lemma stringlengths 8 343 | name stringlengths 1 61 |
|---|---|
: nat_of_bin 0 = 0%nat. | nat_of_bin0 |
p : exists n, nat_of_pos p = S n. | nat_of_pos_s |
p1 p2 : nat_of_pos p1 = nat_of_pos p2 -> p1=p2. | nat_of_pos_inj |
n1 n2 : nat_of_bin n1 = nat_of_bin n2 -> n1=n2. | nat_of_bin_inj |
n m :(N.to_nat n < N.to_nat m)%N ->(n < m)%N. | N2Nat_lt |
(rty : realFieldType) (a : rty) :- a + a = 0. | subrr' |
(rty : realFieldType) (a b : rty) :a - b + b = a. | sub_add_0 |
(a b : nat) :is_true (leq a (S b)) ->a = S b \/ is_true (leq a b). | leq_case |
(y z : nat) :z%:Q = y%:Q + (z%:Q - y%:Q). | s5 |
(rty : numDomainType) (z : nat) :leq (S O) z ->0 < (z%:R^-1 : rty). | s9 |
(rty : numDomainType) (z : nat) :leq (S O) z ->(z%:R : rty) != (0%:R : rty). | s10 |
(y : nat) :leq 2 y ->0 < y%:Q. | sr3 |
(y : nat) :leq 2 y ->3%:Q / 2%:Q <= y%:Q. | sr4 |
i1 i2 l :ord (F i1) (F i2) ->ord (F (getOrd i1 l)) (F (getOrd i2 l)). | getOrd_mono |
i0 (Hi0 : P i0) : P (getOrd_sub i0). | getOrd_sub_hasP |
: [forall (i | P i), max >= F i]. | maxP |
: [forall (i | P i), min <= F i]. | minP |
: min <= max. | min_le_max |
(rty : realFieldType) (I : finType) (f : I -> rty) (def i : I) :f i <= max xpredT f def. | max_ge |
(rty : realFieldType) (I : finType) (f : I -> rty) (def i : I) :min xpredT f def <= f i. | min_le |
:(forall s t, reflect (s = t) (eqf s t)) *(forall a b, reflect (a = b) (eqp a b)). | form_prog_dec |
: (pcancel picklef unpicklef) /\ (pcancel picklep unpicklep). | pickleP |
: pcancel picklef unpicklef. | picklefP |
(M : fmodel) :((forall s (w:M), reflect (eval s w) (evalb s w))*(forall p (w v : M), reflect (reach p w v) (reachb p w v)%type)). | eval_reachP |
(M:fmodel) (w : M) s : reflect (eval s w) (evalb s w). | evalP |
(M:fmodel) : stable (@eval M). | fin_modelP |
p s t : prv (s ---> t) -> prv ([p]s ---> [p]t). | rNorm |
p u s : prv (u ---> [p]u) -> prv (u ---> s) -> prv (u ---> [p^*]s). | rStar_ind |
p s : prv (s ---> [p][p^*]s ---> [p^*]s). | axStar |
p : prv (~~: EX p Bot). | axnEXF |
p s t : prv (s ---> t) -> prv (EX p s ---> EX p t). | rEXn |
p s t : prv (EX p s ---> [p]t ---> EX p (s :/\: t)). | axDBD |
p s : prv (~~:[p]s <--> EX p (~~:s)). | dmAX |
p s t : prv ([p]s :/\: [p]t ---> [p](s :/\: t)). | axABBA |
p s t : prv (EX p (s :\/: t) ---> EX p s :\/: EX p t). | axEOOE |
s: prv s -> valid s. | soundness |
(M:ts) :(forall s, prv s -> forall (w : M), eval s w) -> stable (@eval M). | soundness_classical |
(dn: forall P , ~ ~ P -> P) s:prv s -> forall (M: ts) (v: M), eval s v. | classical_soundness |
p s : prv ([p^*](s ---> [p]s) ---> s ---> [p^*]s). | segerberg |
s : SEG.prv s <-> prv s. | segerberg_vs_inductive |
s : s \in FL s. | FL_refl |
p s : [p]s \in FL0 p s. | FL0_refl |
:(forall s t : form, t \in FL s -> FL t `<=` FL s) *(forall a (s t : form), t \in FL0 a s -> FL t `<=` FL0 a s `|` FL s). | FL_trans_mut |
s t u : t \in FL s -> s \in FL u -> t \in FL u. | FL_trans |
u : sf_closed (FL u). | FL_closed |
:(forall s : form, size (FL s) <= sizef s) *(forall a s, size (FL0 a s) <= sizep a). | FL_size |
a s : isBox_spec a s (isBox a s). | isBoxP |
a s : isCBox_spec a s (isCBox a s). | isCBoxP |
F s : s \in F -> forall b, (s, b) \in flipcl F. | flipcl_refl |
F s : s \in flipcl F -> drop_sign s \in F. | flip_drop_sign |
F : flip_closed (flipcl F). | closed_flipcl |
F : size (flipcl F) <= 2 * size F. | size_flipcl |
s t : fImp s t^+ \in C -> s^- \in C \/ t^+ \in C. | hint_imp_pos |
s t : fImp s t^- \in C -> s^+ \in C /\ t^- \in C. | hint_imp_neg |
p0 p1 s b : ([p0;;p1]s, b) \in C -> ([p0][p1]s, b) \in C. | hint_box_con |
p0 p1 s : [p0 + p1]s^+ \in C -> [p0]s^+ \in C /\ [p1]s^+ \in C. | hint_box_ch |
p0 p1 s : [p0 + p1]s^- \in C -> [p0]s^- \in C \/ [p1]s^- \in C. | hint_dia_ch |
p s : [p^*]s^+ \in C -> s^+ \in C /\ [p][p^*]s^+ \in C. | hint_box_star |
p s : [p^*]s^- \in C -> s^- \in C \/ [p][p^*]s^- \in C. | hint_dia_star |
C a s : (s^+ \in R a C) = ([pV a]s^+ \in C). | RE |
a s C : s^- \notin R a C. | Rpos |
a (C C' : clause) : R a (C `|` C') = (R a C `|` R a C'). | RU |
a (s : sform) :R a [fset s] = if s is [pV b]u^+ then if (a == b) then [fset u^+] else fset0 else fset0. | R1 |
a : R a fset0 = fset0. | R0 |
F a : sf_closed F ->forall C, C \in powerset (flipcl F) -> R a C \in powerset (flipcl F). | RinU |
C a s : (s^+ \in Rc a C) = ([(pV a)^^]s^+ \in C). | RcE |
(C: clause) a : prv ([af C] ---> [pV a][af R a C]). | box_request |
p q s : prv ([p;;q]s <--> [p][q]s). | ax_Con |
p q s : prv (EX (p;;q) s <--> EX p (EX q s)). | ax_ConE |
p q s : prv ([p + q]s <--> ([p]s :/\: [q]s)). | ax_Ch |
p q s : prv (EX (p + q) s <--> ((EX p s) :\/: (EX q s))). | ax_ChE |
s t : prv ([s??]t <--> (s ---> t)). | ax_test |
p s : prv (s ---> [p^^](EX p s)). | ax_convBB |
p s : prv (s ---> [p](EX p^^ s)). | ax_convBF |
p s : prv (EX p [p^^]s ---> s). | ax_convEF |
p s : prv (EX (p^^) [p]s ---> s). | ax_convEB |
p s : prv ([(p^^)^^]s <--> [p]s). | ax_conv |
p q s : prv ([(p;;q)^^]s <--> [q^^;;p^^]s). | ax_conv_con |
p q s : prv ([(p+q)^^]s <--> [p^^ + q^^]s). | ax_conv_ch |
p s t :prv (s ---> t) -> prv (EX p t ---> t) -> prv (EX (p^*) s ---> t). | rStarE_ind |
p s : prv ([(p^*)^^]s <--> [(p^^)^*]s). | ax_conv_star |
s t : prv ([t??^^]s <--> [t??]s). | ax_conv_test |
s p q : (forall t, prv ([p]t <--> [q]t)) -> prv ([p^*]s <--> [q^*]s). | rStar_eq |
:(forall (s : form), prv (cnf s <--> s)) /\(forall (p : prog) (s : form), prv ([cnp true p]s <--> [p^^]s) * prv ([cnp false p]s <--> [p]s)). | cnf_eq_mut |
s : prv (cnf s <--> s). | ax_cnf |
s : prv (cnf s ---> s). | ax_cnfE |
s : prv (s ---> cnf s). | ax_cnfI |
: (forall s, is_cnf (cnf s)) * (forall p b, is_cnp (cnp b p)). | is_cn_mut |
:(forall s, is_cnf s -> forall t, t \in FL s -> is_cnf t)*(forall p s, is_cnp p -> is_cnf s -> forall t, t \in FL0 p s -> is_cnf t). | FL_cnf_mut |
s : is_cnf s -> (forall t, t \in FL s -> is_cnf t). | FL_cnf |
:(forall s, sizef (cnf s) <= 2 * sizef s)*(forall p b, sizep (cnp b p) <= 2 * sizep p). | size_cnf_mut |
s : sizef (cnf s) <= 2 * sizef s. | size_cnf |
s C : prv ([af C] ---> s) -> href (s^- |` C). | refI1n |
s C : href (s^- |` C) -> prv ([af C] ---> s). | refE1n |
C : ~~ lcons C -> prv ([af C] ---> Bot). | ax_lcons |
C s : prv ([af C] ---> [af s^+ |` C] :\/: [af s^- |` C]). | bigxm |
C : C \in PU -> ~~ S |> C -> href C. | adm_P1 |
p s : [p]s \in F -> s \in F. | sfc_box |
C : pref F C -> href C. | href_of |
s : valid s -> prv s. | completeness |
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