hcju/isabelleps huggingface dataset:
assert_bound_unsat: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and as :: "('a \<times> 'b atom) set"   and a :: "'a \<times> 'b atom"   and s' :: "('a, 'b) state" assumes "AssertBound assert_bound"   and "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "index_valid as s"   and "assert_bound a s = s'"   and "\<U> s'" shows "minimal_unsat_state_core s'"
bf_lit_def: fixes v :: "'a" shows "bf_lit v \<equiv> \<lambda>l. l v"
lemma_2_10_30: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "imp_l (imp_r (imp_l b a) a) a = imp_l b a"
simps_8: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and vs :: "(variable \<times> typ) list"   and ns :: "variable set"   and Hs :: "Core.term list"   and t :: "Core.term" shows "replay' thy vs ns Hs (Hyp t) = (if t \<in> set Hs then Some t else None)"
exhaust: fixes y :: "bexp\<^sub>C" obtains x1 :: "var\<^sub>C" and x2 :: "nat" where "y = bexp\<^sub>C.Eq x1 x2"
Data2PData_def: fixes D :: "'b data" shows "Data2PData D = (let (L, DP) = Rep_data D; OL = map Some L in Abs_pdata (OL, DP))"
range_def: fixes k :: "natural" shows "local.range k = iterate (log 2147483561 k) (\<lambda>l. next \<circ>\<rightarrow> (\<lambda>v. Pair (v + l * 2147483561))) 1 \<circ>\<rightarrow> (\<lambda>v. Pair (v mod k))"
extensional_arr: fixes f :: "'s"   and f' :: "'s" assumes "par f f'"   and "\<And>x. \<guillemotleft>x : unity \<rightarrow> local.dom f\<guillemotright> \<Longrightarrow> f \<cdot> x = f' \<cdot> x" shows "f = f'"
differing_vars_def: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "differing_vars mem\<^sub>1 mem\<^sub>2 \<equiv> {x. mem\<^sub>1 x \<noteq> mem\<^sub>2 x}"
contour_integral_shiftpath: fixes g :: "real \<Rightarrow> complex"   and a :: "real"   and f :: "complex \<Rightarrow> complex" assumes "valid_path g"   and "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "contour_integral (shiftpath a g) f = contour_integral g f"
scons_eq: fixes a :: "'c"   and v :: "'c stream"   and w :: "'c stream" shows "(a ## v = w) = (shd w = a \<and> stl w = v)"
eventually_laplace_integrand_le: fixes M :: "real"   and c :: "real"   and f :: "real \<Rightarrow> complex"   and s :: "complex" assumes "exponential_order M c f" shows "\<forall>\<^sub>F t in at_top. cmod (laplace_integrand f s t) \<le> M * exp (- (Re s - c) * t)"
NotI: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "\<Gamma> \<turnstile> \<^bold>\<not> F"
lfp_pair: fixes F :: "('d \<Rightarrow> 'e \<Rightarrow> 'c) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'e" shows "lfp (\<lambda>f (a, b). F (\<lambda>a b. f (a, b)) a b) (a, b) = lfp F a b"
Nml_HcompNml_3: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Cod (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Cod t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod u"
preserves_arr: fixes f :: "'a" assumes "hom.arr f" shows "hom'.arr (e\<^sub>1 \<star> f \<star> d\<^sub>0)"
ptrm_infer_type_swp: fixes a :: "'b"   and b :: "'b"   and X :: "'b ptrm"   and \<Gamma> :: "'b \<Rightarrow> PreSimplyTyped.type option"   and \<tau> :: "PreSimplyTyped.type" assumes "a \<noteq> b"   and "b \<notin> ptrm_fvs X" shows "ptrm_infer_type (\<Gamma>(a \<mapsto> \<tau>)) X = ptrm_infer_type (\<Gamma>(b \<mapsto> \<tau>)) ([a \<leftrightarrow> b] \<bullet> X)"
assms: shows "(M, N) \<in> s_mul_ext Id R"
natural_isomorphism_\<ll>: shows "natural_isomorphism (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) L S.map \<ll>"
nn_integral_0_iff_AE: fixes u :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (AE x in M. u x = 0)"
tiny_dg_in_Vset: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "tiny_digraph \<alpha> \<CC>" shows "\<CC> \<in>\<^sub>\<circ> Vset \<alpha>"
\<tau>move2TryCatch2: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and C :: "String.literal"   and V :: "nat"   and e' :: "'addr expr1" shows "\<tau>move2 P h stk (try e catch(C V) e') (Suc (length (compE2 e))) None"
sorted_list_of_set_sort_remdups: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq (set xs) = linorder.sort_key less_eq (\<lambda>x. x) (remdups xs)"
dual_dual: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual (dual x) = x"
completeness: fixes p :: "'b fm" assumes "\<forall>M s. M, s \<Turnstile> p" shows "\<turnstile> p"
prod_cases6: fixes y :: "'a \<times> 'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'d" and e :: "'e" and f :: "'f" where "y = (a, b, c, d, e, f)"
nAct_def: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and n :: "enat"   and t :: "cnf llist" shows "dynamic_component.nAct tCMP active c n t \<equiv> llength (dynamic_component.proj tCMP active c (ltake n t))"
square_free_multD_1: fixes f :: "'c poly"   and g :: "'c poly"   and h :: "'c poly" assumes "square_free (f * g)"   and "h dvd f"   and "h dvd g" shows "degree h = 0"
r_univ: fixes i :: "nat"   and f :: "recf"   and x :: "nat" assumes "i = encode f"   and "recfn (e_length x) f" shows "eval r_univ [i, x] = eval f (Partial_Recursive.list_decode x)"
pp_init_invar: shows "Labeling c s t pp_init_f pp_init_l"
holomorphic_Hadjicostas_integral: shows "Hadjicostas_integral holomorphic_on {z. - 2 < Re z}"
split_list_long: fixes us :: "'b list"   and u :: "'b" assumes "1 < \<^bold>|us\<^bold>|"   and "u \<in> set us" obtains xs :: "'b list" and ys :: "'b list" where "us = xs \<cdot> [u] \<cdot> ys" and "xs \<cdot> ys \<noteq> \<epsilon>"
finite_proots: fixes p :: "'b poly"   and s :: "'b set" assumes "p \<noteq> 0" shows "finite (proots_within p s)"
Prop1B: fixes c :: "OBJ" assumes "secure c" shows " \<Turnstile> c : Sec (\<lambda>(r, t, \<beta>). \<exists>s. s , c \<Down> r  \<and> s \<equiv>\<^sub>\<beta> t) "
continuous_on_cong: fixes s :: "'d set"   and t :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "s = t"   and "\<And>x. x \<in> t \<Longrightarrow> f x = g x" shows "continuous_on s f = continuous_on t g"
prop_in_f_3_c: fixes v :: "i" shows "[\<^bold>\<not>Indiscriminate O! in v]"
find_sig_reducer_SomeD_red_single: fixes t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and i :: "nat" assumes "t \<in> keys (rep_list p)"   and "find_sig_reducer (map spp_of bs) (lt p) t 0 = Some i" shows "sig_red_single (\<prec>\<^sub>t) (\<preceq>) p (p - monom_mult (lookup (rep_list p) t / punit.lc (rep_list (bs ! i))) (t - punit.lt (rep_list (bs ! i))) (bs ! i)) (bs ! i) (t - punit.lt (rep_list (bs ! i)))"
eqf: shows "{w \<in> space M. \<not> w !! 0} = pseudo_proj_True (Suc 0) -` {False ## sconst True}"
Suc_lessE: fixes i :: "nat"   and k :: "nat" assumes "Suc i < k" obtains j :: "nat" where "i < j" and "k = Suc j"
Sigma_Int_Paircomp1: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and P :: "'a \<Rightarrow> bool" shows "Sigma A B \<inter> {(x, y). P x} = Sigma (A \<inter> {x. P x}) B"
g_def: shows "g \<equiv> \<lambda>x. if x \<in> S then f x else supp_sum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>"
zero_fun_apply: fixes x :: "'c" shows "0 x = (0::'b)"
bind_def: fixes x :: "('c, 'e) state"   and f :: "'e \<Rightarrow> ('c, 'd) state" shows "x \<bind> f = State (\<lambda>s. case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
cong_trans: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a"   and d :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a"   and "unique_euclidean_semiring.cong modulo c d a" shows "unique_euclidean_semiring.cong modulo b d a"
RedTryFail: fixes s :: "heap \<times> sfields \<times> sheap"   and a :: "nat"   and D :: "char list"   and fs :: "Objects.fields"   and C :: "char list"   and P :: "(char list list \<times> char list exp) prog"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and b :: "bool" assumes "hp s a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>try Throw a catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>Throw a,s,b\<rangle>"
typ_of1_weaken: fixes Ts :: "typ list"   and t :: "Core.term"   and T :: "typ"   and Ts' :: "typ list" assumes "typ_of1 Ts t = Some T" shows "typ_of1 (Ts @ Ts') t = Some T"
f_eq_zero_iff: fixes x :: "real" shows "(f x = 0) = (x \<le> 0)"
enatSuc0: fixes x :: "enat" shows "enat (Suc 0) * x = x"
basic_subtype_x_Boolean: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B Boolean" obtains "\<tau> = OclVoid"
rv_plus_rv: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "(\<lambda>w. f w + g w) \<in> rv M"
srtc_transitive: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "relation.srtc r a b"   and "relation.srtc r b c" shows "relation.srtc r a c"
sel_3: fixes x1 :: "'c"   and x2 :: "'d"   and x3 :: "nat"   and x4 :: "('c, 'd) BinomialTree list" shows "rank (Node x1 x2 x3 x4) = x3"
code_unique_ref: fixes \<C> :: "'b list set"   and us :: "'b list list" assumes "code \<C>"   and "us \<in> lists \<langle>\<C>\<rangle>" shows "Ref \<C> us = Dec \<C> concat us"
T: fixes i :: "'a" assumes "i \<in> I" shows "T i \<subseteq> topspace X"
induct: fixes P :: "'a prog"   and x1 :: "String.literal"   and x2 :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option"   and Pa :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option) \<Rightarrow> bool" assumes "P \<turnstile> x1 sees_methods x2"   and "\<And>D fs ms Mm. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms\<rbrakk> \<Longrightarrow> Pa Object Mm"   and "\<And>C D fs ms Mm Mm'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D sees_methods Mm; Pa D Mm; Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)\<rbrakk> \<Longrightarrow> Pa C Mm'" shows "Pa x1 x2"
comp_in_homE: fixes g :: "'a"   and f :: "'a"   and a :: "'a"   and c :: "'a" assumes "\<guillemotleft>g \<cdot>\<^sub>C f : a \<rightarrow>\<^sub>C c\<guillemotright>" obtains b :: "'a" where "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>C c\<guillemotright>"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "connex A less_eq"   and "transitive A less_eq" shows "total_quasi_ordered_set A less_eq"
case_cong: fixes ename :: "ename"   and ename' :: "ename"   and f1 :: "vname \<Rightarrow> 'a"   and g1 :: "vname \<Rightarrow> 'a"   and f2 :: "'a"   and g2 :: "'a" assumes "ename = ename'"   and "\<And>x1. ename' = VNam x1 \<Longrightarrow> f1 x1 = g1 x1"   and "ename' = Res \<Longrightarrow> f2 = g2" shows "(case ename of VNam x1 \<Rightarrow> f1 x1 | Res \<Rightarrow> f2) = (case ename' of VNam x1 \<Rightarrow> g1 x1 | Res \<Rightarrow> g2)"
image_add_0: fixes S :: "'c set" shows "(+) (0::'c) ` S = S"
cnj: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla" shows "cnj \<phi>1 \<phi>2 \<in> fmla"
all_gval_not_false: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "(\<forall>g\<in>set G. gval g s \<noteq> false) = (\<forall>g\<in>set G. gval g s = true) \<or> (\<exists>g\<in>set G. gval g s = invalid)"
distinct_replacefacesAt_norm: fixes i :: "nat"   and Fss :: "face list list"   and "is" :: "nat list"   and newFs :: "face list"   and olfF :: "face" assumes "i < |Fss|"   and "distinct is"   and "distinct (normFaces (Fss ! i))"   and "distinct (normFaces newFs)"   and "set (normFaces (Fss ! i)) \<inter> set (normFaces newFs) \<subseteq> {}" shows "distinct (normFaces (replacefacesAt is olfF newFs Fss ! i))"
i: shows "i < dim_row (mat_of_rows (dim_col A) (map (Matrix.row A) [0..<k]) @\<^sub>r mat_of_rows (dim_col A) (map (Matrix.row A) [k..<dim_row A]))"
size_Diff_singleton_if: fixes A :: "'b multiset"   and x :: "'b" shows "size (remove1_mset x A) = (if x \<in># A then size A - 1 else size A)"
v_nS12: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and cfg :: "'a cfg" assumes "Reachability_Problem K S S1 S2"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "state cfg \<notin> S1"   and "state cfg \<notin> S2" shows "Reachability_Problem.v K S S1 S2 cfg = 0"
monad_alt_stateT': assumes "monad_alt Monad_Overloading.return Monad_Overloading.bind alt" shows "monad_alt Monad_Overloading.return Monad_Overloading.bind alt"
upper_triangular_invertible: fixes A :: "((real, 'a) vec, 'a) vec" assumes "upper_triangular A"   and "\<forall>i. A $ i $ i \<noteq> 0" shows "invertible A"
expand_inv_def: fixes \<phi> :: "'b ltlr"   and n_ns :: "('b, 'c) node_scheme \<times> ('b, 'd) node_scheme set" shows "expand_inv \<phi> n_ns \<equiv> case n_ns of (n, ns) \<Rightarrow> expand_inv_node \<phi> n \<and> expand_inv_result \<phi> ns"
ultra_filter_prime: fixes F :: "'a set" assumes "ultra_filter F" shows "prime_filter F"
bin_last_AND: fixes x :: "int"   and y :: "int" shows "odd (x AND y) = (odd x \<and> odd y)"
morphism_ringed_spaces_axioms: shows "morphism_ringed_spaces S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S is_open \<FF> \<rho> b add_str mult_str zero_str one_str (identity S) (\<lambda>U. identity (\<FF> U))"
fres_galois: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "less_eq_prod (times_prod' x y) z = less_eq_prod x (prod.fres z y)"
simps: fixes x :: "real"   and y :: "real" shows "(Rmax x \<le> Rmax y) = (x \<le> y)"
min_elt_minimal: fixes B :: "'f set"   and y :: "'f" assumes "B \<subseteq> UNIV"   and "B \<noteq> {}"   and "y \<in> UNIV"   and "Prec_F_g D y (min_elt B)" shows "y \<notin> B"
delete_append: fixes x :: "'a"   and l1 :: "('a \<times> 'b) list"   and l2 :: "('a \<times> 'b) list" shows "delete x (l1 @ l2) = delete x l1 @ delete x l2"
abs_mod_less: fixes l :: "int"   and k :: "int" assumes "l \<noteq> 0" shows "\<bar>k mod l\<bar> < \<bar>l\<bar>"
symclp_pointfree: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "symclp r = sup r r\<inverse>\<inverse>"
uadmit_prog_fml_ntadjoint'_1: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<alpha> :: "('sf + 'b, 'sc, 'sz) hp" assumes "\<And>i. dsafe (\<sigma> i)"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>x\<in>{x. Inl (Inr x) \<in> SIGP \<alpha>}. FVT (\<sigma> x))"   and "hpsafe \<alpha>" shows "prog_sem (adjointFO I \<sigma> \<nu>) \<alpha> = prog_sem (adjointFO I \<sigma> \<omega>) \<alpha>"
resOutputInputTrans: fixes x :: "name"   and a :: "name"   and b :: "name"   and P :: "pi"   and c :: "name"   and y :: "name"   and P' :: "pi" assumes "<\<nu>x>a{b}.P \<longmapsto> c<y> \<prec> P'" shows "False"
construct_lhs_vector_drop_consistent_R: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "vec_of_list (mtx_row_R signs I) \<bullet> w_vec_R p qs signs = vec_of_list (mtx_row_R (characterize_consistent_signs_at_roots p qs) I) \<bullet> w_vec_R p qs (characterize_consistent_signs_at_roots p qs)"
config_append: fixes step :: "'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and A :: "('a \<Rightarrow> 'b) \<times> ('a \<times> 'b \<Rightarrow> 'c \<Rightarrow> 'd \<times> 'b)"   and s :: "'a"   and xs :: "'c list"   and ys :: "'c list" shows "On_Off.config' step A (s, fst A s) (xs @ ys) = On_Off.config' step A (On_Off.config' step A (s, fst A s) xs) ys"
length_sort: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "length (sort_key f xs) = length xs"
similar_mat_wit_sym: fixes A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit B A Q P"
iso_incl_partial_order: shows "partial_order_on \<S> (relation_of (\<lesssim>) \<S>)"
map_restrict_set_def: fixes A :: "'c set" shows "map_restrict_set A = map_filter (\<lambda>a. a \<in> A)"
span_image: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and S :: "'e set" assumes "module_hom s1 s2 f" shows "module.span s2 (f ` S) = f ` module.span s1 S"
invar_def: fixes bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.invar bq = (SkewBinomialHeapStruc.queue_invar bq \<and> SkewBinomialHeapStruc.rank_skew_invar bq)"
vector_1_comm: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "is_vector y" shows "(x \<cdot> y) ; z = x ; z \<cdot> y"
Snd: fixes x :: "ZF"   and y :: "ZF" shows "Snd (Opair x y) = y"
run_foldl: fixes \<Delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "run \<Delta> q\<^sub>0 w i = foldl \<Delta> q\<^sub>0 (map w [0..<i])"
map_of_set_eq': fixes a :: "'a set"   and b :: "'a set" shows "(map_of_set a = map_of_set b) = (a = b)"
poly_of_const_over_carrier: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "ring.poly_of_const R ` carrier R = {p \<in> carrier (carrier R [X]\<^bsub>R\<^esub>). degree p = 0}"
distinct_51: fixes x3 :: "nop_type"   and x8 :: "shift_type" shows "nop_type x3 \<noteq> shift_type x8"
\<omega>_limit_set_in_compact_subset: shows "\<omega>_limit_set x \<subseteq> K"
rel_carrier_Eps_in_2: fixes X :: "('a set \<times> 'b) set" assumes "X \<in> carrier_direct_lim" shows "(SOME x. x \<in> X) \<in> Sigma I \<FF>"
valid_ConsE: fixes f :: "recf"   and xs :: "nat list"   and locs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((f, xs, locs) # rest)" shows "valid rest \<and> recfn (length xs) f"
suffix_subseq_join: fixes i :: "nat"   and j :: "nat"   and v :: "nat \<Rightarrow> 'b" assumes "i \<le> j" shows "v [i \<rightarrow> j] \<frown> suffix j v = suffix i v"
higher_differentiable_on_Pair: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and k :: "nat"   and g :: "'b \<Rightarrow> 'd" assumes "open S"   and "higher_differentiable_on S f k"   and "higher_differentiable_on S g k" shows "higher_differentiable_on S (\<lambda>x. (f x, g x)) k"
sum_mod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> f a mod b = (0::'b)" shows "sum f A mod b = (0::'b)"
Bij_def: fixes S :: "'e set" shows "Bij S = extensional S \<inter> {f. bij_betw f S S}"
assms: shows "x - y = 2 * real_of_int k * pi + pi \<or> x + y = 2 * real_of_int k * pi + pi"
extended_Gromov_product_inside: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "extended_Gromov_product_at e (to_Gromov_completion x) (to_Gromov_completion y) = ereal (Gromov_product_at e x y)"
simps_3: fixes x :: "'b" shows "norm_instr (Inca.instr.ILoad x) = Std.instr.ILoad x"
ldropn_eq_LNil: fixes n :: "nat"   and xs :: "'b llist" shows "(ldropn n xs = LNil) = (llength xs \<le> enat n)"
disjunctive_def: shows "Comp.disjunctive = {x. \<forall>y z. x \<circ> sup y z = sup (x \<circ> y) (x \<circ> z)}"
map_iterator_genord_I: fixes l0 :: "('d \<times> 'e) list"   and m :: "'d \<Rightarrow> 'e option"   and R :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and iti :: "('f \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f) \<Rightarrow> 'f \<Rightarrow> 'f" assumes "distinct (map fst l0)"   and "m = map_of l0"   and "sorted_wrt R l0"   and "iti = foldli l0" shows "map_iterator_genord iti m R"
Suc_1: assumes "mod_type_class.to_nat i < k"   and "\<not> is_zero_row i A"   and "k \<le> nrows A" shows "Hermite_of_upt_row_i A k ass res $ i $ (LEAST n. A $ i $ n \<noteq> (0::'b)) \<in> range ass"
x: shows "x \<in> {fs ! i $ j |i j. i < m \<and> j < n} \<union> {d\<mu> fs i j |i j. j < i \<and> i < m} \<union> {d fs i |i. i \<le> m} \<union> {fs.\<mu>' i j |i j. j \<le> i \<and> i < m} \<union> {fs.\<sigma>s l i j |i j l. i < m \<and> j \<le> i \<and> l < j}"
rcos_inv: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "H \<lhd> G"   and "x \<in> carrier G" shows "set_inv\<^bsub>G\<^esub> (H #>\<^bsub>G\<^esub> x) = H #>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x"
int_simps_2: shows "\<turnstile> (\<not> #True) = #False"
E1: shows "E1 = (\<lambda>s. 1 + (4 + 3 * nat (s ''b'')) * (nat (s ''a'') - nat (s ''z'')))"
simps: fixes fs :: "face list"   and n :: "nat"   and f :: "face list list"   and h :: "nat list" shows "countVertices (Graph fs n f h) = n"
traverse_tree_append: fixes path :: "dir list"   and ext :: "dir list"   and t :: "'a tree" shows "traverse_tree (path @ ext) t = traverse_tree ext (traverse_tree path t)"
dis_rews_12: shows "is_EQ\<cdot>GT = FF"
poincare_map_def: fixes P :: "'d set"   and x :: "'d" shows "poincare_map P x = flow0 x (return_time P x)"
surjective: fixes r :: "'a point4d_scheme" shows "r = \<lparr>xc = xc r, yc = yc r, zc = zc r, wc = wc r, \<dots> = point4d.more r\<rparr>"
subcring_iff: fixes R :: "('c, 'd) ring_scheme"   and H :: "'c set" assumes "ring R"   and "H \<subseteq> carrier R" shows "subcring H R = cring (R\<lparr>carrier := H\<rparr>)"
AssertAllState'_sat_atoms_equiv_bounds: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list" assumes "\<triangle> t"   and "\<not> \<U> (assert_all_state t as)" shows "(snd ` set as) \<doteq> \<B> (assert_all_state t as)"
distinct_card: fixes xs :: "'b list" assumes "distinct xs" shows "card (set xs) = length xs"