hcju/isabelleps huggingface dataset:
norm_bound: fixes f :: "'c \<Rightarrow>\<^sub>C 'd"   and b :: "real" assumes "\<And>x. norm (apply_bcontfun f x) \<le> b" shows "norm f \<le> b"
prems_1: fixes i :: "nat" assumes "i < Suc a" shows "length (f i) = d"
hermitean_elems_1: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "hermitean (A, B, C, D)" shows "is_real A"
A: shows "(\<lambda>n. f n * (0::'a) ^ n) = (\<lambda>n. if n = 0 then f n else (0::'a))"
TameEnum_comp: fixes p :: "nat"   and g :: "graph" assumes "Seed\<^bsub>p\<^esub> [next_plane\<^bsub>p\<^esub>]\<rightarrow>* g"   and "final g"   and "tame g" shows "Seed\<^bsub>p\<^esub> [next_tame\<^bsub>p\<^esub>]\<rightarrow>* g"
set_diff_list: fixes xs :: "'b list"   and ys :: "'b list" shows "set (xs -- ys) = set xs - set ys"
assms: shows "\<tau>.P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"
intro: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool" assumes "equivp eq"   and "(\<sim>\<^sub>C) \<le> eq"   and "eq \<le> (\<sim>\<^sub>L)" shows "ltl_equivalence eq"
E: shows "E = (\<lambda>s. 1 + 2 * (nat b + 1 - nat (s a)))"
word_length_lt: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and as :: "'b list"   and A :: "'b set"   and a :: "'b" assumes "class.monoid_add plus zero"   and "as \<in> lists A"   and "monoid_add.sum_list plus zero as = a"   and "\<not> monoid_add.reduced_word_for plus zero A a as" shows "order.greater (length as) (monoid_add.word_length plus zero A a)"
simps_1: fixes P :: "'e \<Rightarrow> bool"   and g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" shows "dropWhile_raw P g (True, s) = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip (True, s') | Yield a s' \<Rightarrow> if P a then Skip (True, s') else Yield a (False, s'))"
the_coext_prop_1: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f" shows "\<guillemotleft>x\<epsilon>.the_coext x' f : x' \<rightarrow>\<^sub>D Go x\<guillemotright>"
similar_sq_mtx_def: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" shows "(A \<sim> B) = (\<exists>P. mtx_invertible P \<and> A = P\<^sup>-\<^sup>1 * B * P)"
length_list_update: fixes xs :: "'f list"   and i :: "nat"   and x :: "'f" shows "length (xs[i := x]) = length xs"
inessential_spheremap_lowdim_gen: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "convex S"   and "bounded S"   and "convex T"   and "bounded T"   and "aff_dim S < aff_dim T"   and "continuous_on (rel_frontier S) f"   and "f ` rel_frontier S \<subseteq> rel_frontier T" obtains c :: "'b" where "homotopic_with_canon (\<lambda>z. True) (rel_frontier S) (rel_frontier T) f (\<lambda>x. c)"
abs_eq: fixes x :: "1 word \<times> 'a word \<times> 'b word" shows "sign (Abs_float x) = (case x of (s, uu_, uua_) \<Rightarrow> unat s)"
e_psp: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and A' :: "'d set"   and B :: "'d set"   and B' :: "'d set" assumes "F \<in> A leadsTo[CC] A'"   and "F \<in> B co B'"   and "\<forall>C\<in>CC. C \<inter> B \<inter> B' \<in> CC" shows "F \<in> (A \<inter> B') leadsTo[CC] (A' \<inter> B \<union> (B' - B))"
antisympD: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "antisymp r"   and "r a b"   and "r b a" shows "a = b"
nostatediff: fixes x :: "variable"   and \<nu> :: "variable \<Rightarrow> real"   and \<omega> :: "variable \<Rightarrow> real" assumes "x \<notin> statediff \<nu> \<omega>" shows "\<nu> x = \<omega> x"
dom_hcomp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"   and "span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu>" shows "partial_magma.dom (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1 \<nu> \<mu>) = span_bicategory.hcomp C prj0 prj1 (partial_magma.dom (span_vertical_category.vcomp C) \<nu>) (partial_magma.dom (span_vertical_category.vcomp C) \<mu>)"
invalid_assn_const: fixes P :: "assn"   and x :: "'c"   and y :: "'d" shows "invalid_assn (\<lambda>_ _. P) x y = \<up> (vassn_tag P) * true"
up_antitone: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (\<up>y \<subseteq> \<up>x)"
fun_relD2: fixes f :: "'g \<Rightarrow> 'h"   and f' :: "'i \<Rightarrow> 'j"   and Ra :: "('g \<times> 'i) set"   and Rr :: "('h \<times> 'j) set"   and x' :: "'i"   and r' :: "'j" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f' x' = r'" shows "\<forall>x. (x, x') \<in> Ra \<longrightarrow> (f x, r') \<in> Rr"
lambda_one: shows "(\<lambda>x. x) = (*) (1::'b)"
A: fixes ys :: "'a list"   and zs :: "'a list" shows "xs \<cong> {ys, zs, \<lambda>v vs. P v (vs @ ws)} \<longrightarrow> xs @ ws \<cong> {ys @ ws, zs, P}"
lex_conv: fixes r :: "('a \<times> 'a) set" shows "lex r = {(xs, ys). length xs = length ys \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y) \<in> r)}"
26: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> (x \<sqinter> z \<squnion> x \<sqinter> (y \<squnion> z)) = x \<sqinter> (y \<squnion> z)"
span_base: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and S :: "'c set" assumes "module scale"   and "a \<in> S" shows "a \<in> module.span scale S"
leg1_simps_3: shows "src r\<^sub>1 = src r\<^sub>0"
rotation_minus_pi_2_x_axis: shows "moebius_circline (moebius_rotation (- pi / 2)) x_axis = y_axis"
cgenideal_minimal: fixes R :: "('e, 'f) ring_scheme"   and J :: "'e set"   and a :: "'e" assumes "ring R"   and "ideal J R"   and "a \<in> J" shows "PIdl\<^bsub>R\<^esub> a \<subseteq> J"
simps_5: fixes procs :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and b :: "expr"   and c :: "cmd"   and ps :: "char list list"   and p :: "char list" shows "containsCall procs (while (b) c) ps p = containsCall procs c ps p"
stermsl_nobigger: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> stermsl p" shows "size q \<le> size p"
upd_set'_aux1: fixes k :: "'c"   and a :: "'b" shows "upd_set' Mapping.empty d f {b. b = k \<or> (a, b) \<in> A} = Mapping.update k d (upd_set' Mapping.empty d f {b. (a, b) \<in> A})"
set_zero_plus': fixes C :: "'b set" shows "(0::'b) +o C = C"
space_density: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" shows "space (density M f) = space M"
assms_3: shows "orient_lit x2 s2 t2 p2"
makespan_def': fixes m :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "LoadBalancing m" shows "f_Max\<^sub>0 T m = Max (T ` {1..m})"
map_disc_iff: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c llist" shows "lnull (lmap f a) = lnull a"
Collect_mem_givenBy: fixes P :: "'e \<Rightarrow> bool"   and v :: "'d \<Rightarrow> 'e" shows "{s. P (v s)} \<in> givenBy v"
algebraicE'_nonzero: fixes x :: "'a" assumes "algebraic x"   and "x \<noteq> (0::'a)" obtains p :: "int poly" where "p \<noteq> 0" and "poly.coeff p 0 \<noteq> 0" and "poly (of_int_poly p) x = (0::'a)"
Cons_3: shows "\<forall>a\<in>set (A # As). square_mat a"
imageE: fixes b :: "'ag"   and f :: "'ah \<Rightarrow> 'ag"   and A :: "'ah set" assumes "b \<in> f ` A" obtains x :: "'ah" where "b = f x" and "x \<in> A"
Midpoint_def: fixes M :: "'p"   and A :: "'p"   and B :: "'p" shows "M Midpoint A B \<equiv> Bet A M B \<and> Cong A M M B"
one: fixes i :: "'a"   and x :: "'b" assumes "i \<in> I"   and "f i x = \<one>\<^bsub>H i\<^esub>"   and "x \<in> carrier (G i)" shows "x = \<one>\<^bsub>G i\<^esub>"
assms_1: shows "A \<in> set (map wordinterval_to_set (getParts rs))"
kD_def: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "kD rt \<equiv> {dip. rt dip \<noteq> None}"
subset_closed_def: fixes U :: "'d set set" shows "subset_closed U \<equiv> \<forall>A\<in>U. Pow A \<subseteq> U"
glbound_in_of_less_eq1: fixes P :: "'a set"   and x :: "'a"   and y :: "'a"   and b :: "'a" assumes "glbound_in_of P x y b" shows "x \<succeq> b"
CondThrow: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>if (Throw r) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
select_convs_1: fixes carrier :: "'d set"   and more :: "'e" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
mult_zero_left: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.mult_zero times zero" shows "times zero a = zero"
suc_def: fixes A :: "(nat + 'a Bonk_Schramm_extension_unfolded set) set" shows "suc wo_aux A \<equiv> minim wo_aux (AboveS wo_aux A)"
simps: fixes x :: "nat"   and y :: "nat"   and u :: "nat" shows "lgR [x, y, u] = (y ^ u \<le> x)"
f3: shows "take i (\<delta> @ \<alpha>) = take i \<beta>"
loop_iterates: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "healthy (wp body)"   and "bd_cts (wp body)"   and "sound P" shows "(\<lambda>i. iterates body G i P s) \<longlonglongrightarrow> wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P s"
Rep_formula': fixes x :: "('b, 'c, 'd) formula" shows "hereditarily_fs (Rep_formula x)"
get_addr_def: fixes op_list :: "inst_operand list"   and s :: "'b sparc_state" shows "get_addr op_list s \<equiv> let rs1 = get_operand_w5 (op_list ! 1); curr_win = ucast (get_CWP (cpu_reg_val PSR s)); rs1_val = user_reg_val curr_win rs1 s in Let (get_operand2 op_list s) ((+) rs1_val)"
spp_inv_spp_of: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" assumes "r \<noteq> 0"   and "r \<in> dgrad_sig_set' (length fs) dgrad" shows "spp_inv (spp_of r)"
OC_wrt_equivalence_iff_indRelTEQ_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "equivalence TRel" shows "(operational_corresponding TRel \<and> weak_reduction_bisimulation TRel Target) = weak_reduction_bisimulation (indRelTEQ TRel) (STCal Source Target)"
int_vertices_def: fixes p :: "('a \<times> 'b \<times> 'a) list" shows "int_vertices p \<equiv> set (map fst p)"
rInput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and C :: "'d"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (M\<lparr>\<lambda>*xvec N\<rparr>.P) (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec]) [] (\<one>)"
hd_remdups_adj: fixes xs :: "'a list" shows "hd (remdups_adj xs) = hd xs"
IH: fixes f :: "'a \<Rightarrow> 'b" assumes "higher_differentiable_on X f (Suc i)" shows "\<exists>f'. ((\<lambda>x. nth_derivative i f x h) has_derivative f') (at x) \<and> f' h = nth_derivative (Suc i) f x h"
sort_key_embedding_injective: fixes x :: "'b sort_key"   and y :: "'b sort_key" assumes "sort_key_embedding x = sort_key_embedding y" shows "x = y"
uminus_mult_left_mat: fixes A :: "'a mat"   and B :: "'a mat" assumes "dim_col A = dim_row B" shows "- A * B = - (A * B)"
ifbb_ifbd_demorgan: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Inf_pres \<phi>" shows "bb\<^sup>-\<^sub>\<F> \<phi> = (bd\<^sup>-\<^sub>\<F> \<circ> \<partial>\<^sub>F) \<phi>"
finsum_cong: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> nat \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> nat \<Rightarrow> 'a" assumes "A = B"   and "(f \<in> B \<rightarrow> carrier P) = True"   and "\<And>i. i \<in> B =simp=> f i = g i" shows "finsum P f A = finsum P g B"
HT_mods_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and mods :: "char list set"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT_mods \<pi> mods P c Q \<equiv> HT \<pi> P c (\<lambda>s\<^sub>0 s. modifies mods s s\<^sub>0 \<and> Q s\<^sub>0 s)"
leg0_in_hom_2: shows "\<guillemotleft>tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 : tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0 \<rightarrow> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0\<guillemotright>"
diff_def: fixes x :: "'b bit0"   and y :: "'b bit0" shows "x - y = Abs_bit0 ((Rep_bit0 x - Rep_bit0 y) mod int CARD('b bit0))"
lapp_fin_infT: fixes s :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "s \<in> A\<^sup>\<star>"   and "t \<in> A\<^sup>\<omega>" shows "s @@ t \<in> A\<^sup>\<omega>"
disjE: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and \<gamma>'' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" assumes "(\<gamma> \<or>\<^sup>c \<gamma>') t n"   and "\<gamma> t n \<Longrightarrow> \<gamma>'' t n"   and "\<gamma>' t n \<Longrightarrow> \<gamma>'' t n" shows "\<gamma>'' t n"
lm068: fixes a :: "('d \<times> 'e set) set"   and G :: "'e set"   and N :: "'d set" assumes "a \<in> allAllocations N G" shows "is_non_overlapping (omega ` a)"
mHom_to_zero: fixes M :: "('f, 'a, 'g) Module_scheme"   and e :: "'h" assumes "R module M" shows "mHom R M (Zm R e) = {mzeromap M (Zm R e)}"
squarefree_mult_imp_coprime: fixes a :: "'a"   and b :: "'a" assumes "squarefree (a * b)" shows "coprime a b"
qfN_div_prime_general: fixes P :: "int"   and A :: "int"   and N :: "int" assumes "prime P"   and "P dvd A"   and "is_qfN A N"   and "is_qfN P N" shows "\<exists>Q. A = Q * P \<and> is_qfN Q N"
iT_Plus_assoc: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" shows "I \<oplus> a \<oplus> b = I \<oplus> a + b"
delete_elt_in_traverse: fixes lt :: "interval_tree"   and x :: "nat idx_interval"   and m :: "nat"   and rt :: "interval_tree" shows "in_traverse (delete_elt_tree (Node lt x m rt)) = in_traverse lt @ in_traverse rt"
trace_callee_resource_of_oracle2: fixes callee :: "'c \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'c) spmf"   and p :: "'c spmf"   and xs :: "('b \<times> 'a) list"   and x :: "'b" shows "trace_callee run_resource (map_spmf (RES callee) p) xs x = trace_callee callee p xs x"
lea_total: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F" shows "A B C LeA D E F \<or> D E F LeA A B C"
mat_mult_scalar: fixes A :: "(('d, 'g) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and B :: "(('d, 'e) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec" shows "A ** B = HMA_Connect.mk_mat (\<lambda>i j. scalar_product (row i A) (column j B))"
Union_upper: fixes B :: "'b set"   and A :: "'b set set" assumes "B \<in> A" shows "B \<subseteq> \<Union> A"
less_eq_Max_atms_of: fixes C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "C \<noteq> {#}"   and "C \<le> D" shows "Max (atms_of C) \<le> Max (atms_of D)"
liftAll_and: fixes phi :: "'d \<Rightarrow> bool"   and chi :: "'d \<Rightarrow> bool"   and inp :: "'e \<Rightarrow> 'd option" shows "liftAll (\<lambda>x. phi x \<and> chi x) inp = (liftAll phi inp \<and> liftAll chi inp)"
length_int_gt0: fixes i :: "int" assumes "0 < i" shows "length_int i = Suc (length_nat (nat i))"
f': fixes x :: "real" assumes "x0 \<le> x" shows "(f has_real_derivative f' x) (at x)"
DsubE: shows "P \<turnstile> D \<preceq>\<^sup>* E"
iprev_inext_infin: fixes I :: "nat set"   and n :: "nat" assumes "infinite I" shows "iprev (inext n I) I = n"
enc_reflects_pred_iff_indRelRPO_reflects_pred: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> bool" shows "enc_reflects_pred Pred = rel_reflects_pred indRelRPO Pred"
starfun_def: fixes f :: "'d \<Rightarrow> 'e" shows "*f* f \<equiv> Ifun (star_of f)"
ba3_export: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - (x \<squnion> y) = x \<squnion> - y"
has_parametrized_subprob_densityD_3: fixes M :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" assumes "has_parametrized_subprob_density M N R f" shows "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)"
delete_correct'_2: fixes m :: "(uint32, ('a, 'b) assoc_list) RBT.rbt"   and k :: "'a" assumes "HashMap_Impl.invar m" shows "HashMap_Impl.invar (HashMap_Impl.delete k m)"
CptnModNone: fixes P :: "'b com"   and s :: "'b"   and t :: "'b"   and xs :: "('b com option \<times> 'b) list" assumes "(Some P, s) -c\<rightarrow> (None, t)"   and "(None, t) # xs \<in> cptn_mod" shows "(Some P, s) # (None, t) # xs \<in> cptn_mod"
elims_3: fixes x :: "nat"   and xa :: "nat list"   and xb :: "cell list \<times> cell list" assumes "\<not> wprepare_loop_start x xa xb" obtains m :: "nat" and lm :: "nat list" and l :: "cell list" and r :: "cell list" where "x = m" and "xa = lm" and "xb = (l, r)" and "\<not> (wprepare_loop_start_on_rightmost m lm (l, r) \<or> wprepare_loop_start_in_middle m lm (l, r))"
covfi: shows "ov O f\<inverse> \<subseteq> b \<union> m \<union> ov"
nlists_eq_empty_iff: fixes A :: "'a set"   and n :: "nat" shows "(nlists A n = {}) = (0 < n \<and> A = {})"
LeastI2_wellorder_ex: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x"   and "\<And>a. \<lbrakk>P a; \<forall>b. P b \<longrightarrow> less_eq a b\<rbrakk> \<Longrightarrow> Q a" shows "Q (ord.Least less_eq P)"
right_uvar_indep_pre_uvar: fixes x :: "'d \<Longrightarrow> 'g"   and y :: "'h \<Longrightarrow> 'e" shows "1-x \<bowtie> y\<^sub><"
next_filter_generator: fixes g :: "('a, 'c) generator"   and s :: "'c" shows "list.next (local.filter_generator g) s = (let (x, s') = list.next g s in if P x then (x, s') else list.next (local.filter_generator g) s')"
max_1: shows "t_max \<in> connected_component_set ((\<lambda>t. x t - y t) -` {0::'a} \<inter> ({t0--t1} \<inter> {t0--t2})) t0"
\<pi>'_def: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and Q :: "'a \<Rightarrow> enat"   and \<pi> :: "'a \<Rightarrow> 'a option"   and u :: "'a"   and v' :: "'a" shows "Prim2.\<pi>' w g r Q \<pi> u v' = (if Prim2.upd_cond w g r Q \<pi> u v' then Some u else \<pi> v')"
homotopic_on_clopen_Union: fixes \<F> :: "'a set set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> closedin (top_of_set (\<Union> \<F>)) S"   and "\<And>S. S \<in> \<F> \<Longrightarrow> openin (top_of_set (\<Union> \<F>)) S"   and "\<And>S. S \<in> \<F> \<Longrightarrow> homotopic_with_canon (\<lambda>x. True) S T f g" shows "homotopic_with_canon (\<lambda>x. True) (\<Union> \<F>) T f g"
xy: shows "y = x @\<^sub>v vec_of_scal (1::'a)"
has_products_iff_admits_tupling: fixes I :: "'a set" shows "has_products I = (I \<noteq> UNIV \<and> admits_tupling I)"
det_swap_rows: fixes A :: "'b mat"   and k :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat (k + n) (k + n)" shows "det A = (- (1::'b)) ^ (k * n) * det (mat (k + n) (k + n) (\<lambda>(i, j). A $$ (if i < k then i + n else i - k, j)))"
in_star_iff_concat: fixes w :: "'a list"   and A :: "'a list set" shows "(w \<in> star A) = (\<exists>ws. set ws \<subseteq> A \<and> w = concat ws)"
drop_bit_of_0: fixes n :: "nat" shows "drop_bit n (0::'a) = (0::'a)"
Sup_trans_upper: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" assumes "t \<in> S"   and "\<forall>u\<in>S. \<forall>P. unitary P \<longrightarrow> unitary (u P)" shows "le_utrans t (Sup_trans S)"
assms: shows "\<forall>x. x \<in> mds AsmNoWrite \<longrightarrow> mem' x = mem x"
preserves_ide: fixes a :: "'t" assumes "S'.ide a" shows "S.ide (\<Psi> a)"
constrains_weaken_R: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B' :: "'b set" assumes "F \<in> A co A'"   and "A' \<subseteq> B'" shows "F \<in> A co B'"
respection: shows "rel_respects_binary_pred (indRelRSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
collinear_triples: fixes a :: "'a"   and b :: "'a"   and S :: "'a set" assumes "a \<noteq> b" shows "collinear (insert a (insert b S)) = (\<forall>x\<in>S. collinear {a, b, x})"
blocks_nempty_alt: shows "\<forall>bl\<in>#\<B>'. bl \<noteq> {}"
as: shows "s \<in> fst (simp (Times r1 r2)) \<rightarrow> v"
sem_eq: shows "dterm_sem I \<theta> (a, b) = dterm_sem I \<theta>' (a, b)"
isolate_variable_sparse_mult: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and v :: "nat"   and n :: "nat" shows "isolate_variable_sparse (p * q) v n = (\<Sum>i\<le>n. isolate_variable_sparse p v i * isolate_variable_sparse q v (n - i))"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "S.awalk u (e # es) w = (e \<in> parcs (subdivide G e w) \<and> u = fst e \<and> S.awalk (snd e) es w)"
nth_drop: fixes i :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "i < length (drop n xs)" shows "drop n xs ! i = xs ! (n + i)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "P (Suc 0)"   and "\<And>n. P (Suc n) \<Longrightarrow> P (Suc (Suc n))" shows "P a0"
smooth_on_add: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x + g x)"
sel_14: fixes x61 :: "'a OCL_Types.type"   and x62 :: "String.literal"   and x63 :: "'a expr list" shows "expr.args (StaticOperationCall x61 x62 x63) = x63"
substc_nrename_comm': fixes x :: "name"   and y :: "name"   and P :: "trm"   and M :: "trm"   and c :: "coname"   and z :: "name" assumes "x \<noteq> y"   and "y \<sharp> P" shows "M{c:=x.P}[y\<turnstile>n>z] = M[y\<turnstile>n>z]{c:=x.P}"
norm_unsigned_append2: fixes xs :: "bit list"   and ys :: "bit list" assumes "norm_unsigned xs = []" shows "norm_unsigned (xs @ ys) = norm_unsigned ys"
ex_le_of_int: 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" assumes "class.archimedean_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<exists>z. less_eq x (ring_1.of_int one plus zero minus z)"
not_beta_reducible_decr: fixes t :: "Core.term"   and n :: "nat" assumes "\<not> beta_reducible t" shows "\<not> beta_reducible (decr n t)"