hcju/isabelleps huggingface dataset:
cindex_polyE_cross: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b" shows "cindex_polyE a b 1 p = real_of_int (cross_alt 1 p a b) / 2"
cindex_eq_cindexE_divide: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "a < b"   and "finite {x. (f x = 0 \<or> g x = 0) \<and> a \<le> x \<and> x \<le> b}"   and "\<forall>x\<in>{a..b}. g x = 0 \<longrightarrow> f x \<noteq> 0"   and "continuous_on {a..b} f"   and "continuous_on {a..b} g" shows "cindexE a b (\<lambda>x. f x / g x) = jumpF (\<lambda>x. f x / g x) (at_right a) + real_of_int (cindex a b (\<lambda>x. f x / g x)) - jumpF (\<lambda>x. f x / g x) (at_left b)"
tranclD: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R \<and> (z, y) \<in> R\<^sup>*"
IH: assumes "term_ok \<Theta> a" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq a b"
cnj_injective_iff: fixes A :: "fmla"   and B :: "fmla"   and A' :: "fmla"   and B' :: "fmla" shows "(A AND B = A' AND B') = (A = A' \<and> B = B')"
indep_var_def: fixes Ma :: "'d measure"   and A :: "'a \<Rightarrow> 'd"   and Mb :: "'d measure"   and B :: "'a \<Rightarrow> 'd" shows "indep_var Ma A Mb B = indep_vars (case_bool Ma Mb) (case_bool A B) UNIV"
vec_hom_smult: fixes ev :: "real"   and v :: "real Matrix.vec" shows "of_real_hom.vec_hom (ev \<cdot>\<^sub>v v) = of_real ev \<cdot>\<^sub>v of_real_hom.vec_hom v"
less_imp_neq: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x \<prec> y" shows "x \<noteq> y"
select_convs_2: fixes carrier :: "'d set"   and more :: "'c" shows "carrier.more \<lparr>carrier = carrier, \<dots> = more\<rparr> = more"
notno: shows "OF_priority_match OF_match_fields_safe oft p \<noteq> NoAction"
topspace_subbase: fixes B :: "'c set \<Rightarrow> bool"   and U :: "'c set" shows "topspace (topology (arbitrary union_of (finite intersection_of B relative_to U))) = U"
assms: shows "mopup_jump_over1 (Suc (2 * n), l, Oc # xs) lm n ires"
assms_12: shows "extendRule S r = (Ps,  \<Gamma> \<oplus> F \<nabla> [x].A \<Rightarrow>* \<Delta>)"
cos_periodic_int: fixes x :: "real"   and i :: "int" shows "cos (x + real_of_int i * (2 * pi)) = cos x"
phl_cons1: fixes x :: "'a"   and w :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> w"   and "w ; y \<le> y ; z" shows "x ; y \<le> y ; z"
length_greater_0_conv: fixes xs :: "'b list" shows "order.greater (length xs) 0 = (xs \<noteq> [])"
rep_eq: fixes x :: "'k"   and xa :: "'k multiset" shows "count (add_mset x xa) = (\<lambda>b. if b = x then Suc (count xa b) else count xa b)"
irreducible_altdef: fixes x :: "'a" shows "irreducible x = (x \<noteq> (0::'a) \<and> \<not> is_unit x \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> is_unit b))"
fres_top_top: shows "\<top> \<leftarrow> \<top> = \<top>"
prod_mset_delta: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and y :: "'c"   and c :: "'b"   and A :: "'c multiset" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one {#if x = y then c else one. x \<in># A#} = power.power one times c (count A y)"
next_plane0_incr: fixes Q :: "graph \<Rightarrow> graph \<Rightarrow> bool"   and P :: "graph \<Rightarrow> graph \<Rightarrow> bool"   and g :: "graph"   and g' :: "graph"   and p :: "nat" assumes "\<And>x y z. \<lbrakk>Q x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>f g. \<lbrakk>f \<in> \<F> g; \<not> final f\<rbrakk> \<Longrightarrow> P g (makeFaceFinal f g)"   and "\<And>g u v f vs. pre_splitFace g u v f vs \<Longrightarrow> Q g (snd (snd (splitFace g u v f vs)))"   and "minGraphProps g"   and "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)" shows "P g g'"
valid_prio: fixes q :: "dir list"   and dq :: "dir"   and p :: "dir list"   and dp :: "dir" assumes "prio (q @ [dq]) < prio (p @ [dp])" shows "take (length p) q \<noteq> p"
inf_complement: fixes x :: "'a" shows "- x \<sqinter> - - x = bot"
of_rat_less_0_iff: fixes r :: "rat" shows "(of_rat r < (0::'a)) = (r < 0)"
IH: fixes s :: "'a rexp" shows "fold (\<lambda>a r. pnorm (deriv a r)) w s = flatten PLUS (fold pderiv_set w {s})"
backflows_wf: fixes N :: "'a set"   and E :: "('a \<times> 'a) set" assumes "wf_graph \<lparr>nodes = N, edges = E\<rparr>" shows "wf_graph \<lparr>nodes = N, edges = backflows E\<rparr>"
W_2: shows "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
append_prefixD: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "prefix (xs @ ys) zs" shows "prefix xs zs"
Ord_omega_sum: fixes ms :: "nat list" shows "Ord (omega_sum ms)"
A2: shows "\<forall>a b p q r s. a b \<equiv> p q \<and> a b \<equiv> r s \<longrightarrow> p q \<equiv> r s"
one_smult: fixes m :: "'v" assumes "m \<in> V" shows "(1::'f) \<cdot> m = m"
vemptyE: fixes A :: "V" assumes "A \<noteq> []\<^sub>\<circ>" obtains x :: "V" where "x \<in>\<^sub>\<circ> A"
UINF_mem_Continuous_closed: fixes H :: "'d upred \<Rightarrow> 'd upred"   and A :: "'e set"   and P :: "'e \<Rightarrow> 'd upred" assumes "Continuous H"   and "\<And>i. i \<in> A \<Longrightarrow> P i is H"   and "\<not> A = {}" shows "(\<Sqinter> i \<in> A \<bullet> P i) is H"
assms: fixes x :: "'a" assumes "x \<in> S" shows "f x = g x"
wf\<^sub>l\<^sub>s\<^sub>t\<^sub>s_def: fixes \<S> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set" shows "wf\<^sub>l\<^sub>s\<^sub>t\<^sub>s \<S> \<equiv> (\<forall>\<A>\<in>\<S>. wf\<^sub>l\<^sub>s\<^sub>t {} \<A>) \<and> (\<forall>\<A>\<in>\<S>. \<forall>\<A>'\<in>\<S>. fv\<^sub>l\<^sub>s\<^sub>t \<A> \<inter> bvars\<^sub>l\<^sub>s\<^sub>t \<A>' = {})"
cong_ex_bigtheta: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f2 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> o[F](g1)) = (f2 \<in> o[F](g2))"
Refl_under_underS: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "Refl r"   and "a \<in> Field r" shows "under r a = underS r a \<union> {a}"
E_of_succ_def: fixes succ :: "'c \<Rightarrow> 'd list" shows "E_of_succ succ \<equiv> {(u, v). v \<in> set (succ u)}"
axioms_1: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah" assumes "isomorphic_categories A B" shows "category A"
ChiRel_SA_OneAncestor: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto"   and U :: "'a" assumes "(S, T) \<in> ChiRel A"   and "T \<in> States SA"   and "U \<in> States SA"   and "SA \<in> SAs A" shows "(S, U) \<in> ChiRel A"
intros_2: fixes u :: "'c \<Rightarrow> 'b"   and c :: "'c"   and d :: "'b" assumes "- u c \<le> d" shows "dbm_entry_val u None (Some c) (Le d)"
lm137: fixes f :: "('d \<times> 'c) set"   and X :: "'d set" shows "MiscTools.sum' f X = MiscTools.sum' f X"
f: fixes x :: "real" assumes "0 < x"   and "x < a" shows "(f has_real_derivative f' x) (at x)"
edomIff: fixes a :: "'d"   and m :: "'d \<Rightarrow> 'e" shows "(a \<in> edom m) = (m a \<noteq> \<bottom>)"
distinct_1: fixes x2 :: "8 word" shows "ProtoAny \<noteq> Proto x2"
\<Phi>_nn: fixes t :: "'a tree" shows "0 \<le> \<Phi> t"
subseq_same_length: fixes xs :: "'b list"   and ys :: "'b list" assumes "subseq xs ys"   and "length xs = length ys" shows "xs = ys"
divide_inverse: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a b = times a (inverse b)"
InvariantVarsMAfterBackjump: fixes M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list"   and Vbl :: "nat set"   and p :: "(Literal \<times> bool) list"   and bLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantVarsM M F0 Vbl"   and "isPrefix p M"   and "var bLiteral \<in> vars F0 \<union> Vbl"   and "M' = p @ [(bLiteral, False)]" shows "InvariantVarsM M' F0 Vbl"
dominant_term_ms_aux_def: fixes xs :: "('a \<times> real) msllist" shows "dominant_term_ms_aux xs = (case xs of MSLNil \<Rightarrow> (0, replicate (Suc (expansion_level TYPE('a))) 0) | MSLCons (C, e) x \<Rightarrow> case dominant_term C of (c, es) \<Rightarrow> (c, e # es))"
snd_def: fixes prod :: "'c \<times> 'b" shows "snd prod = (case prod of (x1, x2) \<Rightarrow> x2)"
UseCertVerify: fixes KA :: "nat"   and nb :: "msg"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list"   and A :: "agent" assumes "Crypt (invKey KA) (Hash \<lbrace>nb, Agent B, Nonce PMS\<rbrace>) \<in> parts (knows Spy evs)"   and "certificate A KA \<in> parts (knows Spy evs)"   and "evs \<in> tls"   and "A \<notin> bad" shows "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"
inclusion: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.arr (fst f) \<and> B.VV.arr (snd f) \<and> src\<^sub>B (fst f) = trg\<^sub>B (fst (snd f))" shows "B.VxVxV.arr f"
CastElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and d :: "Class" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (checkcast d)" shows "\<exists>a Flds ops. h\<down>a = Some (d, Flds) \<and> os = RVal (Loc a) # ops \<and> n = Suc 0 \<and> t = (RVal (Loc a) # ops, S, h) \<and> ll = l + 1"
Collect_conj_eq: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "{x. P x \<and> Q x} = {x. P x} \<inter> {x. Q x}"
rel_spmf_scaleI: fixes r :: "real"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c spmf"   and q :: "'d spmf" assumes "0 < r \<Longrightarrow> rel_spmf A p q" shows "rel_spmf A (scale_spmf r p) (scale_spmf r q)"
additive_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "Transformers.additive t \<equiv> \<forall>P Q. sound P \<and> sound Q \<longrightarrow> t (\<lambda>s. P s + Q s) = (\<lambda>s. t P s + t Q s)"
tangent_spaceI: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "extensional0 manifold_eucl.dest.diff_fun_space X"   and "linear_fun_on manifold_eucl.dest.diff_fun_space X"   and "\<And>f g. \<lbrakk>f \<in> manifold_eucl.dest.diff_fun_space; g \<in> manifold_eucl.dest.diff_fun_space\<rbrakk> \<Longrightarrow> X (f * g) = f p * X g + g p * X f" shows "X \<in> manifold_eucl.dest.tangent_space p"
nlea__lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "\<not> A B C LeA D E F"   and "A \<noteq> B"   and "B \<noteq> C"   and "D \<noteq> E"   and "E \<noteq> F" shows "D E F LtA A B C"
Mapping_update_keys: fixes a :: "'a"   and b :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.update a b m) = Mapping.keys m \<union> {a}"
a: shows "Esigma (a * m) = Esigma a * Esigma m"
nonzero_def: fixes R :: "('c, 'd) ring_scheme" shows "nonzero R = {a \<in> carrier R. a \<noteq> \<zero>\<^bsub>R\<^esub>}"
GiSi': shows "\<GG> ! i \<subseteq> \<GG> ! (i + 1)"
wqo_on_imp_almost_full_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "wqo_on P A" shows "almost_full_on P A"
hyps_2: shows "b \<rightarrow>\<^sub>\<beta> c"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "A.dom (A.map f) = A.map (A.dom f)"
pos_minus_divideC_le_eq: fixes c :: "complex"   and b :: "'a"   and a :: "'a" assumes "0 < c" shows "(- (b /\<^sub>C c) \<le> a) = (- b \<le> c *\<^sub>C a)"
gen_ideal_def: fixes R :: "'b set"   and addition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and multiplication :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and unit :: "'b"   and x :: "'b" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.gen_ideal R multiplication x \<equiv> {y. \<exists>r\<in>R. y = multiplication r x}"
lemma2: fixes t :: "lam"   and u :: "lam" assumes "t \<rightarrow>\<^sub>s u" shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* u"
order_unique_lemma: fixes a :: "'c"   and n :: "nat"   and p :: "'c poly" assumes "[:- a, 1::'c:] ^ n dvd p"   and "\<not> [:- a, 1::'c:] ^ Suc n dvd p" shows "order a p = n"
set_size_axioms: shows "set_size rs.\<alpha> rs.invar rs.size"
dl_def: fixes r :: "('b \<times> 'b) set"   and \<sigma> :: "'b list" shows "r \<down>l \<sigma> = r \<down>s set \<sigma>"
assms: fixes x :: "'a"   and xs :: "'a list"   and a :: "'c \<times> 'd" shows "fst (h x xs a) = fst a"
seq_valid_def: fixes S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "seq_valid S \<equiv> \<forall>I. is_interp I \<longrightarrow> seq_sem I S = UNIV"
pin: shows "\<forall>P\<in>#{#mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2}. b2 \<in># filter_mset ((\<subseteq>) p) (remove1_mset blv \<B>)#}. count P p = 1"
assert_bound_nolhs_index_valid: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "index_valid as s" shows "index_valid (insert a as) (assert_bound a s)"
qt: shows "lookup (tail q) v \<noteq> (0::'b)"
parallelD2: fixes x :: "'a list"   and y :: "'a list" assumes "x \<parallel> y" shows "\<not> prefix y x"
simps_2: fixes k :: "'a"   and v :: "'d"   and k' :: "'a"   and v' :: "'d"   and xs :: "('a \<times> 'd) list" shows "update_by_pair (k, v) ((k', v') # xs) = (case comp k k' of Eq \<Rightarrow> if v = (0::'d) then xs else (k, v) # xs | Lt \<Rightarrow> if v = (0::'d) then (k', v') # xs else (k, v) # (k', v') # xs | Gt \<Rightarrow> (k', v') # update_by_pair (k, v) xs)"
is_shadow_root_ptr_kind_cast: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "is_shadow_root_ptr_kind (cast shadow_root_ptr)"
lass_val_of_red: fixes e :: "char list exp"   and a :: "char list \<times> val"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "lass_val_of e = \<lfloor>a\<rfloor>"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>" shows "e' = unit \<and> h' = h \<and> l' = l(fst a \<mapsto> snd a) \<and> sh' = sh \<and> b = b'"
\<R>_sym: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" shows "sym (\<R> \<Gamma> \<S> P)"
neg_andalso: fixes x :: "tr"   and y :: "tr" shows "neg\<cdot>(x andalso y) = (neg\<cdot>x orelse neg\<cdot>y)"
P4: shows "A' B' Perp Q X' \<and> A' B' OS P Q"
tc_eq: fixes x :: "'a" assumes "x ; x = x" shows "x\<^sup>\<star> ; x = x"
REC_le_RECT: fixes body :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c" shows "REC body x \<le> REC\<^sub>T body x"
emp_star: fixes p :: "assertion" shows "Emp \<star> p = p"
poly_mult_var: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])" shows "p \<otimes>\<^bsub>K [X]\<^esub> X = (if p = [] then [] else p @ [\<zero>])"
same_length_different: fixes xs :: "'g list"   and ys :: "'g list" assumes "xs \<noteq> ys"   and "length xs = length ys" shows "\<exists>pre x xs' y ys'. x \<noteq> y \<and> xs = pre @ [x] @ xs' \<and> ys = pre @ [y] @ ys'"
children_eq_h2: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "h2 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h3 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
Diff_Int_distrib2: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "(A - B) \<inter> C = A \<inter> C - B \<inter> C"
bound: shows "\<forall>\<^sub>F t in at_top. norm (f t) \<le> M * exp (c * t)"
dma_def: fixes m :: "Eg1.var \<Rightarrow> nat"   and x :: "Eg1.var" shows "dma m x \<equiv> if x = buffer then dma_control_var (m control_var) else if x = high_var then High else Low"
chain: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat" assumes "kuhn_simplex p n base upd s"   and "a \<in> s"   and "b \<in> s" shows "a \<le> b \<or> b \<le> a"
simps: fixes p :: "int poly"   and l :: "rat"   and r :: "rat" shows "local.root_pos_1 (p, l, r) = select_correct_factor_int_poly (local.tighten_bound_root (compare_1_rat (p, l, r))) (\<lambda>x. x) (local.initial_lower_bound l, local.initial_upper_bound r) (poly_nth_root n p)"
\<tau>: shows "\<tau> \<in> ptrm (Suc 0)"
snv: fixes \<sigma> :: "'tp" shows "infTp \<sigma> \<or> (\<forall>x\<in>nvT T. tpOfV x \<noteq> \<sigma>)"
supp_fun_def: fixes f :: "'a \<Rightarrow> 'b" shows "supp_fun f = {x. f x \<noteq> (0::'b)}"
cases: fixes x :: "'a \<times> 'b \<times> 'c \<times> 'a" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'a" where "x = (a, b, c, d)"
leq1: fixes i :: "nat" assumes "i < n" shows "index i \<le> k1"
finprod_cong': fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'a" assumes "A = B"   and "g \<in> B \<rightarrow> carrier R"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum R f A = finsum R g B"
assms_2: fixes x' :: "'a"   and y' :: "'a" assumes "x' \<in> set ys"   and "y' \<in> set ys"   and "rel x' y'" shows "x' = y'"
update_convs_2: fixes Map' :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e" shows "Map_update Map' \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = \<lparr>Pol = Pol, Map = Map' Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr>"
finite_Collect_subsets: fixes A :: "'b set" assumes "finite A" shows "finite {B. B \<subseteq> A}"
Gromov_converging_at_boundary_imp_unbounded: fixes u :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "Gromov_converging_at_boundary u" shows "(\<lambda>x. ereal (dist a (u x))) \<longlonglongrightarrow> \<infinity>"
d_meet_closed: fixes x :: "'a"   and y :: "'a" shows "d (d x \<sqinter> d y) = d x \<sqinter> d y"
subprob_spaceI: fixes M :: "'a measure" assumes "emeasure M (space M) \<le> 1"   and "space M \<noteq> {}" shows "subprob_space M"
Cons_1: fixes c :: "('a, 'b) conf" assumes "se_star c ls c'" shows "states c' = {\<sigma>'. \<exists>\<sigma>\<in>states c. ce_star \<sigma> ls \<sigma>'}"
entails_trans: fixes N1 :: "'f set"   and N2 :: "'f set"   and N3 :: "'f set" assumes "N1 \<Turnstile> N2"   and "N2 \<Turnstile> N3" shows "N1 \<Turnstile> N3"
to_nat_suc: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep x + 1 < CARD('a)" shows "mod_type.to_nat Rep (plus x one) = mod_type.to_nat Rep x + 1"
R_mult_right_mono: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a \<le> b"   and "0 \<le> c" shows "a * c \<le> b * c"
smaller_lits_are_false_def: fixes t :: "'a trm"   and C :: "'a literal set"   and S :: "'a eclause set" shows "smaller_lits_are_false t C S = (\<forall>L u v. (L \<in> C \<longrightarrow> orient_lit L u v pos \<longrightarrow> (u, t) \<in> trm_ord \<longrightarrow> (v, t) \<in> trm_ord \<longrightarrow> trm_rep u S \<noteq> trm_rep v S) \<and> (L \<in> C \<longrightarrow> orient_lit L u v neg \<longrightarrow> (u, t) \<in> trm_ord \<longrightarrow> (v, t) \<in> trm_ord \<longrightarrow> trm_rep u S = trm_rep v S))"
1: shows "eval_dioph ks xs mod int n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod int n"
effect__strips_iii_a: fixes s' :: "'c \<Rightarrow> bool option"   and s :: "'c \<Rightarrow> bool option"   and op :: "'c strips_operator"   and v :: "'c" assumes "s' = s \<then> op"   and "v \<in> set (add_effects_of op)" shows "s' v = Some True"
cball_translation: fixes a :: "'a"   and c :: "'a"   and r :: "real" shows "cball (a + c) r = (+) a ` cball c r"
exponential_order_additiveI: fixes M :: "real"   and f :: "real \<Rightarrow> 'a"   and K :: "real"   and c :: "real" assumes "0 < M"   and "\<forall>\<^sub>F t in at_top. norm (f t) \<le> K + M * exp (c * t)"   and "0 \<le> c" obtains M' :: "real" where "exponential_order M' c f"
Mul_Append: fixes n :: "nat" shows "\<turnstile> Mul_Append n  \<lbrace>\<acute>Mul_Proper n\<rbrace>"
fold_simps_2: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list"   and s :: "'d" shows "fold f (x # xs) s = fold f xs (f x s)"
le_funD: fixes f :: "'f \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'g"   and x :: "'f" assumes "f \<le> g" shows "f x \<le> g x"
sorted_map_same: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c list \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f (filter (\<lambda>x. f x = g xs) xs))"
SigmaD1: fixes a :: "'a"   and b :: "'b"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "(a, b) \<in> Sigma A B" shows "a \<in> A"
adhs_def: fixes T :: "'b set set"   and x :: "'b"   and A :: "'b set" shows "(x adh\<^bsub>T\<^esub> A) = (\<forall>U\<in>nhds\<^bsub>T\<^esub> x. U \<inter> A \<noteq> {})"
ext_cont_def: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" shows "ext_cont f a b = (\<lambda>x. f (clamp a b x))"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<exists>!n. \<forall>f. C n f = n \<and> C f n = n" shows "partial_magma C"
part_transitivity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "P x y"   and "P y z" shows "P x z"
continuous_map_product_projection: fixes k :: "'a"   and I :: "'a set"   and X :: "'a \<Rightarrow> 'b topology" assumes "k \<in> I" shows "continuous_map (product_topology X I) (X k) (\<lambda>x. x k)"
that: fixes a :: "'a" assumes "homotopic_with_canon P S S (\<lambda>x. x) (\<lambda>x. a)" shows "thesis"
cases: fixes p :: "'a \<times> 'b"   and a :: "'a" assumes "fstsp p a" obtains "a = fst p"
new_subc_put: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and t :: "'a FOL_Fitting.term"   and m :: "nat" assumes "new_term c s" shows "subc c s (p[t/m]) = subc c s (p[subc_term c s t/m])"
subspace_refl: fixes V :: "'b set" assumes "vectorspace V" shows "Subspace.subspace V V"
ipurge_tr_rev_aux_first: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list"   and x :: "'a"   and ws :: "'a list" assumes "ipurge_tr_rev_aux I D U xs = x # ws" shows "\<exists>ys zs. xs = ys @ x # zs \<and> ipurge_tr_rev_aux I D (sources_aux I D U (x # zs)) ys = [] \<and> (\<exists>v\<in>sources_aux I D U zs. (D x, v) \<in> I)"
in_aux0_le_\<tau>: fixes t :: "nat"   and X :: "'a option list set" assumes "(t, X) \<in> set aux0" shows "t \<le> \<tau> \<sigma> ne"
AF_Suc: fixes k :: "nat"   and i :: "nat" shows "AF k i \<lless> AF k (Suc i)"
path_emptyI: fixes step :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" shows "LTS.path step s [] s"