Interactive Formal Verification Review (17) 1-7.pdfInteractive Formal Verification Review (17) Tjark...
53
Interactive Formal Verification Review (1-7) Tjark Weber Computer Laboratory University of Cambridge
-
Upload
trinhduong -
Category
Documents
-
view
217 -
download
0
Transcript of Interactive Formal Verification Review (17) 1-7.pdfInteractive Formal Verification Review (17) Tjark...
Interactive Formal VerificationReview (17)
Tjark WeberComputer Laboratory
University of Cambridge
Isabelle Theories
theory T imports Main A B
begin
end
Isabelle Theories
theory T imports Main A B
begin
end
Name of the theory
Isabelle Theories
theory T imports Main A B
begin
end
Names of existing theories
Name of the theory
Isabelle Theories
theory T imports Main A B
begin
end
Names of existing theoriesMain: contains all of Isabelle/HOL
Name of the theory
Defining Types
● typedecl ('a,'b) t
Defining Types
● typedecl ('a,'b) tIntroduces an unspecified type
Defining Types
● typedecl ('a,'b) tIntroduces an unspecified type
Optional: type arguments
Defining Types
● typedecl ('a,'b) t
● type_synonym 'a multiset = "'a => nat"
Introduces an unspecified type
Optional: type arguments
Defining Types
● typedecl ('a,'b) t
● type_synonym 'a multiset = "'a => nat"
Introduces a new name for an existing type
Optional: type arguments
Introduces an unspecified type
Defining Types
● typedecl ('a,'b) t
● type_synonym 'a multiset = "'a => nat"
● datatype 'a list = Nil | Cons 'a "'a list"
Optional: type arguments
Introduces an unspecified type
Introduces a new name for an existing type
Defining Types
● typedecl ('a,'b) t
● type_synonym 'a multiset = "'a => nat"
● datatype 'a list = Nil | Cons 'a "'a list"
Optional: type arguments
Defines an inductive datatype
Introduces an unspecified type
Introduces a new name for an existing type
Defining Types
● typedecl ('a,'b) t
● type_synonym 'a multiset = "'a => nat"
● datatype 'a list = Nil | Cons 'a "'a list"
Optional: type arguments
Defines an inductive datatype
Constructor names and argument types
Introduces an unspecified type
Introduces a new name for an existing type
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
For non-recursive definitions
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
For non-recursive definitions Optional: the constant's type
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
For non-recursive definitions Optional: the constant's type
Provides a lemma: even_def
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
● fun even' where "even' 0 = True"| "even' (Suc 0) = False"| "even' n = even' (n-2)"
For non-recursive definitions Optional: the constant's type
Provides a lemma: even_def
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
● fun even' where "even' 0 = True"| "even' (Suc 0) = False"| "even' n = even' (n-2)"
For non-recursive definitions Optional: the constant's type
Provides a lemma: even_defFor recursive functions
Defining Constants
● definition even :: "nat => bool" where "even n = (∃k. n = 2*k)"
● fun even' where "even' 0 = True"| "even' (Suc 0) = False"| "even' n = even' (n-2)"
For non-recursive definitions Optional: the constant's type
Provides a lemma: even_def
Provides even'.simps and even'.induct
For recursive functions
Defining Constants
● inductive_set tclfor R :: "('a*'a) set"where "(x,y):R ==> (x,y):tcl R"| "(x,y):tcl R ==> (y,z):tcl R ==> (x,z):tcl R"
Defining Constants
● inductive_set tclfor R :: "('a*'a) set"where "(x,y):R ==> (x,y):tcl R"| "(x,y):tcl R ==> (y,z):tcl R ==> (x,z):tcl R"
For inductive sets
Defining Constants
● inductive_set tclfor R :: "('a*'a) set"where "(x,y):R ==> (x,y):tcl R"| "(x,y):tcl R ==> (y,z):tcl R ==> (x,z):tcl R"
For inductive sets Optional: the constant's type
Defining Constants
● inductive_set tclfor R :: "('a*'a) set"where "(x,y):R ==> (x,y):tcl R"| "(x,y):tcl R ==> (y,z):tcl R ==> (x,z):tcl R"
For inductive sets
Parameters (types are optional again)
Optional: the constant's type
Defining Constants
● inductive_set tclfor R :: "('a*'a) set"where "(x,y):R ==> (x,y):tcl R"| "(x,y):tcl R ==> (y,z):tcl R ==> (x,z):tcl R"
For inductive sets
Provides tcl.cases, tcl.induct,tcl.intros and tcl.simps
Optional: the constant's type
Parameters (types are optional again)
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proof
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proofOptional: a name and attributes
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proofOptional: a name and attributes
Modifies some subgoal(s)
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proofOptional: a name and attributes
Modifies some subgoal(s)
Finishes a proof
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proofOptional: a name and attributes
Modifies some subgoal(s)
Finishes a proof
Finishes a proof in a single step
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proofOptional: a name and attributes
Modifies some subgoal(s)
Finishes a proof
Finishes a proof in a single step
Aborts a proof attempt
Theorems and Proofs
● lemma add_com [simp]: "x+y = y+x"● apply method● done● by method● oops● sorry
Starts a proofOptional: a name and attributes
Modifies some subgoal(s)
Finishes a proof
Finishes a proof in a single step
Aborts a proof attempt
Finishes a proof (cheating!)
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Simplification
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Simplification
Simplification and some logic
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Simplification
Simplification and some logic
Good for sets and quantifiers
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Simplification
Simplification and some logic
Good for sets and quantifiers
Good for arithmetic goals
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Simplification
Simplification and some logic
Good for sets and quantifiers
Powerful first-order prover
Good for arithmetic goals
Automated Proof Methods
● (induct x y arbitrary: z rule: r.induct)● (simp add: l1 del: l2)● (auto simp add: l1 intro: l2)● (blast intro: l1 elim: l2)● arith● (metis l1 l2 l3)● sledgehammer
Induction
Simplification
Simplification and some logic
Good for sets and quantifiers
Powerful first-order prover
Good for arithmetic goals
Finds lemmas for metis
Basic Methods for Rulesthm: "[| P1; ...; Pn |] ==> Q"
● (rule thm)● (erule thm)● (drule thm)● (frule thm)
● (rule_tac x="..." and y="..." in thm)
Basic Methods for Rulesthm: "[| P1; ...; Pn |] ==> Q"
● (rule thm)● (erule thm)● (drule thm)● (frule thm)
● (rule_tac x="..." and y="..." in thm)
Unifies Q with the conclusion
Basic Methods for Rulesthm: "[| P1; ...; Pn |] ==> Q"
● (rule thm)● (erule thm)● (drule thm)● (frule thm)
● (rule_tac x="..." and y="..." in thm)
Unifies Q with the conclusion
Unifies Q; unifies P1 with some assumption
Basic Methods for Rulesthm: "[| P1; ...; Pn |] ==> Q"
● (rule thm)● (erule thm)● (drule thm)● (frule thm)
● (rule_tac x="..." and y="..." in thm)
Unifies Q with the conclusion
Unifies Q; unifies P1 with some assumption
Unifies P1 with some assumption
Basic Methods for Rulesthm: "[| P1; ...; Pn |] ==> Q"
● (rule thm)● (erule thm)● (drule thm)● (frule thm)
● (rule_tac x="..." and y="..." in thm)
Unifies Q with the conclusion
Unifies Q; unifies P1 with some assumption
Unifies P1 with some assumption
Like drule, but does not delete the assumption
Basic Methods for Rulesthm: "[| P1; ...; Pn |] ==> Q"
● (rule thm)● (erule thm)● (drule thm)● (frule thm)
● (rule_tac x="..." and y="..." in thm)
Unifies Q with the conclusion
Unifies Q; unifies P1 with some assumption
Unifies P1 with some assumption
Manual instantiation of variables
Like drule, but does not delete the assumption
Insiders' Tips
● term "..."● thm name● Find theorems● Isabelle > Settings > Display ...● Isabelle > Show me ...
Insiders' Tips
● term "..."● thm name● Find theorems● Isabelle > Settings > Display ...● Isabelle > Show me ...
Prints a term (with its type)
Insiders' Tips
● term "..."● thm name● Find theorems● Isabelle > Settings > Display ...● Isabelle > Show me ...
Prints a term (with its type)
Prints a specific theorem
Insiders' Tips
● term "..."● thm name● Find theorems● Isabelle > Settings > Display ...● Isabelle > Show me ...
Prints a term (with its type)
Prints a specific theorem
Search for theorems by pattern
Insiders' Tips
● term "..."● thm name● Find theorems● Isabelle > Settings > Display ...● Isabelle > Show me ...
Prints a term (with its type)
Prints a specific theorem
Search for theorems by pattern
Show types, sorts etc.
Insiders' Tips
● term "..."● thm name● Find theorems● Isabelle > Settings > Display ...● Isabelle > Show me ...
Prints a term (with its type)
Prints a specific theorem
Search for theorems by pattern
Show types, sorts etc.
Show all commands, all methods etc.
