hcju/isabelleps huggingface dataset:
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and w :: "'c"   and \<theta> :: "'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and u :: "'c"   and \<nu> :: "'c"   and \<omega> :: "'c"   and g :: "'c"   and r :: "'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<rho> :: "'c" assumes "partial_magma.ide V w \<and> partial_magma.in_hom V \<theta> (H f w) u \<and> partial_magma.in_hom V \<nu> (partial_magma.dom V \<omega>) (H g w) \<and> category.iso V \<nu> \<and> V (H r \<theta>) (V (\<a> r f w) (V (H \<rho> w) \<nu>)) = \<omega>" shows "uw\<theta>\<omega>\<nu>_axioms V H \<a> r \<rho> f g u w \<theta> \<omega> \<nu>"
intros_1: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm (\<lbrakk>S\<rbrakk>)"
R2: fixes v :: "'a set" assumes "v \<in> nhds y"   and "x \<notin> v" shows "R"
circ_unfold_sum: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = x\<^sup>\<star> \<squnion> x\<^sup>\<star> * y * (x \<squnion> y)\<^sup>\<star>"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "rev (x # xs) = rev xs @ [x]"
ord_spmf_bind_pmfI1: fixes p :: "'c pmf"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd spmf"   and q :: "'d spmf" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> ord_spmf R (f x) q" shows "ord_spmf R (p \<bind> f) q"
assms_1: shows "\<forall>\<^sub>F n in sequentially. 2 \<le> f n"
AG_lem1: fixes x :: "'b"   and s :: "'b set" assumes "x \<in> s \<and> x \<in> AX (AG s)" shows "x \<in> AG s"
unfold_tllist_eq_TNil: fixes IS_TNIL :: "'c \<Rightarrow> bool"   and TNIL :: "'c \<Rightarrow> 'b"   and THD :: "'c \<Rightarrow> 'a"   and TTL :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'b" shows "(unfold_tllist IS_TNIL TNIL THD TTL a = TNil b) = (IS_TNIL a \<and> b = TNIL a)"
mker_submodule: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and N :: "('i, 'f, 'j) Module_scheme"   and f :: "'e \<Rightarrow> 'i" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N" shows "submodule R M (ker\<^bsub>M,N\<^esub> f)"
is_extensional: fixes f :: "'a" assumes "\<not> D.J.arr f" shows "\<tau>o\<chi>.map f = D.C.null"
field_poly_in_prime_factorization_imp_prime: fixes p :: "'b poly"   and x :: "'b poly" assumes "p \<in># field_poly.prime_factorization x" shows "prime_elem p"
wfP_empty: shows "wfP (\<lambda>x y. False)"
norm_ell2_vec_of_basis_enum: fixes \<psi> :: "'a" shows "norm \<psi> = (let \<psi>' = vec_of_basis_enum \<psi> in sqrt (\<Sum>i = 0..<dim_vec \<psi>'. let z = \<psi>' $ i in (Re z)\<^sup>2 + (Im z)\<^sup>2))"
null_space_orthogonal_complement_row_space: fixes A :: "(('a, 'rows) vec, 'cols) vec" shows "null_space A = ROWS.v.orthogonal_complement (row_space (\<chi>i j. cnj_class.cnj (A $ i $ j)))"
winding_number_pos_lt: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and e :: "real" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>"   and "0 < e"   and "\<And>x. \<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> e \<le> Im (vector_derivative \<gamma> (at x) * cnj (\<gamma> x - z))" shows "0 < Re (winding_number \<gamma> z)"
mult_nonpos_nonpos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq a zero"   and "less_eq b zero" shows "less_eq zero (times a b)"
assms: shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>e\<^sub>s\<^sub>t A"
beta_reduces_pair_right: fixes B :: "'a trm"   and B' :: "'a trm"   and A :: "'a trm" assumes "B \<rightarrow>\<beta>\<^sup>* B'" shows "SimplyTyped.Pair A B \<rightarrow>\<beta>\<^sup>* SimplyTyped.Pair A B'"
assoc: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "a +\<^sub>o\<^sub>w b +\<^sub>o\<^sub>w c = a +\<^sub>o\<^sub>w (b +\<^sub>o\<^sub>w c)"
simps_2: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and c :: "color"   and r :: "('b \<times> color) tree" shows "invh \<langle>l, (x, c), r\<rangle> = (bheight l = bheight r \<and> invh l \<and> invh r)"
quad_form_def: fixes v :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "quad_form v H = vec_cnj v *\<^sub>v\<^sub>m H *\<^sub>v\<^sub>v v"
row_length_def: fixes xs :: "int poly fract list list" shows "rat_poly.row_length xs \<equiv> if xs = [] then 0 else length (hd xs)"
while_top_def: fixes b :: "'a upred"   and P :: "'a hrel" shows "while\<^sup>\<top> b do P od = (\<nu> X \<bullet> P ;; X \<triangleleft> b \<triangleright>\<^sub>r II)"
integrable_MAX: fixes I :: "'b set"   and M :: "'c measure"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> real" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> integrable M (f i)" shows "integrable M (\<lambda>x. MAX i\<in>I. f i x)"
awalk_verts_append: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list"   and q :: "'c list"   and v :: "'b" assumes "wf_digraph G"   and "pre_digraph.awalk G u (p @ q) v" shows "pre_digraph.awalk_verts G u (p @ q) = pre_digraph.awalk_verts G u p @ tl (pre_digraph.awalk_verts G (last (pre_digraph.awalk_verts G u p)) q)"
futures_def: fixes P :: "'b process"   and xs :: "'b list" shows "futures P xs \<equiv> {(ys, Y). (xs @ ys, Y) \<in> failures P}"
pentagon\<^sub>A\<^sub>W\<^sub>C: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "ide k"   and "sources f \<inter> targets g \<noteq> {}"   and "sources g \<inter> targets h \<noteq> {}"   and "sources h \<inter> targets k \<noteq> {}" shows "(f \<star> \<a>[g, h, k]) \<cdot> \<a>[f, g \<star> h, k] \<cdot> (\<a>[f, g, h] \<star> k) = \<a>[f, g, h \<star> k] \<cdot> \<a>[f \<star> g, h, k]"
vectorspace_axioms: shows "vectorspace class_ring (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})"
bij: shows "bij_betw (\<lambda>(k, d). (d * k, k)) (SIGMA k:{k. 0 < k \<and> real k \<le> x}. {d. 0 < d \<and> real d \<le> x / real k}) (SIGMA k:{k. 0 < k \<and> real k \<le> x}. {d. d dvd k})"
tendsto_diff: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c"   and b :: "'c" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
kat_1_equiv_opp: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y \<cdot> d x \<le> d z \<cdot> y) = (ad z \<cdot> y \<le> y \<cdot> ad x)"
t_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [a] : [] _> [t]"
drop_per_pref: fixes w :: "'b list"   and u :: "'b list" assumes "w \<le>p u\<^sup>\<omega>" shows "drop \<^bold>|u\<^bold>| w \<le>p w"
val2formAppend: fixes valuation1 :: "Literal list"   and valuation2 :: "Literal list" shows "val2form (valuation1 @ valuation2) = val2form valuation1 @ val2form valuation2"
of_nat_prod: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> nat"   and A :: "'b set" assumes "class.comm_semiring_1 times one plus zero" shows "semiring_1.of_nat one plus zero (prod f A) = comm_monoid_mult.prod times one (\<lambda>x. semiring_1.of_nat one plus zero (f x)) A"
TTreeAnalysisCarrier_axioms: shows "TTreeAnalysisCarrier Texp Aexp"
div2_in_bounds: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and n :: "nat" assumes "braun \<langle>l, x, r\<rangle>"   and "n \<in> {1..size \<langle>l, x, r\<rangle>}"   and "1 < n" shows "(odd n \<longrightarrow> n div 2 \<in> {1..size r}) \<and> (even n \<longrightarrow> n div 2 \<in> {1..size l})"
vsv_vdomain_vrange_vsingleton: fixes a :: "V"   and b :: "V" assumes "\<D>\<^sub>\<circ> F = ZFC_in_HOL.set {a}"   and "\<R>\<^sub>\<circ> F = ZFC_in_HOL.set {b}" shows "F = ZFC_in_HOL.set {\<langle>a, b\<rangle>}"
minus_flow: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and g :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "wf_residual_network"   and "flow \<Delta> f"   and "flow \<Delta> g"   and "value_flow \<Delta> g \<le> value_flow \<Delta> f"   and "f (source \<Delta>, x) \<noteq> \<top>"   and "\<And>y. edge \<Delta> (source \<Delta>) y = (y = x)" shows "flow (residual_network g) (f \<ominus> g)"
assms_2: assumes "P = FF" shows "Q \<noteq> \<bottom>"
inferences_from_def: fixes \<Gamma> :: "'b Inference_System.inference set"   and CC :: "'b literal multiset set" shows "inference_system.inferences_from \<Gamma> CC = {\<gamma> \<in> \<Gamma>. old_infer_from CC \<gamma>}"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Arr \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Arr t \<and> Arr u \<and> Arr v)"
l_set_mode_get_shadow_root_axioms_def: fixes set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_shadow_root_axioms set_mode_locs get_shadow_root_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
is_scc_mconn_eqclasses: fixes E :: "('b \<times> 'b) set"   and U :: "'b set" shows "is_scc E U = (U \<in> UNIV // mconn E)"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "swapLeaves (Node w t\<^sub>1 t\<^sub>2) w\<^sub>a a w\<^sub>b b = Node w (swapLeaves t\<^sub>1 w\<^sub>a a w\<^sub>b b) (swapLeaves t\<^sub>2 w\<^sub>a a w\<^sub>b b)"
spaceN_contains_zero: fixes N :: "'c quasinorm" shows "(0::'c) \<in> space\<^sub>N N"
error_free_def: fixes s :: "state" shows "error_free s = (\<nexists>err. fst s = Some (Error err))"
acc_induct_rule: fixes a :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<And>y. (y, x) \<in> r \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
leq_add_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.add_linorder_min plus minus zero less_eq less" shows "less_eq x (plus y x)"
Q_def: fixes p :: "L"   and H :: "Var \<Rightarrow> L" shows "Q p H = (\<lambda>s t. \<forall>x. \<not> LEQ p (H x) \<longrightarrow> t x = s x)"
circ_simulate: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z * x \<le> y * z \<longrightarrow> z * x\<^sup>\<circ> \<le> y\<^sup>\<circ> * z"
freshEnv_swapEnv_exchange2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "goodEnv rho" shows "freshEnv xs x2 (rho &[x2 \<and> x1]_xs) = freshEnv xs x1 rho"
integral_combine_tagged_division_topdown: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and p :: "('a \<times> 'a set) set" assumes "f integrable_on cbox a b"   and "p tagged_division_of cbox a b" shows "integral (cbox a b) f = (\<Sum>(x, k)\<in>p. integral k f)"
round_len_eq: fixes t :: "nat \<times> nat list \<times> 'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "bn_count (fst (snd t)) = foldl (+) 0 (fst (snd t))" shows "length (fst (snd (round index key p q r t))) = foldl (+) 0 (fst (snd t))"
to_nat_less_card: fixes j :: "'b" shows "mod_type_class.to_nat j < CARD('b)"
keys_update: fixes k :: "'c"   and v :: "'d"   and m :: "('c, 'd) mapping" shows "Mapping.keys (Mapping.update k v m) = insert k (Mapping.keys m)"
Union_disjoint: fixes C :: "'a set set"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>A\<in>C. finite A"   and "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" shows "sum g (\<Union> C) = (sum \<circ> sum) g C"
bij_def: fixes f :: "'a \<Rightarrow> 'b" shows "bij f = (inj f \<and> top_pres (\<P> f))"
simps: fixes a :: "'a list"   and b :: "'a list" shows "get_pos (BUF a b) = length a"
continuous_open_vimage: fixes S :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "open S"   and "\<And>x. isCont f x" shows "open (f -` S)"
spec: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<forall>x. P x" shows "P x"
aux7aa: fixes a :: "'b set set"   and b :: "'b set set"   and poo :: "'c"   and p :: "('b set set, 'c) Combinators list"   and c :: "'b set set"   and d :: "'b set set"   and po :: "'c" assumes "AllowPortFromTo a b poo \<in> set p"   and "allNetsDistinct (AllowPortFromTo c d po # p)"   and "a \<noteq> c \<or> b \<noteq> d" shows "twoNetsDistinct a b c d"
block_comp_incomplete: fixes bl :: "'a set" assumes "incomplete_block bl" shows "0 < card (bl\<^sup>c)"
drop_not_empty: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "drop (length xs div 2) xs \<noteq> []"
NS_Spy_analz_staticSecret: fixes m :: "msg"   and A :: "agent"   and evs :: "event list" assumes "m \<in> staticSecret A"   and "evs \<in> ns_public" shows "(m \<in> analz (knows Spy evs)) = (A \<in> bad)"
rotate_3: fixes queue :: "'queue"   and n :: "nat" assumes "\<alpha>queue' @ [t] = queue_\<alpha> queue"   and "local.round_robin_invar (queue, n) (dom (thr_\<alpha> (thr s)))"   and "state_invar s"   and "state_\<alpha> s \<in> I" shows "map_option (apsnd (apsnd round_robin_\<alpha>)) (round_robin_reschedule t0 queue n0 s) = \<alpha>.round_robin_reschedule t0 (queue_\<alpha> queue) n0 (state_\<alpha> s)"
assms_2: assumes "dg" shows "deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v))"
wf_bounded_set: fixes r :: "('b \<times> 'b) set"   and ub :: "'b \<Rightarrow> 'c set"   and f :: "'b \<Rightarrow> 'c set" assumes "\<And>a b. (b, a) \<in> r \<Longrightarrow> finite (ub a) \<and> ub b \<subseteq> ub a \<and> f b \<subseteq> ub a \<and> f a \<subset> f b" shows "wf r"
Neg_I0: fixes i :: "'a"   and A :: "fm"   and H :: "fm set" assumes "\<And>B. atom i \<sharp> B \<Longrightarrow> insert A H \<turnstile> B" shows "H \<turnstile> Neg A"
l1_inv1I: fixes x :: "l1_state" assumes "\<And>Ra A B gny. 0 < signalsInit x (Commit A B (Exp gny (NonceF (Ra $ nx)))) \<Longrightarrow> guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> progress x Ra = Some {xnx, xgnx, xgny, xsk, xEnd} \<and> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))" shows "x \<in> l1_inv1"
not_Interp_imp_not_interp: fixes D' :: "'a literal multiset"   and D :: "'a literal multiset"   and C :: "'a literal multiset" assumes "D' \<le> D"   and "D' \<in> N \<and> S D' = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> Interp D \<Turnstile> D'"   and "D < C" shows "\<not> interp C \<Turnstile> D'"
simps_4: fixes f :: "typ \<Rightarrow> typ"   and T :: "typ"   and t :: "Core.term" shows "map_types f (Abs T t) = Abs (f T) (map_types f t)"
map_mono_suffix: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "suffix xs ys" shows "suffix (map f xs) (map f ys)"
symrun_interp_remdups_absorb: fixes \<Gamma> :: "'a constr list" shows "\<lbrakk>\<lbrakk> \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> remdups \<Gamma> \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
finite_UnI: fixes F :: "'x set"   and G :: "'x set" assumes "finite F"   and "finite G" shows "finite (F \<union> G)"
step_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s')"   and "s = Fault f" shows "s' = Fault f"
simps_11: fixes v :: "'a formula"   and va :: "'a formula" shows "is_lit_plus (v \<^bold>\<rightarrow> va) = False"
lappend_snocL1_conv_LCons2: fixes xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" shows "lappend (lappend xs (LCons y LNil)) ys = lappend xs (LCons y ys)"
map_sq_matrix_diff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "('b, 'd) sq_matrix"   and B :: "('b, 'd) sq_matrix" assumes "\<And>a b. f (a - b) = f a - f b" shows "map_sq_matrix f (A - B) = map_sq_matrix f A - map_sq_matrix f B"
assoc_in_homs_3: shows "\<guillemotleft>\<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<star> \<rho> : ((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho> \<Rightarrow> (\<mu> \<star> \<nu> \<star> \<pi>) \<star> \<rho>\<guillemotright>"
Digamma_half_integer: fixes n :: "nat" shows "Digamma (of_nat n + (1::'a) / (2::'a)) = (\<Sum>k<n. (2::'a) / of_nat (2 * k + 1)) - euler_mascheroni - of_real (2 * ln 2)"
ab: shows "Re (zeta 3) = real_of_int a / real b"
set_tag_name_get_child_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" assumes "l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes get_child_nodes_locs" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
assms_5: fixes t :: "real" assumes "t \<in> T" shows "0 < t"
homology_additivity_axiom_gen: fixes \<U> :: "'b set set"   and X :: "'b topology"   and p :: "int" assumes "disjoint \<U>"   and "\<Union> \<U> = topspace X"   and "\<And>C T. \<lbrakk>compactin X C; path_connectedin X C; T \<in> \<U>; \<not> disjnt C T\<rbrakk> \<Longrightarrow> C \<subseteq> T" shows "(\<lambda>x. gfinprod (homology_group p X) (\<lambda>V. hom_induced p (subtopology X V) {} X {} id (x V)) \<U>) \<in> Group.iso (sum_group \<U> (\<lambda>S. homology_group p (subtopology X S))) (homology_group p X)"
acc_err: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "acc r" shows "acc (Err.le r)"
sgn_if: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sgn :: "'k \<Rightarrow> 'k"   and abs :: "'k \<Rightarrow> 'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and x :: "'k" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "sgn x = (if x = zero then zero else if less zero x then one else uminus one)"
vector_mult_closed: fixes y :: "'c \<times> 'c \<Rightarrow> 'd"   and x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_bounded_idempotent_semiring.vector y" shows "matrix_bounded_idempotent_semiring.vector (x \<odot> y)"
set_induct_1: fixes x :: "'a"   and a :: "('a, 'b) tllist"   and P :: "'a \<Rightarrow> ('a, 'b) tllist \<Rightarrow> bool" assumes "x \<in> tset a"   and "\<And>z1 z2. P z1 (TCons z1 z2)"   and "\<And>z1 z2 xa. \<lbrakk>xa \<in> tset z2; P xa z2\<rbrakk> \<Longrightarrow> P xa (TCons z1 z2)" shows "P x a"
approximation_preproc_int_10: fixes i :: "int" shows "real_of_int \<bar>i\<bar> = \<bar>real_of_int i\<bar>"
simps_3: fixes P :: "'state"   and \<phi> :: "'pred" shows "valid_Tree P (tPred \<phi>) = P \<turnstile> \<phi>"
p2r_conj_hom_var: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "p2r P ; p2r Q = p2r (P \<sqinter> Q)"
lcp_mismatch': fixes u :: "'b list"   and v :: "'b list" assumes "\<not> u \<bowtie> v" shows "u ! \<^bold>|u \<and>\<^sub>p v\<^bold>| \<noteq> v ! \<^bold>|u \<and>\<^sub>p v\<^bold>|"
f_nxt_def: fixes f :: "'a \<Rightarrow> 'a ttree"   and T :: "'a set"   and x :: "'a" shows "f_nxt f T x = (if x \<in> T then f(x := TTree.empty) else f)"
insert_body_preserve_indices: fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" shows "fst ` set (insert_body xs e) = fst ` set xs \<union> {fst e}"
simps_12: fixes v :: "fm"   and va :: "fm" shows "not (And v va) = NOT (And v va)"
code_el_dec: fixes c :: "'a list" assumes "c \<in> \<C>" shows "Dec \<C> c = [c]"
IH: assumes "Subobjs\<^sub>R P (hd Cs') (Cs' @ [D])"   and "Cs' \<noteq> []" shows "Subobjs\<^sub>R P (hd Cs') Cs'"
starlike_compact_projective1: fixes S :: "'c set"   and a :: "'c" assumes "compact S"   and "a \<in> rel_interior S"   and "\<And>x. x \<in> S \<Longrightarrow> open_segment a x \<subseteq> rel_interior S" shows "S - rel_interior S homeomorphic sphere a 1 \<inter> affine hull S"
all_subfrmls__UnionD: fixes A :: "'b ltlr set"   and B :: "'b ltlr set"   and x :: "'b ltlr"   and y :: "'b ltlr" assumes "\<Union> (subfrmlsr ` A) \<subseteq> B"   and "x \<in> A"   and "y \<in> subfrmlsr x" shows "y \<in> B"
path_eq_chain: shows "chain_subdiv_path \<gamma> one_chain"
prime_factor_nat: fixes n :: "nat" assumes "n \<noteq> 1" shows "\<exists>p. prime p \<and> p dvd n"
absolutely_integrable_integrable_bound: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "f integrable_on S"   and "g integrable_on S" shows "f absolutely_integrable_on S"
fst_foldl_ef_k_less: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" assumes "k < ncols A" shows "snd (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k]) \<le> nrows A"
assms: fixes i :: "'a" assumes "i \<in> I" shows "X i \<in> M.events i"
induce_subgraph_def: fixes G :: "('c, 'd) pre_digraph"   and vs :: "'c set" shows "G \<restriction> vs = \<lparr>verts = vs, arcs = {e \<in> arcs G. tail G e \<in> vs \<and> head G e \<in> vs}, tail = tail G, head = head G\<rparr>"
progress_sat_cong: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<sigma>' :: "(String.literal \<times> event_data list) trace"   and i :: "nat"   and v :: "event_data list" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'"   and "i < pprogress \<phi> \<pi>" shows "Formula.sat \<sigma> Map.empty v i \<phi> = Formula.sat \<sigma>' Map.empty v i \<phi>"
abs_eq_iff: fixes x :: "('b \<times> real) nonzero"   and y :: "('b \<times> real) nonzero" shows "(Proj x = Proj y) = proj_rel x y"
p_antimono: fixes x :: "real"   and y :: "real" assumes "x \<le> y"   and "t_stop \<le> x" shows "p y \<le> p x"
vars_s: shows "vars_mset t \<subseteq># vars_mset s"
fds_abs_diverges: fixes s :: "'a"   and f :: "'a fds" assumes "ereal (s \<bullet> (1::'a)) < abs_conv_abscissa f" shows "\<not> fds_abs_converges f s"
ucast_le_ucast: fixes x :: "'c word"   and y :: "'c word" assumes "LENGTH('c) \<le> LENGTH('d)" shows "(UCAST('c \<rightarrow> 'd) x \<le> UCAST('c \<rightarrow> 'd) y) = (x \<le> y)"