hcju/isabelleps huggingface dataset:
timesE2: fixes a :: "hf"   and b :: "hf"   and A :: "hf"   and B :: "hf" assumes "\<langle>a, b\<rangle> \<^bold>\<in> A * B" obtains "a \<^bold>\<in> A" and "b \<^bold>\<in> B"
compl_le_compl_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<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 bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (uminus x) (uminus y) = less_eq y x"
interfree_aux1: fixes c :: "'b ann_com option"   and s :: "'b"   and r :: "'b ann_com option"   and t :: "'b"   and c1 :: "'b ann_com option"   and q1 :: "'b set" assumes "(c, s) -1\<rightarrow> (r, t)"   and "interfree_aux (c1, q1, c)" shows "interfree_aux (c1, q1, r)"
rank_lt: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> b" shows "rank a \<subset>\<^sub>\<circ> rank b"
fst_SUP: fixes f :: "'e \<Rightarrow> 'c \<times> 'd"   and A :: "'e set" shows "fst (Sup (f ` A)) = (SUP x\<in>A. fst (f x))"
Min_in: fixes A :: "'b word set" assumes "finite A"   and "A \<noteq> {}" shows "signed.Min A \<in> A"
finite_INT: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> M" shows "\<Inter> (A ` I) \<in> M"
pp_total: fixes x :: "'a" shows "(- - x \<sqinter> top = top) = dense (x \<sqinter> top)"
simps_2: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" shows "hsize (App s t) = hsize s + hsize t"
ceiling_diff_one: fixes x :: "'b" shows "\<lceil>x - (1::'b)\<rceil> = \<lceil>x\<rceil> - 1"
bot_least: fixes C :: "'a option acom"   and c :: "com" assumes "strip C = c" shows "bot c \<le> C"
simps_12: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and e1' :: "'b aexp"   and e2' :: "'b aexp" shows "GExp_Lexorder.less_gexp_aux (Gt e1 e2) (Eq e1' e2') = False"
foo_sig_not_undeclared_in_Ext: shows "\<not> tprg\<turnstile> mid foo_sig undeclared_in Ext"
y_max: shows "\<forall>x\<in>#M. x < y"
step_def: fixes g :: "'g" shows "uninst.step uses phis g \<equiv> \<lambda>(u, p). (uninst.uses' (uses(g := u)) (phis(g := p)) g, uninst.phis' (uses(g := u)) (phis(g := p)) g)"
a: shows "!x \<cdot> y \<cdot> t z = (0::'a)"
openE: fixes T :: "'b set set"   and m :: "'b set" assumes "m open\<^bsub>T\<^esub>" obtains "m \<in> T"
independent_from_def: fixes d :: "nat"   and Y :: "'b \<times> 'b pdevs" shows "independent_from d Y = msum_aform d (0::'b, zero_pdevs) Y"
ale_shift: fixes x :: "ant"   and y :: "ant"   and z :: "ant" assumes "x \<le> y"   and "y = z" shows "x \<le> z"
has_integral_reflect_real: fixes f :: "real \<Rightarrow> 'b"   and i :: "'b"   and b :: "real"   and a :: "real" shows "((\<lambda>x. f (- x)) has_integral i) {- b..- a} = (f has_integral i) {a..b}"
countable_Un_iff: fixes A :: "'c set"   and B :: "'c set" shows "countable (A \<union> B) = (countable A \<and> countable B)"
c1_on_open_rev: shows "c1_on_open (- f) (- f') X"
arr_Rel_converse_Rel: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "arr_Rel \<alpha> (T\<inverse>\<^sub>R\<^sub>e\<^sub>l)"
preserves_reflects_arr: fixes f :: "'a" shows "Adj.C.arr (Adj.\<epsilon> f) = Adj.C.arr f"
bi_unique_mod_ring_rel_2: shows "left_unique mod_ring_rel"
simps: fixes key_gen :: "('b \<times> 'c, 'd, 'e) gpv"   and encrypt :: "'b \<Rightarrow> 'f \<Rightarrow> ('g, 'd, 'e) gpv"   and valid_plains :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and "oracle" :: "'h \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'h) spmf"   and \<sigma> :: "'h"   and \<A> :: "('b \<Rightarrow> (('f \<times> 'f) \<times> 'i, 'd, 'e) gpv) \<times> ('g \<Rightarrow> 'i \<Rightarrow> (bool, 'd, 'e) gpv)" shows "advantage key_gen encrypt valid_plains (oracle, \<sigma>) \<A> = \<bar>spmf (run_gpv oracle (game key_gen encrypt valid_plains \<A>) \<sigma>) True - 1 / 2\<bar>"
cobounded2: fixes b :: "'b"   and a :: "'b" shows "b \<le> a + b"
cos_even_kpi: fixes k :: "int" assumes "even k" shows "cos (real_of_int k * pi) = 1"
neg: shows "\<integral>\<^sup>+ x. ennreal (- (case x of (x, y, z) \<Rightarrow> Pxz (x, z) * (Pyz (y, z) / Pz z) / Pxyz (x, y, z))) \<partial>density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x)) = 0"
frangeD: fixes b :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> r" shows "\<exists>a. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
f_expand_not_empty_conv: fixes xs :: "'d message_af list"   and k :: "nat" shows "(xs \<odot> k \<noteq> []) = (xs \<noteq> [] \<and> 0 < k)"
has_type_le_env: fixes A :: "type_scheme list"   and e :: "expr"   and t :: "typ"   and B :: "type_scheme list" assumes "A |- e :: t"   and "A \<le> B" shows "B |- e :: t"
sorted_tl: fixes xs :: "'a word list" assumes "signed.sorted xs" shows "signed.sorted (tl xs)"
f_last_message_hold_take: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" shows "xs \<down> n \<longmapsto> k = xs \<longmapsto> k \<down> n"
rank_gt_minor: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat"   and I :: "nat set"   and J :: "nat set" assumes "A \<in> carrier_mat n nc"   and "det (submatrix A I J) \<noteq> (0::'a)" shows "card {j. j < nc \<and> j \<in> J} \<le> vec_space.rank n A"
Sup_least: fixes X :: "real set"   and z :: "real" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> x \<le> z" shows "Sup X \<le> z"
nonrecE: fixes \<Gamma> :: "(Vars.var \<times> exp) list" assumes "nonrec \<Gamma>" obtains x :: "Vars.var" and e :: "exp" where "\<Gamma> = [(x, e)]" and "x \<notin> fv e"
scTd: shows "singular_chain (p - Suc 0) (subtopology X T) (chain_boundary p d)"
distrib_right': fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b"   and z :: "'a \<times> 'b" shows "times_prod' (sup_prod x y) z = sup_prod (times_prod' x z) (times_prod' y z)"
hyps: assumes "local.independent K Us"   and "local.independent K Vs"   and "Span K Us \<inter> Span K Vs = {\<zero>}" shows "local.independent K (Us @ Vs)"
rfn_tool11: fixes b :: "nat"   and a :: "nat" assumes "0 < b"   and "a \<le> b - Suc 0" shows "a < b"
power_minus_Bit0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus x) (numeral (num.Bit0 k)) = power.power one times x (numeral (num.Bit0 k))"
prems_1: shows "f \<in> insert a A \<rightarrow> carrier R"
distinct_take: fixes xs :: "'c list"   and i :: "nat" assumes "distinct xs" shows "distinct (take i xs)"
prime_divisor_sqrt: fixes n :: "nat" shows "prime n = (n \<noteq> 1 \<and> (\<forall>d. d dvd n \<and> d\<^sup>2 \<le> n \<longrightarrow> d = 1))"
a1: shows "low_equal s1 s2 \<and> Some t1 = memory_write_asi 10 addr bm data s1 \<and> Some t2 = memory_write_asi 10 addr bm data s2"
simps: fixes ant :: "'b form multiset"   and suc :: "'b form multiset" shows "succ ( ant \<Rightarrow>* suc) = suc"
real_tendsto_divide_at_top: fixes f :: "'b \<Rightarrow> real"   and c :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g at_top F" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
rel_eq_onp: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool" shows "rel_prod (eq_onp P1) (eq_onp P2) = eq_onp (pred_prod P1 P2)"
num_of_odd_nodes_def: fixes g :: "('c, 'd) graph" shows "num_of_odd_nodes g \<equiv> card (odd_nodes_set g)"
axioms: fixes f :: "'d \<Rightarrow> 'e" assumes "clinear f" shows "Vector_Spaces.linear (*\<^sub>C) (*\<^sub>C) f"
Zfun_add: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "Zfun f F"   and "Zfun g F" shows "Zfun (\<lambda>x. f x + g x) F"
fmempty_of_list: shows "{$$} = fmap_of_list []"
step_p: shows "\<forall>(q, \<tau>)\<in>set (step p (start ! p)). Sorted_Less.sorted (rev (p # rev [0..<length (g_V G)])) \<or> p # rev [0..<length (g_V G)] = start ! q"
next_subseqs_spec_2: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and head :: "'d"   and tail :: "'d iarray"   and prevs :: "(nat \<times> 'c) list"   and out :: "'c list"   and head' :: "'d"   and tail' :: "'d iarray"   and prevs' :: "(nat \<times> 'c) list" assumes "my_subseqs.next_subseqs f (head, tail, prevs) = (out, head', tail', prevs')" shows "set out = (f head \<circ> snd) ` set prevs \<union> snd ` set prevs'"
prems_1: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list"   and x' :: "'a"   and ys' :: "'a list" assumes "ls = xs @ [x] @ ys"   and "ls' = xs @ [x'] @ ys'"   and "x \<noteq> x'" shows "thesis"
fair_run_is_fair': fixes s :: "nat \<Rightarrow> 'a" assumes "is_fair_run s" shows "is_fair' s"
iT_Minus_imirror_conv: fixes k :: "nat"   and I :: "nat set" shows "k \<ominus> I = imirror (I \<down>\<le> k) \<oplus> k \<oplus>- iMin I + Max (I \<down>\<le> k)"
word_bits_size: fixes w :: "32 Word.word" shows "size w = word_bits"
equal_ell2_def: fixes M :: "'a ell2"   and N :: "'a ell2" shows "equal_class.equal M N = (vec_of_ell2 M = vec_of_ell2 N)"
extreme_def: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'c" shows "extreme X r e \<equiv> e \<in> X \<and> (\<forall>x\<in>X. r x e)"
lenL: shows "length (mat_multI ze pl ti nr m1 m2) = nc"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>A) (F \<circ> F'G'.FG.map) F (F \<circ> F'G'.\<epsilon>)"
CFG_edge_Uses_pred_equal: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and a :: "'f"   and s :: "'h"   and s' :: "'h" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val"   and "valid_edge a"   and "pred (kind a) s"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V" shows "pred (kind a) s'"
image_spec__eq: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and P' :: "'a"   and A :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ReflectL Bet Cong P P' A A" shows "P = P'"
contlub_cfun_arg: fixes Y :: "nat \<Rightarrow> 'e"   and f :: "'e \<rightarrow> 'f" assumes "chain Y" shows "f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
absolutely_integrable_zero: fixes S :: "'a set" shows "(\<lambda>x. 0::'b) absolutely_integrable_on S"
composite_simps_8: shows "cod ((f \<star> g) \<star> h) = (cod f \<star> cod g) \<star> cod h"
field_divide_inverse: fixes a :: "'e"   and b :: "'e" shows "a / b = a * inverse b"
total_def: fixes ES :: "'b ES_rec"   and E :: "'b set" shows "EventSystems.total ES E \<equiv> E \<subseteq> E\<^bsub>ES\<^esub> \<and> (\<forall>\<tau>\<in>Tr\<^bsub>ES\<^esub>. \<forall>e\<in>E. \<tau> @ [e] \<in> Tr\<^bsub>ES\<^esub>)"
Standard_HComplex: fixes r :: "real star"   and s :: "real star" assumes "r \<in> Standard"   and "s \<in> Standard" shows "HComplex r s \<in> SComplex"
hm_ops_def: shows "hm_ops \<equiv> hm_basic.dflt_ops"
closedD: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "closed A f"   and "x \<in> A"   and "y \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<in> A"
multiplicity_unit_right: fixes x :: "'a"   and p :: "'a" assumes "is_unit x" shows "multiplicity p x = 0"
stack_not_finished: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and x :: "'c" assumes "DFS_invar G param s"   and "x \<in> set (stack s)" shows "x \<notin> dom (finished s)"
cobounded2: fixes b :: "'c"   and a :: "'c" shows "b \<le> sup a b"
dbm_entry_val_mono_2: fixes u :: "'b \<Rightarrow> 'c"   and c :: "'b"   and b :: "'c DBMEntry"   and b' :: "'c DBMEntry" assumes "dbm_entry_val u None (Some c) b"   and "b \<preceq> b'" shows "dbm_entry_val u None (Some c) b'"
components_are_iso: fixes a :: "'a" assumes "hom.ide a" shows "hom.iso (\<phi>.map a)"
Not_zs: shows "inversion (Some i, map Not zs) = xs"
append_def: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "Execute.append xs ys zs = (zs = xs @ ys)"
simps_3: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Neg \<phi>) j = Monitor.progress \<sigma> \<phi> j"
\<omega>\<^sub>z_as_\<omega>: shows "\<omega>\<^sub>z = zhmset_of \<omega>"
legs_form_cospan_1: shows "C.cospan doms.\<nu>.leg0 doms.\<pi>.leg1"
intrel_def: shows "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
sumAssoc: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" shows "(P \<oplus> Q) \<oplus> R \<sim>\<^sub>l P \<oplus> Q \<oplus> R"
edges: fixes u :: "'v"   and v :: "'v" assumes "\<pi>' Q \<pi> u u = Some v" shows "(u, v) \<in> edges rg"
c_def: shows "c = 1 / (trace (A - f 0) + 1)"
cl1: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> + x \<cdot> nc = x \<cdot> U"
approx_unE: fixes F :: "float interval \<Rightarrow> float interval option"   and f :: "real \<Rightarrow> real"   and e :: "nat \<Rightarrow> real"   and p :: "nat"   and X'err :: "((real \<times> real pdevs) \<times> real) option"   and Ye :: "(real \<times> real pdevs) \<times> real"   and x :: "real" assumes "interval_extension1 F f"   and "e \<in> UNIV \<rightarrow> {- 1..1}"   and "approx_un p F X'err = Some Ye"   and "case X'err of None \<Rightarrow> True | Some X'err \<Rightarrow> x \<in> aform_err e X'err" shows "f x \<in> aform_err e Ye"
2: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (unlabel A')"
degW: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and q_ty :: "'c itself" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "degree (#w + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q q_ty)) (#Knuth_ex_4_6_2_22_main.w' q_ty a b u v w)) = degree w"
eint_minus_ineq'': fixes c :: "eint"   and d :: "eint"   and b :: "eint"   and e :: "eint" assumes "c \<noteq> \<infinity>"   and "d \<le> b"   and "0 < e"   and "e \<noteq> \<infinity>" shows "e * d - e * c \<le> e * (b - c)"
pmf_cond: fixes p :: "'c pmf"   and s :: "'c set"   and x :: "'c" assumes "set_pmf p \<inter> s \<noteq> {}" shows "pmf (cond_pmf p s) x = (if x \<in> s then pmf p x / measure_pmf.prob p s else 0)"
lower0: fixes i :: "nat"   and j :: "nat" shows "lowerPoly i j 0 = 0"
Gt_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Gt Bet Cong A B C D \<equiv> Tarski_neutral_dimensionless.Lt Bet Cong C D A B"
mgu_complete: fixes s :: "('d, 'c) Term.term"   and t :: "('d, 'c) Term.term" assumes "mgu s t = None" shows "unifiers {(s, t)} = {}"
assms: shows "set X \<inter> subst_domain \<theta> = {}"
infin_imp_iNextWeak_eq_iNext: fixes I :: "nat set"   and t0 :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "infinite I"   and "t0 \<in> I" shows "iNextWeak t0 I P = iNext t0 I P"
ireturn_eq: fixes x :: "'c" shows "RETURN x = SPEC ((=) x)"
cases: fixes x :: "nat \<times> nat queue \<times> nat \<times> bool list \<times> bool list \<times> ('a option list, nat) mapping \<times> (nat, ('a option list, nat + nat) mapping) mapping \<times> 'a option list set list \<times> nat" obtains tp :: "nat" and tss :: "nat queue" and len :: "nat" and maskL :: "bool list" and maskR :: "bool list" and a1_map :: "('a option list, nat) mapping" and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping" and "done" :: "'a option list set list" and done_length :: "nat" where "x = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)"
measurable_snth: fixes n :: "nat"   and M :: "'a measure" shows "(\<lambda>\<omega>. \<omega> !! n) \<in> stream_space M \<rightarrow>\<^sub>M M"
extend_set_project_set: fixes C :: "'c set" shows "C \<subseteq> extend_set h (project_set h C)"
A81_DAcc_level1: shows "DAcc level1 sA81 = {sA91, sA92}"
weight_less_iff: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term" assumes "weight_less s t" shows "weight_le s t \<and> weight s < weight t"
cutoff_branch': fixes T :: "tree"   and red :: "bool list \<Rightarrow> bool"   and ds :: "bool list" assumes "anybranch T (\<lambda>p. red (ds @ p))" shows "anybranch (cutoff red ds T) (\<lambda>p. red (ds @ p))"
det_0_iff_vec_prod_zero: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A = (0::'a)) = (\<exists>v. v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> A *\<^sub>v v = 0\<^sub>v n)"
rep_numbers_constant: shows "replication_numbers = {\<Lambda> * (\<v> - 1) div (\<k> - 1)}"
cong_preserves_bet: fixes B :: "'p"   and A' :: "'p"   and A0 :: "'p"   and E :: "'p"   and D' :: "'p"   and D0 :: "'p" assumes "Bet B A' A0"   and "Cong B A' E D'"   and "Cong B A0 E D0"   and "E Out D' D0" shows "Bet E D' D0"
mtx_def: fixes A :: "real list list" shows "mtx A = to_mtx (vector (map vector A))"
Sup_insert: 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 bot :: "'b"   and top :: "'b"   and a :: "'b"   and A :: "'b set" assumes "class.complete_lattice (Inf::'b set \<Rightarrow> 'b) (Sup::'b set \<Rightarrow> 'b) (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (bot::'b) (top::'b)" shows "(Sup::'b set \<Rightarrow> 'b) ((a::'b) \<triangleright> (A::'b set)) = (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) a (Sup A)"
strand_sem_wf_ik_or_assignment_rhs_fun_subterm: fixes A :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and x :: "'var"   and f :: "'fun"   and T :: "('fun, 'var) Term.term list"   and t\<^sub>i :: "('fun, 'var) Term.term" assumes "wf\<^sub>s\<^sub>t {} A"   and "\<lbrakk>{}; A\<rbrakk>\<^sub>c \<I>"   and "Var x \<in> ik\<^sub>s\<^sub>t A"   and "\<I> x = Fun f T"   and "t\<^sub>i \<in> set T"   and "\<not> ik\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t\<^sub>i"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" obtains S :: "('fun, 'var) Term.term list" where "Fun f S \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t ik\<^sub>s\<^sub>t A \<or> Fun f S \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t assignment_rhs\<^sub>s\<^sub>t A" and "Fun f T = Fun f S \<cdot> \<I>"
solve_equation: fixes H :: "'a set set"   and x :: "'a set"   and y :: "'a set" assumes "subgroup H (G Mod N)"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes>\<^bsub>G Mod N\<^esub> x"
disc_7: fixes x21 :: "nat"   and x22 :: "'d"   and x23 :: "'d"   and x24 :: "'e"   and x25 :: "'e"   and x26 :: "'f" shows "isSend (Send x21 x22 x23 x24 x25 x26)"
diff_def: fixes M :: "'b multiset"   and S :: "'b set" shows "M -s S = {#x \<in># M. x \<notin> S#}"
Idl_subset_ideal': fixes a :: "('c multiset \<Rightarrow> 'a) list"   and b :: "('c multiset \<Rightarrow> 'a) list" assumes "a \<in> carrier (poly_ring L)"   and "b \<in> carrier (poly_ring L)" shows "(Idl\<^bsub>poly_ring L\<^esub> {a} \<subseteq> Idl\<^bsub>poly_ring L\<^esub> {b}) = (a \<in> Idl\<^bsub>poly_ring L\<^esub> {b})"
concurrentI: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s1 :: "'c"   and s2 :: "'c" assumes "happens_before hb_weak hb"   and "\<not> hb s1 s2"   and "\<not> hb s2 s1" shows "happens_before.concurrent hb s1 s2"
simps: fixes a :: "('state, 'act, 'out) trans list" shows "valid a = ((\<exists>trn. a = [trn] \<and> validTrans trn) \<or> (\<exists>trn tr. a = trn # tr \<and> validTrans trn \<and> tgtOf trn = srcOf (hd tr) \<and> valid tr))"
Pi_eq_empty: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" shows "(Pi A B = {}) = (\<exists>x\<in>A. B x = {})"
lhd'_def: fixes xs :: "'c llist" shows "lhd' xs = (if lnull xs then None else Some (lhd xs))"
assms_2: shows "comm_monoid_mult_class.coprime f g"
insertion_polyadd: fixes e :: "nat \<Rightarrow> 'a"   and a :: "'a poly"   and b :: "'a poly" shows "insertion e (a +\<^sub>p b) = insertion e a + insertion e b"
gen_keys_def: fixes kts :: "('c \<times> ('c, 'd) rbt) list"   and t :: "('c, 'd) rbt" shows "RBT_Impl.gen_keys kts t = RBT_Impl.keys t @ concat (map (\<lambda>(k, t). k # RBT_Impl.keys t) kts)"
ev_alw_imp_alw_ev: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev (alw \<phi>) xs" shows "alw (ev \<phi>) xs"
disc: shows "\<forall>a. \<not> disc2 (C a)"
assms_1: shows "bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V})"
geom_proc_eq_pseudo_proj_True: fixes n :: "nat"   and x :: "bool stream"   and y :: "bool stream" assumes "\<And>m. m \<le> n \<Longrightarrow> geom_proc m x = geom_proc m y" shows "pseudo_proj_True n x = pseudo_proj_True n y"
HPhase0Read_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d"   and "HInv3 s" shows "HInv3 s'"
B_typ: shows "\<turnstile>\<^sub>\<tau> B : \<tau> \<rightarrow> propT"
finite_entropy_distributed: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real" assumes "finite_entropy S X Px" shows "distributed M S X (\<lambda>x. ennreal (Px x))"
uset_l: fixes p :: "fm" assumes "islin p" shows "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
unique_quotient: fixes a :: "int"   and b :: "int"   and q :: "int"   and r :: "int"   and q' :: "int"   and r' :: "int" assumes "eucl_rel_int a b (q, r)"   and "eucl_rel_int a b (q', r')" shows "q = q'"
valid: shows "PTA.valid_abstraction A \<X> (\<lambda>x. real (k x))"
axioms_2: fixes L :: "('a, 'b) gorder_scheme" assumes "weak_partial_order L" shows "weak_partial_order_axioms L"
LeftDerivation_LeftDerives1: fixes a :: "symbol list"   and S :: "(nat \<times> symbol \<times> symbol list) list"   and y :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and z :: "symbol list" assumes "LeftDerivation a S y"   and "LeftDerives1 y i r z" shows "LeftDerivation a (S @ [(i, r)]) z"
symmetric_condition_1: assumes "\<Lambda> * (\<v> - 1) = \<k> * (\<k> - 1)" shows "\<b> = \<v> \<and> \<Lambda> * (\<v> - 1) div (\<k> - 1) = \<k>"
ball_subset_cball: fixes x :: "'c"   and e :: "real" shows "ball x e \<subseteq> cball x e"
invar_children: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e a r ts)" shows "SkewBinomialHeapStruc.queue_invar ts"
lemma_1_i: fixes s :: "('d, 'e) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" assumes "s \<in> valid_states PROB"   and "a \<in> PROB" shows "state_succ s a \<in> valid_states PROB"
Entry_no_predecessor: fixes g :: "'g" shows "old.predecessors g (Entry g) = []"
ccompatible_subs: fixes g :: "('c, real) cconstraint"   and R :: "('c \<Rightarrow> real) set"   and u :: "'c \<Rightarrow> real" assumes "ccompatible \<R> g"   and "R \<in> \<R>"   and "u \<in> R"   and "u \<turnstile> g" shows "R \<subseteq> {u. u \<turnstile> g}"