hcju/isabelleps huggingface dataset:
foundation6: fixes \<tau> :: "'d state \<times> 'd state"   and P :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P" shows "\<tau> \<Turnstile> \<delta> P"
lex: shows "fst (lex_ext_unbounded kbo (ss1 @ s # ss2) (ss1 @ t # ss2))"
assms: fixes i :: "nat" assumes "i \<in> set_fml fml" shows "i < length ls"
G1: shows "finite {m. dom (lm.\<alpha> m) = dom (lm.\<alpha> (gState.vars g)) \<and> ran (lm.\<alpha> m) \<subseteq> Collect variable_inv}"
is_ground_cls_imp_is_ground_atm: fixes A :: "'a"   and C :: "'a literal multiset" assumes "A \<in> atms_of C"   and "is_ground_cls C" shows "is_ground_atm A"
isolated_verts_def: shows "H.isolated_verts \<equiv> {v \<in> verts (del_vert u). out_arcs (del_vert u) v = {}}"
eq_SomeD: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "x = Eps P"   and "\<exists>x. P x" shows "P x"
cnj_def: fixes A :: "fmla"   and B :: "fmla" shows "A AND B \<equiv> Robinson_Arithmetic.neg (A IMP Robinson_Arithmetic.neg B)"
abs_eq: shows "\<bottom>\<^sub>S = Abs_scene (\<lambda>x y. x)"
cont: fixes i :: "'b"   and n :: "nat" assumes "i \<in> Basis" shows "continuous_on UNIV (g i n)"
bind_lunit: fixes x :: "'n"   and f :: "'n \<Rightarrow> 'm option" shows "Option.bind (Some x) f = f x"
scale_minus_left: fixes a :: "rat"   and x :: "'a" shows "- a *R x = - (a *R x)"
arr_char: fixes f :: "'a1 \<times> 'a2" shows "arr f = (C1.arr (fst f) \<and> C2.arr (snd f))"
is_SDG_path_irs_SDG_path: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'" shows "n irs-ns\<rightarrow>\<^sub>d* n'"
Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex_def: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and to_ns :: "('a \<times> constraint) list \<Rightarrow> ('a \<times> 'b ns_constraint) list"   and from_ns :: "(nat, 'b) mapping \<Rightarrow> 'b ns_constraint list \<Rightarrow> (nat, rat) mapping" shows "Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc to_ns from_ns \<equiv> Incremental_NS_Constraint_Ops init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc \<and> To_ns to_ns from_ns"
rotate_downneg_def: fixes x :: "wall"   and y :: "wall" shows "rotate_downneg x y \<equiv> x = basic [cup, vert] \<circ> basic [vert, brick.under] \<and> y = basic [vert, cup] \<circ> basic [over, vert]"
e: fixes h :: "'a set" assumes "h \<in> F"   and "a h \<bullet> z < b h" shows "ball z e \<subseteq> {w. a h \<bullet> w < b h}"
long_divides_2: shows "r = [] \<or> degree r < degree [\<one>, \<ominus> x]"
on_circline_x_axis: fixes z :: "complex_homo" shows "on_circline x_axis z = ((\<exists>c. is_real c \<and> z = of_complex c) \<or> z = \<infinity>\<^sub>h)"
rcliff_greek_less1: fixes r :: "('a \<times> 'a) set"   and as :: "(accent \<times> 'a) list"   and a :: "'a"   and b :: "'a"   and b' :: "(accent \<times> 'a) list"   and cs :: "(accent \<times> 'a) list"   and a' :: "(accent \<times> 'a) list"   and bs :: "(accent \<times> 'a) list" assumes "trans r"   and "snd ` set as \<subseteq> under r a \<inter> under r b"   and "b' \<in> {[(Grave, b)], []}"   and "snd ` set cs \<subseteq> under r b"   and "a' = [(Macron, a)]"   and "snd ` set bs \<subseteq> under r b" shows "(as @ b' @ cs @ a' @ bs, [(Macron, a), (Grave, b)]) \<in> greek_less r"
pair_spmf_alt_def: fixes p :: "'a spmf"   and q :: "'b spmf" shows "pair_spmf p q = p \<bind> (\<lambda>x. q \<bind> (\<lambda>y. return_spmf (x, y)))"
in_Span_obtain_same_length_coeffs: fixes n :: "'a"   and smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and ms :: "'a list" assumes "n \<in> R_scalar_mult.RSpan UNIV smult ms" shows "\<exists>rs. set rs \<subseteq> UNIV \<and> length rs = length ms \<and> n = scalar_mult.lincomb smult rs ms"
bou: shows "bounded (range (\<lambda>k. integral S (f k)))"
not_Liminf_llist_imp_exists_index: fixes Xs :: "'a set llist"   and x :: "'a"   and i :: "nat" assumes "\<not> lnull Xs"   and "x \<notin> Liminf_llist Xs"   and "enat i < llength Xs" shows "\<exists>j\<ge>i. enat j < llength Xs \<and> x \<notin> lnth Xs j"
rev_append: fixes xs :: "'c list"   and ys :: "'c list" shows "rev (xs @ ys) = rev ys @ rev xs"
wf_prefix: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t V S"
hyps_2: shows "ran (fst fP) \<subseteq> {0..2 * card (nodes A)}"
pred_sum_inject_2: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and b :: "'b" shows "pred_sum P1 P2 (Inr b) = P2 b"
simps_1: shows "sys' 0 = CH (PROC x) (ACT ''b'' (VAR (0::'a)))"
oexp_mono: fixes \<alpha> :: "V"   and \<beta> :: "V"   and \<gamma> :: "V" assumes "Ord \<alpha>"   and "\<alpha> \<noteq> 0"   and "Ord \<beta>"   and "\<gamma> \<sqsubseteq> \<beta>" shows "\<alpha> \<up> \<gamma> \<le> \<alpha> \<up> \<beta>"
rrep_prrep: shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PRrep-:4} \<longrightarrow> dip \<xi> \<in> kD (rt \<xi>) \<and> the (flag (rt \<xi>) (dip \<xi>)) = val)"
l2norm_pos_le: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S" shows "0 \<le> l2norm S f"
compat_low: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i < length cms\<^sub>1"   and "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "dma mem\<^sub>1 x = Low"
exhaust: fixes y :: "msg" obtains x1 :: "agent" where "y = Agent x1"   | x2 :: "nat" where "y = Number x2"   | x3 :: "nat" where "y = Nonce x3"   | x4 :: "nat" where "y = Key x4"   | x5 :: "msg" where "y = Hash x5"   | x61 :: "msg" and x62 :: "msg" where "y = \<lbrace>x61, x62\<rbrace>"   | x71 :: "nat" and x72 :: "msg" where "y = Crypt x71 x72"
design_blocks_nempty: shows "\<B>\<^sup>+ \<noteq> {#}"
is_max: shows "\<forall>a\<in>Field order_of_orders. (max, a) \<in> order_of_orders \<longrightarrow> a = max"
map_G_id: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) G" shows "map_G id id id id id id x = x"
map_sel_1: fixes a :: "('g, 'h) tllist"   and f1 :: "'g \<Rightarrow> 'j"   and f2 :: "'h \<Rightarrow> 'i" assumes "is_TNil a" shows "terminal (tmap f1 f2 a) = f2 (terminal a)"
add_diff_cancel_left': fixes a :: "'k"   and b :: "'k" shows "a + b - a = b"
simps_2: fixes l :: "'c btree"   and a :: "'c"   and r :: "'c btree" shows "bal_up\<^sub>i (Up\<^sub>i l a r) = (height l = height r \<and> bal l \<and> bal r)"
mk_default_def: fixes x :: "'c set" shows "mk_default x = x"
foldl_spine_strict: fixes xs :: "['b]"   and f :: "'c \<rightarrow> 'b \<rightarrow> 'c"   and x :: "'c" assumes "\<not> finite_list xs" shows "foldl\<cdot>f\<cdot>x\<cdot>xs = \<bottom>"
ForallE': fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term"   and B :: "('a, 'b) form" assumes "G \<turnstile> Forall a"   and "a[t/0] # G \<turnstile> B" shows "G \<turnstile> B"
tm_defined_1: shows "finite {s \<in> {t1..<t2}. flow0 x s \<in> {a<--<b}}"
is_aligned_neg_mask2: fixes a :: "'b word"   and n :: "nat" shows "is_aligned (a AND NOT (mask n)) n"
dvd_gcdD2: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a (gcd b c)" shows "dvd.dvd times a c"
borel_measurable_minus_ennreal: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
neg_1: shows "negligible {x \<in> S - T. f x \<noteq> (0::'b)}"
measurable_emeasure: fixes N :: "'c measure"   and A :: "'c \<Rightarrow> 'a set" assumes "\<And>x. x \<in> space N \<Longrightarrow> A x \<subseteq> space M"   and "{x \<in> space (N \<Otimes>\<^sub>M M). snd x \<in> A (fst x)} \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (A x)) \<in> borel_measurable N"
hyps_3: shows "cancels_to_1\<^sup>*\<^sup>* (l @ a @ l') (l @ b @ l')"
topspace_product_topology: fixes T :: "'a \<Rightarrow> 'b topology"   and I :: "'a set" shows "topspace (product_topology T I) = (\<Pi>\<^sub>E i\<in>I. topspace (T i))"
cases: fixes a1 :: "nat"   and a2 :: "'b \<Rightarrow> 'b" assumes "swapidseq a1 a2" obtains "a1 = 0" and "a2 = id"   | n :: "nat" and p :: "'b \<Rightarrow> 'b" and a :: "'b" and b :: "'b" where "a1 = Suc n" and "a2 = Fun.swap a b id \<circ> p" and "swapidseq n p" and "a \<noteq> b"
assms_3: fixes xs :: "'a stream" assumes "\<psi>1 xs" shows "\<psi>2 xs"
tuple_decode_pdec2: fixes a :: "nat"   and es :: "nat" shows "tuple_decode a (pdec2 es) = tl (tuple_decode (Suc a) es)"
head_\<omega>_of_nat: fixes n :: "nat" shows "head_\<omega> (of_nat n) = (if n = 0 then 0 else 1)"
funpow_simps_right_2: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f ^^ Suc n = f ^^ n \<circ> f"
flip_def: fixes a :: "'b"   and b :: "'b" shows "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
simps_1: fixes uu :: "'a word"   and uv :: "'a wordinterval" shows "matching_dsts uu [] uv = Empty_WordInterval"
unit_factor_idem: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "unit_factor (unit_factor a) = unit_factor a"
wf_Top: fixes \<Gamma> :: "binding list" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f Top"
br'_cond_def: shows "br'_cond \<equiv> {(Q, W, rcm). W \<noteq> {}}"
mk_poly_delete: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A" shows "mat_delete (mk_poly A) i (dim_col A - 1) = coeff_lift_hom.mat_hom (mat_delete A i (dim_col A - 1))"
quasi_isometry_betweenI: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'c set" assumes "lambda C -quasi_isometry_on X f"   and "f ` X \<subseteq> Y"   and "\<And>y. y \<in> Y \<Longrightarrow> \<exists>x\<in>X. dist (f x) y \<le> C" shows "lambda C -quasi_isometry_between X Y f"
cmp_simps_2: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "src\<^sub>D (\<Phi> (f, g)) = src\<^sub>D (F g)"
usc_limsup_at: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "usc_at x0 f = (Limsup (at x0) f \<le> f x0)"
openin_subtopology_alt: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set" shows "openin (subtopology X U) S = (S \<in> (\<inter>) U ` Collect (openin X))"
vsubset_vsingleton_right: fixes A :: "V"   and a :: "V" shows "(A \<subseteq>\<^sub>\<circ> ZFC_in_HOL.set {a}) = (A = ZFC_in_HOL.set {a} \<or> A = []\<^sub>\<circ>)"
prime_number_theorem''': shows "(\<lambda>x. real (card {p. prime p \<and> p \<le> x})) \<sim>[sequentially] (\<lambda>n. real n / ln (real n))"
before_in_swap: fixes xs :: "'c list"   and ys :: "'c list"   and n :: "nat"   and x :: "'c"   and y :: "'c" assumes "dist_perm xs ys"   and "Suc n < length xs" shows "x < y in swap n xs = (x < y in xs \<and> \<not> (x = xs ! n \<and> y = xs ! Suc n) \<or> x = xs ! Suc n \<and> y = xs ! n)"
round_robin_reschedule_invar_None: fixes t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and \<sigma>' :: "'t list \<times> nat" assumes "round_robin_reschedule t0 queue n0 s = \<lfloor>(t, None, \<sigma>')\<rfloor>"   and "round_robin_invar (queue, n0) (dom (thr s))"   and "t0 \<in> set queue" shows "round_robin_invar \<sigma>' (dom (thr s))"
msg_defs_3: shows "Nonce \<equiv> map_fun id Ab cNonce"
hyps_3: assumes "finite B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> carrier G" shows "(\<Otimes>a\<in>A. finprod G (g a) B) = (\<Otimes>a\<in>A \<times> B. case a of (a, b) \<Rightarrow> g a b)"
bij_betw_UNION_chain: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and f :: "'d \<Rightarrow> 'e"   and A' :: "'c \<Rightarrow> 'e set" assumes "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> A i \<subseteq> A j \<or> A j \<subseteq> A i"   and "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" shows "bij_betw f (\<Union> (A ` I)) (\<Union> (A' ` I))"
valid_node_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry" shows "CFG.valid_node sourcenode targetnode valid_edge n \<equiv> \<exists>a. valid_edge a \<and> (n = sourcenode a \<or> n = targetnode a)"
mono_step': fixes S :: "'av Abs_State.st option"   and S' :: "'av Abs_State.st option"   and c :: "'av Abs_State.st option acom"   and c' :: "'av Abs_State.st option acom" assumes "S \<sqsubseteq> S'"   and "c \<sqsubseteq> c'" shows "local.step' S c \<sqsubseteq> local.step' S' c'"
simps_2: fixes \<tau> :: "'a basic_type" shows "to_required_type \<tau>[?] = \<tau>[1]"
card_of_Plus_ordLess_infinite_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'b set"   and B :: "'c set" assumes "infinite (Field r)"   and "Card_order r"   and "|A| <o r"   and "|B| <o r" shows "|A <+> B| <o r"
assms: shows "subst_domain \<theta> \<inter> bvars\<^sub>s\<^sub>t A = {}"
nonzero_imp_ex_nonzero_res: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "\<exists>k. x (Suc k) \<noteq> 0"
R'_simps_4: fixes n :: "nat" shows "((Suc n, Suc n) \<in> R' n) = (\<exists>x'>n. \<exists>y'>n. (x', y') \<in> R)"
inv_iv_def: fixes E :: "'b set set"   and S :: "'b set"   and X :: "'b set" assumes "Graph_E E" shows "Graph_E.inv_iv E S X = (independent_vertices E S \<and> X \<subseteq> Graph_E.vertices E \<and> (\<forall>v1\<in>Graph_E.vertices E - X. \<forall>v2\<in>S. {v1, v2} \<notin> E) \<and> S \<subseteq> X)"
lower_eq_zero_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(lower p v = 0) = (\<forall>u. u \<prec>\<^sub>t v \<longrightarrow> lookup p u = (0::'c))"
lexord_asymmetric: fixes R :: "('b \<times> 'b) set"   and a :: "'b list"   and b :: "'b list" assumes "asym R"   and "(a, b) \<in> lexord R" shows "(b, a) \<notin> lexord R"
connected_openin_diff_countable: fixes S :: "'b set"   and T :: "'b set" assumes "connected S"   and "openin (top_of_set (affine hull S)) S"   and "\<not> collinear S"   and "countable T" shows "connected (S - T)"
simps_2: fixes p :: "'b \<Rightarrow> bool"   and i :: "'b"   and ea :: "('a, 'b) Node" shows "FingerTreeStruc.nsplitTree p i (Single ea) = (Empty, ea, Empty)"
noDAAll: fixes p :: "('b, 'c) Combinators list" shows "noDenyAll p = (\<not> memberP DenyAll p)"
det_def: fixes A :: "(('c, 'd) vec, 'd) vec" shows "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A $ i $ p i))"
assms_5: shows "set \<psi>s \<subseteq> fmla"
rep_eq: fixes x :: "'a fset" shows "fNext x = Next (fset x)"
spmf_restrict_spmf: fixes p :: "'c spmf"   and A :: "'c set"   and x :: "'c" shows "spmf (p \<upharpoonleft> A) x = (if x \<in> A then spmf p x else 0)"
eventually_at_top_dense: fixes P :: "'a \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n>N. P n)"
semiring_norm_137: shows "\<not> (0::'d) < (0::'d)"
resCap_cf_impl_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" shows "resCap_cf_impl cf p \<equiv> case p of [] \<Rightarrow> return (0::'a) | e # p \<Rightarrow> cf_get cf e \<bind> (\<lambda>cap. assert (distinct p) \<bind> (\<lambda>_. nfoldli p (\<lambda>_. True) (\<lambda>e cap. cf_get cf e \<bind> (\<lambda>cape. return (min cape cap))) cap))"
when_simps_1: fixes a :: "'b" shows "(a when True) = a"
igSubstIGVar2_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSubstIGVar2 MOD \<equiv> \<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y"
endpow_list_apply_closed: fixes v :: "'v"   and k :: "nat" assumes "v \<in> V" shows "set (map (\<lambda>S. S v) (map endpow [0..<k])) \<subseteq> V"
that: fixes err :: "real" assumes "x = y + err"   and "\<bar>err\<bar> \<le> e" shows "thesis"
new_document_no_child_nodes: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M (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 new_document_ptr) \<rightarrow>\<^sub>r []"
bal_list_def: fixes n :: "nat"   and xs :: "'a list" shows "bal_list n xs = fst (bal n xs)"
intros_1: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<bottom> \<in># \<Gamma>" shows "\<Gamma> \<Rightarrow> \<Delta>"
strict_implies_order: fixes b :: "'d multiset"   and a :: "'d multiset" assumes "b \<subset># a" shows "b \<subseteq># a"
CConcStart: fixes xs :: "((int \<times> int) set set, int) Combinators list"   and a :: "((int \<times> int) set set, int) Combinators"   and x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "xs \<noteq> []"   and "C a x = \<bottom>" shows "C (list2FWpolicy (xs @ [a])) x = C (list2FWpolicy xs) x"
solves_ode_def: fixes y :: "real \<Rightarrow> 'd"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "real set"   and X :: "'d set" shows "(y solves_ode f) T X = ((y has_vderiv_on (\<lambda>t. f t (y t))) T \<and> y \<in> T \<rightarrow> X)"
assms_3: shows "B \<in> obj\<^bsub>C\<^esub>"
jeroslow_godel_second: assumes "consistent" shows "\<not> prv jcons"
coreI: fixes m :: "'a set"   and s :: "'a set"   and x :: "'a" assumes "m open"   and "m \<subseteq> s"   and "x \<in> m" shows "x \<in> core s"
1: shows "fmlookup f \<subseteq>\<^sub>m fmlookup g"
mod_add_cong: 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 a :: "'b"   and c :: "'b"   and a' :: "'b"   and b :: "'b"   and b' :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (plus a b) c = modulo (plus a' b') c"
assms_1: shows "F \<^bold>\<in> hexp B C"
find_md_m_match': fixes mds :: "meth_def list"   and m :: "char list"   and cl :: "cl"   and m' :: "char list"   and vds :: "vd list"   and mb :: "meth_body" assumes "find_meth_def_in_list_f mds m = Some (meth_def_def (meth_sig_def cl m' vds) mb)" shows "m' = m"
wens_def: fixes F :: "'b program"   and act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wens F act B \<equiv> gfp (\<lambda>X. wp act B \<inter> awp F (B \<union> X) \<union> B)"
2: fixes xs :: "nat list" assumes "xs \<noteq> []" shows "sum_list (tl xs) = sum_list xs - hd xs"
simps_3: fixes ODE1 :: "('a, 'b) ODE"   and ODE2 :: "('a, 'b) ODE"   and \<sigma> :: "('a, 'c, 'b) subst" shows "Osubst (OProd ODE1 ODE2) \<sigma> = OProd (Osubst ODE1 \<sigma>) (Osubst ODE2 \<sigma>)"
fresh_2: fixes a :: "name"   and x2 :: "trm"   and x1 :: "trm" shows "a \<sharp> App x2 x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
abs_mult: fixes abs :: "'c \<Rightarrow> 'c"   and sgn :: "'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 a :: "'c"   and b :: "'c" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (times a b) = times (abs a) (abs b)"
simps_7: fixes x3 :: "'b"   and x1 :: "'a" shows "Fault x3 \<noteq> Normal x1"
ge_zero: shows "0 \<le> \<parallel>f\<parallel>\<hyphen>F"
r_mult_eqn: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G"   and "a = b" shows "a \<cdot> c = b \<cdot> c"
removeKey_def: fixes a :: "'c"   and ps :: "('c \<times> 'd) list" shows "removeKey a ps \<equiv> filter (\<lambda>p. a \<noteq> fst p) ps"
apos_in_aug_inf: fixes a :: "ant" assumes "0 \<le> a" shows "a \<in> Z\<^sub>\<infinity>"
dual_adjoint_equivalence: shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg g f (local.inv \<epsilon>) (local.inv \<eta>)"
B: fixes b :: "'a set" assumes "b \<in> B" shows "partition_on b (P b)"
set_mult_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set"   and K :: "'b set" shows "H <#>\<^bsub>G\<^esub> K = (\<Union>h\<in>H. \<Union>k\<in>K. {h \<otimes>\<^bsub>G\<^esub> k})"
Ptl_not_null: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path_no_deadend G P v0 p \<sigma>" shows "\<not> lnull (ltl P)"
distributed_measurable: fixes M :: "'d measure"   and N :: "'e measure"   and X :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> ennreal" assumes "distributed M N X f" shows "X \<in> M \<rightarrow>\<^sub>M N"
open_map_into_discrete_topology: fixes X :: "'c topology"   and U :: "'d set"   and f :: "'c \<Rightarrow> 'd" shows "open_map X (discrete_topology U) f = (f ` topspace X \<subseteq> U)"
coeff_prod_closed: fixes A :: "'a set"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "ring_closed A"   and "\<And>x m. x \<in> X \<Longrightarrow> MPoly_Type.coeff (f x) m \<in> A" shows "MPoly_Type.coeff (prod f X) m \<in> A"
less_supI1: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less x a" shows "less x (sup a b)"
zero_flow_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> cf.E" shows "f' (u, v) = (0::'capacity)"