem 44 *) |
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295 |
lemma "(ALL x. f(x) --> |
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296 |
(EX y. g(y) & h x y & (EX y. g(y) & ~ h x y))) & |
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297 |
(EX x. j(x) & (ALL y. g(y) --> h x y)) |
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298 |
--> (EX x. j(x) & ~f(x))" |
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299 |
by rules |
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300 |
|
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301 |
(* Problem 48 *) |
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302 |
lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c" |
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303 |
by rules |
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304 |
|
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305 |
(* Problem 51 *) |
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306 |
lemma "((EX z w. (ALL x y. (P x y = ((x = z) & (y = w))))) --> |
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307 |
(EX z. (ALL x. (EX w. ((ALL y. (P x y = (y = w))) = (x = z))))))" |
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308 |
by rules |
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309 |
|
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310 |
(* Problem 52 *) |
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311 |
(*Almost the same as 51. *) |
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312 |
lemma "((EX z w. (ALL x y. (P x y = ((x = z) & (y = w))))) --> |
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313 |
(EX w. (ALL y. (EX z. ((ALL x. (P x y = (x = z))) = (y = w))))))" |
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314 |
by rules |
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315 |
|
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316 |
(* Problem 56 *) |
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317 |
lemma "(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) = (ALL x. P(x) --> P(f(x)))" |
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318 |
by rules |
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319 |
|
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320 |
(* Problem 57 *) |
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321 |
lemma "P (f a b) (f b c) & P (f b c) (f a c) & |
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322 |
(ALL x y z. P x y & P y z --> P x z) --> P (f a b) (f a c)" |
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323 |
by rules |
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|
324 |
|
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325 |
(* Problem 60 *) |
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326 |
lemma "ALL x. P x (f x) = (EX y. (ALL z. P z y --> P z (f x)) & P x y)" |
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327 |
by rules |
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328 |
|
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329 |
end |