[PPT] - MAP INTERNATIONAL SPRING SCH L ON FORMALIZATION OF MATHEMATICS PowerPoint Presentation

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L ON FORMALIZATION OF SPRING SCH

SOPHIA ANTIPOLIS, FRANCE / 12-16 MARCH

MATHEMATICS 2012 MAP INTERNATIONAL

SSReflect - Logics & Basic tactics

Laurence Rideau 12 March

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SSR Tactics Structure

SSReflect – Reminder

(SSR = Small Scale Reflection)

SSReflect: extension of Coq developed while formalizing the Four Color Theorem (2004), now used for the Odd Order Theorem.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

SSReflect – Reminder

(SSR = Small Scale Reflection)

SSReflect: extension of Coq developed while formalizing the Four Color Theorem (2004), now used for the Odd Order Theorem. Changes with standard Coq: Vernacular (Commands) and Gallina are mostly unchanged (e.g., Definition, Lemma, forall, match with); standard tactics are still available some tactics are superseded (e.g., apply, rewrite) new libraries are provided (e.g., nat, seq)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Design Decisions

Simplify and generalize the syntax of tactics.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Design Decisions

Simplify and generalize the syntax of tactics. Add some ways to structure the scripts, so that breakages are easier to understand.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Design Decisions

Simplify and generalize the syntax of tactics. Add some ways to structure the scripts, so that breakages are easier to understand. Force the user to explicitly name things.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Design Decisions

Simplify and generalize the syntax of tactics. Add some ways to structure the scripts, so that breakages are easier to understand. Force the user to explicitly name things. Ease the use of boolean reflection.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Outline

1

Logics

2

Tactics, Tacticals

3

Proof Structure

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Outline

1

Logics First Order Logic Booleans

2

Tactics, Tacticals

3

Proof Structure

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic

Propositional variables: P Q R . . . Propositions: (even 4) (x < 10) (7 <= 2) Implication: -> Formulas: (P -> Q) -> (Q -> R) -> P -> R

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic

Propositional variables: P Q R . . . Propositions: (even 4) (x < 10) (7 <= 2) Implication: -> Formulas: (P -> Q) -> (Q -> R) -> P -> R Propositional are of sort Prop : (P : Prop). Declaring variables: Variables P Q R :Prop.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic

Propositional variables: P Q R . . . Propositions: (even 4) (x < 10) (7 <= 2) Implication: -> Formulas: (P -> Q) -> (Q -> R) -> P -> R Propositional are of sort Prop : (P : Prop). Declaring variables: Variables P Q R :Prop. Any term of type P (p : P) is a proof of P.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

State and Proof a theorem

Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

State and Proof a theorem

Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

. . . P : Prop Q : Prop R : Prop          named hypotheses (Context) (P → Q) → (Q → R) → P → R } current goal

Assumptions

Conclusion

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

State and Proof a theorem

Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

. . . P : Prop Q : Prop R : Prop          named hypotheses (Context) (P → Q) → (Q → R) → P → R } current goal

Assumptions

Conclusion Tactic: any operation that allows the simplification, decomposition into subgoals, or resolution of a goal.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq.

P : Prop Q : Prop R : Prop Hpq : (P → Q) (Q → R) → P → R

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq Hqr p.

P : Prop Q : Prop R : Prop Hpq : (P → Q) Hqr : (Q → R) p : P R

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq Hqr p. apply: Hqr.

P : Prop Q : Prop R : Prop Hpq : (P → Q) p : P Q

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq Hqr p. apply: Hqr. apply: (Hpq).

P : Prop Q : Prop R : Prop Hpq : (P → Q) p : P P

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq Hqr p. apply: Hqr. apply: Hpq. exact: p.

Proof completed.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq Hqr p. apply: Hqr. exact: (Hpq p).

Proof completed.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Proof

Theorem command: Lemma imp_trans :(P -> Q) -> (Q -> R) -> P -> R.

Proof. (* start the proof of a Lemma *)

move=> Hpq Hqr p. apply: Hqr. exact: (Hpq p). Qed.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic with universal quantifier

forall (P Q R :Prop), (P ->Q)-> (Q -> R) -> P -> R

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic with universal quantifier

forall (P Q R :Prop), (P ->Q)-> (Q -> R) -> P -> R as a goal: move=> P Q R.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic with universal quantifier

forall (P Q R :Prop), (P ->Q)-> (Q -> R) -> P -> R as a goal: move=> P Q R. as an hypothesis named H: apply: H. apply: (H A B). or . . .

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic with universal quantifier

forall (P Q R :Prop), (P ->Q)-> (Q -> R) -> P -> R as a goal: move=> P Q R. as an hypothesis named H: apply: H. apply: (H A B). or . . . forall n:nat, 0 <= n

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic with universal quantifier

forall (P Q R :Prop), (P ->Q)-> (Q -> R) -> P -> R as a goal: move=> P Q R. as an hypothesis named H: apply: H. apply: (H A B). or . . . forall n:nat, 0 <= n move=> n.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Minimal Propositional Logic with universal quantifier

forall (P Q R :Prop), (P ->Q)-> (Q -> R) -> P -> R as a goal: move=> P Q R. as an hypothesis named H: apply: H. apply: (H A B). or . . . forall n:nat, 0 <= n move=> n. apply: H. apply: (H a).

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Conjunction

Conjunction : A /\ B

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Conjunction

Conjunction : A /\ B

case: ab. (* Break the (ab : A /\ B) hypothesis *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Conjunction

Conjunction : A /\ B

case: ab. (* Break the (ab : A /\ B) hypothesis *) ab : A /\ B G → A -> B -> G

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Conjunction

Conjunction : A /\ B

case: ab. (* Break the (ab : A /\ B) hypothesis *) ab : A /\ B G → A -> B -> G

split. (* Prove a conjunction :A /\B *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Conjunction

Conjunction : A /\ B

case: ab. (* Break the (ab : A /\ B) hypothesis *) ab : A /\ B G → A -> B -> G

split. (* Prove a conjunction :A /\B *)

A /\ B → A B

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Disjunction

Disjunction : A \/ B

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Disjunction

Disjunction : A \/ B

case: ab. (* Break the (ab : A \/ B) hypothesis *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Disjunction

Disjunction : A \/ B

case: ab. (* Break the (ab : A \/ B) hypothesis *) ab : A \/ B G → A -> G B -> G

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Disjunction

Disjunction : A \/ B

case: ab. (* Break the (ab : A \/ B) hypothesis *) ab : A \/ B G → A -> G B -> G

left. (* Prove a disjunction :A \/ B *)

(* by choosing the left part *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Disjunction

Disjunction : A \/ B

case: ab. (* Break the (ab : A \/ B) hypothesis *) ab : A \/ B G → A -> G B -> G

left. (* Prove a disjunction :A \/ B *)

(* by choosing the left part *) A \/ B → A

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Disjunction

Disjunction : A \/ B

case: ab. (* Break the (ab : A \/ B) hypothesis *) ab : A \/ B G → A -> G B -> G

left. (* Prove a disjunction :A \/ B *)

(* by choosing the left part *) A \/ B → A

right. (* Prove a disjunction :A \/ B *)

(* by choosing the right part *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Negation

Negation : ~B

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Negation

Negation : ~B

~B is defined as (B -> False)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Negation

Negation : ~B

~B is defined as (B -> False) move=> B. (* To prove the goal (~B)*)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Negation

Negation : ~B

~B is defined as (B -> False) move=> B. (* To prove the goal (~B)*) . . . ~B → b : B False

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Propositional Logic, Negation

Negation : ~B

~B is defined as (B -> False) move=> B. (* To prove the goal (~B)) . . . ~B → b : B False Then apply: H. ( for a (H :~C) in the context*)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Existential Quantifier

Existential: exists n:nat, P n

(* P is a predicate on nat (P :nat ->Prop)*)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Existential Quantifier

Existential: exists n:nat, P n

(* P is a predicate on nat (P :nat ->Prop)) exists 2. (To prove an exists, give a witness *)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Existential Quantifier

Existential: exists n:nat, P n

(* P is a predicate on nat (P :nat ->Prop)) exists 2. (To prove an exists, give a witness *)

. . .

exists n:nat, P n

→ . . .

P 2

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Existential Quantifier

Existential: exists n:nat, P n

(* P is a predicate on nat (P :nat ->Prop)) exists 2. (To prove an exists, give a witness *)

. . .

exists n:nat, P n

→ . . .

P 2 case: Hex. (* To break the (Hex:exists n, P n)hypothesis ) ( combined with (move=>n Hn.)*)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Existential Quantifier

Existential: exists n:nat, P n

(* P is a predicate on nat (P :nat ->Prop)) exists 2. (To prove an exists, give a witness *)

. . .

exists n:nat, P n

→ . . .

P 2 case: Hex. (* To break the (Hex:exists n, P n)hypothesis ) ( combined with (move=>n Hn.)*) Hex: exists n, P n G

→

n : nat Hn : P n

G

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Outline

1

Logics First Order Logic Booleans

2

Tactics, Tacticals

3

Proof Structure

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans

Inductive bool := true | false.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans

Inductive bool := true | false.

Operators: ”&&”, ”||”, ”~~”, ”==>”, ”(+)”.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans

Inductive bool := true | false.

Operators: ”&&”, ”||”, ”~~”, ”==>”, ”(+)”. b1 b2 b1 && b2 b1 || b2 b1 ==> b2 b1 (+) b2 T T T T T F T F F T F T F T F T T T F F F F T F

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans

Inductive bool := true | false.

Operators: ”&&”, ”||”, ”~~”, ”==>”, ”(+)”. Some notations

"[ && b1 , b2 , .. , bn & c ]" := (b1 && (b2 && .. (bn && c).. )) "[ || b1 , b2 , .. , bn | c ]" := (b1 || (b2 || .. (bn || c).. ))

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans

Inductive bool := true | false.

Operators: ”&&”, ”||”, ”~~”, ”==>”, ”(+)”. is true : bool -> Prop.

fun b : bool => b = true. Notation : "x ’is_true’" := (is_true x)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans in proofs

Reason by case on a boolean:

case: a.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans in proofs

Reason by case on a boolean:

case: a.

. . . a : bool b : bool

a (+) b = (a && b)|| (a && b)

→ . . . b : bool

true (+)b = (true && b)|| (true && b)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans in proofs

Reason by case on a boolean:

case: a.

. . . a : bool b : bool

a (+) b = (a && b)|| (a && b)

→ . . . b : bool

true (+)b = (true && b)|| (true && b)

. . . b : bool

false (+)b = (false && b)|| (false && b)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans in proofs(2)

Compute, simplify:

rewrite /=.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure FOL Bool

Booleans in proofs(2)

Compute, simplify:

rewrite /=.

. . . b : bool

true (+)b = (true && b)|| ( true && b)

→ . . . b : bool

b = b || false

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Outline

1

Logics First Order Logic Booleans

2

Tactics, Tacticals

3

Proof Structure Forward reasoning Proof control flow Subgoal selectors

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics / Tacticals

Tactic: any operation that allows the simplification, decomposition into subgoals, or resolution of a goal. Tactical: any function of tactics (eg. ; the composition of two tactics).

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics and Tacticals

move=> by tactical apply: exact: case: elim: rewrite

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Introduction Tactic

move=> a b c.

pops the top 3 elements of the goal, and it puts them into the context with names a, b, and c.

move=> _.

pops the first top element of the goal, without putting it in the context.

move=> a _ c.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactical by and Tactics apply / exact

”by []” tries to solve the current goal by some trivial means; it fails if it doesn’t succeed.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactical by and Tactics apply / exact

”by []” tries to solve the current goal by some trivial means; it fails if it doesn’t succeed. ”by any tactic” applies the argument tactic, then tries to solve the current goal. ”apply: H” applies H to the goal.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactical by and Tactics apply / exact

”by []” tries to solve the current goal by some trivial means; it fails if it doesn’t succeed. ”by any tactic” applies the argument tactic, then tries to solve the current goal. ”apply: H” applies H to the goal. . . . H: P -> Q Q → . . . P

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactical by and Tactics apply / exact

”by []” tries to solve the current goal by some trivial means; it fails if it doesn’t succeed. ”by any tactic” applies the argument tactic, then tries to solve the current goal. ”apply: H” applies H to the goal. . . . H: P -> Q Q → . . . P "exact:H” performs ”by apply: H”

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics case: / elim:

Performs a case analysis / inductive elimination on the element given as an argument.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics case: / elim:

Performs a case analysis / inductive elimination on the element given as an argument.

Inductive nat := O | S of nat

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics case: / elim:

Performs a case analysis / inductive elimination on the element given as an argument.

Inductive nat := O | S of nat Lemma P_of_n forall n : nat , P n. move=>n.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics case: / elim:

Performs a case analysis / inductive elimination on the element given as an argument.

Inductive nat := O | S of nat Lemma P_of_n forall n : nat , P n. move=>n. case:n.

1. P 0 2. forall n : nat, P (S n)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Tactics case: / elim:

Performs a case analysis / inductive elimination on the element given as an argument.

Inductive nat := O | S of nat Lemma P_of_n forall n : nat , P n. move=>n. elim:n.

1. P 0 2. forall n, P n -> P (S n)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Basic Rewriting tactic

Tactic ”rewrite items . . .” modifies subterms of the goal: ”/name” unfolds a definition ”-/name” folds a definition ”term” rewrites (left to right) with a lemma or an hypothesis which conclusion is an equality

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Basic Rewriting tactic

Tactic ”rewrite items . . .” modifies subterms of the goal: ”/name” unfolds a definition ”-/name” folds a definition ”term” rewrites (left to right) with a lemma or an hypothesis which conclusion is an equality

Eqab: a = b

P a →

Eqab: a = b

P b

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Basic Rewriting tactic

Tactic ”rewrite items . . .” modifies subterms of the goal: ”/name” unfolds a definition ”-/name” folds a definition ”term” rewrites (left to right) with a lemma or an hypothesis which conclusion is an equality

Eqab: a = b

P a →

Eqab: a = b

P b ”-term” rewrites right to left

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times.

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times. rewrite -{number}term

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times. rewrite -{number}term

Lemma dbl a b : 2 * (a + b) = (b + a) + (a + b).

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times. rewrite -{number}term

Lemma dbl a b : 2 * (a + b) = (b + a) + (a + b). Proof.

a : nat b : nat 2 * (a + b) = (b + a) + (a + b)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times. rewrite -{number}term

Lemma dbl a b : 2 * (a + b) = (b + a) + (a + b). rewrite

!addnA.

a : nat b : nat 2 * (a + b) = b + (a + (a + b))

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times. rewrite -{number}term

Lemma dbl a b : 2 * (a + b) = (b + a) + (a + b). rewrite {2} addnC.

a : nat b : nat 2 * (a + b) = (b + a) + (b + a)

Laurence Rideau SSReflect - Logics & Basic tactics

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SSR Tactics Structure

Multiple Rewriting and Occurrence selection

rewrite -multiplicityterm ”?”: as many times as possible, possibly none, ”!”: as many times as possible, at least once, ”n?”: at most n times, ”n!”: exactly n times. rewrite -{number}term

Lemma dbl a b : 2 * (a + b) = (b + a) + (a + b). rewrite

!addnA {2} addnC.

a : nat b : nat 2 * (a + b) = (b + (a + (b + a)

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SSR Tactics Structure Forward Control flow Subgoals

Outline

1

Logics

2

Tactics, Tacticals

3

Proof Structure Forward reasoning Proof control flow Subgoal selectors

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SSR Tactics Structure Forward Control flow Subgoals

Forward Reasoning

have suffices (suff)

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SSR Tactics Structure Forward Control flow Subgoals

Forward Reasoning: have / suffices

have H : intermediate goal performs a logical cut.

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SSR Tactics Structure Forward Control flow Subgoals

Forward Reasoning: have / suffices

have H : intermediate goal performs a logical cut.

Variable f : nat -> nat. Variable P : nat -> Prop. Lemma P_of_3: P 3.

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 88

SSR Tactics Structure Forward Control flow Subgoals

Forward Reasoning: have / suffices

have H : intermediate goal performs a logical cut.

Variable f : nat -> nat. Variable P : nat -> Prop. Lemma P_of_3: P 3. Proof. have H: exists x, f x = 3.

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 89

SSR Tactics Structure Forward Control flow Subgoals

Forward Reasoning: have / suffices

have H : intermediate goal performs a logical cut.

Variable f : nat -> nat. Variable P : nat -> Prop. Lemma P_of_3: P 3. Proof. have H: exists x, f x = 3.

1. exists x, f x = 3 2. H: exists x, f x = 3 P 3

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 90

SSR Tactics Structure Forward Control flow Subgoals

Forward Reasoning: have / suffices

have H : intermediate goal performs a logical cut.

Variable f : nat -> nat. Variable P : nat -> Prop. Lemma P_of_3: P 3. Proof. have H: exists x, f x = 3.

Tactic ”suff” also performs a logical cut, but it produces the two subgoals in the opposite order.

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 91

SSR Tactics Structure Forward Control flow Subgoals

Proof control flow

Tabulation (depending on the number of subgoals number)

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 92

SSR Tactics Structure Forward Control flow Subgoals

Proof control flow

Tabulation (depending on the number of subgoals number) Bullets -, +, *

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 93

SSR Tactics Structure Forward Control flow Subgoals

Proof control flow

Tabulation (depending on the number of subgoals number) Bullets -, +, * Proof terminators : by , exact:

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 94

SSR Tactics Structure Forward Control flow Subgoals

A proof example

Laurence Rideau SSReflect - Logics & Basic tactics

SLIDE 95

SSR Tactics Structure Forward Control flow Subgoals

Subgoal Selectors

Solving one subgoal with a single tactic:

tactic ; first by tactic tactic ; last by tactic

Laurence Rideau SSReflect - Logics & Basic tactics