hcju/isabelleps huggingface dataset:
is_derivation_def: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" shows "l.is_derivation X p = (linear_on l.diff_fun_space UNIV (*\<^sub>R) (*\<^sub>R) X \<and> (\<forall>f g. f \<in> l.diff_fun_space \<longrightarrow> g \<in> l.diff_fun_space \<longrightarrow> X (f * g) = f p * X g + g p * X f))"
prj_chine_composite_1: shows "cods.Prj\<^sub>1\<^sub>1 \<cdot> \<mu>\<nu>_\<pi>.chine = \<mu>.chine \<cdot> doms.Prj\<^sub>1\<^sub>1"
Collect_case_prodD: fixes x :: "'a \<times> 'b"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "x \<in> _Collect (x, y) (A x y)" shows "A (fst x) (snd x)"
distinct_indices_ns_def: fixes ns :: "('a \<times> 'b ns_constraint) set" shows "distinct_indices_ns ns = (\<forall>n1 n2 i. (i, n1) \<in> ns \<longrightarrow> (i, n2) \<in> ns \<longrightarrow> poly n1 = poly n2 \<and> ns_constraint_const n1 = ns_constraint_const n2)"
that: fixes r2' :: "'d list" assumes "path B (w || r2') p2 \<and> length w = length r2'" shows "thesis"
simps_3: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "ichosen \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = ichosen"
RECT_unfold: fixes B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" assumes "trimono B" shows "REC\<^sub>T B = B (REC\<^sub>T B)"
NG: shows "subgroup N (G\<lparr>carrier := H\<rparr>)"
relpow_empty: fixes n :: "nat" assumes "0 < n" shows "{} ^^ n = {}"
assms_2: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "X xs ys"   and "\<not> lnull xs" shows "\<not> lnull ys \<and> (\<not> lnull ys \<longrightarrow> r (lhd xs) (lhd ys) \<or> lhd xs = lhd ys \<and> (X (ltl xs) (ltl ys) \<or> llexord r (ltl xs) (ltl ys)))"
Collect_empty_eq: fixes P :: "'e \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
cos_30: shows "cos (pi / 6) = sqrt 3 / 2"
hom_0: fixes x :: "int" assumes "of_int x = (0::'a)" shows "x = 0"
r_ifz_def: shows "r_ifz \<equiv> let ifzero = Cn 3 r_mul [r_dummy 2 r_not, recf.Id 3 1]; ifnzero = Cn 3 r_mul [r_dummy 2 r_sign, recf.Id 3 2] in Cn 3 r_add [ifzero, ifnzero]"
ThrowThrow: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>throw (Throw r),s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
size_Diff_subset_Int: fixes M :: "'c multiset"   and M' :: "'c multiset" shows "size (M - M') = size M - size (M \<inter># M')"
1: fixes i :: "nat" assumes "i \<in> liset A (lselect s w)" shows "enat i < esize s"
FF_def: shows "F.FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (local.map (fst \<mu>\<nu>), local.map (snd \<mu>\<nu>)) else D.VV.null"
locks_ticket_def: fixes s :: "thread"   and k :: "nat" shows "locks_ticket s k \<equiv> s = HOLD k \<or> s = REL k"
simps: fixes l :: "nat"   and i :: "nat"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'b Heap"   and s :: "'b" shows "imp_for_down' l i f s = (let i = i - 1 in f i s \<bind> (\<lambda>s. if l < i then imp_for_down' l i f s else return s))"
open_closed: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "open S = topological_space.closed open (- S)"
mon_pl_of_\<alpha>nl: fixes fg :: "('d, 'b, 'e, 'a, 'f) flowgraph_rec_scheme"   and w :: "('b, 'c) label list el_step list" shows "mon_pl (map (\<alpha>nl fg) w) = mon_loc fg w \<union> mon_env fg w"
c_pair_inj': fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" shows "(c_pair x1 y1 = c_pair x2 y2) = (x1 = x2 \<and> y1 = y2)"
octo_of_int_of_nat: fixes n :: "nat" shows "octo_of_int (int n) = octo_of_nat n"
h': shows "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"
code: fixes z :: "complex" shows "cnj z = Complex (Re z) (- Im z)"
transfer: shows "pcr_complex_homo \<infinity>\<^sub>h\<^sub>c \<infinity>\<^sub>h"
reflcl_of_refl_rel: fixes Rel :: "('c \<times> 'c) set" assumes "refl Rel" shows "Rel\<^sup>= = Rel"
rcis_mult_cis: fixes n :: "real"   and a :: "real"   and b :: "real" shows "rcis n a * cis b = rcis n (a + b)"
add: fixes xa :: "int"   and ya :: "int" shows "xa \<oplus>\<^bsub>\<Z>\<^esub> ya = xa + ya"
Inf_notin_open: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and "open" :: "'b set \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linear_continuum_topology Inf Sup inf less_eq less sup open"   and "open A"   and "\<forall>a\<in>A. less x a" shows "Inf A \<notin> A"
env_eq_def: fixes \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" shows "\<rho> \<approx> \<rho>' \<equiv> \<forall>x. lookup \<rho> x = lookup \<rho>' x"
insert_UNIV: fixes x :: "'c" shows "insert x UNIV = UNIV"
abs_eq: shows "zero_pdevs = Abs_pdev (\<lambda>_. 0::'a)"
intros_1: fixes \<C> :: "t_context"   and v :: "v" shows "\<C> \<turnstile> [C v] : [] _> [typeof v]"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Dom \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = Dom t \<^bold>\<otimes> Dom u \<^bold>\<otimes> Dom v"
ys'_def: shows "ys' = map (\<lambda>i. ys ! f i) [0..<length ys]"
nipd_1: shows "\<forall>n\<in>{k..<l}. \<pi> n \<noteq> ipd (\<pi> k)"
mk_eqcl'_len: fixes xs :: "nat option list"   and i :: "nat"   and j :: "nat"   and l :: "nat"   and T :: "bool list list" shows "length (mk_eqcl' xs i j l T) = length xs"
ask_nondet_def: fixes ask :: "('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> ('g, 'h) nondetT" shows "nondetM_base.ask_nondet ask f = NondetT (ask (\<lambda>r. run_nondet (f r)))"
degree_mult_eq_left_unit: fixes p :: "'a poly"   and q :: "'a poly" assumes "lead_coeff p dvd (1::'a)"   and "q \<noteq> 0" shows "degree (p * q) = degree p + degree q"
eval_wp_SetDC: fixes p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "'a \<Rightarrow> 'b set"   and R :: "'a \<Rightarrow> real"   and s :: "'a" shows "wp (SetDC p S) R s = (INF a\<in>S s. wp (p a) R s)"
linear_simps_3: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "f (0::'a) = (0::'b)"
elims: fixes x :: "exp"   and y :: "nat set" assumes "FV x = y" obtains xa :: "nat" where "x = EVar xa__" and "y = {xa__}"   | n :: "nat" where "x = ENat n" and "y = {}"   | xa :: "nat" and e :: "exp" where "x = ELam xa e" and "y = FV e - {xa}"   | e1 :: "exp" and e2 :: "exp" where "x = EApp e1a__ e2a__" and "y = FV e1a__ \<union> FV e2a__"   | f :: "nat \<Rightarrow> nat \<Rightarrow> nat" and e1 :: "exp" and e2 :: "exp" where "x = EPrim f e1b__ e2b__" and "y = FV e1b__ \<union> FV e2b__"   | e1 :: "exp" and e2 :: "exp" and e3 :: "exp" where "x = EIf e1 e2 e3" and "y = FV e1 \<union> FV e2 \<union> FV e3"
fds_ln_1: fixes l :: "'b" shows "fds_ln l 1 = fds_const l"
is_zero_row_upt_k_le: fixes i :: "'d"   and k :: "nat"   and A :: "(('e, 'f) vec, 'd) vec" assumes "is_zero_row_upt_k i (Suc k) A" shows "is_zero_row_upt_k i k A"
subst_codomain_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and V :: "'b set" shows "subst_codomain \<sigma> V = {x. \<exists>y. Var y \<lhd> \<sigma> = Var x \<and> y \<in> V}"
G_arcs_1: shows "(u, v) \<in> parcs G"
hull_Un_subset: fixes S :: "'d set \<Rightarrow> bool"   and s :: "'d set"   and t :: "'d set" shows "S hull s \<union> S hull t \<subseteq> S hull (s \<union> t)"
basis_max_indep: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and X :: "'b set"   and x :: "'b" assumes "indep_system carrier indep"   and "indep_system.basis carrier indep X"   and "x \<in> carrier - X" shows "\<not> indep (insert x X)"
prev_pair_less: fixes j :: "nat"   and i :: "nat"   and ji' :: "nat \<times> nat" assumes "prev j i = Some ji'" shows "(ji', j, i) \<in> pair_less"
Rat_normalize_idempotent: fixes x :: "int \<times> int" shows "Rat.normalize (Rat.normalize x) = Rat.normalize x"
elims_1: fixes x :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xa :: "'a list"   and xb :: "'b list"   and y :: "bool" assumes "list_ex2 x xa xb = y" obtains P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and xs :: "'b list" where "x = P__" and "xa = []" and "xb = xsa__" and "\<not> y"   | P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and v :: "'a" and va :: "'a list" where "x = Pa__" and "xa = v # va" and "xb = []" and "\<not> y"   | P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" and xc :: "'a" and xs :: "'a list" and ya :: "'b" and ys :: "'b list" where "x = P" and "xa = xc # xs" and "xb = ya # ys" and "y = (P xc ya \<or> list_ex2 P xs ys)"
coerceQuant_eq_iff: fixes x :: "'c['a, 'd]" assumes "QD('a) = QD('b)" shows "QCOERCE['b] x \<cong>\<^sub>Q x"
normalize_primitive_extract_preserves_unrelated_normalized_n_primitive: fixes m :: "'a match_expr"   and disc2 :: "'a \<Rightarrow> bool"   and sel2 :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool"   and disc1 :: "'a \<Rightarrow> bool"   and sel1 :: "'a \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'a"   and f :: "'c negation_type list \<Rightarrow> 'c list" assumes "normalized_nnf_match m"   and "normalized_n_primitive (disc2, sel2) P m"   and "wf_disc_sel (disc1, sel1) C"   and "\<forall>a. \<not> disc2 (C a)" shows "\<forall>mn\<in>set (normalize_primitive_extract (disc1, sel1) C f m). normalized_n_primitive (disc2, sel2) P mn"
tendsto_cos: fixes a :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. cos (f x)) \<longlongrightarrow> cos a) F"
vreal_uminus_def: shows "vreal_uminus = (\<lambda>x\<in>\<^sub>\<circ>\<real>\<^sub>\<circ>. (- real_of_vreal x)\<^sub>\<real>)"
lincomb_def: fixes smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and rs :: "'b list"   and ms :: "'a list" shows "scalar_mult.lincomb smult rs ms = sum_list (map2 smult rs ms)"
B: shows "u \<in> sinks_aux I D (U \<union> V) (xs @ [x])"
to_complex_zero_zero: shows "to_complex 0\<^sub>h = 0"
vsubset_vdoubleton_leftD_1: fixes a :: "V"   and b :: "V"   and A :: "V" assumes "set {a, b} \<subseteq>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> A"
cong_1_p: fixes p :: "nat" assumes "noninert_gauss_int_prime p" shows "[p = 1] (mod 4)"
floor_divide_upper: fixes q :: "'b"   and p :: "'b" assumes "(0::'b) < q" shows "p < (of_int \<lfloor>p / q\<rfloor> + (1::'b)) * q"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" shows "remove_all_cycles (x # xs) ys = remove_all_cycles xs (start_remove ys x [])"
general_1: fixes na :: "nat"   and ma :: "nat" assumes "(p ^^ na) x = (p ^^ ma) x"   and "ma < na" shows "(p ^^ (na - ma)) x = x"
pseudofunctor_axioms: shows "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C H \<Phi>\<^sub>H"
card_finite_spectrum_1: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "finite (spectrum A)"
lvl_Suc_iff: fixes t :: "('c \<times> nat) tree"   and n :: "nat" shows "(lvl t = Suc n) = (\<exists>l a r. t = \<langle>l, (a, Suc n), r\<rangle>)"
steps_c1: shows "\<Gamma>\<turnstile> ([c\<^sub>1], (cs, c\<^sub>2 # cs) # css, Normal s) \<rightarrow>\<^sup>* ([], (cs, c\<^sub>2 # cs) # css, s')"
path_inv_Branching_Right: fixes p :: "bool list"   and l :: "tree"   and r :: "tree" shows "path (Right # p) (Branching l r) = path p r"
Quotient_to_Domainp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "Quotient R Abs Rep T" shows "Domainp T = (\<lambda>x. R x x)"
vsubset_VPow_VUnion: fixes A :: "V" shows "A \<subseteq>\<^sub>\<circ> VPow (\<Union>\<^sub>\<circ> A)"
ample_set_def: fixes q :: "'state"   and A :: "'action set" shows "ample_set q A \<equiv> A \<subseteq> {a. en a q} \<and> (A \<subset> {a. en a q} \<longrightarrow> A \<noteq> {}) \<and> (\<forall>a. A \<subset> {a. en a q} \<longrightarrow> a \<in> A \<longrightarrow> scut\<inverse>\<inverse> (ex a q) q) \<and> (A \<subset> {a. en a q} \<longrightarrow> A \<subseteq> invisible) \<and> (\<forall>w. A \<subset> {a. en a q} \<longrightarrow> path w q \<longrightarrow> A \<inter> set w = {} \<longrightarrow> Ind A (set w))"
normalT: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> G) \<longrightarrow> \<box>F \<longrightarrow> \<box>G"
writes_singleton: fixes f :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and a :: "'o"   and h :: "'l"   and h' :: "'l" shows "writes (all_args f) (f a) h h'"
conseq_exploit_pre: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> ({s} \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
v11_axioms: shows "v11 (r\<inverse>\<^sub>\<circ>)"
axis_def: fixes k :: "'e"   and x :: "'d" shows "axis k x = (\<chi>i. if i = k then x else (0::'d))"
absI: fixes M :: "msg"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and H :: "msg set" assumes "M \<in> payload"   and "implem (Chan x A B M) \<in> H" shows "Chan x A B M \<in> local.abs H"
norm_fstzero_zerosnd_2: shows "norm no_real_inner.zerosnd = 1"
transition_def: fixes stateA :: "State"   and stateB :: "State"   and decisionVars :: "nat set" shows "transition stateA stateB decisionVars \<equiv> appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB \<or> appliedLearn stateA stateB \<or> appliedBackjump stateA stateB"
ssubst_mem: fixes t :: "'d"   and s :: "'d"   and X :: "'d set" assumes "t = s"   and "s \<in> X" shows "t \<in> X"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "even (f x) = even (g x)"
fs_cont: fixes k :: "nat" assumes "k \<le> 2 * N + 1" shows "continuous_on {real a..} (fs k)"
mcompos_inj_inj: fixes L :: "('i, 'b, 'j) Module_scheme"   and N :: "('k, 'b, 'l) Module_scheme"   and f :: "'i \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'k" assumes "R module L"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N"   and "injec\<^bsub>L,M\<^esub> f"   and "injec\<^bsub>M,N\<^esub> g" shows "injec\<^bsub>L,N\<^esub> compos L g f"
delete_eq_empty_lookup_other_fail: fixes ks :: "'c list"   and t :: "('c, 'd) trie"   and ks' :: "'c list" assumes "delete_trie ks t = Trie None []"   and "ks' \<noteq> ks" shows "lookup_trie t ks' = None"
col_add: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and j :: "nat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "j < nc" shows "col (A + B) j = col A j + col B j"
sup_inf_distrib1: fixes x :: "'d multiset"   and y :: "'d multiset"   and z :: "'d multiset" shows "x \<union># (y \<inter># z) = x \<union># y \<inter># (x \<union># z)"
distinct_51: fixes x9 :: "('a, String.literal) phantom" shows "Real \<noteq> Enum x9"
m_rcancel: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<noteq> \<zero>\<^bsub>K [X]\<^esub>"   and "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "c \<in> carrier (K [X])" shows "(b \<otimes>\<^bsub>K [X]\<^esub> a = c \<otimes>\<^bsub>K [X]\<^esub> a) = (b = c)"
simps: fixes S :: "'node SDG_node set"   and n :: "'node SDG_node"   and a :: "'var" shows "relevant_varsp S n a = (\<exists>as n' V. a = V \<and> parent_node n -as\<rightarrow>\<^sub>\<iota>* parent_node n' \<and> n' \<in> HRB_slice S \<and> V \<in> Use\<^bsub>SDG\<^esub> n' \<and> (\<forall>n''. valid_SDG_node n'' \<and> parent_node n'' \<in> set (sourcenodes as) \<longrightarrow> V \<notin> Def\<^bsub>SDG\<^esub> n''))"
diag_mat_def: fixes f :: "'c \<Rightarrow> 'b" shows "diag_mat f = (\<chi>i j. if i = j then f i else (0::'b))"
rep_eq: fixes x :: "'b"   and xa :: "('b, 'c) mapping" shows "rep (delete x xa) = (rep xa)(x := None)"
lincomb_list_carrier: fixes vs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "set vs \<subseteq> carrier_vec n" shows "n.lincomb_list c vs \<in> carrier_vec n"
dvd_add_right_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b" shows "(a dvd b + c) = (a dvd c)"
wfactorsE: fixes G :: "('c, 'd) monoid_scheme"   and fs :: "'c list"   and a :: "'c" assumes "wfactors G fs a" obtains "\<forall>f\<in>set fs. irreducible G f" and "foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<sim>\<^bsub>G\<^esub> a"
sum_dce: fixes A :: "'a"   and B :: "'a"   and C :: "'a" shows "(A + B \<rightarrow>\<^sub>\<E> C) = (A \<rightarrow>\<^sub>\<E> C \<or> B \<rightarrow>\<^sub>\<E> C)"
run_complete: fixes c :: "'b Heap"   and \<sigma> :: "heap option" obtains \<sigma>' :: "heap option" and r :: "'b" where "run c \<sigma> \<sigma>' r"
defs'_def: fixes g :: "'g"   and n :: "'node" shows "defs' g n \<equiv> (\<lambda>v. (v, n, SimpleDef)) ` defs g n"
no_loop: fixes x :: "'v" shows "\<not> edge \<Delta> x x"
nth_take_lemma: fixes k :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "k \<le> length xs"   and "k \<le> length ys"   and "\<And>i. i < k \<longrightarrow> xs ! i = ys ! i" shows "take k xs = take k ys"
indRelRSTPO_impl_SRel_and_TRel_preserve_barbs_2: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_preserves_binary_pred (indRelRSTPO SRel TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
ccpo_Sup_singleton: fixes Sup :: "'a set \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.ccpo Sup less_eq less" shows "Sup {x} = x"
Fix1: fixes \<phi> :: "(((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set" assumes "Monotone \<phi>"   and "FIX \<phi> (s, t, \<beta>)" shows "\<phi> (FIX \<phi>) (s, t, \<beta>)"
le_cases: fixes less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and x :: "'q"   and y :: "'q" assumes "class.linorder less_eq less" obtains "less_eq x y"   | "less_eq y x"
app_def: fixes t1 :: "('c, 'd) FingerTreeStruc"   and t2 :: "('c, 'd) FingerTreeStruc" shows "app t1 t2 = app3 t1 [] t2"
similar_mat_witD2_7: fixes A :: "'c mat"   and n :: "nat"   and m :: "nat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "Q \<in> carrier_mat n n"
expand: fixes status :: "status"   and status' :: "status" assumes "is_failed status = is_failed status'"   and "is_success status = is_success status'" shows "status = status'"
Conf_LabelFunSem: fixes S :: "'d"   and ST :: "('d, 'e, 'f) status" shows "(IN S \<in> the (LabelFunSem (HA ST) ST)) = (S \<in> Conf ST)"
lemma_tan_total1: fixes y :: "real" shows "\<exists>x>- (pi / 2). x < pi / 2 \<and> tan x = y"
slow_median_def: fixes xs :: "'a list" shows "slow_median xs = slow_select ((length xs - 1) div 2) xs"
assms_5: shows "occurs_on ev \<noteq> occurs_on ev'"
rep_eq: fixes x :: "\<kappa> \<Rightarrow> \<o>" shows "IsProperInX x = (\<forall>xa v. (\<exists>a. \<nu>\<upsilon> a = \<nu>\<upsilon> xa \<and> eval\<o> (x (a\<^sup>P)) dj v) = eval\<o> (x (xa\<^sup>P)) dj v)"
inject_6: fixes x6 :: "(char list \<times> trm) list"   and y6 :: "(char list \<times> trm) list" shows "(Rcd x6 = Rcd y6) = (x6 = y6)"
FormP_Neg: fixes x :: "tm" shows "{FormP x} \<turnstile> FormP (Q_Neg x)"
vec_lambda_unique: fixes f :: "('c, 'b) Finite_Cartesian_Product.vec"   and g :: "'b \<Rightarrow> 'c" shows "(\<forall>i. f $h i = g i) = (vec_lambda g = f)"
projection_commute: fixes l :: "'b list"   and X :: "'b set"   and Y :: "'b set" shows "l \<upharpoonleft> X \<upharpoonleft> Y = l \<upharpoonleft> Y \<upharpoonleft> X"
poincare_distance_pred_def: fixes u :: "complex_homo"   and v :: "complex_homo"   and d :: "real" shows "poincare_distance_pred u v d = (u = v \<and> d = 0 \<or> u \<noteq> v \<and> (\<forall>i1 i2. ideal_points (poincare_line u v) = {i1, i2} \<longrightarrow> d = calc_poincare_distance u i1 v i2))"
op_is_valid: shows "op \<in> set (\<Pi>\<^sub>\<O>)"
unify_unification: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "\<sigma> \<in> unifiers (set E)" shows "\<exists>\<theta>. is_imgu \<theta> (set E)"
elims_2: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a list" assumes "sorted_wrt x xa" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "xa__ = P__" and "xa = []"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and x :: "'a" and ys :: "'a list" where "xa__ = P" and "xa = x # ys" and "Ball (set ys) (P x) \<and> sorted_wrt P ys"
pred_False_is_False: fixes mem :: "'Var \<Rightarrow> 'Val" shows "\<not> ev\<^sub>B mem pred_False"
b0: fixes s :: "'a" assumes "s \<in> S" shows "0 \<le> b s"
idom_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "idom_hom hom \<equiv> comm_ring_hom hom"
convex_hull_subset_affine_hull: fixes s :: "'a set" shows "convex hull s \<subseteq> affine hull s"
fic_a_star_reduce: fixes M :: "trm"   and x :: "coname"   and M' :: "trm" assumes "fic M x"   and "M \<longrightarrow>\<^sub>a* M'" shows "fic M' x"
extendf: fixes i :: "nat" assumes "int i \<le> aff_dim T" shows "\<exists>g. continuous_on (\<Union> (\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i})) g \<and> g ` \<Union> (\<G> \<union> {D. \<exists>C\<in>\<F>. D face_of C \<and> aff_dim D < int i}) \<subseteq> rel_frontier T \<and> (\<forall>x\<in>\<Union> \<G>. g x = f x)"
delete_twist: fixes x :: "'c"   and y :: "'c"   and al :: "('c \<times> 'd) list" shows "delete x (delete y al) = delete y (delete x al)"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c pmf" shows "map_pmf g (map_pmf f v) = map_pmf (g \<circ> f) v"
closed_compactin: fixes X :: "'b topology"   and K :: "'b set"   and C :: "'b set" assumes "compactin X K"   and "C \<subseteq> K"   and "closedin X C" shows "compactin X C"
wf_trm_subst_rangeI: fixes \<delta> :: "'c \<Rightarrow> ('fun, 'c) Term.term" assumes "\<And>x. wf\<^sub>t\<^sub>r\<^sub>m (\<delta> x)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
simps: fixes a1 :: "s"   and a2 :: "s" shows "store_extension a1 a2 = (\<exists>insts insts' fs fs' tclss tclss' bss bss' gs gs'. a1 = \<lparr>inst = insts, funcs = fs, tab = tclss, mem = bss, globs = gs\<rparr> \<and> a2 = \<lparr>inst = insts', funcs = fs', tab = tclss', mem = bss', globs = gs'\<rparr> \<and> insts = insts' \<and> fs = fs' \<and> tclss = tclss' \<and> list_all2 (\<lambda>bs bs'. mem_size bs \<le> mem_size bs') bss bss' \<and> gs = gs')"
l0c: shows "e * c / (f2 * (c - 1)) \<le> real l0"