hcju/isabelleps huggingface dataset:
update_satisfies_tableau: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and c :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<langle>\<V> s\<rangle> \<Turnstile>\<^sub>t \<T> s \<longrightarrow> \<langle>\<V> (update x c s)\<rangle> \<Turnstile>\<^sub>t \<T> s"
igSwapClsSTR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igSwapClsSTR MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbsSTR MOD \<and> igSwapIGOpSTR MOD"
the_eq: shows "(THE f. \<forall>i. v $ i = (\<Sum>x\<in>UNIV. f x * X $ x $ i)) = (THE f. \<forall>i. w $ i = (\<Sum>x\<in>UNIV. f x * X $ x $ i))"
stars: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) (a @ b) = length a - 1 + Partial_Cost_Model.T_on' (rTS h0) ([x, y], rev a @ h) b"
length_rotate1: fixes xs :: "'d list" shows "length (rotate1 xs) = length xs"
wlp_mono_and_close: fixes S :: "com"   and P :: "complex mat"   and Q :: "complex mat" assumes "well_com S"   and "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "P \<le>\<^sub>L Q" shows "is_quantum_predicate (wlp S P) \<and> wlp S P \<le>\<^sub>L wlp S Q"
log2_of_power_le: fixes m :: "nat"   and n :: "nat" assumes "m \<le> 2 ^ n"   and "0 < m" shows "log 2 (real m) \<le> real n"
list_relD: fixes x :: "'b list"   and y :: "'c list"   and a :: "'b \<Rightarrow> 'c"   and I :: "'b \<Rightarrow> bool" assumes "(x, y) \<in> \<langle>br a I\<rangle>list_rel" shows "y = map a x"
OF_unsafe_safe_match3_eq: fixes oft :: "(of_match_field, 'b) flow_entry_match list" assumes "list_all (all_prerequisites \<circ> ofe_fields) oft" shows "OF_priority_match OF_match_fields_unsafe oft = OF_priority_match OF_match_fields_safe oft"
T_on'_append: fixes step :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'b"   and t :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and A :: "('b \<Rightarrow> 'e) \<times> ('b \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'e)"   and s :: "'b \<times> 'e"   and xs :: "'c list"   and ys :: "'c list" shows "On_Off.T_on' step t A s (xs @ ys) = On_Off.T_on' step t A s xs + On_Off.T_on' step t A (On_Off.config' step A s xs) ys"
foldSetD_closed: fixes A :: "'b set"   and z :: "'c"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and e :: "'c" assumes "(A, z) \<in> foldSetD D f e" shows "z \<in> D"
HMA_V_def: shows "Mod_Type_Connect.HMA_V = (\<lambda>v w. v = Mod_Type_Connect.from_hma\<^sub>v w)"
strictly_generalizes_irrefl: fixes C :: "'a literal multiset" shows "\<not> C \<prec>\<cdot> C"
matrix_row_length: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and nr :: "nat"   and nc :: "nat"   and M :: "'a list list" assumes "Matrix_Tensor.mult id f"   and "Matrix_Legacy.mat nr nc M" shows "Matrix_Legacy.mat (mult.row_length M) (length M) M"
continuous_at_imp_continuous_on: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>s. isCont f x" shows "continuous_on s f"
partition_on_mset_def: fixes A :: "'b multiset"   and P :: "'b multiset multiset" shows "partition_on_mset A P = (\<Sum>\<^sub># P = A \<and> {#} \<notin># P)"
assoc_oo: fixes f :: "'e \<rightarrow> 'd"   and g :: "'f \<rightarrow> 'e"   and h :: "'c \<rightarrow> 'f" shows "f oo g oo h = (f oo g) oo h"
hol0: shows "(\<lambda>z. f (a + z)) holomorphic_on cball 0 r"
measurable_piecewise_restrict_iff: fixes C :: "'c set set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "countable C"   and "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M"   and "space M \<subseteq> \<Union> C" shows "(f \<in> M \<rightarrow>\<^sub>M N) = (\<forall>\<Omega>\<in>C. f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N)"
rel_with_target_impl_TRel_weakly_reflects_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_reflects_barbs Rel (STCalWB SWB TWB)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel" shows "rel_weakly_reflects_barbs TRel TWB"
kcdj': shows "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> \<pi> m)"
pderivs_lang_Plus: fixes r1 :: "'a rexp"   and r2 :: "'a rexp" shows "pderivs_lang UNIV1 (Plus r1 r2) = pderivs_lang UNIV1 r1 \<union> pderivs_lang UNIV1 r2"
bis_imp_4: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>01 c2" shows "c1 \<approx>w c2"
gla_1: fixes x :: "'a"   and y :: "'a" assumes "ad x \<cdot> y = (0::'a)" shows "ad x \<le> ad y"
map_upt_len_same_len_conv: fixes xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'c" assumes "length xs = length ys" shows "map (\<lambda>i. f (xs ! i)) [0..<length ys] = map f xs"
exhaust: fixes y :: "('f, 'g, 'd, 'e, 'h) qAbs" obtains x1 :: "'d" and x2 :: "'e" and x3 :: "('f, 'g, 'd, 'e, 'h) qTerm" where "y = qAbs x1 x2 x3"
word_not_simps_1: fixes x :: "'b word" shows "\<not> x < 0"
monom_numeral: fixes n :: "num" shows "MPoly_Type.monom 0 (numeral n) = numeral n"
cos_angle_opposite1: fixes H :: "ocircline"   and H' :: "ocircline" shows "cos_angle (opposite_ocircline H) H' = - cos_angle H H'"
pd: shows "\<pi> l pd\<rightarrow> \<pi> k"
IH: fixes accu :: "('a \<times> 'a) list" assumes "\<forall>a\<in>set M. case a of (a, b) \<Rightarrow> SecurityInvariant_complies_formal_def a b"   and "wf_list_graph G"   and "set Es \<subseteq> set (edgesL G)"   and "set accu \<subseteq> set (edgesL G)"   and "distinct (Es @ accu)" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G (get_impl M) accu Es = TopoS_Stateful_Policy_Algorithm.filter_IFS_no_violations_accu (list_graph_to_graph G) (get_spec M) accu Es"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "hom'.in_hom f a b = (hom'.arr a \<and> hom'.arr b \<and> hom'.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
strand_sem_Send_map_1: fixes T :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<And>t. t \<in> set T \<Longrightarrow> \<lbrakk>M; [send\<langle>t\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>c \<I>" shows "\<lbrakk>M; map Send T\<rbrakk>\<^sub>c \<I>"
2: shows "tfr\<^sub>s\<^sub>t (unlabel A')"
fin: shows "integral\<^sup>N M g \<noteq> \<infinity>"
return_def: fixes a :: "'c" shows "return a \<equiv> \<lambda>s. ({(a, s)}, False)"
f5: shows "rk {a, c, A, C, \<beta>} = 3"
obj_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "D.obj (trg\<^sub>D \<mu>)"
1: shows "ChamberComplex.chamber_distance A D' C = chamber_distance D' C"
iFROM_subset: fixes n :: "nat"   and n' :: "nat" assumes "n \<le> n'" shows "[n'\<dots>] \<subseteq> [n\<dots>]"
simps_9: fixes I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset"   and X :: "WS1S_Nameful.so"   and \<phi> :: "WS1S_Nameful.ws1s" shows "satisfies I1 I2 (All2 X \<phi>) = (\<forall>N. satisfies I1 (I2(X := N)) \<phi>)"
holomorphic_sqrt: fixes f :: "complex \<Rightarrow> complex" assumes "f holomorphic_on S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0"   and "\<And>f. \<lbrakk>f holomorphic_on S; \<forall>z\<in>S. f z \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = exp (g z))" shows "\<exists>g. g holomorphic_on S \<and> (\<forall>z\<in>S. f z = (g z)\<^sup>2)"
step_3_def: fixes A :: "'b mat" shows "step_3 A = step_3_main (dim_row A) 1 A"
Nmlize_Nmlize: fixes t :: "'a E.term" assumes "E.Arr t" shows "\<^bold>\<lfloor>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
mgp_nextVertex_face_ex2: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "\<exists>f'\<in>set (facesAt g (f \<bullet> v)). f' \<bullet> (f \<bullet> v) = v"
isubexpr_reduct: fixes d1 :: "exp"   and e1 :: "exp"   and CT :: "nat \<Rightarrow> classDef option"   and d2 :: "exp" assumes "d1 \<in> isubexprs(e1)"   and "CT \<turnstile> d1 \<rightarrow> d2" shows "\<exists>e2. CT \<turnstile> e1 \<rightarrow> e2"
fic_nrename: fixes M :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname" assumes "fic (M[x\<turnstile>n>y]) c" shows "fic M c"
concat_butlast_last: fixes ws :: "'b list list" assumes "ws \<noteq> \<epsilon>" shows "concat (butlast ws) \<cdot> last ws = concat ws"
cases: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" assumes "g3mod1p a1 a2" obtains A :: "(C, BD) form" where "a1 = [ \<Empt> \<Rightarrow>* \<LM> Aa__  \<RM>]" and "a2 = ( \<Empt> \<Rightarrow>* \<LM> \<box> Aa__  \<RM>)"   | A :: "(C, BD) form" where "a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>]" and "a2 = ( \<LM> \<diamond> A  \<RM> \<Rightarrow>* \<Empt>)"
upper_asymptotic_densityI: fixes A :: "nat set"   and l :: "real" assumes "\<forall>\<^sub>F n in sequentially. real (card (A \<inter> {..<n})) \<le> l * real n" shows "upper_asymptotic_density A \<le> l"
equality: fixes r :: "'b alpha_d_scheme"   and r' :: "'b alpha_d_scheme" assumes "ok r = ok r'"   and "more r = more r'" shows "r = r'"
sorted_insort_key: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "sorted (map f (insort_key f x xs)) = sorted (map f xs)"
lst_at_ball: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (\<forall>C<f x0. \<exists>d>0. \<forall>y\<in>ball x0 d. C < f y)"
subst_del_id: fixes x :: "'e \<Longrightarrow> 'f" assumes "vwb_lens x" shows "id -\<^sub>s x = id"
ml_def: shows "ml = max_list_non_empty (map (\<lambda>i. norm (coeff p i)) [0..<degree p]) / norm (lead_coeff p)"
special_nsg_G1: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "\<natural>H \<triangleright> H"
sigma_finite_incseq: obtains A :: "nat \<Rightarrow> 'a set" where "range A \<subseteq> sets M" and "\<Union> (range A) = space M" and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" and "incseq A"
orderE: fixes a :: "'b"   and b :: "'b" assumes "order.greater_eq b a" obtains "a = inf a b"
Cd_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and d :: "'e" assumes "Contracts Xd Xh Pd Ch" shows "Contracts.Cd Pd d \<equiv> set_option \<circ> MaxR.MaxR_opt (Pd d)"
SN_nat_gt: shows "SN {(a, b). b < a}"
psubst_def: shows "psubst \<equiv> \<lambda>\<theta>. trm_rec (lookup \<theta>) (\<lambda>x e. Lam x) (\<lambda>e\<^sub>1 e\<^sub>2. App)"
not_nonzero_memI: fixes a :: "nat \<Rightarrow> int" assumes "a = \<zero>" shows "a \<notin> nonzero Zp"
simps_2: fixes x :: "'d"   and \<theta> :: "('b, 'd) trm"   and \<sigma> :: "'e \<Rightarrow> ('b, 'c, 'd) formula" shows "PPsubst (x := \<theta>) \<sigma> = (x := \<theta>)"
scast_eq1: shows "sint (SCAST(32 \<rightarrow> 64) w1) = sint w1"
quadratic_change_coefficients: fixes A1 :: "complex"   and qf :: "complex \<Rightarrow> complex \<Rightarrow> complex"   and b1 :: "complex"   and b2 :: "complex"   and c :: "complex"   and x :: "complex"   and y :: "complex"   and qf_1 :: "complex \<Rightarrow> complex \<Rightarrow> complex" assumes "A1 \<noteq> 0"   and "qf = (\<lambda>x y. A1 * x\<^sup>2 + A1 * y\<^sup>2 + b1 * x + b2 * y + c)"   and "qf x y = 0"   and "qf_1 = (\<lambda>x y. x\<^sup>2 + y\<^sup>2 + b1 / A1 * x + b2 / A1 * y + c / A1)" shows "qf_1 x y = 0"
f1: fixes a :: "'a"   and aa :: "'a" shows "ad (a \<cdot> ad (ad aa)) = ad (a \<cdot> aa)"
elims: fixes x :: "'d strand_label \<times> ('e, 'f) stateful_strand_step"   and xa :: "'f \<Rightarrow> ('e, 'f) Term.term"   and y :: "'d strand_label \<times> ('e, 'f) stateful_strand_step" assumes "x \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p xa = y" obtains l :: "'d strand_label" and s :: "('e, 'f) stateful_strand_step" and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" where "x = (l, s)" and "xa = \<theta>" and "y = (l, s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)"
linear_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'f \<Rightarrow> 'f"   and scale\<^sub>1 :: "'g \<Rightarrow> 'f \<Rightarrow> 'f"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'h \<Rightarrow> 'h"   and scale\<^sub>2 :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and f :: "'f \<Rightarrow> 'h" shows "linear_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 f \<equiv> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 f"
pOp_assoc_cancel: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> c) = a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> c"
A: shows "\<forall>x xs. P x (filter Q xs) = P x xs"
per_union_memD: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "(x, y) \<in> per_union R a b" shows "(x, y) \<in> R \<or> (x, a) \<in> R \<and> (b, y) \<in> R \<or> (x, b) \<in> R \<and> (a, y) \<in> R"
BseqI': fixes X :: "nat \<Rightarrow> 'b"   and K :: "real" assumes "\<And>n. norm (X n) \<le> K" shows "Bseq X"
Bex_set: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "Bex (set xs) P = list_ex P xs"
iUntil_imp: fixes I :: "nat set"   and P1 :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool"   and P2 :: "nat \<Rightarrow> bool" assumes "P1 t1. t1 \<U> t2 I. Q t2"   and "\<box> t I. P1 t \<longrightarrow> P2 t" shows "P2 t1. t1 \<U> t2 I. Q t2"
mk_poly2_col: fixes i :: "nat"   and A :: "'b mat"   and pv :: "'b poly vec"   and j :: "nat" assumes "i < dim_row A"   and "dim_row A = dim_vec pv" shows "mk_poly2_col A pv j $ i = pv $ i + (\<Sum>j'<j. monom (A $$ (i, dim_col A - Suc j')) j')"
P2: shows "Bet A X Q \<and> (X = B \<or> B Out X C)"
Ll_def: shows "Ll = map Lli [0..<length (prems_of \<iota>)]"
subword_nth2: fixes i :: "nat"   and w :: "'c list" assumes "i + 1 \<le> length w" shows "subword w i 1 = [w ! i]"
lookup_default_neutral_combine_with_key: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "'c"   and d :: "'b"   and m1 :: "('c, 'b) mapping"   and m2 :: "('c, 'b) mapping" assumes "\<And>x. f k d x = x"   and "\<And>x. f k x d = x" shows "lookup_default d (combine_with_key f m1 m2) k = f k (lookup_default d m1 k) (lookup_default d m2 k)"
prod_prime_factors': fixes n :: "nat" assumes "0 < n" shows "g n = (\<Prod>p\<in>prime_factors n. if multiplicity p n = 2 then - 1 else 0)"
IH: fixes B :: "'a multiset" assumes "\<Prod>\<^sub># A dvd \<Prod>\<^sub># B"   and "\<And>p. p \<in># A \<Longrightarrow> prime p"   and "\<And>p. p \<in># B \<Longrightarrow> prime p" shows "A \<subseteq># B"
Compl_lessThan: fixes k :: "'c" shows "- {..<k} = {k..}"
r_distr: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "z \<otimes>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = z \<otimes>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> z \<otimes>\<^bsub>P\<^esub> y"
empty_def: shows "RBT_Set.empty = \<langle>\<rangle>"
thin2: fixes A1 :: "fmla"   and H :: "fmla set"   and B :: "fmla"   and A2 :: "fmla" assumes "insert A1 H \<turnstile> B" shows "insert A1 (insert A2 H) \<turnstile> B"
metric_LIM_equal2: fixes g :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a"   and R :: "real"   and f :: "'a \<Rightarrow> 'b" assumes "g \<midarrow>a\<rightarrow> l"   and "0 < R"   and "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x" shows "f \<midarrow>a\<rightarrow> l"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "D (D h g) f = D h (D g f)"
path_defs_2: fixes g :: "real \<Rightarrow> 'a" shows "pathstart g = g 0"
assms_4: fixes y :: "'a"   and x :: "'a" assumes "y \<in> Y"   and "x \<in> X" shows "y \<le> x"
find_None: fixes f :: "'b \<Rightarrow> 'a option"   and xs :: "'b list" shows "(Preliminaries.find f xs = None) = (\<forall>x\<in>set xs. f x = None)"
even_mask_iff: fixes n :: "nat" shows "even ((2::'a) ^ n - (1::'a)) = (n = 0)"
split_cong: fixes q :: "'c \<times> 'd"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and p :: "'c \<times> 'd" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
b_le_i: shows "b \<le> mod_type_class.from_nat i"
lower_mult_float_interval: fixes p :: "nat"   and x :: "float interval"   and y :: "float interval" shows "lower (mult_float_interval p x y) = fst (bnds_mult p (lower x) (upper x) (lower y) (upper y))"
nhds_countable: fixes x :: "'a" obtains X :: "nat \<Rightarrow> 'a set" where "decseq X" and "\<And>n. open (X n)" and "\<And>n. x \<in> X n" and "nhds x = (INF n. principal (X n))"
nop': fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "cmod (p' x) \<le> L"
distinct_9: fixes x21 :: "'a"   and x22 :: "'a form list" shows "Compound x21 x22 \<noteq> Em"
weak_le_antisym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> x"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x .= y"
simps_2: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> b"   and x :: "'a"   and y :: "'a"   and ys :: "'a list"   and z :: "'a" shows "is_path' r x (y # ys) z = (r x y = T \<and> is_path' r y ys z)"
lspace_mono: fixes f :: "'a \<Rightarrow> real"   and q :: "ennreal"   and S :: "'a set"   and p :: "ennreal" assumes "f \<in> lspace (lebesgue_on S) q"   and "S \<in> lmeasurable"   and "0 < p"   and "p \<le> q"   and "q < \<infinity>" shows "f \<in> lspace (lebesgue_on S) p"
Id_SPEC_refine: fixes S :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "S \<le> SPEC \<Phi>" shows "S \<le> \<Down> Id (SPEC \<Phi>)"
cos_two_le_zero: shows "cos 2 \<le> 0"
Fun_\<Phi>: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and f :: "'b"   and x :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.arr S f"   and "x \<in> set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)" shows "set_category_given_img.Fun S' (set_category.some_img S' \<AA>) (two_set_categories_bij_betw_Univ.\<Phi> S S' \<AA> \<phi> f) (\<phi> x) = two_set_categories_bij_betw_Univ.\<Phi>a S \<AA> \<phi> f (\<phi> x)"
image_eqI: fixes b :: "'f"   and f :: "'g \<Rightarrow> 'f"   and x :: "'g"   and A :: "'g set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
std\<^sub>1_falsifies: fixes C\<^sub>1 :: "fterm literal set"   and G :: "bool list" shows "falsifies\<^sub>c G C\<^sub>1 = falsifies\<^sub>c G (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''1'' @ x)))"
signed_take_bit_eq_iff_take_bit_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b"   and b :: "'b" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "(ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n b) = (take_bit (Suc n) a = take_bit (Suc n) b)"
abs_eq: fixes x :: "int" shows "Uint (integer_of_int x) = Abs_uint (word_of_int x)"
less_eq_admS_def: fixes x :: "'a admS"   and y :: "'a admS" shows "x \<le> y \<equiv> unlr x \<subseteq> unlr y"
finsum_ldistr: fixes A :: "'e set"   and a :: "'a"   and f :: "'e \<Rightarrow> 'a" assumes "finite A"   and "a \<in> carrier R"   and "f \<in> A \<rightarrow> carrier R" shows "finsum R f A \<otimes> a = (\<Oplus>i\<in>A. f i \<otimes> a)"
replace_def: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "replace k v m = (if k \<in> keys m then update k v m else m)"
underS_E: fixes i :: "'b"   and R :: "('b \<times> 'b) set"   and j :: "'b" assumes "i \<in> Order_Relation.underS R j" shows "i \<noteq> j \<and> (i, j) \<in> R"
in_cong_bigtheta: fixes f :: "real \<Rightarrow> real"   and F :: "real filter"   and g :: "real \<Rightarrow> real"   and h :: "real \<Rightarrow> real" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> L F h) = (g \<in> L F h)"
divide_eq_minus_1_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = uminus one) = (b \<noteq> zero \<and> a = uminus b)"
simps: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and xs :: "'d list"   and i :: "nat"   and n :: "nat" shows "subseqs_i_n_main f b xs i n = (if i = 0 then [(b, [])] else if i = n then [(foldr f xs b, xs)] else case xs of y # ys \<Rightarrow> map (\<lambda>(c, zs). (c, y # zs)) (subseqs_i_n_main f (f y b) ys (i - 1) (n - 1)) @ subseqs_i_n_main f b ys i (n - 1))"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "i < n - m" shows "map f [m..<n] ! i = f (m + i)"
cop: fixes q :: "real poly" assumes "q \<in> set (removeAt i fss)" shows "comm_monoid_mult_class.coprime (fss ! i) q"
mignotte_coeff_helper: fixes h :: "int poly"   and i :: "nat" shows "real_of_int \<bar>poly.coeff h i\<bar> \<le> real (degree h - 1 choose i) * mahler_measure h + real_of_int (int (min i 1 * (degree h - 1 choose (i - 1))) * \<bar>lead_coeff h\<bar>)"