hcju/isabelleps huggingface dataset:
brec_invar_final: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist"   and Qi :: "('a, unit) HashMap.hashmap" assumes "rqrm_invar rqrm"   and "rqrm_prop (ls.\<alpha> \<delta>) rqrm"   and "ls.invar \<delta>"   and "hs.invar Qi" shows "\<forall>sc. sc \<in> wa_invar (det_wa_wa (brec_det_algo rqrm Qi \<delta>)) \<and> sc \<notin> wa_cond (det_wa_wa (brec_det_algo rqrm Qi \<delta>)) \<longrightarrow> (hs.\<alpha> Qi \<inter> dom (hm.\<alpha> (fst sc)) = {}) = (hs.\<alpha> Qi \<inter> b_accessible (ls.\<alpha> \<delta>) = {}) \<and> witness_prop (ls.\<alpha> \<delta>) (hm.\<alpha> (fst sc))"
normalize_coeff: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "ring.coeff R p = ring.coeff R (ring.normalize R p)"
maxU12: shows "wmin (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2) \<equiv>\<^sub>L min (rl1 * rl2) (ru1 * rl2)"
monom_eq_iff': fixes c :: "'a"   and n :: "nat"   and d :: "'a"   and m :: "nat" shows "(monom c n = monom d m) = (c = d \<and> (c = (0::'a) \<or> n = m))"
less_not_refl: fixes n :: "nat" shows "\<not> order.greater n n"
standard_uniqueness_automorphs: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and C :: "'a set" assumes "ChamberComplexAutomorphism X f"   and "ChamberComplexAutomorphism X g"   and "folding_g.chamber C"   and "fun_eq_on f g C" shows "f = g"
sinv: shows "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> Q"
greatest_def: fixes L :: "('c, 'd) gorder_scheme"   and g :: "'c"   and A :: "'c set" shows "greatest L g A = (A \<subseteq> carrier L \<and> g \<in> A \<and> (\<forall>x\<in>A. x \<sqsubseteq>\<^bsub>L\<^esub> g))"
ipurge_tr_aux_single_dom: fixes I :: "('d \<times> 'd) set"   and D :: "'c \<Rightarrow> 'd"   and u :: "'d"   and xs :: "'c list" shows "ipurge_tr_aux I D {u} xs = ipurge_tr I D u xs"
idx_sequence_inj: fixes idx :: "nat \<Rightarrow> nat"   and m :: "nat"   and n :: "nat" assumes "idx_sequence idx"   and "idx m = idx n" shows "m = n"
OUT_plus_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g" shows "d_OUT (plus_current f g) x = (if x \<in> RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) then d_OUT f x else d_OUT g x)"
max_bot2: fixes x :: "'b" shows "max x (0::'b) = x"
simps_4: fixes x2 :: "String.literal \<times> int"   and x1 :: "String.literal" shows "Var x2 \<noteq> Free x1"
image_affinity_cbox: fixes m :: "real"   and c :: "'b"   and a :: "'b"   and b :: "'b" shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b = (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c) else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c))"
gbm_nodes_list_it_def: fixes m1_ops :: "('d, 'e, 'f, 'g) map_ops_scheme"   and g :: "'f" shows "GraphByMapDefs.gbm_nodes_list_it m1_ops g \<equiv> map_iterator_dom (poly_map_iteratei_defs.iteratei (map_op_list_it m1_ops) g)"
language_coinduct_upto_Plus: fixes R :: "'b language \<Rightarrow> 'b language \<Rightarrow> bool"   and L :: "'b language"   and K :: "'b language" assumes "R L K"   and "\<And>L K. R L K \<Longrightarrow> \<oo> L = \<oo> K \<and> (\<forall>x. Plus_cong R (\<dd> L x) (\<dd> K x))" shows "L = K"
poly_pCons: fixes a :: "'a"   and p :: "'a Polynomial.poly"   and x :: "'a" shows "poly (pCons a p) x = a + x * poly p x"
freshBinp_def: fixes xs :: "'c"   and x :: "'d"   and binp :: "'e \<Rightarrow> ('f, 'g, 'c, 'd, 'h) qAbs set option" shows "freshBinp xs x binp \<equiv> liftAll (freshAbs xs x) binp"
ass: shows "ys <\<^sub>r\<^sub>l\<^sub>e\<^sub>x xs"
sublens_greatest: fixes X :: "'a \<Longrightarrow> 'b" assumes "vwb_lens X" shows "X \<subseteq>\<^sub>L 1\<^sub>L"
cont_por1: fixes y :: "ValD" shows "cont (\<lambda>x. x por y)"
dist_fs_triangle_finite: fixes a :: "complex"   and b :: "complex"   and c :: "complex" shows "cmod (a - b) / (sqrt (1 + (cmod a)\<^sup>2) * sqrt (1 + (cmod b)\<^sup>2)) \<le> cmod (a - c) / (sqrt (1 + (cmod a)\<^sup>2) * sqrt (1 + (cmod c)\<^sup>2)) + cmod (c - b) / (sqrt (1 + (cmod b)\<^sup>2) * sqrt (1 + (cmod c)\<^sup>2))"
S\<^sub>4_complete: fixes w :: "alphabet list" shows "length (filter (\<lambda>x. x = a) w) = length (filter (\<lambda>x. x = b) w) \<longrightarrow> w \<in> S\<^sub>4"
ipresCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Iteration.model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
spanning_subset_independent: fixes B :: "('a, 'b) vec set"   and A :: "('a, 'b) vec set" assumes "B \<subseteq> A"   and "vec.independent A"   and "A \<subseteq> vec.span B" shows "A = B"
is_pure_alt_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> assn" shows "is_pure R = (\<exists>Ri. \<forall>x y. R x y = \<up> ((y, x) \<in> Ri))"
cproper_interval_integer_def: shows "cproper_interval = proper_interval"
rel_prod_sel: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and p :: "'c \<times> 'e"   and q :: "'d \<times> 'f" shows "rel_prod R1 R2 p q = (R1 (fst p) (fst q) \<and> R2 (snd p) (snd q))"
C_eq_s: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "p \<noteq> []" shows "Cp (list2FWpolicy (separate p)) = Cp (list2FWpolicy p)"
measurable_simple_function: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "simple_function M f" shows "f \<in> M \<rightarrow>\<^sub>M count_space UNIV"
composable: shows "src (local.dom \<mu>) = trg (local.dom \<nu>)"
finite: fixes g :: "'g" assumes "G_invar g" shows "finite (WGraph.edges (G_\<alpha> g))"
assms_5: shows "\<guillemotleft>\<psi> : G f \<cong>\<^sub>D G f'\<guillemotright>"
assms_2: shows "lfinite (\<pi>\<^bsub>c\<^esub>inf_llist t)"
con_rews_1: shows "WriterT\<cdot>\<bottom> = \<bottom>"
accepting_pair\<^sub>R_def: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and P :: "('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set"   and w :: "nat \<Rightarrow> 'c" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 P w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> fst P = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> snd P \<noteq> {}"
expectation_gpv1_def: shows "expectation_gpv1 \<equiv> expectation_gpv fail \<I> f"
ab_group_add_hom_axioms: shows "ab_group_add_hom hom"
isMinimalState_def: fixes stateMin :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set" shows "isMinimalState stateMin F0 decisionVars \<equiv> \<forall>state. (state, stateMin) \<notin> terminationLess F0 decisionVars"
sfv_cons: fixes a :: "form"   and list :: "form list" shows "sfv (a # list) = fv a @ sfv list"
sumsg: fixes w :: "complex" assumes "w \<in> ball \<xi> r" shows "(\<lambda>i. (deriv ^^ (i + n)) f \<xi> / fact (i + n) * (w - \<xi>) ^ i) sums g w"
elims: fixes x :: "real \<times> real \<times> real"   and y :: "atomUni fmUni" assumes "allZero' x = y" obtains a :: "real" and b :: "real" and c :: "real" where "x = (a, b, c)" and "y = AndUni (AndUni (AtomUni (EqUni (0, 0, a))) (AtomUni (EqUni (0, 0, b)))) (AtomUni (EqUni (0, 0, c)))"
conv_abscissa_mono: fixes g :: "'b fds"   and f :: "'b fds" assumes "\<And>s. fds_converges g s \<Longrightarrow> fds_converges f s" shows "conv_abscissa f \<le> conv_abscissa g"
perp_per_1: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C A" shows "Tarski_neutral_dimensionless.Per Bet Cong B A C"
elims_3: fixes x :: "'b gexp" assumes "\<not> contains_bool x" obtains g1 :: "'b gexp" and g2 :: "'b gexp" where "x = Nor g1 g2" and "\<not> (contains_bool g1 \<or> contains_bool g2)"   | v :: "'b aexp" and va :: "'b aexp" where "x = Eq v__ va__"   | v :: "'b aexp" and va :: "'b aexp" where "x = Gt vb__ vaa__"   | v :: "'b" and va :: "value list" where "x = In v va"
Unique_def: fixes ev :: "event"   and evs :: "event list" shows "(Unique ev on evs) = (ev \<notin> set (tl (dropWhile (\<lambda>z. z \<noteq> ev) evs)))"
assms: shows "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))"
anycast_weakenE: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a seq_action"   and Q :: "'a \<Rightarrow> bool" assumes "anycast P a"   and "\<And>m. P m \<Longrightarrow> Q m" shows "anycast Q a"
Sup_finite_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Squnion> (insert a A) = a \<squnion> \<Squnion> A"
append_rows_split: fixes k :: "nat"   and A :: "'a mat" assumes "k \<le> dim_row A" shows "A = mat_of_rows (dim_col A) (map (Matrix.row A) [0..<k]) @\<^sub>r mat_of_rows (dim_col A) (map (Matrix.row A) [k..<dim_row A])"
eq_diff_iff: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k \<le> m"   and "k \<le> n" shows "(m - k = n - k) = (m = n)"
all_dupE: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x. P x" obtains "P x" and "\<forall>x. P x"
sum_mset_sum_list: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'a list" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum_mset plus zero (mset xs) = monoid_add.sum_list plus zero xs"
map_2: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and f3 :: "'f \<Rightarrow> 'c"   and x21 :: "'e"   and x22 :: "'f" shows "map_generat f1 f2 f3 (IO x21 x22) = IO (f2 x21) (f3 x22)"
cil_ileq: fixes w :: "'a list"   and w1 :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and w2 :: "'a list" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "w1 \<preceq> w \<and> w2 \<preceq> w"
simps_2: fixes k :: "nat"   and u :: "nat" shows "c_unfold (Suc k) u = (if k = 0 then [u] else c_fst u # c_unfold k (c_snd u))"
s_cond_def_2: fixes t :: "real"   and a :: "real"   and v :: "real"   and s0 :: "real" assumes "0 \<le> t"   and "t \<le> movement.t_stop a v" shows "movement.s a v s0 t = movement.p a v s0 t"
conseq: fixes \<Gamma> :: "'f \<Rightarrow> ('g, 'f, 'h) com option"   and \<Theta> :: "('g set \<times> 'f \<times> 'g set \<times> 'g set) set"   and F :: "'h set"   and P' :: "'e \<Rightarrow> 'g set"   and c :: "('g, 'f, 'h) com"   and Q' :: "'e \<Rightarrow> 'g set"   and A' :: "'e \<Rightarrow> 'g set"   and P :: "'g set"   and Q :: "'g set"   and A :: "'g set" assumes "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)"   and "\<forall>s. s \<in> P \<longrightarrow> (\<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
staticSecret_synth_eq: fixes m :: "msg"   and A :: "agent"   and H :: "msg set" assumes "m \<in> staticSecret A" shows "(m \<in> synth H) = (m \<in> H)"
simps: fixes uu :: "nat"   and a :: "'b"   and ts :: "'b tree list" shows "mset_tree (Node uu a ts) = {#a#} + \<Sum>\<^sub># (image_mset mset_tree (mset ts))"
open_Union_compact_subsets: fixes S :: "'a set" assumes "open S" obtains C :: "nat \<Rightarrow> 'a set" where "\<And>n. compact (C n)" and "\<And>n. C n \<subseteq> S" and "\<And>n. C n \<subseteq> interior (C (Suc n))" and "\<Union> (range C) = S" and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n\<ge>N. K \<subseteq> C n"
left_idem: fixes a :: "'d"   and b :: "'d" shows "inf a (inf a b) = inf a b"
Radj_repd1: fixes x :: "'sz"   and y :: "'sz"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and r :: "real" shows "Radj x y (repd \<nu> y r) = repd (Radj x y \<nu>) x r"
simps_1: fixes x11 :: "'a"   and x12 :: "bool"   and y11 :: "'a"   and y12 :: "bool" shows "(<x11, inM x12> = <y11, inM y12>) = (x11 = y11 \<and> x12 = y12)"
disjoint_iff_not_equal: fixes A :: "'k set"   and B :: "'k set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
homomthm1: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "gbij\<^bsub>(F / gker\<^bsub>F,G\<^esub> f),Img\<^bsub>F,G\<^esub> f\<^esub> (f\<dieresis>\<^bsub>F,Img\<^bsub>F,G\<^esub> f\<^esub>)"
run_scons_elim: fixes a :: "'action"   and r :: "'action stream"   and p :: "'state" assumes "R.run (a ## r) p" obtains "ren a p" and "R.run r (ex a p)"
inf_cancel_left1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "x \<cdot> a \<cdot> (- x \<cdot> b) = 0"
cases: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and a :: "(('b, 'c, 'a) formula list \<times> ('b, 'c, 'a) formula list) \<times> (nat \<times> ('b, 'c, 'a) step) list" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4"   and "ids.proof_ok vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 a" obtains D :: "('b, 'c, 'a) formula list \<times> ('b, 'c, 'a) formula list" and S :: "(nat \<times> ('b, 'c, 'a) step) list" where "a = (D, S)" and "ids.deriv_ok vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 (ids.start_proof D) S"
axioms_1: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "primeideal I R" shows "ideal I R"
connected_openin: fixes S :: "'b set" shows "connected S = (\<nexists>E1 E2. openin (top_of_set S) E1 \<and> openin (top_of_set S) E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
K4_def: fixes upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list"   and LV :: "char list set"   and C :: "acom"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "K4 upds LV C Q = (K C LV Q \<and> K3 upds C Q \<and> (\<forall>s1 s2. s1 = s2 on kdeps C \<longrightarrow> time C s1 = time C s2))"
hyps_2: shows "x \<notin>\<^sub>\<circ> F"
sa_impl_rel_eext_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rl :: "('h \<times> 'k) set" shows "\<langle>Rm, Rv, Rl\<rangle>sa_impl_rel_eext \<equiv> \<langle>Rm, Rv \<rightarrow> Rl\<rangle>gen_sa_impl_rel_eext"
unsat_imp_se_unsat: fixes c :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c' :: "('c, 'd) conf" assumes "se c l c'"   and "\<not> Conf.sat c" shows "\<not> Conf.sat c'"
joinable_def: fixes R :: "('b \<times> 'c) set" shows "joinable R = {(x, y). \<exists>z. (x, z) \<in> R \<and> (y, z) \<in> R}"
assms_3: shows "R x' y' \<equiv> R' x' y'"
bacc_mono: fixes n :: "nat"   and m :: "nat"   and r :: "('a \<times> 'a) set" assumes "n \<le> m" shows "bacc r n \<subseteq> bacc r m"
has_derivative_iff_norm: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
split_of_bool: fixes one :: "'d"   and zero :: "'d"   and P :: "'d \<Rightarrow> bool"   and p :: "bool" assumes "class.zero_neq_one one zero" shows "P (zero_neq_one.of_bool one zero p) = ((p \<longrightarrow> P one) \<and> (\<not> p \<longrightarrow> P zero))"
that: fixes q :: "'b list"   and r :: "'b list" assumes "p = q @ r"   and "awalk w (r @ q) w"   and "set (awalk_verts w (r @ q)) = set (awalk_verts u p)" shows "thesis"
changes_all_nonneg: fixes xs :: "'a list" assumes "\<And>i. (0::'a) \<le> nth_default (0::'a) xs i" shows "changes xs = 0"
chainI: fixes Y :: "nat \<Rightarrow> 'c" assumes "\<And>i. Y i \<sqsubseteq> Y (Suc i)" shows "chain Y"
div_pos_pos_trivial: fixes k :: "int"   and l :: "int" assumes "0 \<le> k"   and "k < l" shows "k div l = 0"
inverse_le_1_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse x) one = (less_eq x zero \<or> less_eq one x)"
scale_right_diff_distrib: fixes a :: "complex"   and x :: "'a"   and y :: "'a" shows "a *\<^sub>C (x - y) = a *\<^sub>C x - a *\<^sub>C y"
Map_ide_1: fixes a :: "('a E.term, 'a) arr" assumes "ide a" shows "Map a = EVAL (Dom a)"
Pd'_tax_x: shows "x \<in> Field (Pd'_tax (Xd x))"
fdom_make_fmap: fixes ks :: "'a fset"   and v :: "'b" shows "fmdom [ ks |=> v ] = ks"
l7_13: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P' :: "'a"   and P :: "'a"   and Q' :: "'a"   and Q :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P' P"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A Q' Q" shows "Cong P Q P' Q'"
continuous_on_sum: fixes I :: "'a set"   and S :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>i. i \<in> I \<Longrightarrow> continuous_on S (f i)" shows "continuous_on S (\<lambda>x. \<Sum>i\<in>I. f i x)"
monad_commute_id: shows "monad_commute return_id bind_id"
prems: fixes n :: "nat" assumes "n < length (ia # ias)" shows "interrupt_action_ok (redT_updIs is (take n (ia # ias))) ((ia # ias) ! n)"
subsumees_conv: fixes subs :: "(('b \<times> nat) \<times> 'b \<times> nat) set" shows "subsumees subs = {v. \<exists>v'. (v, v') \<in> subs}"
do_logE: fixes op :: "lop"   and v :: "v"   and e :: "exp0"   and res :: "exp_or_val option" assumes "do_log op v e = res" obtains "op = And" and "res = (case v of Litv x \<Rightarrow> Map.empty x | Conv None l2 \<Rightarrow> None | Conv (Some (s1, t1)) l2 \<Rightarrow> if s1 = ''true'' then case t1 of TypeId (Short s2) \<Rightarrow> if s2 = ''bool'' then case l2 of [] \<Rightarrow> Some (Exp e) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else if s1 = ''false'' then case t1 of TypeId (Short s4) \<Rightarrow> if s4 = ''bool'' then case l2 of [] \<Rightarrow> Some (Val v) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else None | Closure x xa xb \<Rightarrow> Map.empty xb | Recclosure x xa xb \<Rightarrow> Map.empty xb | Loc x \<Rightarrow> Map.empty x | Vectorv x \<Rightarrow> Map.empty x)"   | "op = Or" and "res = (case v of Litv x \<Rightarrow> Map.empty x | Conv None l6 \<Rightarrow> None | Conv (Some (s8, t0)) l6 \<Rightarrow> if s8 = ''false'' then case t0 of TypeId (Short s9) \<Rightarrow> if s9 = ''bool'' then case l6 of [] \<Rightarrow> Some (Exp e) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else if s8 = ''true'' then case t0 of TypeId (Short s11) \<Rightarrow> if s11 = ''bool'' then case l6 of [] \<Rightarrow> Some (Val v) | a # x \<Rightarrow> Map.empty x else None | TypeId (Long xa xb) \<Rightarrow> Map.empty xb | TypeExn x \<Rightarrow> Map.empty x else None | Closure x xa xb \<Rightarrow> Map.empty xb | Recclosure x xa xb \<Rightarrow> Map.empty xb | Loc x \<Rightarrow> Map.empty x | Vectorv x \<Rightarrow> Map.empty x)"
coprime_rewrites: shows "comm_monoid_mult.coprime (*) (1::'a) = coprime"
distinct_59: fixes x51 :: "MIR.num"   and x52 :: "MIR.num"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "Add x51 x52 \<noteq> CF x91 x92 x93"
Abs_ivl_induct: fixes P :: "ivl \<Rightarrow> bool"   and x :: "ivl" assumes "\<And>y. y \<in> {c. \<exists>x. eq_ivl x x \<and> c = Collect (eq_ivl x)} \<Longrightarrow> P (Abs_ivl y)" shows "P x"
encode_kind_n_3: fixes m :: "nat"   and n :: "nat" shows "encode_kind (encode (recf.Id m n)) = 2"
assms: shows "list_all (\<lambda>a. varset_action a (prob_dom PROB - vs)) as"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>F 'e" shows "fmdom' x = dom (fmlookup x)"
Suc: fixes r :: "'a poly"   and d :: "'a poly"   and dr :: "nat" shows "local.divide_poly_main (0::'a) 0 r d dr n_ = 0"
nn_integral_pmf': fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and p :: "'b pmf" assumes "inj_on f A" shows "\<integral>\<^sup>+ x. ennreal (pmf p (f x)) \<partial>count_space A = emeasure (measure_pmf p) (f ` A)"
Re_complex_div_eq_0: fixes a :: "complex"   and b :: "complex" shows "(Re (a / b) = 0) = (Re (a * cnj b) = 0)"
elims: fixes x :: "variable \<Rightarrow> 'a"   and xa :: "object \<Rightarrow> 'a"   and xb :: "PDDL_STRIPS_Semantics.term"   and y :: "'a" assumes "ty_term x xa xb = y" obtains varT :: "variable \<Rightarrow> 'a" and objT :: "object \<Rightarrow> 'a" and v :: "variable" where "x = varT__" and "xa = objT__" and "xb = term.VAR v" and "y = varT__ v"   | varT :: "variable \<Rightarrow> 'a" and objT :: "object \<Rightarrow> 'a" and c :: "object" where "x = varT" and "xa = objT" and "xb = term.CONST c" and "y = objT c"
face_cycle_sets_def: shows "H.face_cycle_sets = H.face_cycle_set ` arcs H"
length_Cons: fixes x :: "'c"   and xs :: "'c list" shows "length (x # xs) = Suc (length xs)"
irred_0: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and F :: "('c \<Rightarrow>\<^sub>0 'f) set" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "\<not> ordered_term.is_red pair_of_term term_of_pair ord_term F 0"
qf3_cube_prime_impl_cube_form: fixes a :: "int"   and b :: "int"   and P :: "int" assumes "coprime a b"   and "P ^ 3 = a\<^sup>2 + 3 * b\<^sup>2"   and "prime P \<and> odd P" shows "is_cube_form a b"
is_iso_class_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b set" assumes "category C" shows "category.is_iso_class C F \<equiv> \<exists>f. f \<in> F \<and> F = category.iso_class C f"
base: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "\<phi> xs" shows "ev \<phi> xs"
8: shows "{} \<turnstile> local.P AND Neg Q\<^sub>1 IMP Q\<^sub>2"
lNactLe: fixes c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> n"
ifp_compatible_with_sp_spec: shows "\<forall>a b. Policy.sp_spec_subj_subj a b \<longrightarrow> Policy.ifp a b \<and> Policy.ifp b a"
append_Nil: fixes ys :: "'f list" shows "[] @ ys = ys"
arr_iff_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.arr (free_monoidal_category.comp C) f = free_monoidal_category.ARR C f"
uint64_simps_4: fixes x :: "uint64"   and y :: "uint64" shows "int_of_uint64 (x * y) = int_of_uint64 x * int_of_uint64 y mod 18446744073709551616"
negate_eq2a: fixes x :: "'a" assumes "x \<in> V" shows "- 1 \<cdot> x = - x"
I_qe_eps1: fixes \<phi> :: "atom fm"   and xs :: "'a list" assumes "nqfree \<phi>" shows "DLO.I (qe_eps\<^sub>1 \<phi>) xs = (\<exists>x. DLO.I \<phi> (x # xs))"