Thm* hyp,M,N:Formula List, q,r:Formula. |= < q.hyp,r.(M @ N) > |= < hyp,M @ (qr.N) > formula_imp_right_sound

Thm* hyp,M,N:Formula List, q,r:Formula, a:Assignment. a | < q.hyp,r.(M @ N) > a | < hyp,M @ (qr.N) > formula_imp_right_falsifiable

Thm* hyp,M,N:Formula List, q,r:Formula, a:Assignment. a |= < hyp,M @ (qr.N) > a |= < q.hyp,r.(M @ N) > formula_imp_right_sat

Thm* concl,M,N:Formula List, q,r:Formula. |= < M @ N,q.concl > |= < r.(M @ N),concl > |= < M @ (qr.N),concl > formula_imp_left_sound

Thm* concl,M,N:Formula List, q,r:Formula, a:Assignment. a | < r.(M @ N),concl > a | < M @ N,q.concl > a | < M @ (qr.N),concl > formula_imp_left_falsifiable

Thm* concl,M,N:Formula List, q,r:Formula, a:Assignment. a |= < r.(M @ N),concl > & a |= < M @ N,q.concl > a |= < M @ (qr.N),concl > formula_imp_left_sat

Thm* concl,M,N:Formula List, q,r:Formula. |= < q.(M @ N),concl > |= < r.(M @ N),concl > |= < M @ (qr.N),concl > formula_or_left_sound

Thm* concl,M,N:Formula List, q,r:Formula, a:Assignment. a | < q.(M @ N),concl > a | < r.(M @ N),concl > a | < M @ (qr.N),concl > formula_or_left_falsifiable

Thm* concl,M,N:Formula List, q,r:Formula, a:Assignment. a |= < q.(M @ N),concl > & a |= < r.(M @ N),concl > a |= < M @ (qr.N),concl > formula_or_left_sat

Thm* hyp,M,N:Formula List, q,r:Formula. |= < hyp,[q; r/ M @ N] > |= < hyp,M @ (qr.N) > formula_or_right_sound

Thm* hyp,M,N:Formula List, q,r:Formula, a:Assignment. a | < hyp,[q; r/ M @ N] > a | < hyp,M @ (qr.N) > formula_or_right_falsifiable

Thm* hyp,M,N:Formula List, q,r:Formula, a:Assignment. a |= < hyp,[q; r/ M @ N] > a |= < hyp,M @ (qr.N) > formula_or_right_sat

Thm* hyp,M,N:Formula List, q,r:Formula. |= < hyp,q.(M @ N) > |= < hyp,r.(M @ N) > |= < hyp,M @ (qr.N) > formula_and_right_sound

Thm* hyp,M,N:Formula List, q,r:Formula, a:Assignment. a | < hyp,q.(M @ N) > a | < hyp,r.(M @ N) > a | < hyp,M @ (qr.N) > formula_and_right_falsifiable

Thm* hyp,M,N:Formula List, q,r:Formula, a:Assignment. a |= < hyp,q.(M @ N) > & a |= < hyp,r.(M @ N) > a |= < hyp,M @ (qr.N) > formula_and_right_sat

Thm* concl,M,N:Formula List, q,r:Formula. |= < [q; r/ M @ N],concl > |= < M @ (qr.N),concl > formula_and_left_sound

Thm* concl,M,N:Formula List, q,r:Formula, a:Assignment. a | < [q; r/ M @ N],concl > a | < M @ (qr.N),concl > formula_and_left_falsifiable

Thm* concl,M,N:Formula List, q,r:Formula, a:Assignment. a |= < [q; r/ M @ N],concl > a |= < M @ (qr.N),concl > formula_and_left_sat

Thm* hyp,M,N:Formula List, f2:Formula. |= < f2.hyp,M @ N > |= < hyp,M @ (f2.N) > formula_not_right_sound

Thm* hyp,M,N:Formula List, f2:Formula, a:Assignment. a | < f2.hyp,M @ N > a | < hyp,M @ (f2.N) > formula_not_right_falsifiable

Thm* hyp,M,N:Formula List, f:Formula, a:Assignment. a |= < f.hyp,M @ N > a |= < hyp,M @ (f.N) > formula_not_right_sat

Thm* concl,M,N:Formula List, f2:Formula. |= < M @ N,f2.concl > |= < M @ (f2.N),concl > formula_not_left_sound

Thm* concl,M,N:Formula List, f2:Formula, a:Assignment. a | < M @ N,f2.concl > a | < M @ (f2.N),concl > formula_not_left_falsifiable

Thm* concl,M,N:Formula List, f2:Formula, a:Assignment. a |= < M @ N,f2.concl > a |= < M @ (f2.N),concl > formula_not_left_sat

In prior sections: formula valuation formula satisfaction formula falsification formula rank formula equality sat lemmas full assignment formula validity sequent formula list sequent rank sequent valid