hcju/isabelleps huggingface dataset:
to_fun_sub: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "a \<in> carrier R" shows "to_fun (Cring_Poly.compose R f g) a = to_fun f (to_fun g a)"
outPortsRule_def: fixes r :: "prop_rule" shows "task.outPortsRule r = ffUnion ((\<lambda>a. (\<lambda>h. Hyp h a) |`| a_hyps a) |`| predicate.f_antecedent r) |\<union>| Reg |`| predicate.f_consequent r"
sup_top_def: fixes x :: "'a top"   and y :: "'a top" shows "sup x y = (case x of top.Value v \<Rightarrow> case y of top.Value v' \<Rightarrow> top.Value (sup v v') | Top \<Rightarrow> Top | Top \<Rightarrow> Top)"
accessible_on_downward: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and b :: "'b"   and a :: "'b" assumes "accessible_on P A b"   and "a \<in> A"   and "P a b" shows "accessible_on P A a"
simps: fixes w1 :: "32 Word.word"   and w2 :: "32 Word.word" shows "wmax w1 w2 = (if w1 <s w2 then w2 else w1)"
tauLaw4SimRight: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi> P Q. \<Psi> \<rhd> P \<sim> Q \<Longrightarrow> (\<Psi>, P, Q) \<in> Rel"   and "\<And>\<Psi> P. (\<Psi>, P, P) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<leadsto><Rel> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
Enc_Sign: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Enc X Y \<noteq> Sign X' Y'"
lyn: shows "Lyndon (u \<cdot> v \<cdot> u)"
prj_chine_3: shows "\<lbrakk>\<lbrakk>g\<^sub>0h\<^sub>1.p\<^sub>0 \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0 \<star> TTfgh_TfTgh.chine\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>fg\<^sub>0h\<^sub>1.p\<^sub>0\<rbrakk>\<rbrakk>"
closest_pair_rec_eq_val_closest_pair_rec_tm: fixes xs :: "(int \<times> int) list" shows "Time_Monad.val (closest_pair_rec_tm xs) = closest_pair_rec xs"
make_mat_carrier: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and B :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op" shows "Smith_Impl.make_mat n k B \<in> carrier_mat n n"
det_0_iff_vec_prod_zero_field: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A = (0::'a)) = (\<exists>v. v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> A *\<^sub>v v = 0\<^sub>v n)"
compose_eq: fixes x :: "'b"   and A :: "'b set"   and g :: "'d \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'd" assumes "x \<in> A" shows "compose A g f x = g (f x)"
starfun_Im: fixes f :: "'b \<Rightarrow> complex" shows "*f* (\<lambda>x. Im (f x)) = (\<lambda>x. hIm ((*f* f) x))"
reflect_poly_mult: fixes p :: "'b poly"   and q :: "'b poly" shows "reflect_poly (p * q) = reflect_poly p * reflect_poly q"
comp_def: fixes f :: "'j \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j" shows "f \<circ> g = (\<lambda>x. f (g x))"
pmat_extension_def: fixes m :: "'b mat" shows "b.pmat_extension m = b.ptensor_mat m (1\<^sub>m c.d2)"
A: fixes a :: "'a"   and x :: "'b" shows "flat_ge (f a x) (f' a x)"
poly_entire: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b list"   and q :: "'b list" assumes "class.idom_char_0 one times plus zero minus uminus" shows "(semiring_0.poly plus zero times (semiring_0.pmult plus zero times p q) = semiring_0.poly plus zero times []) = (semiring_0.poly plus zero times p = semiring_0.poly plus zero times [] \<or> semiring_0.poly plus zero times q = semiring_0.poly plus zero times [])"
less_asym: fixes x :: "'c word"   and y :: "'c word"   and P :: "bool" assumes "x <s y"   and "\<not> P \<Longrightarrow> y <s x" shows "P"
symKey_neq_priEK: fixes K :: "nat"   and A :: "agent" assumes "K \<in> symKeys" shows "K \<noteq> priEK A"
poly_prod_0: fixes ps :: "'a set"   and f :: "'a \<Rightarrow> 'b poly"   and x :: "'b" assumes "finite ps" shows "(poly (prod f ps) x = (0::'b)) = (\<exists>p\<in>ps. poly (f p) x = (0::'b))"
sorted_snoc_iff: fixes xs :: "'b list"   and x :: "'b" shows "sorted_less (xs @ [x]) = (sorted_less xs \<and> (\<forall>y\<in>set xs. y < x))"
Compl_eq_Compl_iff: fixes A :: "'b set"   and B :: "'b set" shows "(- A = - B) = (A = B)"
abs: fixes s :: "Core.term"   and t :: "Core.term"   and T :: "typ" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "Abs T s \<rightarrow>\<^sub>\<eta> Abs T t"
CatDom: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "PreFunctor F" shows "Category (CatDom F)"
sets: fixes i :: "'b" assumes "i \<in> I" shows "{x \<in> space M. P i x} \<in> events"
Quotient_generator: shows "Quotient (eq_onp terminates) Generator generator cr_generator"
induct: fixes P :: "'b list \<times> 'b list \<times> 'b list \<Rightarrow> bool"   and a0 :: "'b list \<times> 'b list \<times> 'b list" assumes "\<And>x xs ys. P (xs, [], ys @ [x]) \<Longrightarrow> P (x # xs, ys, [])"   and "\<And>x xs ys z zs. \<lbrakk>x \<le> z \<Longrightarrow> P (xs, [], ys @ x # z # zs); \<not> x \<le> z \<Longrightarrow> P (x # xs, ys @ [z], zs)\<rbrakk> \<Longrightarrow> P (x # xs, ys, z # zs)"   and "\<And>ys zs. P ([], ys, zs)" shows "P a0"
antisym: fixes b :: "'b"   and a :: "'b" assumes "b \<sqsubseteq> a"   and "a \<sqsubseteq> b" shows "a = b"
is_hsum_cases: fixes z :: "hf" shows "(\<exists>x. z = HF.Inl x \<or> z = HF.Inr x) \<or> \<not> is_hsum z"
that_7: fixes \<Gamma>' :: "char list \<Rightarrow> secLevel option"   and T :: "secLevel"   and b :: "expr"   and c :: "com" assumes "x = \<Gamma>'"   and "xa = T"   and "xb = while (b) c"   and "\<Gamma>' \<turnstile> b : T"   and "\<Gamma>',T \<turnstile> c" shows "thesis"
middle_1: fixes x :: "'b"   and y :: "'b" assumes "complete_space_with_middle TYPE('b)" shows "dist x (complete_space_with_middle.middle x y) = dist x y / 2"
eq_fract_1: fixes b :: "'c"   and d :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<noteq> (0::'c)"   and "d \<noteq> (0::'c)" shows "(Fraction_Field.Fract a b = Fraction_Field.Fract c d) = (a * d = c * b)"
IH1_4: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M1" shows "the (Mapping.lookup M1 f) < length slp1"
dfree_Differential_simps: fixes x :: "('a, 'b) trm" shows "dfree (Differential x) = False"
consistent: shows "sign \<in> set (characterize_consistent_signs_at_roots_copr p qs)"
in_strong_fusion: fixes z :: "i" shows "P z (\<sigma>x. z = x)"
image_image: fixes f :: "'c::type \<Rightarrow> 'b::type"   and g :: "'d::type \<Rightarrow> 'c::type"   and A :: "'d::type set" shows "(f::'c::type \<Rightarrow> 'b::type) ` (g::'d::type \<Rightarrow> 'c::type) ` (A::'d::type set) = (\<lambda>x::'d::type. f (g x)) ` A"
append_eq_conv_conj: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys = zs) = (xs = take \<^bold>|xs\<^bold>| zs \<and> ys = drop \<^bold>|xs\<^bold>| zs)"
not_eventually: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<not> eventually P F) = (\<exists>\<^sub>F x in F. \<not> P x)"
inv_eq_one_eq: fixes x :: "'a" assumes "x \<in> Units G" shows "(inv x = \<one>) = (x = \<one>)"
inject_5: fixes x51 :: "'a"   and x52 :: "bexp"   and x53 :: "'a"   and x54 :: "'a acom"   and x55 :: "'a"   and y51 :: "'a"   and y52 :: "bexp"   and y53 :: "'a"   and y54 :: "'a acom"   and y55 :: "'a" shows "({x51} WHILE x52 DO {x53} x54 {x55} = {y51} WHILE y52 DO {y53} y54 {y55}) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53 \<and> x54 = y54 \<and> x55 = y55)"
m: shows "(p, r, q, m) \<in> network (rho n)"
power2_eq_iff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.idom times one plus zero minus uminus" shows "(power.power one times x 2 = power.power one times y 2) = (x = y \<or> x = uminus y)"
assms_6: fixes x :: "real" assumes "x \<in> {a<..<b} - s" shows "(f has_vector_derivative f' x) (at x)"
assms_2: shows "\<beta> \<in> {0<..<1}"
nb: fixes k :: "nat" shows "norm_bound (of_real_hom.mat_hom A ^\<^sub>m k) (c1 + c2 * real k ^ (d - 1))"
length_append_singleton: fixes xs :: "'b list"   and x :: "'b" shows "length (xs @ [x]) = Suc (length xs)"
simps_1: fixes f :: "'b \<Rightarrow> 'a option set" shows "map_option_set f None = {None}"
right_move_to_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.lgroup minus plus zero uminus inf less_eq less sup"   and "less_eq a (plus b (uminus c))" shows "less_eq (plus a c) b"
zip_map2: fixes xs :: "'c list"   and f :: "'e \<Rightarrow> 'd"   and ys :: "'e list" shows "zip xs (map f ys) = map2 (\<lambda>x y. (x, f y)) xs ys"
inj_comm_semiring_hom_axioms: shows "inj_comm_semiring_hom hom"
dist_les_11: fixes nat :: "nat" shows "ValN\<cdot>nat \<notsqsubseteq> ValTT"
\<Q>_converges: fixes k :: "nat" assumes "length Doc \<le> k" shows "\<Q> k = \<PP>"
induct: fixes P :: "'b tree23 \<Rightarrow> 'b \<Rightarrow> 'b upD \<Rightarrow> 'b \<Rightarrow> 'b tree23 \<Rightarrow> bool"   and a0 :: "'b tree23"   and a1 :: "'b"   and a2 :: "'b upD"   and a3 :: "'b"   and a4 :: "'b tree23" assumes "\<And>t1 a t2 b t3. P t1 a (TD t2) b t3"   and "\<And>t1 a t2 b t3 c t4. P t1 a (UF t2) b \<langle>t3, c, t4\<rangle>"   and "\<And>t1 a t2 b t3 c t4 d t5. P t1 a (UF t2) b \<langle>t3, c, t4, d, t5\<rangle>"   and "\<And>a b v d. P a b (UF v) d \<langle>\<rangle>" shows "P a0 a1 a2 a3 a4"
hyp_newmethod: fixes Ia :: "qtname"   and i :: "iface"   and newa :: "mhead" assumes "iface G Ia = Some i"   and "table_of (imethods i) sig = Some newa" shows "G\<turnstile>resTy (Ia, newa)\<preceq>resTy old \<and> is_static (Ia, newa) = is_static old"
absolutely_integrable_mult_Fejer_kernel: fixes f :: "real \<Rightarrow> real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>x. Fejer_kernel n x * f x) absolutely_integrable_on {- pi..pi}"
max_word_wrap: fixes x :: "'b word" assumes "x + 1 = 0" shows "x = - 1"
simps_8: fixes \<sigma> :: "'a basic_type" shows "String \<squnion> \<sigma> = (if String \<le> \<sigma> then \<sigma> else if \<sigma> \<le> String then String else OclAny)"
borel_measurable_has_bochner_integral: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "has_bochner_integral M f x" shows "f \<in> borel_measurable M"
norm_triangle_ineq: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (plus x y) \<le> norm x + norm y"
mor_incl_min_alg: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e" assumes "alg B1 B2 s1 s2" shows "mor (min_alg1 s1 s2) (min_alg2 s1 s2) s1 s2 B1 B2 s1 s2 id id"
intro: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<And>x y z. F (F x y) z = F x z"   and "\<And>x y z. F x (F y z) = F x z" shows "overrider F"
rep_eq: fixes x :: "('c \<times> 'd) list" shows "fmlookup (fmap_of_list x) = map_of x"
measurable_measure_Diff: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> fmeasurable M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A - Sigma_Algebra.measure M B"
sym_sym: fixes A :: "'a set" shows "sym (A \<times> A)"
nprv_ptrmE0_uni0: fixes \<sigma> :: "'fmla"   and n :: "nat"   and F :: "'fmla set"   and t :: "'trm"   and \<psi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "\<sigma> \<in> F"   and "nprv F (subst \<sigma> t out)"   and "nprv (insert (eql (Var out) t) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t \<in> trm" shows "nprv F \<psi>"
discr_app: fixes f :: "'b discr \<Rightarrow> 'a"   and x :: "'b" shows "(\<Lambda> s. f s)\<cdot>(Discr x) = f (Discr x)"
floor_half_interval: fixes d :: "nat"   and n :: "nat" assumes "d \<noteq> 0" shows "real (n div d) - real (2 * (n div 2 div d)) = (if odd (n div d) then 1 else 0)"
zero_diff_eq: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "0 = j - i" shows "j = i"
preserves_arr: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "S.arr (Hom.map f)"
dom_char: fixes f :: "'a" shows "V.dom f = (if f \<in> Collect arr then f else V.null)"
LIMSEQ_power_zero: fixes x :: "'b" assumes "norm x < 1" shows "(^) x \<longlonglongrightarrow> (0::'b)"
hoare_valid_def: fixes P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool" shows "(\<Turnstile> {P} c {Q}) = (\<forall>s t. s -c\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t)"
lex_pm_alt: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "lex_pm s t = (s = t \<or> (\<exists>x. lookup s x < lookup t x \<and> (\<forall>y<x. lookup s y = lookup t y)))"
abc_acd_bcd: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[a c d]]" shows "[[b c d]]"
\<X>_length_bound: fixes t :: "symbol"   and c :: "character list"   and k :: "nat" assumes "(t, c) \<in> \<X> k" shows "k + length c \<le> length Doc"
that: assumes "\<sigma>' = Some (Ref.set p x (the_state \<sigma>))"   and "r = ()" shows "thesis"
rev_eq: fixes xs :: "'a list"   and ys :: "'a list" assumes "rev xs = rev ys" shows "xs = ys"
Smith_1xn_aux_Q_carrier: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "(S', Q') = Smith_Impl.Smith_1xn_aux Smith_1x2 i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n" shows "Q' \<in> carrier_mat n n"
lang_samequot: fixes n :: "nat"   and r :: "'b rexp"   and a :: "'a" assumes "local.wf n r"   and "a \<in> \<Sigma> n" shows "lang n (samequot a r) = SAMEQUOT a (lang n r)"
HBL2: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and P :: "'c"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "HBL1_2_3 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}" shows "bprv (imp (cnj (HBL1.PP var subst P (enc \<phi>)) (HBL1.PP var subst P (enc (imp \<phi> \<chi>)))) (HBL1.PP var subst P (enc \<chi>)))"
finite_keys: fixes f :: "'d \<Rightarrow>\<^sub>0 'e" shows "finite (keys f)"
in_set_prefixes: fixes xs :: "'a list"   and ys :: "'a list" shows "(xs \<in> set (prefixes ys)) = prefix xs ys"
less_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "(f < g) = (f \<le> g \<and> \<not> g \<le> f)"
sat_prefix_conv_gen: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<sigma>' :: "(String.literal \<times> event_data list) trace"   and i :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and V' :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'"   and "i < Monitor.progress \<sigma> P \<phi> (plen \<pi>)"   and "dom V = dom V'"   and "dom P = dom V"   and "pred_mapping (\<lambda>x. x \<le> plen \<pi>) P"   and "\<And>p i \<phi>. \<lbrakk>p \<in> dom V; i < the (P p)\<rbrakk> \<Longrightarrow> the (V p) i = the (V' p) i" shows "Formula.sat \<sigma> V v i \<phi> = Formula.sat \<sigma>' V' v i \<phi>"
prefix_upt_0: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "prefix [0..<i] [0..<j]"
insert_def: fixes x :: "'a"   and t :: "'a bro" shows "insert x t = tree (ins x t)"
supp_is_least_supports: fixes S :: "atom set"   and x :: "'a" assumes "S supports x"   and "finite S"   and "\<And>S'. \<lbrakk>finite S'; S' supports x\<rbrakk> \<Longrightarrow> S \<subseteq> S'" shows "supp x = S"
dioid_sum_prod_var: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "\<Sum> (f ` A) \<cdot> \<Sum> (g ` A) = \<Sum> {f x \<cdot> g y |x y. x \<in> A \<and> y \<in> A}"
sup_compl_top: fixes x :: "'b upred" shows "(x \<or> \<not> x) = true"
p_dist_sup: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) = - x \<sqinter> - y"
r_comp_rtrancl_eq: fixes r :: "('a \<times> 'a) set" shows "r O r\<^sup>* = r\<^sup>* O r"
par_commute: fixes par :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and skip :: "'b::refinement_lattice"   and a :: "'b::refinement_lattice"   and b :: "'b::refinement_lattice" assumes "parallel (par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (skip::'b::refinement_lattice)" shows "(par::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (a::'b::refinement_lattice) (b::'b::refinement_lattice) = par b a"
pos_mat_lim_is_pos_aux: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat" assumes "limit_mat X A m"   and "\<exists>k. \<forall>n\<ge>k. positive (X n)" shows "positive A"
ostep_invariantE: fixes A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "('d \<times> 'e) \<times> 'f \<times> 'd \<times> 'e \<Rightarrow> bool"   and \<sigma> :: "'d"   and s :: "'e"   and a :: "'f"   and \<sigma>' :: "'d"   and s' :: "'e" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" obtains "P ((\<sigma>, s), a, \<sigma>', s')"
that: fixes k :: "V" assumes "n = ZFC_in_HOL.succ k" shows "thesis"
Span_eq_combine_set_length_version: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "Span K Us = {combine Ks Us |Ks. length Ks = length Us \<and> set Ks \<subseteq> K}"
Abs_Pg_inverse: fixes y :: "game set" assumes "y \<in> Pg" shows "Rep_Pg (Abs_Pg y) = y"
that: fixes u :: "'a \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "thesis"
less_max_iff_disj: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(z < max x y) = (z < x \<or> z < y)"
CondSwap: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P1 :: "'e set"   and c1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and P2 :: "'e set"   and c2 :: "('e, 'd, 'f) com"   and P :: "'e set"   and b :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P1 c1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P2 c2 Q,A"   and "P \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P1) \<and> (s \<notin> b \<longrightarrow> s \<in> P2)}" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Cond b c1 c2 Q,A"
insert_commute: fixes x :: "'b"   and y :: "'b"   and A :: "'b set" shows "insert x (insert y A) = insert y (insert x A)"
0: shows "{m'. coeff (monom m a) m' \<noteq> (0::'a)} = {}"
Units_int_pow_closed: fixes x :: "'a"   and n :: "int" assumes "x \<in> Units G" shows "x [^] n \<in> Units G"
lmap_unfold_llist: fixes f :: "'b \<Rightarrow> 'a"   and IS_LNIL :: "'c \<Rightarrow> bool"   and LHD :: "'c \<Rightarrow> 'b"   and LTL :: "'c \<Rightarrow> 'c"   and b :: "'c" shows "lmap f (unfold_llist IS_LNIL LHD LTL b) = unfold_llist IS_LNIL (f \<circ> LHD) LTL b"
path_is_path: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c" assumes "IFC prob" shows "IFC_def.is_path prob (IFC_def.path prob \<sigma>)"
HSem_def': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" shows "\<lbrace>\<Gamma>\<rbrace>\<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
bind_put: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and get :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and put :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'f"   and m :: "'e"   and f :: "'d \<Rightarrow> 'e" assumes "monad_state return bind get put" shows "bind (put s m) f = put s (bind m f)"
Card_order_wo_rel: fixes r :: "('b \<times> 'b) set" assumes "Card_order r" shows "wo_rel r"
create_name_gba_def: fixes \<phi> :: "'a ltlr" shows "create_name_gba \<phi> \<equiv> create_gba \<phi> \<bind> (\<lambda>G. ASSERT (nds_invars (g_V G)) \<bind> (\<lambda>_. RETURN (gba_rename name G)))"
uminus_fds_def: fixes f :: "'b fds" shows "- f = fds (\<lambda>n. - fds_nth f n)"
finite_sysI: assumes "finite bl" shows "finite_incidence_system bl \<B>\<^sup>D"
strict_monoI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less"   and "\<And>x y. less x y \<Longrightarrow> f x < f y" shows "order.strict_mono less f"
member_class_to_member_class: fixes x :: "'a \<times> 'a"   and r :: "'a"   and s :: "'a"   and y :: "'a \<times> 'a"   and r' :: "'a"   and s' :: "'a" assumes "x \<in> (r |\<^bsub>rel\<^esub> s)"   and "y \<in> (r' |\<^bsub>rel\<^esub> s')" shows "(fst x \<otimes> fst y |\<^bsub>rel\<^esub> snd x \<otimes> snd y) \<in> set_class_of\<^bsub>rel\<^esub>"
PP_oalist_def: fixes xs :: "('c, 'd) oalist_tc" shows "PP_oalist xs = pp_of_fun (OAlist_tc_lookup xs)"
sat_push_action_bound': fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation' list"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts'" shows "length (filter is_SAT_PUSH' p) \<le> 4 * card V * card E"
eSuc_eint_iff: fixes x :: "eint"   and y :: "int" shows "(x + 1 = eint y) = (\<exists>n. y = n + 1 \<and> x = eint n)"