hcju/isabelleps huggingface dataset:
TComp_var_and_subterm_instance_closed_has_subterms_instances: fixes M :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and \<delta> :: "('fun, 'atom) Term.term \<times> nat \<Rightarrow> ('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "is_TComp_var_instance_closed \<Gamma> M"   and "has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set M)) (set M)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"   and "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set M"   and "s \<sqsubseteq> t \<cdot> \<delta>"   and "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "\<exists>m\<in>set M. \<exists>\<theta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>) \<and> s = m \<cdot> \<theta>"
nth_Cons_numeral_1: fixes x :: "'b"   and xs :: "['b]" shows "(x : xs) !! 0 = x"
B: shows "insert (All2 i t A) (insert (A(i::=u)) H) \<turnstile> B"
vinsert_not_empty: fixes a :: "V"   and A :: "V" shows "vinsert a A \<noteq> 0"
mat_tensor_prod_2_col: fixes v :: "complex Matrix.mat"   and w :: "complex Matrix.mat" assumes "state 1 v"   and "state 1 w" shows "Matrix.col (v \<Otimes> w) 0 = Matrix.col v 0 \<otimes> Matrix.col w 0"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 \<mu> (\<nu> \<star> \<pi>)"
append_eq_append_conv_if: fixes xs\<^sub>1 :: "'g list"   and xs\<^sub>2 :: "'g list"   and ys\<^sub>1 :: "'g list"   and ys\<^sub>2 :: "'g list" shows "(xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>1 @ ys\<^sub>2) = (if length xs\<^sub>1 \<le> length ys\<^sub>1 then xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1 \<and> xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2 else take (length ys\<^sub>1) xs\<^sub>1 = ys\<^sub>1 \<and> drop (length ys\<^sub>1) xs\<^sub>1 @ xs\<^sub>2 = ys\<^sub>2)"
f_Exec_Comp_Stream_Acc_Output_Init_def: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun xs c \<equiv> output_fun c # f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c"
all_coarser_partitions_with_def: fixes elem :: "'b"   and Ps :: "'b set set set" shows "all_coarser_partitions_with elem Ps = \<Union> (coarser_partitions_with elem ` Ps)"
not_mod_2_eq_0_eq_1: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(modulo a (numeral.numeral one plus (num.Bit0 num.One)) \<noteq> zero) = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = one)"
evaln_elim_cases_23: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and n :: "nat"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (Throw e)\<succ>\<midarrow>n\<rightarrow> (x, s')" obtains a' :: "val" and s1 :: "state" where "x = \<diamondsuit>" and "s' = abupd (throw a') s1" and "G\<turnstile>Norm s \<midarrow>e-\<succ>a'\<midarrow>n\<rightarrow> s1"
assms: shows "\<not> is_Prim\<^sub>0 t"
simps_1: fixes D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" shows "tr\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s [] D = [[]]"
IH_1: fixes w :: "nat \<Rightarrow> 'a set" assumes "w \<Turnstile> \<phi>" shows "\<exists>j. \<G>\<^sub>F\<^sub>G \<phi> w \<Turnstile>\<^sub>P af\<^sub>G \<phi> (prefix j w)"
pred_set: fixes P :: "'a \<Rightarrow> bool" shows "pred_option P = (\<lambda>x. Ball (set_option x) P)"
homeomorphism_arc: fixes g :: "real \<Rightarrow> 'c" assumes "arc g" obtains h :: "'c \<Rightarrow> real" where "homeomorphism {0..1} (path_image g) g h"
filter_min_relE: fixes rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" assumes "transp rel"   and "reflp rel"   and "x \<in> set xs" obtains y :: "'c" where "y \<in> set (filter_min rel xs)" and "rel y x"
compact_below_lub_iff: fixes x :: "'a"   and Y :: "nat \<Rightarrow> 'a" assumes "compact x"   and "chain Y" shows "(x \<sqsubseteq> (\<Squnion>i. Y i)) = (\<exists>i. x \<sqsubseteq> Y i)"
pivot_and_update_code_def: shows "pivot_and_update_code = PivotUpdate.pivot_and_update pivot_code update_code"
to_tree_order_pure: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_to_tree_order to_tree_order" shows "pure (to_tree_order ptr) h"
f': fixes x :: "complex"   and T :: "complex set" assumes "x \<in> A" shows "(f has_field_derivative deriv f x) (at x within T)"
intros_2: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "t1 \<lesssim> t2" shows "VFun t1 \<sqsubseteq> VFun t2"
swapc: shows "basis_reduction_swap_cost i state' = (state'', c_swap)"
finitesum_mOp_closed: fixes f :: "'f \<Rightarrow> 'a set"   and I :: "'f set"   and a :: "'a" assumes "f \<in> I \<rightarrow> {X. submodule R M X}"   and "I \<noteq> {}"   and "a \<in> finitesum M I f" shows "-\<^sub>a a \<in> finitesum M I f"
broadcastT: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" shows "((\<xi>, {l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p), broadcast (s\<^sub>m\<^sub>s\<^sub>g \<xi>), \<xi>, p) \<in> seqp_sos \<Gamma>"
smult_eq_0_iff: fixes a :: "'c"   and p :: "'c poly" shows "(smult a p = 0) = (a = (0::'c) \<or> p = 0)"
single_subset_iff: fixes a :: "'b"   and M :: "'b multiset" shows "(a, {#} \<subseteq># M) = (a \<in># M)"
power_mult: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m * n) = power.power one times (power.power one times a m) n"
map2_val_compatI: fixes f :: "('a \<times> 'd) list \<Rightarrow> ('a \<times> 'e) list" assumes "\<And>zs. oalist_inv_raw zs \<Longrightarrow> oalist_inv_raw (f zs)"   and "\<And>zs. oalist_inv_raw zs \<Longrightarrow> fst ` set (f zs) \<subseteq> fst ` set zs" shows "map2_val_compat f"
isSym_sessionK: fixes nonces :: "(nat \<times> nat \<times> nat) \<times> role" shows "sessionK nonces \<in> symKeys"
ANR_sphere: fixes a :: "'b"   and r :: "real" shows "ANR (sphere a r)"
wtE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wtE \<xi>" shows "GE.wtE \<xi>"
continuous: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and F :: "'e filter"   and f :: "'e \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c" assumes "bounded_bilinear prod"   and "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. prod (f x) (g x))"
elims: fixes x :: "('b, 'c, 'a) namespace"   and xa :: "'c"   and y :: "'a option" assumes "cupcake_nsLookup x xa = y" obtains v1 :: "('c \<times> 'a) list" and uu_ :: "('b \<times> ('b, 'c, 'a) namespace) list" and n :: "'c" where "x = Bind v1 uu_" and "xa = n" and "y = map_of v1 n"
x_eqn: shows "x ^ 3 - 3 * x - 1 = 0"
columnvector_row_def: fixes v :: "('a, 'c) vec"   and i :: "'c" shows "columnvector_row v i = (\<chi>j. v $ i)"
lcmI: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b c"   and "\<And>d. \<lbrakk>dvd.dvd times a d; dvd.dvd times b d\<rbrakk> \<Longrightarrow> dvd.dvd times c d"   and "normalize c = c" shows "c = lcm a b"
pfun_app_minus: fixes x :: "'a"   and g :: "('a, 'b) pfun"   and f :: "('a, 'b) pfun" assumes "x \<notin> pdom g" shows "(f - g)(x)\<^sub>p = f(x)\<^sub>p"
normalize_alt_simps_10: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and call :: "'a call_expr"   and b :: "'a expr" shows "(\<Gamma> \<turnstile> Call src SafeArrowCall call \<Rrightarrow> b) = (\<exists>src\<^sub>2 \<tau> \<sigma> \<rho> call\<^sub>2. b = Call (TypeOperationCall src\<^sub>2 ArrowCall SelectByKindOp (to_required_type \<sigma>)) ArrowCall call\<^sub>2 \<and> \<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2 \<and> \<Gamma> \<turnstile>\<^sub>E src\<^sub>2 : \<tau> \<and> element_type \<tau> \<sigma> \<and> OclVoid[?] \<le> \<sigma> \<and> \<Gamma> \<turnstile>\<^sub>E TypeOperationCall src\<^sub>2 ArrowCall SelectByKindOp (to_required_type \<sigma>) : \<rho> \<and> (\<Gamma>, \<rho>) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2)"
rep_eq: fixes x :: "'a mat" shows "dim_row x = fst (Rep_mat x)"
fmupd_lookup: fixes a :: "'d"   and b :: "'c"   and m :: "'d \<rightharpoonup>\<^sub>f 'c"   and a' :: "'d" shows "fmlookup (m(a \<mapsto>\<^sub>f b)) a' = (if a = a' then Some b else fmlookup m a')"
equivalence_def: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equivalence A equiv \<equiv> symmetric A equiv \<and> reflexive A equiv \<and> transitive A equiv"
induct: fixes x1 :: "ccs"   and x2 :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and P :: "ccs \<Rightarrow> ccs \<Rightarrow> bool" assumes "(x1, x2) \<in> bangRel Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P !P !Q"   and "\<And>R T P Q. \<lbrakk>(R, T) \<in> Rel; (P, Q) \<in> bangRel Rel; P P Q\<rbrakk> \<Longrightarrow> P (R \<parallel> P) (T \<parallel> Q)" shows "P x1 x2"
nn_integral_fst: fixes f :: "'b \<times> 'a \<Rightarrow> ennreal"   and M1 :: "'b measure" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M N)" shows "\<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>N \<partial>M1 = integral\<^sup>N (M1 \<Otimes>\<^sub>M N) f"
Q: fixes n :: "nat"   and a :: "'a"   and k :: "nat" assumes "(n, a) \<in> set n_as" shows "Q n a k c"
fps_oner: fixes f :: "('c, 'd) fps" shows "f \<cdot> 1 = f"
parallel_reduces_diamond: fixes A :: "'a trm"   and B :: "'a trm"   and C :: "'a trm" assumes "A >>\<^sup>* B"   and "A >>\<^sup>* C" shows "\<exists>D. B >>\<^sup>* D \<and> C >>\<^sup>* D"
Cons_1: fixes state :: "State"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "InvariantConsistent (getM state)"   and "\<forall>c. c \<in> set Wl' \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "literalFalse literal (elements (getM state))"   and "uniq Wl'" shows "let state' = notifyWatches_loop literal Wl' newWl state in getConflictFlag state' = (getConflictFlag state \<or> (\<exists>clause. clause \<in> set Wl' \<and> clauseFalse (getF state ! clause) (elements (getM state))))"
I: shows "I \<subseteq> {..<brn (cl ! n)}"
axioms_1: fixes r :: "V" assumes "vsv r" shows "vbrelation r"
inv_solve_left: fixes a :: "'a \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'b"   and c :: "'a \<Rightarrow> 'b" assumes "a \<in> carrier (DirProds G I)"   and "b \<in> carrier (DirProds G I)"   and "c \<in> carrier (DirProds G I)" shows "(a = inv\<^bsub>DirProds G I\<^esub> b \<otimes>\<^bsub>DirProds G I\<^esub> c) = (c = b \<otimes>\<^bsub>DirProds G I\<^esub> a)"
summable_power_series: fixes f :: "nat \<Rightarrow> real"   and z :: "real" assumes "\<And>i. f i \<le> 1"   and "\<And>i. 0 \<le> f i"   and "0 \<le> z"   and "z < 1" shows "summable (\<lambda>i. f i * z ^ i)"
fid: fixes x :: "'a" assumes "x \<in> T" shows "f x = x"
u_ts_i: shows "u_ts ! i = (u_p, u_is, u_tmps, u_x, u_dirty, u_owns, u_rels)"
vsv_VLambda_cong: fixes f :: "V \<Rightarrow> V" assumes "\<And>a. a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<Longrightarrow> r\<lparr>a\<rparr> = f a" shows "VLambda (\<D>\<^sub>\<circ> r) f = r"
compl_sets: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and a :: "'b set" assumes "algebra \<Omega> M"   and "a \<in> M" shows "\<Omega> - a \<in> M"
pderiv_eval_deriv: fixes R :: "('a, 'b) ring_scheme"   and f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "a \<in> carrier R" shows "derivative R f a = UP_cring.to_fun R (UP_cring.pderiv R f) a"
inf_sup_aci_6: fixes 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 x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (sup (x::'b) (y::'b)) (z::'b) = sup x (sup y z)"
rep_eq: fixes x :: "'a mat" shows "dim_col x = fst (snd (Rep_mat x))"
contains_def: fixes f :: "nat \<Rightarrow> nat \<times> (nat \<times> form) list"   and n :: "nat"   and nf :: "nat \<times> form" shows "contains f n nf = (nf \<in> set (snd (f n)))"
divide_nonneg_pos: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less zero y" shows "less_eq zero (divide x y)"
base_in_hom_1: shows "\<guillemotleft>r \<star> s : src (r \<star> s) \<rightarrow> trg (r \<star> s)\<guillemotright>"
card_greaterThanLessThan: fixes l :: "nat"   and u :: "nat" shows "card {l<..<u} = u - Suc l"
w0_rewrite: fixes u0 :: "nat"   and v0 :: "nat" shows "\<^bold>g [^] (r * u0 + v0) = (\<^bold>g [^] r) [^] u0 \<otimes> \<^bold>g [^] v0"
eval_red2_naturality: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u" shows "\<lbrace>Cod t \<^bold>\<Down> Cod u\<rbrace> \<cdot> (\<lbrace>t\<rbrace> \<otimes> \<lbrace>u\<rbrace>) = \<lbrace>t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u\<rbrace> \<cdot> \<lbrace>Dom t \<^bold>\<Down> Dom u\<rbrace>"
cases: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a \<times> 'a" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize" obtains a :: "'a" and b :: "'a" where "x = (a, b)"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "fds_nth (f x) 1 \<noteq> (0::'b)"
eqp: fixes x :: "'a \<times> nat" assumes "x \<in> set (zip xs [0..<length xs])" shows "(snd x \<in> B) = (snd x \<in> A \<inter> B)"
lset_lnths: fixes xs :: "'c llist"   and I :: "nat set" shows "lset (lnths xs I) = {lnth xs i |i. enat i < llength xs \<and> i \<in> I}"
inf_Int_eq: fixes R :: "'g set"   and S :: "'g set" shows "inf (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<inter> S)"
poly_mult_comm: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_mult p1 p2 = poly_mult p2 p1"
A11: shows "[And D (abstr g (\<lambda>i j. \<infinity>) v)]\<^bsub>v,n\<^esub> = [D]\<^bsub>v,n\<^esub> \<inter> {u. u \<turnstile> g}"
coeff_not_in_monomials: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and m :: "'b mpoly" assumes "x \<notin> monomials m" shows "coeff m x = (0::'b)"
rec_iter_def: fixes f :: "recf" shows "rec_iter f = PR (recf.Id 1 0) (CN f [recf.Id 3 1])"
cong: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "f = g"   and "x = y" shows "f x = g y"
F_eq: shows "{f. \<exists>s\<in>simplices. face f s} = (\<Union>s\<in>simplices. \<Union>a\<in>s. {s - {a}})"
eq_conv_compare: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering"   and x :: "'b"   and y :: "'b" assumes "class.Ord_linear eq compare" shows "eq\<cdot>x\<cdot>y = is_EQ\<cdot>(compare\<cdot>x\<cdot>y)"
rel_fun_eq_onp_rel: fixes R :: "'b \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (eq_onp R) S = (\<lambda>f g. \<forall>x. R x \<longrightarrow> S (f x) (g x))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F F F\<eta>'o\<epsilon>'F.map"
is_supI: fixes x :: "'c"   and sup :: "'c"   and y :: "'c" assumes "x \<sqsubseteq> sup"   and "y \<sqsubseteq> sup"   and "\<And>z. \<lbrakk>x \<sqsubseteq> z; y \<sqsubseteq> z\<rbrakk> \<Longrightarrow> sup \<sqsubseteq> z" shows "is_sup x y sup"
vsv_vimage_eqI: fixes a :: "V"   and Aa :: "V"   and b :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> Aa)"   and "(r \<restriction>\<^sup>l\<^sub>\<circ> Aa)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> A" shows "b \<in>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> Aa) `\<^sub>\<circ> A"
fresh_atm_4: fixes a :: "W.var"   and b :: "W.tvar" shows "a \<sharp> b"
dirichlet_prod_neutral_left_neutral: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex" assumes "0 < n" shows "dirichlet_prod dirichlet_prod_neutral f n = f n"
tendsto_upperbound: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b"   and F :: "'d filter"   and a :: "'b" assumes "class.linorder_topology less_eq less open"   and "filterlim f (topological_space.nhds open x) F"   and "\<forall>\<^sub>F i in F. less_eq (f i) a"   and "F \<noteq> bot" shows "less_eq x a"
log_powr_cancel: fixes a :: "real"   and y :: "real" assumes "0 < a"   and "a \<noteq> 1" shows "Transcendental.log a (a powr y) = y"
diagonal_step_JNF_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat"   and d :: "'b"   and v :: "'b" shows "diagonal_step_JNF A i j d v = Matrix.mat (dim_row A) (dim_col A) (\<lambda>(a, b). if a = i \<and> b = i then d else if a = j \<and> b = j then v * A $$ (j, j) else A $$ (a, b))"
nonzero_Reals_divide: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<real>"   and "b \<in> \<real>"   and "b \<noteq> (0::'b)" shows "a / b \<in> \<real>"
mset_un_cases: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" assumes "a \<in># A + B" obtains "a \<in># A"   | "a \<in># B"
norm_Pair2: fixes x :: "'d" shows "norm (x, 0::'e) = norm x"
assms: shows "\<exists>x. \<alpha>\<^sub>1 - \<alpha>\<^sub>2 = 2 * real_of_int x * pi"
states_of_se_assume: fixes c :: "('c, 'd) conf"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and c' :: "('c, 'd) conf" assumes "se c (Assume e) c'" shows "states c' = {\<sigma> \<in> states c. e \<sigma>}"
plus_Ord_le: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<in> elts \<omega>"   and "Ord \<beta>" shows "\<alpha> + \<beta> \<le> \<beta> + \<alpha>"
not_inj_on: shows "\<not> inj_on f {0..<n}"
select_convs_2: fixes view :: "'f \<Longrightarrow> 'e"   and coview :: "'d \<Longrightarrow> 'e"   and more :: "'g" shows "\<C>\<^bsub>\<lparr>view = view, coview = coview, \<dots> = more\<rparr>\<^esub> = coview"
\<phi>_in_terms_of_\<eta>: fixes y :: "'b"   and f :: "'a"   and x :: "'a" assumes "Adj.D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi>' y f = G' f \<cdot>\<^sub>D Adj'.\<eta> y"
fn: fixes A :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "NF A" shows "NF (Fn x T A)"
lconcat_llist_of: fixes xs :: "'a list list" shows "lconcat (llist_of (map llist_of xs)) = llist_of (concat xs)"
sub_inter_left2: fixes B :: "funty"   and C :: "funty"   and A :: "funty" assumes "B <:: C" shows "A \<sqinter> B <:: C"
frequently_at: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and S :: "'b set" shows "frequently P (at a within S) = (\<forall>d>0. \<exists>x\<in>S. x \<noteq> a \<and> dist x a < d \<and> P x)"
sturm_seq_sturm_squarefree: fixes p :: "real poly" assumes "p \<noteq> 0" shows "sturm_seq (sturm_squarefree p) (p div gcd p (pderiv p))"
call_mono: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'b" shows "monotone (fun_ord ord) ord (\<lambda>f. f t)"
allDefs_var_disjoint: fixes n :: "'b"   and g :: "'a"   and v :: "'e"   and v' :: "'e" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> allDefs g n"   and "v' \<in> allDefs g n"   and "v \<noteq> v'" shows "var g v' \<noteq> var g v"
tllist_ord_partial_function_definitions: fixes b :: "'g" shows "partial_function_definitions (tllist_ord b) (tSup b)"
magQ_qinverse: fixes x :: "'e['f, 'g]" shows "\<lbrakk>x\<^sup>-\<^sup>\<one>\<rbrakk>\<^sub>Q = inverse \<lbrakk>x\<rbrakk>\<^sub>Q"
mcont: fixes x :: "'b \<Rightarrow> 'a" assumes "x \<in> M" shows "mcont \<Or> (\<sqsubseteq>) Sup (\<le>) x"
lta__nlea: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LtA D E F" shows "\<not> D E F LeA A B C"
assms_3: shows "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans p"
lmeasurable_ball: fixes a :: "'b"   and r :: "real" shows "ball a r \<in> lmeasurable"
subst_set_map: fixes x :: "('a, 'b) Term.term"   and X :: "('a, 'b) Term.term list"   and s :: "'b \<Rightarrow> ('a, 'c) Term.term" assumes "x \<in> set X" shows "x \<cdot> s \<in> set (X \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t s)"
RObject_ext_def: fixes nothing :: "unit"   and more :: "'a" shows "\<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons RObject_ext_Tuple_Iso nothing more"
finite_INT: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "\<exists>x\<in>I. finite (A x)" shows "finite (\<Inter> (A ` I))"
plusU_Nothing_right: fixes xs :: "udom\<cdot>maybe" shows "plusU\<cdot>xs\<cdot>Nothing = xs"
component_def: fixes F :: "'c program"   and H :: "'c program" shows "(F \<le> H) = (\<exists>G. F \<squnion> G = H)"
closedin_closed_trans: fixes T :: "'c set"   and S :: "'c set" assumes "closedin (top_of_set T) S"   and "closed T" shows "closed S"
d: fixes x :: "real" assumes "\<bar>x\<bar> < d" shows "f x = g x"
ccBinds_strict: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "ccBinds \<Gamma>\<cdot>\<bottom> = \<bottom>"
set_cinsert: fixes x :: "'c"   and A :: "'c cset" assumes "x \<in>\<^sub>c A" obtains B :: "'c cset" where "A = cinsert x B" and "\<not> x \<in>\<^sub>c B"
dimv: shows "dim_vec (non_pivot_base A pp qj) = nc"
no_tick_before_suc: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and k :: "nat"   and c :: "clock" assumes "dilating f sub r"   and "f n < k \<and> k < f (Suc n)" shows "\<not> hamlet (Rep_run r k c)"