hcju/isabelleps huggingface dataset:
nadds_red_lt: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>q. q \<in> B \<Longrightarrow> \<not> lt q adds\<^sub>t lt p"   and "red B p r" shows "lt r = lt p"
simps_28: fixes vb :: "bexp"   and vc :: "'a acom"   and vd :: "'a acom"   and ve :: "'a"   and v :: "'a acom"   and va :: "'a acom" shows "(IF vb THEN vc ELSE vd {ve} \<le> v;; va) = False"
induct: fixes P :: "char list \<Rightarrow> bool"   and a0 :: "char list" assumes "P []"   and "\<And>s. P [s]"   and "\<And>uu_ v va. P (v # va) \<Longrightarrow> P (uu_ # v # va)" shows "P a0"
Zp_defs_4: shows "(\<otimes>) = padic_mult p"
split_asm: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'g \<Rightarrow> 'd"   and generat :: "('e, 'f, 'g) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. generat = Pure x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. generat = IO x21 x22 \<and> \<not> P (f2 x21 x22))))"
topology_finer_open_id: fixes X :: "'c topology"   and X' :: "'c topology" shows "(\<forall>S. openin X S \<longrightarrow> openin X' S) = open_map X X' id"
Setcompr_eq_image: fixes f :: "'i \<Rightarrow> 'h"   and A :: "'i set" shows "{f x |x. x \<in> A} = f ` A"
Factor_Node_1: fixes u :: "int poly factor_tree"   and f :: "int poly" assumes "u \<in> sub_trees (product_factor_tree p l)"   and "factor_node_info u = f"   and "\<And>x. x \<in># factors_of_factor_tree l \<Longrightarrow> Mp x = x" shows "Mp f = f \<and> f = Mp (\<Prod>\<^sub># (factors_of_factor_tree u)) \<and> factors_of_factor_tree (product_factor_tree p l) = factors_of_factor_tree l"
awalk_verts_reachable_to: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "H.awalk u p v"   and "w \<in> set (H.awalk_verts u p)" shows "w \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
Cons_2: shows "(s, x # xs) \<in> conv ars"
inject_2: fixes x3 :: "'b formula"   and y3 :: "'b formula" shows "(\<^bold>\<not> x3 = \<^bold>\<not> y3) = (x3 = y3)"
simps_4: fixes r :: "'a rexp"   and s :: "'a rexp" shows "lang (Plus r s) = lang r \<union> lang s"
lookup_some_append: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and ys :: "'c list"   and e :: "('a, 'b, 'c) env" assumes "lookup env (xs @ ys) = Some e" shows "\<exists>e. lookup env xs = Some e"
sinks_interference_eq: fixes u :: "'a"   and D :: "'b \<Rightarrow> 'a"   and x :: "'b"   and I :: "('a \<times> 'a) set"   and xs :: "'b list" shows "((u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I)) = (D x \<in> sinks I D u (xs @ [x]))"
minus_mult_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring plus zero minus uminus times" shows "uminus (times a b) = times (uminus a) b"
B: shows "M - 4 * delta + dist x' y' \<le> dist py x'"
row_def: fixes A :: "'a mat"   and i :: "nat" shows "row A i = vec (dim_col A) (\<lambda>j. A $$ (i, j))"
retraction_char: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and g :: "'b" assumes "set_category S \<AA>" shows "category.retraction S g = (partial_magma.arr S g \<and> set_category_given_img.Fun S (set_category.some_img S \<AA>) g ` set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S g) = set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.cod S g))"
prebicategory_with_homs_axioms: shows "prebicategory_with_homs (\<cdot>) (\<star>) \<a> some_src some_trg"
nprt_neg: fixes x :: "'a" shows "nprt (- x) = - pprt x"
merge_conv': fixes xs :: "('c \<times> 'd) list"   and ys :: "('c \<times> 'd) list" shows "map_of (merge xs ys) = map_of xs ++ map_of ys"
b1: shows "S holds_for steps0 (1, l', r') B n3"
R_bound: fixes x :: "real" assumes "0 \<le> x" shows "R x \<in> {0..sqrt x}"
span_mul: fixes x :: "'b"   and S :: "'b set"   and c :: "real" assumes "x \<in> span S" shows "c *\<^sub>R x \<in> span S"
vsv_ex1_app1: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>)" shows "(b = \<BB>\<lparr>Dom\<rparr>\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> \<BB>\<lparr>Dom\<rparr>)"
closure_halfspace_gt: fixes a :: "'b"   and b :: "real" assumes "a \<noteq> (0::'b)" shows "closure {x. b < a \<bullet> x} = {x. b \<le> a \<bullet> x}"
FGModuleIso_restrict0_GSubspace: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "FGModuleIso G (\<cdot>) U (\<star>) (T \<down> U) (T ` U)"
unat_drop_bit_eq: fixes n :: "nat"   and w :: "'b word" shows "unat (drop_bit n w) = drop_bit n (unat w)"
prj_simps_3: shows "Maps.arr SPN_fgh.Prj\<^sub>0"
shift_closed: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "f \<in> carrier (UP Zp)" shows "shift n f \<in> carrier (UP Zp)"
sconst_collapse: fixes y :: "'b" shows "y ## sconst y = sconst y"
fcomp_fid_on_idem: fixes A :: "V" shows "fid_on A \<circ>\<^sub>\<bullet> fid_on A = fid_on A"
ik_append_subset_1: fixes S :: "('c, 'd) strand_step list"   and S' :: "('c, 'd) strand_step list" shows "ik\<^sub>s\<^sub>t S \<subseteq> ik\<^sub>s\<^sub>t (S @ S')"
nn_integral_multc: fixes f :: "'h \<Rightarrow> ennreal"   and M :: "'h measure"   and c :: "ennreal" assumes "f \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * c \<partial>M = integral\<^sup>N M f * c"
rtranclp_unfold: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" shows "r\<^sup>*\<^sup>* a b \<equiv> a = b \<or> r\<^sup>+\<^sup>+ a b"
scaleRat_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.rational_vector plus zero minus uminus scaleRat" shows "scaleRat 1 x = x"
comp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and g :: "'b \<Rightarrow> 'c" assumes "inj_on f A"   and "inj_on g (f ` A)" shows "inj_on (g \<circ> f) A"
vint_commutative_law_multiplication: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x *\<^sub>\<int> y = y *\<^sub>\<int> x"
delay: fixes st :: "'s \<times> ('c \<Rightarrow> real) set"   and l :: "'s"   and R :: "('c \<Rightarrow> real) set"   and R' :: "('c \<Rightarrow> real) set" assumes "st \<in> \<S>"   and "st = (l, R)"   and "R' \<in> Succ \<R> R"   and "R' \<subseteq> {v. v \<turnstile> PTA.inv_of A l}" shows "return_pmf (l, R') \<in> \<K> st"
inf_sup_distrib1: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "x \<inter># (y \<union># z) = x \<inter># y \<union># (x \<inter># z)"
wgeq_lemma: fixes w1 :: "32 Word.word"   and r1 :: "real"   and w2 :: "32 Word.word"   and r2 :: "real" assumes "w1 \<equiv>\<^sub>L r1"   and "w2 \<equiv>\<^sub>U r2"   and "wgeq w1 w2" shows "r2 \<le> r1"
CSP2_meet: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "x is CSP2 healthy"   and "y is CSP2 healthy" shows "x \<and> y is CSP2 healthy"
ucast_down_wi: fixes x :: "int" assumes "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) (word_of_int x) = word_of_int x"
uncross_negative_flip_def: fixes x :: "wall"   and y :: "wall" shows "uncross_negative_flip x y \<equiv> x = right_under \<and> y = left_under"
sum: fixes g :: "'d \<Rightarrow> 'a"   and S :: "'d set" shows "frechet_derivative g (at x) (sum g S) = (\<Sum>a\<in>S. frechet_derivative g (at x) (g a))"
the1_equality': fixes P :: "'d \<Rightarrow> bool"   and a :: "'d" assumes "\<exists>\<^sub>\<le>\<^sub>1 x. P x"   and "P a" shows "(THE x. P x) = a"
take_drop_append: fixes i :: "int"   and xs :: "['b]" shows "take\<cdot>(MkI\<cdot>i)\<cdot>xs ++ Data_List.drop\<cdot>(MkI\<cdot>i)\<cdot>xs = xs"
g_size_abort_def: fixes b :: "nat"   and m :: "'s" shows "g_size_abort b m \<equiv> iteratei m (\<lambda>s. s < b) (\<lambda>_. Suc) 0"
le_iff_sup: fixes y :: "'a"   and x :: "'a" shows "(y \<le> x) = (x \<sqinter> y = y)"
enum_all_finite_5_def: fixes P :: "finite_5 \<Rightarrow> bool" shows "enum_all P = (P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3 \<and> P a\<^sub>4 \<and> P a\<^sub>5)"
acc_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "acc r = wf {(y, x). x \<sqsubset>\<^bsub>r\<^esub> y}"
degree_lessI: fixes p :: "'b poly"   and n :: "nat" assumes "p \<noteq> 0 \<or> 0 < n"   and "\<forall>k\<ge>n. coeff p k = (0::'b)" shows "degree p < n"
assertion_def: fixes p :: "'a" assumes "test p" shows "p\<^sup>o = !p \<cdot> \<top> + (1::'a)"
construct_lhs_vector_drop_consistent: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and I :: "nat list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "list_constr I (length qs)" shows "vec_of_list (mtx_row signs I) \<bullet> w_vec p qs signs = vec_of_list (mtx_row (characterize_consistent_signs_at_roots_copr p qs) I) \<bullet> w_vec p qs (characterize_consistent_signs_at_roots_copr p qs)"
inf_lower: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier (fpl L f)"   and "x \<in> A" shows "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<sqsubseteq>\<^bsub>fpl L f\<^esub> x"
emeasure_in_N: fixes x1 :: "real" assumes "x1 \<in> N" shows "emeasure (lborel_f n) {x \<in> space (lborel_f n). x(n := x1) \<in> A} = emeasure (lborel_f n) (space (lborel_f n))"
dvd_div_neg: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.idom_divide times one plus zero minus uminus divide"   and "dvd.dvd times b a" shows "divide a (uminus b) = uminus (divide a b)"
Await: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and r :: "'e set"   and b :: "'e set"   and P :: "('e, 'd, 'f) ann"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (r \<inter> b) P c Q, A"   and "atom_com c" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnRec r P) (Await b c) Q, A"
fix_least_below: fixes F :: "'c \<rightarrow> 'c"   and x :: "'c" assumes "F\<cdot>x \<sqsubseteq> x" shows "fix\<cdot>F \<sqsubseteq> x"
compress_normalize_primitive_Some: fixes m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and m' :: "'c match_expr"   and \<gamma> :: "('c \<Rightarrow> 'e \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'e \<Rightarrow> bool)"   and a :: "action"   and p :: "'e" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "compress_normalize_primitive (disc, sel) C f m = Some m'"   and "\<And>as as_pos as_neg. f as = Some (as_pos, as_neg) \<Longrightarrow> matches \<gamma> (alist_and (NegPos_map C (map Pos as_pos @ map Neg as_neg))) a p = matches \<gamma> (alist_and (NegPos_map C as)) a p" shows "matches \<gamma> m' a p = matches \<gamma> m a p"
power_set_add: fixes A :: "'a set"   and n :: "nat"   and m :: "nat" shows "A *^ (n + m) = A *^ n ** A *^ m"
assoc_comp: fixes U :: "'a set"   and V :: "'a set"   and W :: "'a set"   and x :: "'b" assumes "is_open U"   and "is_open V"   and "is_open W"   and "V \<subseteq> U"   and "W \<subseteq> V"   and "x \<in> \<FF> U" shows "\<rho> U W x = (\<rho> V W \<circ> \<rho> U V) x"
set_lebesgue_integral_def: fixes M :: "'e measure"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'd" shows "set_lebesgue_integral M A f \<equiv> LINT x|M. indicat_real A x *\<^sub>R f x"
is_parallel_solution_for_problem_def: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<psi> :: "('a, 'b) sas_plus_operator list list" shows "is_parallel_solution_for_problem \<Psi> \<psi> \<equiv> let G = \<Psi>\<^sub>G\<^sub>+; I = \<Psi>\<^sub>I\<^sub>+; Ops = \<Psi>\<^sub>\<O>\<^sub>+ in G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I \<psi> \<and> list_all (list_all (\<lambda>op. ListMem op Ops)) \<psi>"
ball_divide_subset_numeral: fixes x :: "'c"   and e :: "real"   and w :: "num" shows "ball x (e / numeral w) \<subseteq> ball x e"
eq_imp_below: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "x = y" shows "below x y"
r'_def: shows "r' = cardSuc (natLeq_on n)"
col_out2_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and AA :: "'a"   and CC :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.Out Bet B A AA"   and "Tarski_neutral_dimensionless.Out Bet B C CC" shows "Tarski_neutral_dimensionless.Col Bet AA B CC"
f_aggregate_length: fixes xs :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" shows "length (f_aggregate xs k ag) = length xs div k"
intro: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme"   and W :: "('g, 'k, 'l) module_scheme"   and T :: "'i \<Rightarrow> 'k" assumes "vectorspace K V"   and "vectorspace K W"   and "mod_hom K V W T" shows "linear_map K V W T"
simps_27: fixes x61 :: "msg"   and x62 :: "msg"   and x2 :: "nat" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Number x2"
vt_e': fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<sigma> \<in> space M" shows "val_type (expr_sem_rf \<sigma> e') = t'"
put_eq: fixes x :: "'b \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and k :: "'b"   and u :: "'b"   and \<rho> :: "'c"   and v :: "'b" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
eq_divide_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a = b / c) = (if c \<noteq> (0::'a) then a * c = b else a = (0::'a))"
der: fixes n :: "nat" shows "(deriv ^^ n) f z = 0"
rComm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"
STRIPS_sema_sound: fixes \<Sigma> :: "ground_action set"   and M\<^sub>0 :: "object atom formula set"   and s\<^sub>0 :: "predicate \<times> object list \<Rightarrow> bool"   and f :: "ground_action \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option"   and \<alpha>s :: "ground_action list"   and M' :: "object atom formula set" assumes "sound_system \<Sigma> M\<^sub>0 s\<^sub>0 f"   and "set \<alpha>s \<subseteq> \<Sigma>"   and "ground_action_path M\<^sub>0 \<alpha>s M'" obtains s' :: "predicate \<times> object list \<Rightarrow> bool" where "compose_actions (map f \<alpha>s) s\<^sub>0 = Some s'" and "\<forall>fmla\<in>close_world M'. close_eq s' \<Turnstile> fmla"
subprob_space_axioms: fixes s :: "'a" shows "subprob_space (K1.T s)"
islimpt_approachable: fixes x :: "'a"   and S :: "'a set" shows "x islimpt S = (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
update_convs_17: fixes sip' :: "nat \<Rightarrow> nat"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and pre :: "nat set"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip_update sip' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip' sip, \<dots> = more\<rparr>"
execn_to_execn_merge_guards: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>merge_guards c,s\<rangle> =n\<Rightarrow> t"
dom_closed: fixes f :: "'a \<times> 'a" assumes "fst f \<star> snd f \<noteq> null" shows "fst (VxV.dom f) \<star> snd (VxV.dom f) \<noteq> null"
measure_countable_Union_approachable: fixes \<D> :: "'b set set"   and e :: "real"   and B :: "real" assumes "countable \<D>"   and "0 < e"   and "\<And>d. d \<in> \<D> \<Longrightarrow> d \<in> lmeasurable"   and "\<And>D'. \<lbrakk>D' \<subseteq> \<D>; finite D'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure lebesgue (\<Union> D') \<le> B" obtains D' :: "'b set set" where "D' \<subseteq> \<D>" and "finite D'" and "Sigma_Algebra.measure lebesgue (\<Union> \<D>) - e < Sigma_Algebra.measure lebesgue (\<Union> D')"
1: shows "\<forall>y. (y, x) \<in> measure Rep_hf \<longrightarrow> P y"
cong_trans: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b" 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"
converse_galois: fixes f :: "'d \<Rightarrow> ('f \<times> 'e) set"   and g :: "'d \<Rightarrow> ('e \<times> 'f) set" shows "(\<smile> \<circ> f = g) = (\<smile> \<circ> g = f)"
min_minus: fixes m :: "nat"   and k :: "nat" shows "min m (m - k) = m - k"
simps_5: fixes ODE :: "('b, 'a) ODE"   and \<phi> :: "('b, 'c, 'a) formula" shows "BVP (EvolveODE ODE \<phi>) = BVO ODE"
rec_def: fixes e :: "'b"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'a" shows "rec e r x = (THE y. Rec e r x y)"
sup_state_def: fixes P :: "'a prog" shows "sup_state P \<equiv> Product.le (Listn.le (subtype P)) (Listn.le (sup_ty_opt P))"
distinct_3: fixes x31 :: "'c"   and x32 :: "'d" shows "Done \<noteq> Yield x31 x32"
Sup_Inf: fixes A :: "'b set"   and inf :: "'b" assumes "is_Sup {b. \<forall>a\<in>A. b \<sqsubseteq> a} inf" shows "is_Inf A inf"
simps_4: fixes b :: "nat"   and \<phi> :: "'c MFOTL.formula"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (formula.Or \<phi> \<psi>) = MFOTL.fvi b \<phi> \<union> MFOTL.fvi b \<psi>"
quantifiable_T8: fixes forall :: "('a \<Rightarrow> \<o>) \<Rightarrow> \<o>"   and w :: "i"   and \<psi> :: "'a \<Rightarrow> \<o>" assumes "class.quantifiable forall" shows "[forall \<psi> in w] = (\<forall>x. [\<psi> x in w])"
rtrancl_O_push: shows "S\<^sup>* O R \<subseteq> R O S\<^sup>*"
w2_def: shows "w2 = geodesic_segment_param Gxz x (Gromov_product_at x y z)"
poly_as_sum: fixes p :: "'c poly"   and x :: "'c" shows "poly p x = (\<Sum>i\<le>degree p. x ^ i * coeff p i)"
inf_right_idem: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<inter># y \<inter># y = x \<inter># y"
diff_numeral_special_3: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus one (uminus (numeral.numeral one plus n)) = numeral.numeral one plus (num.One + n)"
OK_le_err_OK: fixes x :: "'b"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b" shows "(OK x \<sqsubseteq>\<^bsub>le r\<^esub> OK y) = (x \<sqsubseteq>\<^bsub>r\<^esub> y)"
compactE: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set"   and \<T> :: "'c set set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "S \<subseteq> \<Union> \<T>"   and "\<And>B. B \<in> \<T> \<Longrightarrow> open B" obtains \<T>' :: "'c set set" where "\<T>' \<subseteq> \<T>" and "finite \<T>'" and "S \<subseteq> \<Union> \<T>'"
covering_space_imp_surjective: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "covering_space c p S" shows "p ` c = S"
beta_reduction_prm: fixes M :: "'a trm"   and M' :: "'a trm"   and \<pi> :: "'a prm" assumes "M \<rightarrow>\<beta> M'" shows "\<pi> \<cdot> M \<rightarrow>\<beta> \<pi> \<cdot> M'"
flatten_in_Polys: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set"   and Y :: "'a set" assumes "p \<in> P[X]"   and "lookup p ` keys p \<subseteq> P[Y]" shows "flatten p \<in> P[X \<union> Y]"
rel_resource_coinduct: fixes X :: "('c, 'd) resource \<Rightarrow> ('e, 'f) resource \<Rightarrow> bool"   and res1 :: "('c, 'd) resource"   and res2 :: "('e, 'f) resource"   and A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'f \<Rightarrow> bool" assumes "X res1 res2"   and "\<And>res1 res2. X res1 res2 \<Longrightarrow> rel_fun A (rel_spmf (rel_prod B (\<lambda>res1 res2. X res1 res2 \<or> rel_resource A B res1 res2))) (run_resource res1) (run_resource res2)" shows "rel_resource A B res1 res2"