hcju/isabelleps huggingface dataset:
cases: fixes x :: "('b \<times> nat) tree" obtains "x = \<langle>\<rangle>"   | uu_ :: "('b \<times> nat) tree" and uv_ :: "'b \<times> nat" where "x = \<langle>uu_, uv_, \<langle>\<rangle>\<rangle>"   | uw_ :: "('b \<times> nat) tree" and ux_ :: "'b" and lva :: "nat" and uy_ :: "('b \<times> nat) tree" and uz_ :: "'b" and lvb :: "nat" and va_ :: "('b \<times> nat) tree" where "x = \<langle>uw_, (ux_, lva), \<langle>uy_, (uz_, lvb), va_\<rangle>\<rangle>"
Un_empty_right: fixes A :: "'d set" shows "A \<union> {} = A"
abs_eq: fixes x :: "rat" shows "of_rat_real_alg x = Real_Alg_Quotient (of_rat_3 x)"
interior_inside_frontier: fixes S :: "'b set" assumes "bounded S" shows "interior S \<subseteq> inside (frontier S)"
subset_singletonD: fixes A :: "'b set"   and x :: "'b" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
deduction: fixes p :: "'b pl"   and H :: "'b pl set"   and q :: "'b pl" assumes "insert p H |- q" shows "H |- p -> q"
ThinishChamberComplexFolding_axioms: shows "ThinishChamberComplexFolding X f"
adequacy: fixes fs :: "formula list" assumes "validS fs" shows "fs \<in> deductions CutFreePC"
iso_incl_antisym: fixes A :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and B :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) ring" assumes "A \<in> \<S>"   and "B \<in> \<S>"   and "A \<lesssim> B"   and "B \<lesssim> A" shows "A = B"
vinf\<^sub>2: shows "v \<in> \<V> f\<^sub>2"
cases: fixes a1 :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and a2 :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" assumes "(a1, a2) \<in> timpl_closure'_step TI" obtains a :: "'c set" and b :: "'c set" and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term" where "a1 = t" and "a2 = s" and "(a, b) \<in> TI" and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t s"
stable_on_blocking_onD: fixes ds :: "'b set"   and X :: "'a set"   and X'' :: "'a set"   and h :: "'c" assumes "Bossy''.stable_on ds X"   and "X'' = BCh' h (X \<union> X'')"   and "X'' \<subseteq> Bossy''.CD_on ds (X \<union> X'')" shows "X'' = BCh' h X"
of_nat_eq_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
SystemDESubst_def: shows "SystemDESubst = \<lparr>SFunctions = \<lambda>f. if f = fid1 then Some ($f (Inl fid1) local.empty) else None, SPredicates = \<lambda>p. if p = vid2 then Some (TT && Geq ($f (Inl fid1) local.empty) (Const 0)) else None, SContexts = \<lambda>C. if C = pid1 then Some (Geq (Differential (Var vid1)) (Differential (Const 0))) else None, SPrograms = Map.empty, SODEs = Map.empty\<rparr>"
lens_put_irr1: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'f" assumes "X \<bowtie> Y" shows "get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>"
bigo_lesseq2: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(h)"   and "\<forall>x. \<bar>g x\<bar> \<le> f x" shows "g \<in> O(h)"
insert_before_def: shows "Shadow_DOM.insert_before \<equiv> l_insert_before\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_insert_before Shadow_DOM.get_parent Shadow_DOM.get_child_nodes Shadow_DOM.set_child_nodes get_ancestors_di Shadow_DOM.adopt_node set_disconnected_nodes get_disconnected_nodes Shadow_DOM.get_owner_document"
Object_neq_SXcpt: fixes xn :: "xname" shows "Object \<noteq> SXcpt xn"
has_fields_Object: fixes P :: "'b prog"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "FDTs = map (\<lambda>(F, b, T). ((F, Object), b, T)) fs" shows "P \<turnstile> Object has_fields FDTs"
fst_quot_of_fract_0_imp: fixes x :: "'b fract" assumes "fst (quot_of_fract x) = (0::'b)" shows "snd (quot_of_fract x) = (1::'b)"
simps: fixes \<phi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_letter_opt \<phi> \<nu> = Unf (step \<phi> \<nu>)"
simps_2: fixes x :: "'b"   and y :: "'b"   and xs :: "['b]" shows "init\<cdot>(x : y : xs) = x : init\<cdot>(y : xs)"
can_take_2: fixes t :: "'a transition_scheme"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "can_take_transition t i r = can_take (Arity t) (Guards t) i r"
1: shows "regular v \<and> regular (v * v\<^sup>T)"
l2_inv5E: fixes x :: "l2_state" assumes "x \<in> l2_inv5" obtains "\<And>A B M. Confid A B M \<in> chan x \<or> Secure A B M \<in> chan x \<Longrightarrow> M \<in> dy_fake_msg (bad x) (ik x) (chan x)"
is_sup_fun: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "is_sup f g (\<lambda>x. f x \<squnion> g x)"
linear_asymmetric: fixes x :: "'a" assumes "linear x" shows "asymmetric (- x)"
multiset_add_sub_el_shuffle: fixes c :: "'b"   and B :: "'b multiset"   and b :: "'b" assumes "c \<in># B"   and "b \<noteq> c" shows "b, B - (c, {#}) = (b, B) - (c, {#})"
ncols_Diagonal_to_Smith_row_i: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "ncols (Diagonal_to_Smith_row_i A i bezout) = ncols A"
finite_psubset_def: shows "finite_psubset = rel_ext (\<lambda>(A, B). A \<subset> B \<and> finite B)"
HMA_I_def: shows "Mod_Type_Connect.HMA_I = (\<lambda>i a. i = mod_type_class.to_nat a)"
p_nonneg: fixes K :: "'a \<Rightarrow> 'a pmf"   and x :: "'a"   and y :: "'a"   and n :: "nat" shows "0 \<le> MC_syntax.p K x y n"
sep_conj_ac_1: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "P ** Q = Q ** P"
poly_dvd_antisym: fixes p :: "'b poly"   and q :: "'b poly" assumes "lead_coeff p = lead_coeff q"   and "p dvd q"   and "q dvd p" shows "p = q"
assms_2: assumes "V \<inter> L \<noteq> {}" shows "\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> L \<noteq> {}"
assms_1: shows "c1 \<sqsubseteq> c2"
is_acc_class_def: fixes G :: "prog"   and P :: "pname"   and C :: "qtname" shows "is_acc_class G P C = (is_class G C \<and> G \<turnstile> Class C accessible_in P)"
that: assumes "P,t \<turnstile> \<sigma> -\<lbrace>\<rbrace>-jvm\<rightarrow> \<sigma>'"   and "\<tau>Move2 P \<sigma>" shows "thesis"
mono_neutral_cong_left: fixes T :: "'b set"   and S :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. h i = (1::'c)"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "prod g S = prod h T"
sorted_wrt_iff_nth_less: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "sorted_wrt P xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> P (xs ! i) (xs ! j))"
conjunctiveD: fixes x :: "'b \<Rightarrow> 'b"   and y :: "'b \<Rightarrow> 'b"   and z :: "'b \<Rightarrow> 'b" assumes "x \<in> Comp.conjunctive" shows "x \<circ> y \<sqinter> z = (x \<circ> y) \<sqinter> (x \<circ> z)"
id0: shows "{id} \<subseteq> {p. p permutes {0..<n}}"
NCSaux4: fixes p :: "('b, 'c) Combinators list"   and a :: "'b"   and b :: "'b"   and l :: "'b set list"   and c :: "'c"   and u :: "'b"   and v :: "'b"   and w :: "'c" assumes "noDenyAll p"   and "{a, b} \<in> set l"   and "all_in_list p l"   and "singleCombinators p"   and "FWNormalisationCore.sorted (AllowPortFromTo a b c # p) l"   and "{a, b} \<noteq> firstList p"   and "AllowPortFromTo u v w \<in> set p" shows "{a, b} \<noteq> {u, v}"
Gauss_Jordan_iarrays_PA_def: fixes A :: "'d iarray iarray" shows "Gauss_Jordan_iarrays_PA A = Gauss_Jordan_upt_k_iarrays_PA A (ncols_iarray A - 1)"
lm019: fixes X :: "'c set"   and y :: "'c" assumes "1 \<le> card X"   and "\<forall>x\<in>X. x < y" shows "Max X < y"
zero_in_float_intervalI: shows "0 \<in>\<^sub>r 0"
sep_conj_sep_emptyI: fixes P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "P h" shows "(P \<and>* \<box>) h"
infinite: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "infinite A" shows "sum g A = (0::'c)"
remdupsAppend: fixes a :: "'b list"   and b :: "'b list" shows "remdups (a @ b) = remdups (list_diff a b) @ remdups b"
det_identical_rows: fixes i :: "'a"   and j :: "'a"   and A :: "(('b, 'a) vec, 'a) vec" assumes "i \<noteq> j"   and "row i A = row j A" shows "det A = (0::'b)"
is_red_addsE: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_red F p" obtains f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" and v :: "'x \<Rightarrow>\<^sub>0 nat" where "f \<in> F" and "v \<in> keys p" and "f \<noteq> 0" and "local.punit.adds_term (lpp f) v"
intros_2: fixes i :: "nat"   and Ts :: "typ list" assumes "i < length Ts" shows "Ts \<turnstile>\<^sub>\<tau> Bv i : Ts ! i"
cPar2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and Q' :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<alpha>"   and "A\<^sub>P \<sharp>* Q'"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>P"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<alpha> \<prec> P \<parallel> Q'"
affine_imp_convex: fixes s :: "'c set" assumes "affine s" shows "convex s"
powers_8: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a + (b - c) = a + b - c"
breakdown: shows "w = w1 @ w2 \<and> LeftDerivation u (take n D) w1 \<and> derivation_ge (drop n D) (length w1) \<and> LeftDerivation v (derivation_shift (drop n D) (length w1) 0) w2"
wordinterval_Union: fixes ws :: "'b wordinterval list" shows "wordinterval_to_set (wordinterval_Union ws) = \<Union> (wordinterval_to_set ` set ws)"
integrableE: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" obtains r :: "real" and q :: "real" where "0 \<le> r" and "0 \<le> q" and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal r" and "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M = ennreal q" and "f \<in> borel_measurable M" and "integral\<^sup>L M f = r - q"
assms_2: shows "\<forall>j. no (\<sigma> j) \<le> no (\<sigma>' j)"
hcomplex_hcnj_mult: fixes w :: "complex star"   and z :: "complex star" shows "hcnj (w * z) = hcnj w * hcnj z"
simps_3: fixes f :: "'a \<Rightarrow> 'b"   and v :: "'a"   and vb :: "'a"   and vc :: "'a list" shows "mergesort f (v # vb # vc) = (let n = length (v # vb # vc) div 2; (l, r) = split_at n (v # vb # vc) in Common.merge f (mergesort f l) (mergesort f r))"
elims_2: fixes x :: "nat"   and xa :: "'b btree" assumes "root_order x xa" obtains k :: "nat" where "x = ka__" and "xa = Leaf"   | k :: "nat" and ts :: "('b btree \<times> 'b) list" and t :: "'b btree" where "x = k" and "xa = Node ts t" and "0 < length ts \<and> length ts \<le> 2 * k \<and> Ball (set (subtrees ts)) (order k) \<and> order k t"
drop_bit_take_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
axioms_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "arrow_of_spans C \<mu>"
rtrancl_eta_subst'': fixes s :: "dB"   and t :: "dB"   and u :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "u[s/i] \<rightarrow>\<^sub>\<eta>\<^sup>* u[t/i]"
mkCone_def: fixes e :: "'c" shows "D.mkCone e \<equiv> \<lambda>j. if J.arr j then if j = J.Zero then e else f0 \<cdot> e else C.null"
currentLevel_def: fixes t :: "('a \<times> bool) list" shows "currentLevel t = length (markedElements t)"
split_sel: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'d"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = ((bool = True \<longrightarrow> P f1) \<and> (bool = False \<longrightarrow> P f2))"
neg_def: fixes \<phi> :: "fm" shows "neg \<phi> = \<phi> IMP Fls"
assoc: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Dsemi prod" shows "prod (prod x y) z = prod x (prod y z)"
m_s_h: fixes X :: "char list set"   and S :: "'av st" assumes "finite X" shows "m\<^sub>s S X \<le> h * card X"
inSet_not_MT: fixes a :: "'b"   and p :: "'b list" assumes "a \<in> set p" shows "p \<noteq> []"
run_axioms_def: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" shows "run_axioms s\<^sub>0 lstep s \<equiv> df_system.is_run s\<^sub>0 lstep s"
finite_induct: fixes F :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
transfer: shows "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (*) (*)"
retraction_char: fixes g :: "'s" shows "retraction g = (arr g \<and> Img g = Cod g)"
\<F>: fixes S :: "'a set" assumes "S \<in> \<F>" shows "open S"
rel_mono_strong0: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3 :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and x :: "('d, 'f, 'h) F2"   and y :: "('e, 'g, 'i) F2"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R3a :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "F2rel R1 R2 R3 x y"   and "\<forall>z1\<in>F2set1 x. \<forall>y1\<in>F2set1 y. R1 z1 y1 \<longrightarrow> R1a z1 y1"   and "\<forall>z2\<in>F2set2 x. \<forall>y2\<in>F2set2 y. R2 z2 y2 \<longrightarrow> R2a z2 y2"   and "\<forall>z3\<in>F2set3 x. \<forall>y3\<in>F2set3 y. R3 z3 y3 \<longrightarrow> R3a z3 y3" shows "F2rel R1a R2a R3a x y"
lunit_naturality: fixes f :: "'a" assumes "arr f" shows "\<l>[cod f] \<cdot> (\<one> \<otimes> f) = f \<cdot> \<l>[local.dom f]"
sel_3: fixes x31 :: "'b local.term"   and x32 :: "'b local.term" shows "un_Tensor2 (x31 \<^bold>\<otimes> x32) = x32"
subspaceI: fixes S :: "'b set" assumes "(0::'b) \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c *s x \<in> S" shows "subspace S"
r1_def: shows "r1 = (\<lambda>(x, y). flow0 x y) ` ({flow0 x t1<--<b} \<times> {0<..<d})"
hfunction_def: fixes r :: "hf" shows "hfunction r = (\<forall>x y. \<langle>x, y\<rangle> \<^bold>\<in> r \<longrightarrow> (\<forall>y'. \<langle>x, y'\<rangle> \<^bold>\<in> r \<longrightarrow> y = y'))"
finite_has_maximal2: fixes A :: "'b word set"   and a :: "'b word" assumes "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. a \<le>s m \<and> (\<forall>b\<in>A. m \<le>s b \<longrightarrow> m = b)"
ASum_eqTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant"   and g :: "nat \<Rightarrow> ant" shows "(\<forall>j\<le>n. f j \<in> Z\<^sub>\<infinity>) \<and> (\<forall>j\<le>n. g j \<in> Z\<^sub>\<infinity>) \<and> (\<forall>j\<le>n. f j = g j) \<longrightarrow> ASum f n = ASum g n"
inject_2: fixes x21 :: "'b Trace"   and x22 :: "'b"   and y21 :: "'b Trace"   and y22 :: "'b" shows "(x21 \<leadsto> x22 = y21 \<leadsto> y22) = (x21 = y21 \<and> x22 = y22)"
le_imp_diff_is_add: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "a \<subseteq># b"   and "a \<subseteq># b" shows "(b - a = c) = (b = c + a)"
winding_number_homotopic_paths: fixes z :: "complex"   and g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex" assumes "homotopic_paths (- {z}) g h" shows "winding_number g z = winding_number h z"
assms_1: shows "\<forall>l\<in>labels \<Gamma>\<^sub>T\<^sub>O\<^sub>Y p. P l p"
idem_DT: fixes P :: "'a process" shows "P \<sqsubseteq>\<^sub>D\<^sub>T P"
endlink: assumes "bs \<noteq> vs" shows "\<exists>xss. flip_altsublist_chain (us # xss @ [b # bs])"
g: shows "g = h \<circ> (f \<circ> g)"
rmember_def: fixes A :: "'c set"   and x :: "'c" shows "Nunchaku.rmember A x = (x \<in> A)"
less_eq_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<le> ys) = subseq xs ys"
adds_term_alt: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_term pair_of_term u v = (\<exists>t. v = term_powerprod.splus pair_of_term term_of_pair t u)"
Cauchy_theorem_triangle: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex"   and b :: "complex"   and c :: "complex" assumes "f holomorphic_on convex hull {a, b, c}" shows "(f has_contour_integral 0) (linepath a b +++ linepath b c +++ linepath c a)"
set2list_def: fixes A :: "'a set" shows "set2list A \<equiv> fst (while (\<lambda>(Es, S). S \<noteq> {}) (\<lambda>(Es, S). let x = pick S in (x # Es, S - {x})) ([], A))"
pair_in_swap_image: fixes y :: "'a"   and x :: "'b"   and A :: "('b \<times> 'a) set" shows "((y, x) \<in> prod.swap ` A) = ((x, y) \<in> A)"
not_Some_eq2: fixes v :: "('a \<times> 'b) option" shows "(\<forall>x y. v \<noteq> Some (x, y)) = (v = None)"
hyps_3: assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F"   and "I \<noteq> {}" shows "((\<lambda>x. MIN i\<in>I. f i x) \<longlongrightarrow> Min (a ` I)) F"
hyp2_cltn2_compose: fixes H :: "cltn2"   and p :: "hyp2"   and J :: "cltn2" assumes "is_K2_isometry H" shows "hyp2_cltn2 (hyp2_cltn2 p H) J = hyp2_cltn2 p (cltn2_compose H J)"
assms: shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
monic_map_poly_hom: fixes p :: "int poly" assumes "monic p" shows "monic (of_int_poly p)"
can_Ide_self: fixes t :: "'a E.term" assumes "E.Ide t" shows "can t t = \<lbrace>t\<rbrace>"
of_int_minus: 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 z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- z) = uminus (ring_1.of_int one plus zero minus z)"
fds_converges_cmult_right: fixes f :: "'a fds"   and s :: "'a"   and c :: "'a" assumes "fds_converges f s" shows "fds_converges (f * fds_const c) s"
WTrtVal: fixes h :: "heap"   and v :: "val"   and T :: "ty"   and P :: "(char list list \<times> char list exp) prog"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option" assumes "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>" shows "P,E,h,sh \<turnstile> Val v : T"
ipresCons_unique: fixes f :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and fA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) model_scheme"   and ig :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and igA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "ipresCons f fA MOD"   and "ipresCons ig igA MOD" shows "(wls s X \<longrightarrow> f X = ig X) \<and> (wlsAbs (us, s') A \<longrightarrow> fA A = igA A)"
is_processT7_S: fixes s :: "'b event list"   and t :: "'b event list"   and P :: "'b process" assumes "tickFree s"   and "front_tickFree t"   and "s \<in> D P" shows "s @ t \<in> D P"
1: shows "ide f\<^sub>0gh\<^sub>1.p\<^sub>1"
integer_mod_group_def: fixes n :: "nat" shows "Z n \<equiv> if n = 0 then integer_group else \<lparr>carrier = {0..<int n}, monoid.mult = \<lambda>x y. (x + y) mod int n, one = 0\<rparr>"
assms: shows "get_S w1 = 0 \<and> w2 AND 128 = 0"
span_base: fixes a :: "'a poly"   and S :: "'a poly set" assumes "a \<in> S" shows "a \<in> poly_vs.span S"
tLength_0_conv: fixes t :: "'b Trace" shows "(tLength t = 0) = (\<exists>s. t = tInit s)"
elims: fixes x :: "('d, 'e) ctxt"   and xa :: "('d, 'e) Term.term"   and y :: "('d, 'e) Term.term" assumes "x\<langle>xa\<rangle> = y" obtains s :: "('d, 'e) Term.term" where "x = \<box>" and "xa = s__" and "y = s__"   | f :: "'d" and ss1 :: "('d, 'e) Term.term list" and C :: "('d, 'e) ctxt" and ss2 :: "('d, 'e) Term.term list" and s :: "('d, 'e) Term.term" where "x = More f ss1 C ss2" and "xa = s" and "y = Fun f (ss1 @ C\<langle>s\<rangle> # ss2)"
usolution_def: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "usolution A m n x = (solution2 A m n x \<and> (\<forall>y. solution2 A m n y \<longrightarrow> (\<forall>j<m. y j = x j)))"
assoc: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" shows "a \<inter># b \<inter># c = a \<inter># (b \<inter># c)"
NTCompDefinedE: fixes \<eta>1 :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and \<eta>2 :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" obtains "NatTrans \<eta>1" and "NatTrans \<eta>2" and "NTCatDom \<eta>2 = NTCatDom \<eta>1" and "NTCatCod \<eta>2 = NTCatCod \<eta>1" and "NTCod \<eta>1 = NTDom \<eta>2"
divergences_def: fixes P :: "'a process" shows "divergences P \<equiv> snd (Rep_process P)"
sparsegridE_2: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat" assumes "p \<in> sparsegrid dm lm" shows "level p < lm"
fun_range_witness: fixes f :: "ZF"   and y :: "ZF" assumes "isFun f"   and "Elem y (HOLZF.Range f)" shows "\<exists>x. Elem x (HOLZF.Domain f) \<and> f \<acute> x = y"
pdevs_val_sum_less_degree: fixes X :: "'a pdevs"   and d :: "nat"   and e :: "nat \<Rightarrow> real" assumes "degree X \<le> d" shows "pdevs_val e X = (\<Sum>i<d. e i *\<^sub>R pdevs_apply X i)"
compact_eq_bounded_closed: fixes S :: "'a set" shows "compact S = (bounded S \<and> closed S)"
orbit_conv_funpow_dist1: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "x \<in> orbit f x" shows "orbit f x = (\<lambda>n. (f ^^ n) x) ` {0..<funpow_dist1 f x x}"
rk_ABPa: fixes A :: "Points"   and B :: "Points"   and P :: "Points"   and A' :: "Points"   and Q :: "Points"   and a :: "Points"   and C :: "Points"   and B' :: "Points"   and C' :: "Points"   and R :: "Points" assumes "rk {A, B, P} = 3"   and "rk {A, A'} = 2"   and "rk {A, A', P} = 2"   and "rk {Q, A', a} = 2"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {A, B, C, A', B', C', P} = 3"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {A', B', P, Q} = 4"   and "rk {R, A, a} = 2" shows "4 \<le> rk {A, B, P, a}"
Ax_in_SNa: fixes x :: "name"   and a :: "coname" shows "SNa (Ax x a)"