hcju/isabelleps huggingface dataset:
rScope: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C \<Psi> P (\<alpha> \<prec> P') A\<^sub>P \<Psi>\<^sub>P"   and "x \<sharp> \<Psi>"   and "x \<sharp> \<alpha>"   and "x \<sharp> A\<^sub>P"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* P'"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* C"   and "x \<sharp> C"   and "bn \<alpha> \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) (\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P') (x # A\<^sub>P) \<Psi>\<^sub>P"
equals0I: fixes A :: "'d set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
types_stable_mode_update: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and upd :: "'Var ModeUpd" assumes "types_stable \<Gamma> \<S>"   and "types_wellformed \<Gamma>"   and "anno_type_stable \<Gamma> \<S> upd" shows "types_stable \<Gamma> \<oplus>\<^sub>\<S> upd (add_anno_stable \<S> upd)"
add_decreasing: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq a zero"   and "less_eq c b" shows "less_eq (plus a c) b"
mirror_point_snd_last_eq_lowest: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "mirror_point (fst X) (last (map snd (half_segments_of_aform X))) = lowest_vertex (fst X, nlex_pdevs (snd X))"
eq: shows "(\<Sum>m | m \<in> {1..k} \<and> coprime m k. \<Phi> m * unity_root d (int m)) = (\<Sum>m | m \<in> carrier G. \<Phi> m * unity_root d (int m))"
const_type': shows "\<forall>c\<in>\<C>. \<exists>a\<in>\<TT>\<^sub>a. \<forall>X. \<Gamma> (Fun c X) = Var a"
bisim_simple_def: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set" shows "bisim_simple \<R>\<^sub>A \<equiv> \<forall>c\<^sub>1\<^sub>A mds mem\<^sub>1\<^sub>A c\<^sub>2\<^sub>A mem\<^sub>2\<^sub>A. (\<langle>c\<^sub>1\<^sub>A, mds, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>A, mds, mem\<^sub>2\<^sub>A\<rangle>\<^sub>A) \<in> \<R>\<^sub>A \<longrightarrow> c\<^sub>1\<^sub>A = c\<^sub>2\<^sub>A"
elims_3: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "\<not> successively x xa" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x :: "'a" and y :: "'a" and xs :: "'a list" where "xa__ = P" and "xa = x # y # xs" and "\<not> (P x y \<and> successively P (y # xs))"
add_pos_pos: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "{#} \<subset># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
fold_A_min_gallery_from2_map: fixes F :: "'a set"   and Fs :: "'a set list" assumes "chamber F"   and "F \<in> opp_fold_A \<turnstile> A"   and "min_gallery (D # Fs @ [F])" shows "min_gallery (C # fold_A \<Turnstile> (Fs @ [F]))"
trcl_rev_cases: fixes c :: "'a"   and w :: "'b list"   and c' :: "'a"   and S :: "('a \<times> 'b \<times> 'a) set" assumes "(c, w, c') \<in> trcl S" obtains "w = []" and "c = c'"   | ch :: "'a" and e :: "'b" and wh :: "'b list" where "w = wh @ [e]" and "(c, wh, ch) \<in> trcl S" and "(ch, e, c') \<in> S"
select_convs_5: fixes id :: "nat"   and no :: "nat"   and nhid :: "nat"   and msg :: "msg"   and num :: "nat"   and sid :: "nat"   and more :: "'a" shows "num \<lparr>state.id = id, no = no, nhid = nhid, msg = msg, num = num, sid = sid, \<dots> = more\<rparr> = num"
DERIV_isCont: fixes f :: "'c \<Rightarrow> 'c"   and D :: "'c"   and x :: "'c" assumes "(f has_field_derivative D) (at x)" shows "isCont f x"
linear_injective_on_subspace_0: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "linear f"   and "subspace s" shows "inj_on f s = (\<forall>x\<in>s. f x = (0::'c) \<longrightarrow> x = (0::'b))"
distinguish_attach: fixes d :: "('i, 'j) distinguisher"   and conv :: "('i, 'j, 'k, 'l) converter"   and res :: "('k, 'l) resource" shows "connect d (conv \<rhd> res) = connect (absorb d conv) res"
pureI: fixes P :: "bool"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and c :: "com"   and R :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" assumes "P \<Longrightarrow> \<turnstile>\<^sub>3 {Q} c { R}" shows "\<turnstile>\<^sub>3 {\<up> P \<and>* Q} c { R}"
nxt_search_diverg: fixes b :: "nat" assumes "nxt b \<up>" shows "search b \<up>"
finite_in_span: fixes S :: "'c set"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier V"   and "v \<in> local.module.span S" shows "\<exists>a. a \<in> S \<rightarrow> carrier K \<and> lincomb a S = v"
surjective_section_eq_homeomorphic_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "(section_map X Y f \<and> f ` topspace X = topspace Y) = homeomorphic_map X Y f"
Diagonal_to_Smith_row_i_PQ_JNF_dim: fixes P' :: "'b mat"   and m :: "nat"   and A' :: "'b mat"   and n :: "nat"   and Q' :: "'b mat"   and i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" assumes "P' \<in> carrier_mat m m \<and> A' \<in> carrier_mat m n \<and> Q' \<in> carrier_mat n n"   and "Diagonal_to_Smith_row_i_PQ_JNF i bezout (P', A', Q') = (P, A, Q)" shows "P \<in> carrier_mat m m \<and> A \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
iso_iff3: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "Well_order r'" shows "iso r r' f = (bij_betw f (Field r) (Field r') \<and> compat r r' f)"
T2: fixes a :: "letter"   and b :: "letter"   and ws :: "letter list list"   and zs :: "letter list list"   and w :: "letter list" assumes "a \<noteq> b"   and "T a ws zs" shows "T a ws ((b # w) # zs)"
fP: shows "(\<circ>) g ` {\<pi>. \<pi> permutes {0..<n}} = Z_good_fun"
body_generalize_fundef: fixes F_empty :: "'a"   and F_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'g, 'h, 'i) instr fundef option"   and F_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'f, 'g, 'h, 'i) instr fundef \<Rightarrow> 'a"   and F_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'f, 'g, 'h, 'i) instr fundef) list"   and heap_empty :: "'j"   and heap_get :: "'j \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'j \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'j"   and heap_to_list :: "'j \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and box_ubx1 :: "'h \<Rightarrow> 'c"   and unbox_ubx1 :: "'c \<Rightarrow> 'h option"   and box_ubx2 :: "'i \<Rightarrow> 'c"   and unbox_ubx2 :: "'c \<Rightarrow> 'i option"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'f \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'f option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'f \<Rightarrow> 'e"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'g \<Rightarrow> ('c, 'h, 'i) unboxed list \<Rightarrow> ('c, 'h, 'i) unboxed option"   and \<UU>\<bb>\<xx> :: "'f \<Rightarrow> Unboxed.type option list \<Rightarrow> 'g option"   and \<BB>\<oo>\<xx> :: "'g \<Rightarrow> 'f"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'g \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and fd :: "('c, 'k, 'l, 'm, 'f, 'g, 'h, 'i) instr fundef" assumes "ubx F_empty F_get F_add F_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp>" shows "body (ubx.generalize_fundef box_ubx1 box_ubx2 \<BB>\<oo>\<xx> fd) = map (ubx.generalize_instr box_ubx1 box_ubx2 \<BB>\<oo>\<xx>) (body fd)"
dist_poly_mapping_def: shows "dist \<equiv> \<lambda>x y. \<Sum>n\<in>Poly_Mapping.keys x \<union> Poly_Mapping.keys y. dist (poly_mapping.lookup x n) (poly_mapping.lookup y n)"
cases: fixes s :: "('g, 'h) locks \<times> (('h \<Rightarrow> ('x \<times> 'g \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'j wait_set_status option) \<times> 'h set"   and t :: "'h"   and ta :: "('g, 'h, 'k, 'i, 'j, 'l) thread_action" assumes "actions_ok s t ta" obtains "lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" and "cond_action_oks s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>" and "interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
un_Yield2_def: fixes step :: "('d, 'c) step" shows "un_Yield2 step = (case step of Yield x31 x32 \<Rightarrow> x32)"
GENPRE: shows "gen_dfs_pre E U {} reds u0"
C1: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel"   and "guarded S"   and "guarded T" shows "(\<Psi>', U \<parallel> !S, U \<parallel> !T) \<in> Rel''"
weak_put_eq: fixes \<sigma> :: "'b"   and k :: "'a"   and u :: "'a"   and \<rho> :: "'b"   and v :: "'a" assumes "\<sigma> \<in> \<S>"   and "get \<sigma> = k"   and "put \<sigma> u = put \<rho> v" shows "put \<rho> k = \<sigma>"
int_nat_pow: fixes a :: "nat"   and b :: "int" assumes "0 \<le> a" shows "(\<^bold>g [^] int a) [^] b = \<^bold>g [^] (int a * b)"
finite_subset_wf: fixes A :: "'a set" assumes "finite A" shows "wf {(X, Y). X \<subset> Y \<and> Y \<subseteq> A}"
rel_generatI: fixes x :: "('d, 'e, 'f) generat"   and y :: "('g, 'h, 'i) generat"   and A :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and Out :: "'e \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'i \<Rightarrow> bool" assumes "is_Pure x = is_Pure y"   and "\<lbrakk>is_Pure x; is_Pure y\<rbrakk> \<Longrightarrow> A (generat.result x) (generat.result y)"   and "\<lbrakk>\<not> is_Pure x; \<not> is_Pure y\<rbrakk> \<Longrightarrow> Out (generat.output x) (generat.output y) \<and> R (continuation x) (continuation y)" shows "rel_generat A Out R x y"
pos: shows "0 < bs ! i * x + (hs ! i) x"
assms_1: shows "4 * ln (1 + 2 ^ j) + 3 \<le> d * ln 2 * (1 + 2 ^ j)"
backward: assumes "csa \<in> set (characterize_consistent_signs_at_roots (coprime_r (cast_rat_list fs)) (cast_rat_list fs))" shows "csa \<in> consistent_sign_vectors fs UNIV \<and> has_no_zeros csa"
no_change_if_ge_length_t: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and j :: "nat" assumes "trace init t final"   and "length t \<le> i"   and "i \<le> j" shows "S t i = S t j"
elims_3: fixes x :: "('b ivl \<times> 'b) tree"   and xa :: "'b ivl" assumes "\<not> search x xa" obtains x :: "'b ivl" where "xa__ = \<langle>\<rangle>" and "xa = xb__"   | l :: "('b ivl \<times> 'b) tree" and a :: "'b ivl" and m :: "'b" and r :: "('b ivl \<times> 'b) tree" and x :: "'b ivl" where "xa__ = \<langle>l, (a, m), r\<rangle>" and "xa = x" and "\<not> (if overlap x a then True else if l \<noteq> \<langle>\<rangle> \<and> low x \<le> max_hi l then search l x else search r x)"
isom_Gchains_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> ('e set, 'f) Group_scheme"   and h :: "nat \<Rightarrow> ('e set, 'f) Group_scheme" shows "isom_Gchains n f g h = (\<forall>i\<le>n. g i \<cong> h (f i))"
intro: fixes f :: "nat \<Rightarrow> 'b" assumes "f 0 = (0::'b)"   and "f 1 = (1::'b)"   and "\<And>a b. \<lbrakk>1 < a; 1 < b; coprime a b\<rbrakk> \<Longrightarrow> f (a * b) = f a * f b" shows "multiplicative_function f"
continuous_on_representation: fixes B :: "'a set"   and b :: "'a" assumes "finite B"   and "independent B"   and "b \<in> B"   and "pairwise orthogonal B" shows "continuous_on (span B) (\<lambda>x. representation B x b)"
set: fixes x1 :: "'a form multiset"   and x2 :: "'a form" shows "set_sequent ( x1 \<Rightarrow>* x2) = \<Union> (set_form ` set_mset x1) \<union> set_form x2"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>src\<^sub>C f : src\<^sub>C a \<rightarrow>\<^sub>C src\<^sub>C b\<guillemotright>"
unat_lt2p: fixes x :: "'c word" shows "unat x < 2 ^ LENGTH('c)"
dist_midpoint_3: fixes a :: "'b"   and b :: "'b" shows "dist (midpoint a b) a = dist a b / 2"
col2_os__os: fixes C :: "'p"   and D :: "'p"   and A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "C \<noteq> D"   and "Col A B C"   and "Col A B D"   and "A B OS X Y" shows "C D OS X Y"
ccard_sum: fixes n'' :: "nat"   and n' :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "n'' \<le> n'"   and "n \<le> n''" shows "card {i. n < i \<and> i \<le> n' \<and> P i} = card {i. n < i \<and> i \<le> n'' \<and> P i} + card {i. n'' < i \<and> i \<le> n' \<and> P i}"
approximating_bigstep_fun_singleton_prepend: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rsB :: "'b rule list"   and s :: "state"   and rsC :: "'b rule list"   and r :: "'b rule" assumes "approximating_bigstep_fun \<gamma> p rsB s = approximating_bigstep_fun \<gamma> p rsC s" shows "approximating_bigstep_fun \<gamma> p (r # rsB) s = approximating_bigstep_fun \<gamma> p (r # rsC) s"
wt_subst_Var: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t Var"
real_root_eq_0_iff: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "(root n x = 0) = (x = 0)"
circ_mult_upper_bound: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z\<^sup>\<circ>"   and "y \<le> z\<^sup>\<circ>" shows "x * y \<le> z\<^sup>\<circ>"
E: shows "\<forall>\<^sub>F x in at x\<^sub>0. (\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x) = (\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x\<^sub>0) \<and> (\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x) = sign_changes ps x"
iia: fixes A :: "'a set" shows "iia_on A linear_cf"
moebius_pt_moebius_rotation: fixes \<phi> :: "real"   and z :: "complex" shows "moebius_pt (moebius_rotation \<phi>) (of_complex z) = of_complex (cis \<phi> * z)"
snd_conv: fixes x1 :: "'d"   and x2 :: "'a" shows "snd (x1, x2) = x2"
d_x_nc: fixes x :: "'a" shows "d (x \<cdot> nc) = d x"
scalsquare_vv_zero: fixes v :: "complex \<times> complex" shows "(vec_cnj v *\<^sub>v\<^sub>v v = 0) = (v = vec_zero)"
simps_1: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and s :: "state" shows "approximating_bigstep_fun \<gamma> p [] s = s"
sup_continuous_const: fixes c :: "'b" shows "sup_continuous (\<lambda>x. c)"
gval_fold_equiv_fold_rev: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gAnd G (Bc True)) s = gval (fold gAnd (rev G) (Bc True)) s"
intro: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>C. SimplicialComplex.maxsimp X C \<Longrightarrow> SimplicialComplex.maxsimp Y (f ` C)"   and "\<And>C. SimplicialComplex.maxsimp X C \<Longrightarrow> card (f ` C) = card C" shows "ChamberComplexMorphism_axioms X Y f"
val_no_step: fixes e :: "char list exp"   and v :: "val"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "val_of e = \<lfloor>v\<rfloor>" shows "((e, s, b), e', s', b') \<notin> red P"
simps_1: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "max_stutter_sampler \<sigma> 0 = 0"
piecewise_C1_differentiable_on_subset: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set"   and T :: "real set" assumes "f piecewise_C1_differentiable_on S"   and "T \<subseteq> S" shows "f piecewise_C1_differentiable_on T"
mircfr_def: shows "mircfr = DJ cooper \<circ> ferrack01 \<circ> simpfm \<circ> exsplit"
simps: fixes M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" shows "And M1 M2 = (\<lambda>i j. min (M1 i j) (M2 i j))"
cases: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" assumes "Lfilled a1 a2 a3 a4" obtains vs :: "e list" and lholed :: "Lholed" and es' :: "e list" and es :: "e list" where "a1 = 0" and "a2 = lholed__" and "a3 = es__" and "a4 = vs__ @ es__ @ es'__" and "const_list vs__" and "lholed__ = LBase vs__ es'__"   | vs :: "e list" and lholed :: "Lholed" and n :: "nat" and es' :: "e list" and l :: "Lholed" and es'' :: "e list" and k :: "nat" and es :: "e list" and lfilledk :: "e list" where "a1 = k + 1" and "a2 = lholed" and "a3 = es" and "a4 = vs @ [Label n es' lfilledk] @ es''" and "const_list vs" and "lholed = LRec vs n es' l es''" and "Lfilled k l es lfilledk"
LIM_I: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and L :: "'b" assumes "\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r" shows "f \<midarrow>a\<rightarrow> L"
discrete: fixes default :: "'a"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and power_mono :: "bool"   and discrete :: "bool"   and y :: "'a"   and x :: "'a" assumes "poly_order_carrier default gt power_mono discrete"   and "discrete"   and "y \<le> x" shows "\<exists>k. x = ((+) (1::'a) ^^ k) y"
rk_\<alpha>\<beta>\<gamma>_special_case_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {C', P} = 1" shows "rk {\<alpha>, \<beta>, \<gamma>} \<le> 2"
ssap: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {xx}" shows "ssap \<phi> \<in> F"
un2: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..u} = {l..m} \<union> {m<..u}"
inv: shows "optoy i \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<sigma>, l). l \<in> {PToy-:1} \<longrightarrow> msg_ok \<sigma> (msg (\<sigma> i)))"
Q: fixes i :: "nat" shows "distr Q (Pi\<^sub>M (J i) M) (\<lambda>x. restrict x (J i)) = Pi\<^sub>M (J i) M"
lprefix_ltake_same: fixes n :: "enat"   and xs :: "'a llist"   and m :: "enat" shows "lprefix (ltake n xs) (ltake m xs) = (n \<le> m \<or> llength xs \<le> m)"
assms_5: shows "quicksort_post R h lo (p - Suc 0) xs' xs''"
h3: shows "\<forall>csa. List.member (find_sgas (map real_of_rat_poly (fst (factorize_polys qs)))) csa = (List.member (find_sgas_aux (cast_rat_list (fst (factorize_polys qs)))) csa \<or> List.member (find_consistent_signs_at_roots (coprime_r (cast_rat_list (fst (factorize_polys qs)))) (cast_rat_list (fst (factorize_polys qs)))) csa)"
path_approx_polynomial_function: fixes g :: "real \<Rightarrow> 'a"   and e :: "real" assumes "path g"   and "0 < e" obtains p :: "real \<Rightarrow> 'a" where "polynomial_function p" and "pathstart p = pathstart g" and "pathfinish p = pathfinish g" and "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (p t - g t) < e"
advantage_advantage1: fixes \<A> :: "(unit, bool list + bool list \<times> 'hash, 'hash + unit) gpv"   and q :: "nat" assumes "interaction_bounded_by' (Not \<circ> isl) \<A> q" shows "UPF.advantage \<A> \<le> UPF.advantage1 (guessing_many_one.reduction q (\<lambda>_. \<A>) ()) * real q"
inv: shows "I \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"
composite_is_tabulation: shows "tabulation (\<cdot>) (\<star>) \<a> \<i> src trg (r \<star> s) tab (s\<^sub>0 \<star> p\<^sub>0) (r\<^sub>1 \<star> p\<^sub>1)"
summable: fixes X :: "nat \<Rightarrow> 'a" assumes "summable X" shows "summable (\<lambda>n. f (X n))"
nextl_state: fixes q :: "hf"   and xs :: "'a list" assumes "q \<in> states M" shows "nextl q xs \<in> states M"
fcs_last_x_eq: fixes x :: "'b" assumes "x \<in> segment G.face_cycle_succ a' a"   and "G.face_cycle_succ x = a" shows "H.face_cycle_succ x = G.face_cycle_succ a'"
assms_2: shows "b1 \<union> b2 \<subseteq> carrier M"
associatedD: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "a \<sim>\<^bsub>G\<^esub> b" shows "a divides\<^bsub>G\<^esub> b"
disjoint_affine_hull: fixes s :: "'b set"   and t :: "'b set"   and u :: "'b set" assumes "\<not> affine_dependent s"   and "t \<subseteq> s"   and "u \<subseteq> s"   and "t \<inter> u = {}" shows "affine hull t \<inter> affine hull u = {}"
residue_1_mult: fixes x :: "int"   and y :: "int" assumes "x \<in> carrier (residue_ring 1)"   and "y \<in> carrier (residue_ring 1)" shows "x \<otimes>\<^bsub>residue_ring 1\<^esub> y = 0"
independent_imp_trivial_combine: fixes K :: "'a set"   and Us :: "'a list"   and Ks :: "'a list" assumes "subfield K R"   and "R.independent K Us"   and "set Ks \<subseteq> K"   and "R.combine Ks Us = \<zero>" shows "set (take (length Us) Ks) \<subseteq> {\<zero>}"
"2_2": fixes x :: "'t list" assumes "x = new_syz_sigs ss bs p"   and "sig_crit bs x p" shows "snd (snd (fst (rb_aux ((bs, x, ps), z)))) = []"
set_empty2: fixes xs :: "'c list" shows "({} = set xs) = (xs = [])"
sorted_wrt_gen2: fixes A :: "nat"   and m :: "nat"   and B :: "nat"   and n :: "nat" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2) (gen2 A B m n)"
idem: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type" assumes "semilattice (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) a = a"
a: shows "R[a\<turnstile>c>b] = OrR1 <c>.M e"
abs_summable_on_subset: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'a set"   and A :: "'a set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "f abs_summable_on A"
iteratei_def: fixes list_it :: "'e \<Rightarrow> (('c \<times> 'd) list \<Rightarrow> bool) \<Rightarrow> ('c \<times> 'd \<Rightarrow> ('c \<times> 'd) list \<Rightarrow> ('c \<times> 'd) list) \<Rightarrow> ('c \<times> 'd) list \<Rightarrow> ('c \<times> 'd) list"   and S :: "'e" shows "poly_map_iteratei_defs.iteratei list_it S \<equiv> it_to_it (list_it S)"
infinite_finite_times_lepoll_self: fixes A :: "'b set"   and B :: "'c set" assumes "infinite A"   and "finite B" shows "A \<times> B \<lesssim> A"
intro: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "galois_connection G"   and "retract_axioms G" shows "retract G"
simps_2: shows "Ubx2 \<noteq> Ubx1"
\<G>\<F>_subformulas\<^sub>\<mu>: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> \<phi> w \<subseteq> subformulas\<^sub>\<mu> \<phi>"
oalist_inv_raw_map_val_pair: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "tc.oalist_inv_raw xs" shows "tc.oalist_inv_raw (map_val_pair f xs)"