hcju/isabelleps huggingface dataset:
matrix_compose_cond_def: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list"   and i :: "nat"   and j :: "nat" shows "matrix_compose_cond A1 A2 B1 B2 i j \<equiv> mat (row_length A1) (length A1) A1 \<and> mat (row_length A2) (length A2) A2 \<and> mat (row_length B1) (length B1) B1 \<and> mat (row_length B2) (length B2) B2 \<and> length A1 = row_length A2 \<and> length B1 = row_length B2 \<and> A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> [] \<and> i < row_length A1 * row_length B1 \<and> j < length A2 * length B2"
the_inv_f_f: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "inj f" shows "the_inv f (f x) = x"
lipschitz_on_compose2: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and D :: "real"   and g :: "'c \<Rightarrow> 'd" assumes "C-lipschitz_on U f"   and "D-lipschitz_on (f ` U) g" shows "(D * C)-lipschitz_on U (\<lambda>x. g (f x))"
raw_match'_map_le: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'" shows "subs \<subseteq>\<^sub>m subs'"
fls_pow_conv_regpart: fixes f :: "'a fls"   and n :: "nat" assumes "0 \<le> fls_subdegree f" shows "fls_regpart (f ^ n) = fls_regpart f ^ n"
cong: fixes n :: "nat"   and m :: "nat" shows "LLL.D n m \<equiv> LLL.D n m"
qAbs_alphaAbs_iff: fixes xs :: "'f"   and x :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and A' :: "('d, 'e, 'f, 'g, 'h) qAbs" shows "(qAbs xs x X $= A') = (\<exists>x' y X'. A' = qAbs xs x' X' \<and> y \<notin> {x, x'} \<and> qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
that: fixes q :: "'val" assumes "phiArg g p q"   and "unnecessaryPhi g q"   and "\<not> def_dominates g q p" shows "thesis"
lower_higher_commute: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and s :: "'t"   and t :: "'t" shows "higher (lower p s) t = lower (higher p t) s"
not_call_simps_9: fixes l :: "'d"   and fmsg :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}receive(fmsg) . p)"
shd_def: fixes stream :: "'c stream" shows "shd stream = (case stream of x1 ## x2 \<Rightarrow> x1)"
elims: fixes x :: "('c, 'd option) insert_message list"   and xa :: "nat"   and xb :: "'d"   and xc :: "'c"   and y :: "String.literal + ('c, 'd) message" assumes "create_insert x xa xb xc = y" obtains s :: "('c, 'd option) insert_message list" and n :: "nat" and \<sigma>' :: "'d" and i :: "'c" where "x = s" and "xa = n" and "xb = \<sigma>'" and "xc = i" and "y = nth_visible s n \<bind> (\<lambda>p. nth_visible s (n + 1) \<bind> (\<lambda>q. Inr (Insert (InsertMessage p i q \<sigma>'))))"
elims: fixes x :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and xa :: "'at Literal set set"   and y :: "'at Literal set set" assumes "saturation x xa = y" obtains R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" and S :: "'at Literal set set" where "x = R" and "xa = S" and "y = {C. \<exists>N. \<forall>M\<ge>N. C \<in> inferred_clause_sets R S M}"
prems_5: shows "indexed_eval_aux (P # Ps) i = indexed_eval_aux Qs i"
simple_function_def: fixes M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" shows "simple_function M g = (finite (g ` space M) \<and> (\<forall>x\<in>g ` space M. g -` {x} \<inter> space M \<in> sets M))"
initState_Server: shows "initState Kas = {Key (priEK Kas), Key (priSK Kas)} \<union> Key ` range pubK \<union> Key ` range pubSK \<union> Key ` range shrK"
phase_rel_invD: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state" assumes "phase_rel_inv s" shows "(\<forall>m. sys_ghost_hs_in_sync m s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, filter is_mw_Phase (sys_mem_store_buffers gc s)) \<in> phase_rel"
linorder_on_axioms: shows "linorder_on D2 cmp2"
has_integral_integrable: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and s :: "'c set" assumes "(f has_integral i) s" shows "f integrable_on s"
CCexp_simps_1: fixes x :: "Vars.var"   and a :: "Arity" shows "ccExp_syn a (Var x) = \<bottom>"
left_unique_alt_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "left_unique R = (R OO R\<inverse>\<inverse> \<le> (=))"
that: fixes T :: "'a set" assumes "finite T"   and "T \<subseteq> S"   and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. P t s" shows "thesis"
of_char_of: fixes a :: "'a" shows "of_char (char_of a) = a mod (256::'a)"
\<phi>_\<psi>: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and t' :: "'c" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "t' \<in> Collect (category.terminal S')" shows "\<phi> (inv_into (Collect (category.terminal S)) \<phi> t') = t'"
exsplitnum: fixes x :: "real"   and y :: "real"   and bs :: "real list"   and t :: "MIR.num" shows "Inum (x # y # bs) (exsplitnum t) = Inum ((x + y) # bs) t"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "lcm (lcm a b) c = lcm a (lcm b c)"
assms_2: fixes n :: "nat" shows "u n \<le> C"
abs_eq: fixes x :: "'a list" shows "pdevs_of_list x = Abs_pdev (\<lambda>i. if i < length x then x ! i else (0::'a))"
abs_ge_minus_self: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus a) (abs a)"
RNext_fair: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> \<box>\<diamond>S1 rmhist p \<longrightarrow> WF(RNext memCh mm (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)"
card_Inv_swaps_le: fixes xs :: "'b list"   and sws :: "nat list" assumes "distinct xs" shows "card (Inv xs (swaps sws xs)) \<le> length sws"
mono_exp_of_derived: fixes K :: "'a set"   and H :: "'a set"   and n :: "nat" assumes "K \<subseteq> H" shows "(derived G ^^ n) K \<subseteq> (derived G ^^ n) H"
has_vderiv_on_def: fixes f :: "real \<Rightarrow> 'd"   and f' :: "real \<Rightarrow> 'd"   and S :: "real set" shows "(f has_vderiv_on f') S = (\<forall>x\<in>S. (f has_vector_derivative f' x) (at x within S))"
mult_1: fixes a :: "'c" shows "(1::'c) * a = a"
mult_divide_mult_cancel_left_if: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (times c a) (times c b) = (if c = zero then zero else divide a b)"
wf_Pred: fixes x :: "('idx, 'pred, 'act) formula"   and \<phi> :: "'pred" assumes "weak_formula x" shows "weak_formula \<langle>\<langle>\<tau>\<rangle>\<rangle>Conj (binsert (Pred \<phi>) (bsingleton x))"
fls_X_intpow_times_conv_shift_1: fixes i :: "int"   and f :: "'a fls" shows "fls_X_intpow i * f = fls_shift (- i) f"
consistent_imp_is_solution_solve: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "consistent A b" shows "is_solution (fst (the (solve A b))) A b"
partitionListLeft: fixes xs :: "'d list"   and ys :: "'d list"   and xs' :: "'d list"   and y :: "'d"   and ys' :: "'d list" assumes "xs @ ys = xs' @ y # ys'"   and "y mem xs"   and "distinct (xs @ ys)" obtains zs :: "'d list" where "xs = xs' @ y # zs" and "ys' = zs @ ys"
arith_simps_45: fixes m :: "num"   and n :: "num" shows "numeral m + numeral n = numeral (m + n)"
M_def: shows "M \<equiv> dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) \<rho>"
finite_is_class: fixes 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 "finite {C. is_class P C}"
simps_2: fixes l\<^sub>1 :: "dag"   and v\<^sub>1 :: "ref"   and r\<^sub>1 :: "dag"   and v :: "ref"   and l\<^sub>2 :: "dag"   and v\<^sub>2 :: "ref"   and r\<^sub>2 :: "dag" shows "ordered (Node (Node l\<^sub>1 v\<^sub>1 r\<^sub>1) v (Node l\<^sub>2 v\<^sub>2 r\<^sub>2)) = (\<lambda>var. (var v\<^sub>1 < var v \<and> var v\<^sub>2 < var v) \<and> ordered (Node l\<^sub>1 v\<^sub>1 r\<^sub>1) var \<and> ordered (Node l\<^sub>2 v\<^sub>2 r\<^sub>2) var)"
inject_2: fixes x2 :: "('e, 'f) atm"   and y2 :: "('e, 'f) atm" shows "(Neg x2 = Neg y2) = (x2 = y2)"
assms: shows "P \<or>\<^sub>A Q \<Longrightarrow>\<^sub>A R"
assms: shows "\<forall>x\<in>lset xs. \<forall>Y\<in>lset xs. g x \<inter> h Y = {}"
subfrmlsr_subset: fixes \<psi> :: "'b ltlr"   and \<phi> :: "'b ltlr" assumes "\<psi> \<in> subfrmlsr \<phi>" shows "subfrmlsr \<psi> \<subseteq> subfrmlsr \<phi>"
pair_imageI: fixes a :: "'e"   and b :: "'f"   and A :: "('e \<times> 'f) set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
ne_imp_pos: fixes u :: "nat" assumes "nat_to_set u \<noteq> {}" shows "0 < u"
gx3_ge_x1: shows "x\<^sub>1 \<le> gx3"
dnso4: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rdom (x \<union> y) = rdom x \<union> rdom y"
lim_increasing_cl: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>n m. m \<le> n \<Longrightarrow> f m \<le> f n" obtains l :: "'a" where "f \<longlonglongrightarrow> l"
msetext_huet_hd_or_tl: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and y :: "'b"   and x :: "'b" assumes "\<forall>z y x. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"   and "\<forall>y x. gt y x \<or> gt x y \<or> y = x"   and "length ys = length xs"   and "msetext_huet gt (y # ys) (x # xs)" shows "gt y x \<or> msetext_huet gt ys xs"
d_2: shows "poincare_distance v w = poincare_distance v' w'"
cmp_in_hom_2: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b"   and "src a = trg b" shows "\<guillemotleft>S.cmp\<^sub>U\<^sub>P (a, b) : S.UP a \<star>\<^sub>S S.UP b \<Rightarrow>\<^sub>S S.UP (a \<star> b)\<guillemotright>"
fs_deriv: fixes k :: "nat"   and x :: "real" assumes "k < N"   and "x \<in> {real_of_int a..real_of_int b} - Y" shows "(fs k has_vector_derivative fs (Suc k) x) (at x)"
that: fixes F' :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and q :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite F'"   and "F' \<subseteq> F"   and "p = (\<Sum>f\<in>F'. q f * f)"   and "\<And>f. poly_deg (q f * f) \<le> poly_deg p"   and "\<And>f. f \<notin> F' \<Longrightarrow> q f = 0" shows "thesis"
emp_not_per: fixes x :: "'b list" shows "\<not> x \<le>p \<epsilon>\<^sup>\<omega>"
eval_e\<^sub>2: shows "eval_expr (t + used_tmps e\<^sub>1) e\<^sub>2 = (D\<^sub>2, g\<^sub>2)"
bj: shows "j_to_jb_index n_as j = (bj, lj)"
compute_float_divr: fixes prec :: "nat"   and x :: "float"   and y :: "float" shows "float_divr prec x y = - float_divl prec (- x) y"
dg_def: shows "\<CC> = [\<CC>\<lparr>Obj\<rparr>, \<CC>\<lparr>Arr\<rparr>, \<CC>\<lparr>Dom\<rparr>, \<CC>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
inconsistent: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "inconsistent A b = (\<nexists>x. is_solution x A b)"
similar_matrix_def: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "similar_matrix A B = (\<exists>P Q. similar_matrix_wit A B P Q)"
similar_diag_mat_block_mat: fixes Ms :: "('b mat \<times> 'b mat) list" assumes "\<And>A B. (A, B) \<in> set Ms \<Longrightarrow> similar_mat A B" shows "similar_mat (diag_block_mat (map fst Ms)) (diag_block_mat (map snd Ms))"
sub_assoc_monom: fixes f :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P" shows "Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) (Cring_Poly.compose R q r) = Cring_Poly.compose R (Cring_Poly.compose R (monom P (f (deg R f)) (deg R f)) q) r"
mono_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
a_ptr_disconnected_node_rel_document: fixes h :: "(_) heap"   and node_ptr :: "(_) node_ptr"   and document :: "(_) document_ptr" assumes "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_document node_ptr \<rightarrow>\<^sub>r document" shows "(cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document, cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<in> local.a_ptr_disconnected_node_rel h"
filter_sort: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "filter P (sort_key f xs) = sort_key f (filter P xs)"
pgwt_fun: fixes t :: "('fun, 'var) Term.term" assumes "public_ground_wf_term t" shows "\<exists>f T. t = Fun f T"
power_commutes: fixes a :: "'a"   and n :: "nat" shows "a * power.power (1::'a) (\<lambda>a b. b * a) a n = power.power (1::'a) (\<lambda>a b. b * a) a n * a"
cod_simps_3: shows "trg_f.cod.leg1 = trg_f.leg1"
8: shows "[a] \<preceq>\<^sub>F\<^sub>I u"
z: shows "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
gauge_reflect: fixes \<gamma> :: "'a \<Rightarrow> 'a set" assumes "gauge \<gamma>" shows "gauge (\<lambda>x. uminus ` \<gamma> (- x))"
weak_coinduct: fixes a :: "'a"   and X :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "a \<in> X"   and "X \<subseteq> f X" shows "a \<in> gfp f"
traverse_tree_map_tree: fixes path :: "dir list"   and f :: "'c \<Rightarrow> 'b"   and t :: "'c tree" shows "traverse_tree path (map_tree f t) = map_tree f (traverse_tree path t)"
False: shows "\<not> (x = 0 \<or> y = 0)"
single_division_zero: shows "\<forall>X. (B * X = 0) = (X = 0)"
simps_36: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and x1 :: "const" shows "map_val f1 f2 f3 (CV x1) = CV x1"
dims_A'_pow_2: fixes ws :: "nat \<Rightarrow> real" shows "dim_col (A' ws) = last rs ^ N_half"
times_dimvec_def: fixes x :: "('c, 'd) dimvec"   and y :: "('c, 'd) dimvec" shows "x \<cdot> y = dim_lambda (\<lambda>i. dim_nth x i + dim_nth y i)"
exec1_i_Piii_Pooo_def: shows "exec1_i_Piii_Pooo \<equiv> \<lambda>x xa. pred.Pred (exec1 x xa)"
same_snapshot_state_implies_same_result_snapshot_state: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and c :: "('d, 'e, 'f) configuration"   and p :: "'d"   and d :: "('d, 'e, 'f) configuration"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ev :: "('d, 'e, 'f) event"   and c' :: "('d, 'e, 'f) configuration"   and d' :: "('d, 'e, 'f) configuration" assumes "distributed_system channel"   and "ps c p = ps d p"   and "states c p = states d p"   and "distributed_system.next channel trans send recv c ev c'"   and "distributed_system.next channel trans send recv d ev d'" shows "ps d' p = ps c' p"
length_cofactor_list_syz: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "length (cofactor_list_syz n p) = n"
simps_1: fixes i :: "'i"   and x :: "nat"   and c :: "'a"   and s :: "('i, 'a) state" shows "assert_bound (i, Leq x c) s = assert_bound' Positive i x c s"
invertibleI: fixes u :: "'a set"   and v :: "'a set" assumes "u [\<cdot>] v = Class \<one>"   and "v [\<cdot>] u = Class \<one>"   and "u \<in> Partition"   and "v \<in> Partition" shows "quotient.invertible u"
Cidr: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "f ;;\<^bsub>C\<^esub> id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f) = f"
a1: shows "(\<Sum>\<^sub>ai\<in>F. norm (x i * x i)) \<le> (\<Sum>\<^sub>ai\<in>A. norm (x i * x i))"
rev_ImageI: fixes a :: "'b"   and A :: "'b set"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "a \<in> A"   and "(a, b) \<in> r" shows "b \<in> r `` A"
d1_a: fixes x :: "'a" shows "d x \<cdot> x = x"
c2: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) A = c 2"
cbox_sub_2: shows "cbox u v \<subseteq> box a b"
dim_eq: fixes X :: "(('c \<Rightarrow> real) \<Rightarrow> real) set"   and p :: "'c" assumes "X \<subseteq> manifold_eucl.dest.tangent_space p" shows "manifold_eucl.dest.tangent_space.dim p X = dim X"
dist1: shows "distinct (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))"
scale_zero_right: fixes a :: "real" shows "a *\<^sub>R (0::'c) = (0::'c)"
uniformly_convergent_add: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" assumes "uniformly_convergent_on A f"   and "uniformly_convergent_on A g" shows "uniformly_convergent_on A (\<lambda>k x. f k x + g k x)"
i1: shows "rk X \<le> rk (X \<union> {y, z})"
fg: fixes x :: "'a"   and y :: "'b" assumes "x \<in> B"   and "y \<in> C" shows "f x y = g x y"
is_done_impl_def: fixes SBIP :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list"   and v :: "'b" shows "GS.is_done_impl SBIP v \<equiv> case GS.I SBIP v of None \<Rightarrow> False | Some (STACK nat) \<Rightarrow> False | Some DONE \<Rightarrow> True"
subst_of_map_def: fixes d :: "'b \<Rightarrow> 'a"   and \<sigma> :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "subst_of_map d \<sigma> x = (case \<sigma> x of None \<Rightarrow> d x | Some t \<Rightarrow> t)"
BEGIN_def: fixes i :: "'b" shows "BEGIN i = {j |j. (j, i) \<in> ov \<union> s \<union> m \<union> f\<inverse> \<union> d\<inverse> \<union> e \<union> s\<inverse>}"
distrib_right': fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and y :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and z :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" shows "x \<oplus> y \<otimes> z = (x \<otimes> z) \<oplus> (y \<otimes> z)"
cblinfun_eq_gen_eqI: fixes G :: "'a set"   and A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" assumes "\<And>x. x \<in> G \<Longrightarrow> A *\<^sub>V x = B *\<^sub>V x"   and "ccspan G = \<top>" shows "A = B"
assms_5: shows "last (msgs (S t (i + 1)) cid) \<noteq> Marker"
L_transform_def: fixes x :: "('e, 'f, 'g, 'h) FL_Formula.formula" shows "L_transform x = Formula.formula.Abs_formula (L_transform_formula x)"
intros_1: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "\<one>\<^bsub>G\<^esub> \<in> generate G H"
l_new_character_data_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<equiv> l_new_character_data type_wf \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_new_character_data_get_shadow_root_axioms get_shadow_root_locs"
reachable_imp_reachable_project: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and z :: "'c" assumes "reachable (extend h F \<squnion> G) \<subseteq> C"   and "z \<in> reachable (extend h F \<squnion> G)" shows "f z \<in> reachable (F \<squnion> project h C G)"
c0: shows "BVG (x := \<theta>) \<subseteq> {x}"
alpha_Tree_permute_rep_commute: fixes p :: "perm"   and t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "p \<bullet> rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha> =\<^sub>\<alpha> rep_Tree\<^sub>\<alpha> (p \<bullet> t\<^sub>\<alpha>)"
P2: shows "Col P X' Y' \<and> X' Y' Perp C D \<and> X' Y' Perp E F"
enat_le_Sup_iff: fixes k :: "nat"   and M :: "enat set" shows "(enat k \<le> Sup M) = (k = 0 \<or> (\<exists>m\<in>M. enat k \<le> m))"
is_imgu_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) set" shows "is_imgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<tau> = \<sigma> \<circ>\<^sub>s \<tau>))"
cdov: shows "d O ov \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
skip_P1: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> \<P>" shows "(\<lambda>x. f (x + n)) \<in> \<P>"
inext_atMost: fixes t :: "nat"   and n :: "nat" assumes "t < n" shows "inext t {..n} = Suc t"
esl_def: fixes P :: "'b prog" shows "SemiType.esl P \<equiv> (types P, subtype P, SemiType.sup P)"
fmpred_mono_strong: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>fmlookup m x = Some y; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "fmpred P m" shows "fmpred Q m"
gR_rel_def: fixes x :: "'c"   and y :: "'c" shows "(x \<preceq>\<^sub>R y) = (\<exists>z. D x z \<and> x \<oplus> z = y)"
lt_of_nat_term_order_alt: fixes to :: "'a nat_term_order" shows "lt_of_nat_term_order to = (\<lambda>u v. ko.lt (key_order_of_nat_term_order_inv to) v u)"
simps_7: fixes bs :: "real list"   and a :: "Ferrack.num" shows "Ifm bs (Eq a) = (Inum bs a = 0)"
steps_z_alpha_alt: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set"   and l'' :: "'d"   and Z'' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l'', Z''\<rangle>"
Pair_def: fixes a :: "'c"   and b :: "'d" shows "(a, b) = Abs_prod (Pair_Rep a b)"
prems_2: shows "\<forall>x\<in>set (x # xs). x \<le> y"
Geqo_undef: fixes \<theta> :: "trm option"   and \<eta> :: "trm option" shows "(Geqo \<theta> \<eta> = undeff) = (\<theta> = undeft \<or> \<eta> = undeft)"
adjoint_transpose_left_1: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src f = trg v"   and "src g' = trg u" shows "A'.trnl\<^sub>\<eta> v \<in> hom (f \<star> v) u \<rightarrow> hom v (g' \<star> u)"
worklist_def: fixes G :: "nat \<times> ('d \<times> nat \<times> nat) set"   and Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set" shows "worklist G Rs \<equiv> let G = graph_of G in {(N, R, f). R \<in> Rs \<and> graph_homomorphism (fst R) G f \<and> N = nextMax (Range f) \<and> \<not> extensible R G f}"
eq_right_lang_def: shows "eq_right_lang \<equiv> {(u, v). u \<in> states M \<and> v \<in> states M \<and> right_lang u = right_lang v}"