XII Summer Workshop in Mathematics Interactively Proving Mathematical Theorems Section 2: Predicate Logic Thaynara Arielly de Lima (IME) Mauricio Ayala-Rinc´ on (CIC-MAT) In collaboration with: Ariane Alves de Almeida and Gabriel Ferreira Silva and Thiago Mendon¸ ca Ferreira Ramos Funded by FAPDF DE grant 00193.0000.2144/2018-81, CNPq Research Grant 307672/2017-4 February 10 - 14, 2020 T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Talk’s Plan Section 2 1 Deduction ` a la Gentzen: predicate rules Exercises - predicate logic Gentzen Deductive Rules vs PVS Proof Commands T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Deduction ` a la Gentzen: predicate rules Gentzen Calculus Table: Rules of deduction ` a la Gentzen for predicate logic Left rules Right rules Axioms: Γ , ϕ ) ϕ , ∆ ( Ax ) ? , Γ ) ∆ ( L ⊥ ) Structural rules: Γ ) ∆ Γ ) ∆ ϕ , Γ ) ∆ ( LW eakening ) Γ ) ∆ , ϕ ( RW eakening ) ϕ , ϕ , Γ ) ∆ Γ ) ∆ , ϕ , ϕ ( LC ontraction ) ( RC ontraction ) ϕ , Γ ) ∆ Γ ) ∆ , ϕ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Deduction ` a la Gentzen: predicate rules Gentzen Calculus Table: Rules of deduction ` a la Gentzen for predicate logic Left rules Right rules Logical rules: ϕ i 2 { 1 , 2 } , Γ ) ∆ Γ ) ∆ , ϕ Γ ) ∆ , ψ ( L ^ ) ( R ^ ) ϕ 1 ^ ϕ 2 , Γ ) ∆ Γ ) ∆ , ϕ ^ ψ Γ ) ∆ , ϕ i 2 { 1 , 2 } ϕ , Γ ) ∆ ψ , Γ ) ∆ ( L _ ) ( R _ ) ϕ _ ψ , Γ ) ∆ Γ ) ∆ , ϕ 1 _ ϕ 2 Γ ) ∆ , ϕ ψ , Γ ) ∆ ϕ , Γ ) ∆ , ψ ( L ! ) ( R ! ) ϕ ! ψ , Γ ) ∆ Γ ) ∆ , ϕ ! ψ ϕ [ x/t ] , Γ ) ∆ Γ ) ∆ , ϕ [ x/y ] ( L 8 ) ( R 8 ) , y 62 fv ( Γ , ∆ ) 8 x ϕ , Γ ) ∆ Γ ) ∆ , 8 x ϕ ϕ [ x/y ] , Γ ) ∆ Γ ) ∆ , ϕ [ x/t ] ( L 9 ) , y 62 fv ( Γ , ∆ ) ( R 9 ) 9 x ϕ , Γ ) ∆ Γ ) ∆ , 9 x ϕ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Deduction ` a la Gentzen: predicate rules Gentzen Calculus Derivation of: ` 9 x ¬ ϕ ) ¬ 8 x ϕ ϕ [ x/t ] ) ϕ [ x/t ] ( L ∀ ) 8 x ϕ ) ϕ [ x/t ] (c-equiv) ¬ ϕ [ x/t ] , 8 x ϕ ) (c-equiv) ¬ ϕ [ x/t ] ) ¬ 8 x ϕ ( L ∃ ) 9 x ¬ ϕ ) ¬ 8 x ϕ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Deduction ` a la Gentzen: predicate rules Some inference rules in PVS Predicate: Deduction rule PVS command ϕ [ x/y ] , Γ ) ∆ 9 x ϕ , Γ ` ∆ ( L 9 ) , y 62 fv ( Γ , ∆ ) ϕ [ x/y ] , Γ ` ∆ ( skolem ) , y 62 fv ( Γ , ∆ ) 9 x ϕ , Γ ) ∆ ϕ [ x/t ] , Γ ) ∆ 8 x ϕ , Γ ` ∆ ( L 8 ) ϕ [ x/t ] , Γ ` ∆ ( inst ) 8 x ϕ , Γ ) ∆ [ − 1] ∀ x : T : P ( x ) [ − 1] ∀ x : T : P ( x ) [ − 2] ∃ x : T : ¬ P ( x ) ( skolem − 2 “ z ”) |--- |--- [1] P ( z ) [ − 1] ∀ x : T : P ( x ) [ − 1] P ( z ) ( inst − 1 “ z ”) |--- |--- Q . E . D . [1] P ( z ) [1] P ( z ) T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Exercises - predicate logic Exercises - predicate logic See the file pred algebra.pvs in Exercises directory T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Gentzen Deductive Rules vs PVS Proof Commands Summary - Gentzen Deductive Rules vs Proof Commads Table: Structural Left Rules vs Proof Commands Structural left rules PVS commands ϕ , Γ ` ∆ Γ ) ∆ ϕ , Γ ) ∆ ( LW eakening ) ( hide ) Γ ` ∆ ϕ , ϕ , Γ ) ∆ ϕ , Γ ` ∆ ( LC ontraction ) ϕ , ϕ , Γ ` ∆ ( copy ) ϕ , Γ ) ∆ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Gentzen Deductive Rules vs PVS Proof Commands Summary - Gentzen Deductive Rules vs Proof Commads Table: Structural Right Rules vs Proof Commands Structural right rules PVS commands Γ ` ∆ , ϕ Γ ) ∆ Γ ) ∆ , ϕ ( RW eakening ) ( hide ) Γ ` ∆ Γ ) ∆ , ϕ , ϕ Γ ` ∆ , ϕ ( RC ontraction ) Γ ` ∆ , ϕ , ϕ ( copy ) Γ ) ∆ , ϕ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Gentzen Deductive Rules vs PVS Proof Commands Summary - Gentzen Deductive Rules vs Proof Commads Table: Logical Left Rules vs Proof Commands Left rules PVS commands ϕ 1 , ϕ 2 , Γ ) ∆ ϕ 1 ^ ϕ 2 , Γ ` ∆ ( L ^ ) ( flatten ) ϕ 1 ^ ϕ 2 , Γ ) ∆ ϕ i 2 { 1 , 2 } , Γ ` ∆ ϕ , Γ ) ∆ ψ , Γ ) ∆ ϕ _ ψ , Γ ` ∆ ( L _ ) ( split ) ϕ _ ψ , Γ ) ∆ ϕ , Γ ` ∆ ψ , Γ ` ∆ Γ ) ∆ , ϕ ψ , Γ ) ∆ ϕ ! ψ , Γ ` ∆ ( L ! ) ( split ) ϕ ! ψ , Γ ) ∆ Γ ` ∆ , ϕ ψ , Γ ` ∆ 8 x ϕ , Γ ` ∆ ϕ [ x/t ] , Γ ) ∆ ( L 8 ) ( inst ) 8 x ϕ , Γ ) ∆ ϕ [ x/t ] , Γ ` ∆ ϕ [ x/y ] , Γ ) ∆ 9 x ϕ , Γ ` ∆ ( L 9 ) , y 62 fv ( Γ , ∆ ) ( skolem ) , y 62 fv ( Γ , ∆ ) 9 x ϕ , Γ ) ∆ ϕ [ x/y ] , Γ ` ∆ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Gentzen Deductive Rules vs PVS Proof Commands Summary - Gentzen Deductive Rules vs Proof Commads Table: Logical Right Rules vs Proof Commands Right rules PVS commands Γ ) ∆ , ϕ Γ ) ∆ , ψ Γ ` ∆ , ϕ ^ ψ ( R ^ ) ( split ) Γ ) ∆ , ϕ ^ ψ Γ ` ∆ , ϕ Γ ` ∆ , ψ Γ ) ∆ , ϕ i 2 { 1 , 2 } Γ ` ∆ , ϕ 1 _ ϕ 2 ( R _ ) ( flatten ) Γ ) ∆ , ϕ 1 _ ϕ 2 Γ ` ∆ , ϕ 1 , ϕ 2 ϕ , Γ ) ∆ , ψ Γ ` ∆ , ϕ ! ψ ( R ! ) ( flatten ) Γ ) ∆ , ϕ ! ψ ϕ , Γ ` ∆ , ψ Γ ) ∆ , ϕ [ x/y ] Γ ` ∆ , 8 x ϕ ( R 8 ) , y 62 fv ( Γ , ∆ ) ( skolem ) , y 62 fv ( Γ , ∆ ) Γ ) ∆ , 8 x ϕ Γ ` ∆ , ϕ [ x/y ] Γ ) ∆ , ϕ [ x/t ] Γ ` ∆ , 9 x ϕ ( R 9 ) ( inst ) Γ ) ∆ , 9 x ϕ Γ ` ∆ , ϕ [ x/t ] T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
Section 2 Gentzen Deductive Rules vs PVS Proof Commands Summary - Completing the GC vs PVS rules (hide) (copy) (flatten) (split) (skolem) (inst) (lemma) (case) ⇥ (LW) ⇥ (LC) ⇥ (L ∧ ) ⇥ (L ∨ ) ⇥ ⇥ (L → ) ⇥ (L ∀ ) ⇥ (L ∃ ) ⇥ (RW) ⇥ (RC) ⇥ (R ∧ ) ⇥ (R ∨ ) ⇥ (R → ) ⇥ (R ∀ ) ⇥ (R ∃ ) ⇥ (Cut) ⇥ T. A. de Lima & M. Ayala-Rinc´ on Interactively Proving Mathematical Theorems XII Summer Workshop in Mathematics - UnB
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