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; Depends On: th_lra.plf
;; Proof (from predicates on linear polynomials) that the following imply bottom
;
; -x - 1/2 y + 2 >= 0
; x + y - 8 >= 0
; x - y + 0 >= 0
;
(check
; Variables
(% x var_real
(% y var_real
; linear monomials (combinations)
(@ m1 (lmonc (~ 1/1) x (lmonc (~ 1/2) y lmonn))
(@ m2 (lmonc 1/1 x (lmonc 1/1 y lmonn))
(@ m3 (lmonc 1/1 x (lmonc (~ 1/1) y lmonn))
; linear polynomials (affine)
(@ p1 (polyc 2/1 m1)
(@ p2 (polyc (~ 8/1) m2)
(@ p3 (polyc 0/1 m3)
(% pf_nonneg_1 (th_holds (>=0_poly p1))
(% pf_nonneg_2 (th_holds (>=0_poly p2))
(% pf_nonneg_3 (th_holds (>=0_poly p3))
(:
(holds cln)
(lra_contra_>=
_
(lra_add_>=_>= _ _ _
(lra_mul_c_>= _ _ 4/1 pf_nonneg_1)
(lra_add_>=_>= _ _ _
(lra_mul_c_>= _ _ 3/1 pf_nonneg_2)
(lra_add_>=_>= _ _ _
(lra_mul_c_>= _ _ 1/1 pf_nonneg_3)
(lra_axiom_>= 0/1))))))
)))))
))))
))
)
;; Proof (from predicates on real terms) that the following imply bottom
;
; -x - 1/2 y >= 2
; x + y >= 8
; x - y >= 0
;
(check
; Declarations
; Variables
(% x var_real
(% y var_real
; real predicates
(@ f1 (>=_Real (+_Real (*_Real (a_real (~ 1/1)) (a_var_real x)) (*_Real (a_real (~ 1/2)) (a_var_real y))) (a_real (~ 2/1)))
(@ f2 (>=_Real (+_Real (*_Real (a_real 1/1) (a_var_real x)) (*_Real (a_real 1/1) (a_var_real y))) (a_real 8/1))
(@ f3 (>=_Real (+_Real (*_Real (a_real 1/1) (a_var_real x)) (*_Real (a_real (~ 1/1)) (a_var_real y))) (a_real 0/1))
; proof of real predicates
(% pf_f1 (th_holds f1)
(% pf_f2 (th_holds f2)
(% pf_f3 (th_holds f3)
; Normalization
; real term -> linear polynomial normalization witnesses
(@ n1 (poly_formula_norm_>= _ _ _
(pn_- _ _ _ _ _
(pn_+ _ _ _ _ _
(pn_mul_c_L _ _ _ (~ 1/1) (pn_var x))
(pn_mul_c_L _ _ _ (~ 1/2) (pn_var y)))
(pn_const (~ 2/1))))
(@ n2 (poly_formula_norm_>= _ _ _
(pn_- _ _ _ _ _
(pn_+ _ _ _ _ _
(pn_mul_c_L _ _ _ 1/1 (pn_var x))
(pn_mul_c_L _ _ _ 1/1 (pn_var y)))
(pn_const 8/1)))
(@ n3 (poly_formula_norm_>= _ _ _
(pn_- _ _ _ _ _
(pn_+ _ _ _ _ _
(pn_mul_c_L _ _ _ 1/1 (pn_var x))
(pn_mul_c_L _ _ _ (~ 1/1) (pn_var y)))
(pn_const 0/1)))
; proof of linear polynomial predicates
(@ pf_n1 (poly_form _ _ n1 pf_f1)
(@ pf_n2 (poly_form _ _ n2 pf_f2)
(@ pf_n3 (poly_form _ _ n3 pf_f3)
; derivation of a contradiction using farkas coefficients
(:
(holds cln)
(lra_contra_>= _
(lra_add_>=_>= _ _ _
(lra_mul_c_>= _ _ 4/1 pf_n1)
(lra_add_>=_>= _ _ _
(lra_mul_c_>= _ _ 3/1 pf_n2)
(lra_add_>=_>= _ _ _
(lra_mul_c_>= _ _ 1/1 pf_n3)
(lra_axiom_>= 0/1))))))
)))
)))
)))
)))
))
)
;; Term proof, 2 (>=), one (not >=)
;; Proof (from predicates on real terms) that the following imply bottom
;
; -x + y >= 2
; x + y >= 2
; not[ y >= -2] => [y < -2] => [-y > 2]
;
(check
; Declarations
; Variables
(% x var_real
(% y var_real
; real predicates
(@ f1 (>=_Real
(+_Real (*_Real (a_real (~ 1/1)) (a_var_real x)) (a_var_real y))
(a_real 2/1))
(@ f2 (>=_Real
(+_Real (a_var_real x) (a_var_real y))
(a_real 2/1))
(@ f3 (not (>=_Real (a_var_real y) (a_real (~ 2/1))))
; Normalization
; proof of real predicates
(% pf_f1 (th_holds f1)
(% pf_f2 (th_holds f2)
(% pf_f3 (th_holds f3)
; real term -> linear polynomial normalization witnesses
(@ n1 (poly_formula_norm_>= _ _ _
(pn_- _ _ _ _ _
(pn_+ _ _ _ _ _
(pn_mul_c_L _ _ _ (~ 1/1) (pn_var x))
(pn_var y))
(pn_const 2/1)))
(@ n2 (poly_formula_norm_>= _ _ _
(pn_- _ _ _ _ _
(pn_+ _ _ _ _ _
(pn_var x)
(pn_var y))
(pn_const 2/1)))
(@ n3 (poly_formula_norm_>= _ _ _
(pn_- _ _ _ _ _
(pn_var y)
(pn_const (~ 2/1))))
; proof of linear polynomial predicates
(@ pf_n1 (poly_form _ _ n1 pf_f1)
(@ pf_n2 (poly_form _ _ n2 pf_f2)
(@ pf_n3 (poly_flip_not_>= _ _ (poly_form_not _ _ n3 pf_f3))
; derivation of a contradiction using farkas coefficients
(:
(holds cln)
(lra_contra_> _
(lra_add_>=_> _ _ _
(lra_mul_c_>= _ _ 1/1 pf_n1)
(lra_add_>=_> _ _ _
(lra_mul_c_>= _ _ 1/1 pf_n2)
(lra_add_>_>= _ _ _
(lra_mul_c_> _ _ 2/1 pf_n3)
(lra_axiom_>= 0/1))))))
)))
)))
)))
)))
))
)
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