hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "('d, 'e, 'f) psi"   and x2 :: "('d, 'e, 'f) psi"   and P :: "('d, 'e, 'f) psi \<Rightarrow> ('d, 'e, 'f) psi \<Rightarrow> bool" assumes "x1 \<equiv>\<^sub>s x2"   and "\<And>P. P P P"   and "\<And>P Q. \<lbrakk>P \<equiv>\<^sub>s Q; P P Q\<rbrakk> \<Longrightarrow> P Q P"   and "\<And>P Q R. \<lbrakk>P \<equiv>\<^sub>s Q; P P Q; Q \<equiv>\<^sub>s R; P Q R\<rbrakk> \<Longrightarrow> P P R"   and "\<And>P Q. P (P \<parallel> Q) (Q \<parallel> P)"   and "\<And>P Q R. P (P \<parallel> Q \<parallel> R) (P \<parallel> (Q \<parallel> R))"   and "\<And>P. P (P \<parallel> \<zero>) P"   and "\<And>x. P (\<lparr>\<nu>x\<rparr>\<zero>) (\<zero>)"   and "\<And>x y P. P (\<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P)) (\<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P))"   and "\<And>x P Q. x \<sharp> P \<Longrightarrow> P (\<lparr>\<nu>x\<rparr>(P \<parallel> Q)) (P \<parallel> \<lparr>\<nu>x\<rparr>Q)"   and "\<And>x M xvec N P. \<lbrakk>x \<sharp> M; x \<sharp> xvec; x \<sharp> N\<rbrakk> \<Longrightarrow> P (\<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P)) (M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P)"   and "\<And>x M N P. \<lbrakk>x \<sharp> M; x \<sharp> N\<rbrakk> \<Longrightarrow> P (\<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P)) (M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P)"   and "\<And>x Cs. x \<sharp> map fst Cs \<Longrightarrow> P (\<lparr>\<nu>x\<rparr>(Cases Cs)) (Cases map (\<lambda>(\<phi>, P). (\<phi>, \<lparr>\<nu>x\<rparr>P)) Cs)"   and "\<And>P. guarded P \<Longrightarrow> P (!P) (P \<parallel> !P)" shows "P x1 x2"
has_parametrized_subprob_density_dens_AE: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> ennreal"   and R :: "'b measure"   and N :: "'a \<Rightarrow> 'b measure" assumes "\<And>x. x \<in> space M \<Longrightarrow> AE y in R. f x y = f' x y"   and "(\<lambda>(x, y). f' x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)"   and "has_parametrized_subprob_density M N R f" shows "has_parametrized_subprob_density M N R f'"
Dlist_list_of_dlist: fixes dxs :: "'d dlist" shows "Dlist (list_of_dlist dxs) = dxs"
simps_1: fixes f :: "'a \<Rightarrow> 'a" shows "map_hd f [] = []"
t8: shows "typ_of (Abs_fv x \<tau>' t $ Fv x \<tau>') = typ_of t"
map_ide_simp: fixes a :: "'c" assumes "Cop.ide a" shows "\<Psi>.map a = S.inv (\<Phi> a)"
EX: fixes P1 :: "'b multiset"   and P2 :: "'b multiset" assumes "P = P1 + P2"   and "f `# P1 \<subseteq># M1"   and "f `# P2 \<subseteq># M2" shows "Q"
RNext_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "RNext ch mm rs p \<equiv> Read ch mm rs p \<or> (\<exists>l. Write ch mm rs p l) \<or> MemReturn ch rs p"
C\<^sub>3_def: shows "C\<^sub>3 = max (C\<^sub>1 / (1 - a - c)) ((C\<^sub>1 * real n\<^sub>0' + C\<^sub>2 + C\<^sub>4) / ((1 - a - c) * real n\<^sub>0' - max 0 b - max 0 d - 2))"
ide_cod: fixes f :: "'a" assumes "Right_a.arr f" shows "Right_a.ide (Right_a.cod f)"
M: fixes x :: "'a" shows "mono_Heap (\<lambda>f. cB f x)"
difference_supset: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> set ys" shows "xs - ys = []"
density_unique_iff: fixes f :: "'a \<Rightarrow> ennreal"   and f' :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M" shows "(density M f = density M f') = (AE x in M. f x = f' x)"
R_lim_def: shows "R_lim = EM_remainder 1 (\<lambda>x. (- 1) ^ 1 * pochhammer s 1 * complex_of_real (x + a) powr (- s - of_nat 1)) 0"
m_in_carrier: fixes i :: "nat" assumes "i \<in> N" shows "m i \<in> carrier"
FrameTransmission_def: fixes n :: "nat"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and nC :: "nat \<Rightarrow> Config" shows "FrameTransmission n nStore nReturn nGet nC \<equiv> \<forall>t k. k < n \<longrightarrow> (let s = t mod cycleLength (nC k) in s mem schedule (nC k) \<longrightarrow> nGet k t = [s] \<and> (\<forall>j. j < n \<and> j \<noteq> k \<longrightarrow> nStore j t = nReturn k t))"
a1: shows "low_equal s1 s2 \<and> t1 = cpu_reg_mod w cr s1 \<and> t2 = cpu_reg_mod w cr s2"
of_int_poly: fixes k :: "int" shows "of_int k = [:of_int k:]"
symKeys_invKey_iff: fixes K :: "nat" shows "(invKey K \<in> symKeys) = (K \<in> symKeys)"
lfinite_lappend: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lfinite (lappend xs ys) = (lfinite xs \<and> lfinite ys)"
h_def: shows "h = sum_list (concat (map (\<lambda>i. if 0 < i then [i] else []) is))"
geometric_sum_nat_dvd: fixes a :: "nat"   and k :: "nat" assumes "1 < a"   and "0 < k" shows "a - 1 dvd a ^ k - 1"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<nexists>x1 x2. prod = (x1, x2) \<and> \<not> P (f x1 x2))"
nat_mult_1_right: fixes n :: "nat" shows "n * 1 = n"
Polys_closed_times: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "p \<in> P[X]"   and "q \<in> P[X]" shows "p * q \<in> P[X]"
zero_reorient: fixes x :: "'e" shows "((0::'e) = x) = (x = (0::'e))"
bij_betw_ide_Obj_3: fixes A :: "'j \<Rightarrow> 'c" assumes "A \<in> Collect (functor (\<cdot>\<^sub>J) (\<cdot>))" shows "J_C.Dom (J_C.MkIde A) = A"
simps_1: fixes k :: "nat" shows "\<T> k 0 = {}"
stored_reused_def: fixes t' :: "transition"   and t :: "transition" shows "stored_reused t' t = stored_reused_aux t' t (max (Transition.total_max_reg t) (Transition.total_max_reg t')) (max (length (Outputs t)) (length (Outputs t')))"
mask_eq_sum_exp: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one = comm_monoid_add.sum plus zero (power.power one times (numeral.numeral one plus (num.Bit0 num.One))) {q. q < n}"
case_prod_conv: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and a :: "'c"   and b :: "'d" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
cotrace_all_have_snapshotted: fixes t :: "('a, 'b, 'c) event llist" assumes "cotrace init t" shows "\<exists>i. enat i \<le> llength t \<and> (\<forall>p. ps (coS t i) p \<noteq> None)"
finsum_singleton: fixes G :: "('a, 'b) ring_scheme"   and i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "abelian_monoid G"   and "i \<in> A"   and "finite A"   and "f \<in> A \<rightarrow> carrier G" shows "(\<Oplus>\<^bsub>G\<^esub>j\<in>A. if i = j then f j else \<zero>\<^bsub>G\<^esub>) = f i"
lastn_def: fixes n :: "nat"   and x :: "'a list" shows "lastn n x = drop (length x - n) x"
code: fixes b :: "bool"   and n :: "real" shows "arctan_series_stream_aux b n = MSSCons (if b then - inverse n else inverse n) (arctan_series_stream_aux (\<not> b) (n + 2))"
pdevs_val_perm: fixes xs :: "'b list"   and ys :: "'b list"   and e :: "nat \<Rightarrow> real"   and I :: "real set" assumes "xs <~~> ys"   and "e \<in> UNIV \<rightarrow> I" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> UNIV \<rightarrow> I" and "pdevs_val e (pdevs_of_list xs) = pdevs_val e' (pdevs_of_list ys)"
lUnionM_return: fixes A :: "'a list" shows "lUnionM (map (\<lambda>x. return [x]) A) = return A"
oDeriv_def: fixes F :: "ordinal \<Rightarrow> ordinal" shows "oDeriv F = ordinal_rec (oFix F 0) (\<lambda>p x. oFix F (oSuc x))"
congruent_triangle_sas_21: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "angle a1 b1 c1 = angle a2 b2 c2" shows "angle a1 c1 b1 = angle b2 c2 a2"
subgroup_of_additive_group_of_ring_def: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "subgroup_of_additive_group_of_ring I R addition multiplication zero unit \<equiv> subgroup I R addition zero \<and> ring R addition multiplication zero unit"
left_hom_with_unit_axioms: shows "left_hom_with_unit (\<cdot>) (\<star>) \<a> \<i>[trg \<mu>] (trg \<mu>)"
g_def: fixes i :: "nat"   and p :: "'a" shows "g i p \<equiv> if p \<in> reach A w i then Some (f' (i, p)) else None"
eq_1: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq f g"
add_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
mcont: shows "mcont Sup (\<le>) lub ord (\<lambda>x. f x (eSuc x))"
circ_decompose_6: fixes x :: "'b"   and y :: "'b" shows "x\<^sup>\<star> * (y * x\<^sup>\<star>)\<^sup>\<star> = y\<^sup>\<star> * (x * y\<^sup>\<star>)\<^sup>\<star>"
det_fold_set_def: fixes R :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and \<sigma> :: "'h"   and result :: "'g set \<Rightarrow> 'h" shows "det_fold_set R c f \<sigma> result \<equiv> \<forall>l. distinct l \<and> sorted_wrt R l \<longrightarrow> foldli l c f \<sigma> = result (set l)"
interior_bijective_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f"   and "bij f" shows "interior (f ` S) = f ` interior S"
depthSubst: fixes P :: "pi"   and a :: "name"   and b :: "name" shows "depth (P[a::=b]) = depth P"
set_insort_key: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "set (insort_key f x xs) = insert x (set xs)"
h4: shows "satisfy_equation p qs [[], [0]] [[1], [- 1]]"
cmult_in_bigo_iff': fixes f :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. f x * c) \<in> O[F](g)) = (c = (0::'b) \<or> f \<in> O[F](g))"
iTILL_inext_diff_const: fixes t :: "nat"   and n :: "nat" assumes "t < n" shows "inext t [\<dots>n] - t = Suc 0"
effect__strips_iii_c: fixes s' :: "'c \<Rightarrow> bool option"   and s :: "'c \<Rightarrow> bool option"   and op :: "'c strips_operator"   and v :: "'c" assumes "s' = s \<then> op"   and "v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)" shows "s' v = s v"
cfs_zero: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat" assumes "UP_ring R" shows "\<zero>\<^bsub>UP R\<^esub> n = \<zero>\<^bsub>R\<^esub>"
check_add_def: fixes A :: "(nat, int mpoly) fmap"   and \<V> :: "nat set"   and p :: "nat"   and q :: "nat"   and i :: "nat"   and r :: "int mpoly" shows "check_add A \<V> p q i r = SPEC (\<lambda>b. b \<longrightarrow> p \<in># dom_m A \<and> q \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars r \<subseteq> \<V> \<and> the (fmlookup A p) + the (fmlookup A q) - r \<in> More_Modules.ideal polynomial_bool)"
lpd_fv_regex: fixes s :: "'d regex"   and test :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and r :: "'d regex"   and fv :: "'d \<Rightarrow> 'f set" assumes "s \<in> lpd test i r" shows "fv_regex fv s \<subseteq> fv_regex fv r"
Exists_imp_hfthm: fixes e :: "(name, hf) finfun"   and A :: "fm"   and y :: "hf" assumes "Exists \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e y"   and "{} \<turnstile> A" shows "\<exists>B. y = \<lbrakk>\<guillemotleft>B\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> B"
fresh_subst: fixes z :: "name"   and s :: "trm"   and y :: "name"   and t :: "trm" assumes "z \<sharp> s"   and "z = y \<or> z \<sharp> t" shows "z \<sharp> t[y::=s]"
strict_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "lt a b"   and "lt b c" shows "lt a c"
oracle_hash_def: fixes k :: "'upf_key" shows "UPF.oracle_hash k = (\<lambda>L y. let t = upf_fun k y; L = insert y L in return_spmf (t, L))"
prime_elem_divprod_pow: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "algebraic_semidom.coprime times one a b"   and "dvd.dvd times (power.power one times p n) (times a b)" shows "dvd.dvd times (power.power one times p n) a \<or> dvd.dvd times (power.power one times p n) b"
lowner_lub_unique: shows "Ex1 local.lowner_is_lub"
reachable1_in_verts_1: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>+\<^bsub>G\<^esub> v" shows "u \<in> verts G"
sterm_to_pterm_frees: fixes t :: "sterm" shows "frees (sterm_to_pterm t) = frees t"
divides_rexp: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x dvd y" shows "x dvd y ^ Suc n"
vfsequence_vcard_in_omega: fixes xs :: "V" assumes "vfsequence xs" shows "vcard xs \<in>\<^sub>\<circ> \<omega>"
oo: fixes T :: "'b set" assumes "T \<subseteq> f ` S" shows "openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T"
digit_to_char_simps_7: shows "digit_to_char 6 = CHR ''6''"
finite_roots_unity: fixes n :: "nat" assumes "1 \<le> n" shows "finite {z. z ^ n = (1::'b)}"
mult_mat_vec_hom: fixes A :: "'a mat"   and nr :: "nat"   and n :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "vec_hom (A *\<^sub>v v) = mat_hom A *\<^sub>v vec_hom v"
has_integral_restrict_closed_subintervals_eq: fixes c :: "'c"   and d :: "'c"   and a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and i :: "'d" assumes "cbox c d \<subseteq> cbox a b" shows "((\<lambda>x. if x \<in> cbox c d then f x else (0::'d)) has_integral i) (cbox a b) = (f has_integral i) (cbox c d)"
of_nat_less_0_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (semiring_1.of_nat one plus zero m) zero"
submoduleE_1: fixes H :: "'c set" assumes "Module.submodule H R M" shows "H \<subseteq> carrier M"
meval_sched_appI: fixes ns :: "nat list"   and cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>1'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1'' :: "'Var \<Rightarrow> 'Val"   and ms :: "nat list"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns\<^esub> (cms\<^sub>1'', mem\<^sub>1'') \<and> (cms\<^sub>1'', mem\<^sub>1'') \<rightarrow>\<^bsub>ms\<^esub> (cms\<^sub>1', mem\<^sub>1')" shows "(cms\<^sub>1, mem\<^sub>1) \<rightarrow>\<^bsub>ns @ ms\<^esub> (cms\<^sub>1', mem\<^sub>1')"
cases: fixes a :: "('b com option \<times> 'b) list" assumes "a \<in> cptn" obtains P :: "'b com option" and s :: "'b" where "a = [(P__, s__)]"   | P :: "'b com option" and t :: "'b" and xs :: "('b com option \<times> 'b) list" and s :: "'b" where "a = (Pa__, sa__) # (Pa__, t__) # xs__" and "(Pa__, t__) # xs__ \<in> cptn"   | P :: "'b com option" and s :: "'b" and Q :: "'b com option" and t :: "'b" and xs :: "('b com option \<times> 'b) list" where "a = (P, s) # (Q, t) # xs" and "(P, s) -c\<rightarrow> (Q, t)" and "(Q, t) # xs \<in> cptn"
set_insert: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" obtains B :: "'b set" where "A = insert x B" and "x \<notin> B"
O_2: shows "tree_invar (Node e2 a2 r ts2)"
alpha_out_var: fixes x :: "'d \<Longrightarrow> 'f" shows "x ;\<^sub>L snd\<^sub>L = $x\<acute>"
trans_relcl: fixes L :: "'b set set" assumes "lattice L" shows "trans (relcl L)"
set_empty2: fixes xs :: "'b list" shows "({} = set xs) = (xs = [])"
ev_Stream: fixes P :: "'c stream \<Rightarrow> bool"   and x :: "'c"   and s :: "'c stream" shows "ev P (x ## s) = (P (x ## s) \<or> ev P s)"
connectedin_subset_topspace: fixes X :: "'b topology"   and S :: "'b set" assumes "connectedin X S" shows "S \<subseteq> topspace X"
BuildSeq_imp_Seq: fixes B :: "hf \<Rightarrow> bool"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and s :: "hf"   and k :: "hf"   and y :: "hf" assumes "BuildSeq B C s k y" shows "Seq s (succ k)"
intro_elim_6_d: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>\<equiv> \<psi> in v]"   and "[\<^bold>\<not>\<psi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
map_mono_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "prefix xs ys" shows "prefix (map f xs) (map f ys)"
kop_inj_iff: fixes f :: "'d \<Rightarrow> 'c set"   and g :: "'d \<Rightarrow> 'c set" shows "(op\<^sub>K f = op\<^sub>K g) = (f = g)"
cd_2: shows "(\<lambda>a. \<gamma> a \<bullet> i) ` {0..1} = {c..d}"
simps: fixes a :: "loc" shows "the_Loc (Loc a) = a"
2: assumes "\<phi> \<pi>l"   and "wuntil \<phi> \<psi> (map stl \<pi>l)" shows "\<chi>"
mset_merge: fixes q1 :: "'a tree"   and q2 :: "'a tree" assumes "heap q1"   and "heap q2" shows "mset_tree (merge q1 q2) = mset_tree q1 + mset_tree q2"
triangle_paths_unique: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "\<exists>!ab. ab \<in> \<P> \<and> a \<in> ab \<and> b \<in> ab \<and> a \<noteq> b"
multrow_mat_Unit: fixes k :: "nat"   and n :: "nat"   and a :: "'c"   and b :: "'d" assumes "k < n"   and "a \<noteq> (0::'c)" shows "multrow_mat n k a \<in> Units (ring_mat TYPE('c) n b)"
suminf: fixes f :: "'a \<Rightarrow> 'b"   and X :: "nat \<Rightarrow> 'a" assumes "bounded_linear f"   and "summable X" shows "f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
prems: shows "wf\<^sub>s\<^sub>t V ((x # S) @ S')"
Elem_affine_of_ivl_le: fixes e :: "nat \<Rightarrow> real"   and l :: "'b"   and u :: "'b" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "l \<le> u" shows "l \<le> aform_val e (aform_of_ivl l u)"
init_def: shows "init = ((l1, r1, sr1), l2, r2, sr2)"
minus_mod_eq_div_mult: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (modulo a b) = times (divide a b) b"
synth_synthD: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> synth (synth H)" shows "X \<in> synth H"
torder_on_restrictPI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set" assumes "torder_on A r" shows "torder_on (A \<inter> B) (r |` B)"
simps_7: fixes a :: "'a poly"   and n :: "nat" shows "num_params (Pw a n) = num_params a"
eq': shows "(h\<^sub>0 ++ h) x = (h\<^sub>1 ++ h) x"
has_bochner_integralI_AE: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and g :: "'c \<Rightarrow> 'd" assumes "has_bochner_integral M f x"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "has_bochner_integral M g x"
A: fixes x :: "'a" assumes "x \<in> A" shows "B x \<approx> C x"
finite_UNIV_finfun: shows "finite UNIV = (finite UNIV \<and> finite UNIV \<or> CARD('b) = 1)"
scaleC_conv_of_complex: fixes r :: "complex"   and x :: "'a" shows "r *\<^sub>C x = of_complex r * x"
subgroupI: fixes H :: "cltn2 set" assumes "H \<subseteq> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> b \<in> H" shows "subgroup H \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"
seq_char: fixes f :: "'b"   and g :: "'b" shows "J.seq f g = (J.ide f \<and> f = g)"
arctan_coeffs_def: fixes n :: "nat" shows "arctan_coeffs n = (if odd n then (- (1::'a)) ^ (n div 2) / of_nat n else (0::'a))"
fold_set_union: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set"   and z :: "'b" assumes "comp_fun_idem f"   and "finite A"   and "finite B" shows "Finite_Set.fold f z (A \<union> B) = Finite_Set.fold f (Finite_Set.fold f z A) B"
take_bit_eq_mod: 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 n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit n a = modulo a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)"
simps_2: fixes v :: "'b" shows "head (v)\<^sub>p = (v)\<^sub>p"
fst_mono: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" assumes "x \<le> y" shows "fst x \<le> fst y"
F_unfold: fixes i :: "name" shows "F' i = (if i \<in> insert i__ V then Var (p' \<bullet> i) else Var i)"
normalize_match_preserves_nodisc: fixes disc :: "'c \<Rightarrow> bool"   and m :: "'c match_expr"   and m' :: "'c match_expr" assumes "\<not> has_disc disc m"   and "m' \<in> set (normalize_match m)" shows "\<not> has_disc disc m'"
map_of_zip_Some: fixes v :: "nat"   and xs :: "'a list" assumes "v < length xs" shows "map_of (zip [0..<length xs] xs) v = Some (xs ! v)"
l1: shows "z \<notin> path_image (linepath lb (Complex (Re ub) (Im lb)))"
opn: fixes G :: "'a set" assumes "G \<in> \<G>" shows "open G"
grun_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "grun A w V \<equiv> transition_system.run (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V)"
lcf_of_sub_in_ltrm: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "deg R f = n"   and "0 < deg R g" shows "Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g (deg R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) g)) = f (deg R f) \<otimes> g (deg R g) [^] n"
tendsto_eq_rhs: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b"   and F :: "'c filter"   and y :: "'b" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open x) F"   and "x = y" shows "filterlim f (topological_space.nhds open y) F"