hcju/isabelleps huggingface dataset:
ring_hom_trans: fixes f :: "'d \<Rightarrow> 'e"   and R :: "('d, 'f) ring_scheme"   and S :: "('e, 'g) ring_scheme"   and g :: "'e \<Rightarrow> 'h"   and T :: "('h, 'i) ring_scheme" assumes "f \<in> ring_hom R S"   and "g \<in> ring_hom S T" shows "g \<circ> f \<in> ring_hom R T"
disjSD2aux: fixes a :: "('a set set, 'b) Combinators"   and b :: "('a set set, 'c) Combinators" assumes "disjSD_2 a b"   and "\<not> member DenyAll a"   and "\<not> member DenyAll b" shows "disjSD_2 (DenyAllFromTo (first_srcNet a) (first_destNet a) \<oplus> DenyAllFromTo (first_destNet a) (first_srcNet a) \<oplus> a) b"
closure_act_1b: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>\<^bold>\<A>\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
xy: shows "\<forall>w. P w (x \<otimes> y) = (P w x \<and> P w y)"
empty_Union_conv: fixes A :: "'a set set" shows "({} = \<Union> A) = (\<forall>x\<in>A. x = {})"
set_map_filter: fixes P :: "'b \<Rightarrow> 'a option"   and xs :: "'b list" shows "set (List.map_filter P xs) = the ` (P ` set xs - {None})"
acc_induct: fixes a :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> bool" assumes "a \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>x \<in> Wellfounded.acc r; \<forall>y. (y, x) \<in> r \<longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
abs_update_def: fixes k :: "'a"   and v :: "'b"   and l :: "('a \<times> 'b) list list" shows "abs_update k v l = l[bounded_hashcode_nat (length l) k := fst (ls_update k v (l ! bounded_hashcode_nat (length l) k))]"
INT_iff: fixes b :: "'k"   and B :: "'l \<Rightarrow> 'k set"   and A :: "'l set" shows "(b \<in> \<Inter> (B ` A)) = (\<forall>x\<in>A. b \<in> B x)"
seqp_assignTE: fixes \<xi> :: "'e"   and l :: "'h"   and u :: "'e \<Rightarrow> 'e"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}\<lbrakk>u\<rbrakk> p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "a = \<tau>" and "\<xi>' = u \<xi>" and "q = p"
dualization_unl_3: fixes A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c"   and w :: "nat \<Rightarrow> 'b" shows "(\<not> (\<box>[A]_v) w) = (\<diamond>\<langle>\<not> A\<rangle>_v) w"
conv_radius_mult_power: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "conv_radius (\<lambda>n. c ^ n * f n) = conv_radius f / ereal (norm c)"
uconnected_def: fixes E :: "'a uprod set" shows "uconnected E \<equiv> {(u, v). \<exists>p. epath E u p v}"
irreducible_altdef: fixes x :: "'b" shows "irreducible x = (x \<noteq> (0::'b) \<and> \<not> x dvd (1::'b) \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> b dvd (1::'b)))"
NextRecvCapD_1: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and t :: "'loc \<times> 't" assumes "NextRecvCap' c0 c1 p t" shows "cri.next_recvcap' (exchange_config c0) (exchange_config c1) p t"
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes a :: "('d, 'c) stateful_strand_step"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) = bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
le_utrans_equiv_rsp: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and v :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_utrans t u" shows "le_utrans t v = le_utrans u v"
assms_2: assumes "xs = []\<^sub>\<circ>" shows "P"
4: shows "integral Dy_pair (partial_vector_derivative (\<lambda>a. F a \<bullet> i) j) = integral (cbox a b) (\<lambda>x. integral (cbox (g2 x) (g1 x)) (\<lambda>y. partial_vector_derivative (\<lambda>a. F a \<bullet> i) j (x, y)))"
iso_runit': fixes f :: "'a" assumes "B.ide f" shows "B.iso \<r>\<^sup>-\<^sup>1[f]"
cancel_comm_monoid_add_ow_axioms: shows "cancel_comm_monoid_add_ow U (+\<^sub>o\<^sub>w) (-\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w"
Gramian_determinant_ge1: fixes k :: "nat" assumes "k \<le> length fs" shows "(1::'a) \<le> d k"
graph_plan_lemma_9: fixes vs :: "'d set"   and s :: "('d, 'e) fmap"   and as :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) list" shows "fmrestrict_set vs (exec_plan s (as_proj as vs)) = exec_plan (fmrestrict_set vs s) (as_proj as vs)"
sign_extend_def: fixes n :: "nat"   and w :: "'b word" shows "sign_extend n w \<equiv> if w !! n then w OR NOT (mask n) else w AND mask n"
monad_reader_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind" shows "monad_reader (return_env return) (bind_env bind) ask_env"
bounded_by_def: fixes b :: "real"   and P :: "'a \<Rightarrow> real" shows "bounded_by b P \<equiv> \<forall>x. P x \<le> b"
a1: shows "x \<le> word_of_int (uint (word_of_int (2 ^ LENGTH('b) - 1)))"
S_I: fixes l :: "'s"   and u :: "'c \<Rightarrow> real" assumes "l \<in> L"   and "u \<in> V"   and "u \<turnstile> PTA.inv_of A l" shows "(l, u) \<in> S"
assms_6: fixes x :: "real" assumes "a \<le> x" shows "0 \<le> g' x"
assms: fixes x :: "nat" shows "(x = n) = (s x = n)"
has_bochner_integral_eq_AE: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and g :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "has_bochner_integral M f x"   and "has_bochner_integral M g y"   and "AE x in M. f x = g x" shows "x = y"
is_cupcake_nsE: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and ns :: "(char list, char list, v) namespace" assumes "cakeml_static_env static_cenv"   and "cakeml_static_env.is_cupcake_ns static_cenv ns" obtains v :: "(char list \<times> v) list" where "ns = Bind v []" and "list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) v"
cut_less_mem_iff: fixes x :: "'b"   and I :: "'b set"   and t :: "'b" shows "(x \<in> I \<down>< t) = (x \<in> I \<and> x < t)"
rawpsubst_fls: fixes txs :: "('trm \<times> 'var) list" assumes "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubst fls txs = fls"
elims_2: fixes x :: "ty" assumes "is_Class x" obtains C :: "String.literal" where "x = Class C"
sectionI: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.ide C (C g f)" shows "category.section C f"
prj1_ext: fixes f :: "'a"   and g :: "'a" assumes "\<not> cospan f g" shows "\<p>\<^sub>1[f, g] = null"
ccExp_Let: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "ccHeap \<Gamma> e\<cdot>a G|` (- domA \<Gamma>) \<sqsubseteq> \<G>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e)"
minus_sum: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "\<ominus> (a \<oplus> b) = \<ominus> a \<oplus> \<ominus> b"
F_T: fixes s :: "'c event list"   and X :: "'c event set"   and P :: "'c process" assumes "(s, X) \<in> F P" shows "s \<in> T P"
lborelD: fixes A :: "'a set" assumes "A \<in> sets borel" shows "A \<in> sets lborel"
in_vimage_algebra: fixes A :: "'c set"   and M :: "'c measure"   and f :: "'d \<Rightarrow> 'c"   and X :: "'d set" assumes "A \<in> sets M" shows "f -` A \<inter> X \<in> sets (vimage_algebra X f M)"
infinite: fixes A :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "infinite A" shows "sum g A = (0::'a)"
map_simp_ide: fixes a :: "'a" assumes "\<tau>.A.ide a" shows "\<mu>o\<nu>_\<sigma>o\<tau>.map a = D ((\<mu> \<circ> \<nu>) a) ((\<sigma> \<circ> \<tau>) a)"
frag_cmul_minus_one: fixes x :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_cmul (- 1) x = - x"
dom_reachable: fixes G :: "nat graph_rec"   and n :: "nat"   and d :: "nat" assumes "cfg_doms G"   and "cfg_doms.reachable G n"   and "cfg_doms.dominate G d n" shows "cfg_doms.reachable G d"
radj_ladj_dual: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'d" shows "radj f x = \<partial> (ladj (\<partial>\<^sub>F f) (\<partial> x))"
QTrans: shows "Q \<longmapsto> a<x> \<prec> Q'"
targets: shows "target (w || map fst tr) qA = pA \<and> target (w || map snd tr) qB = pB"
pairwise_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> R x y)"
integrable_cos_cx: fixes c :: "real" shows "integrable (lebesgue_on {- pi..pi}) (\<lambda>x. cos (x * c))"
cnf_lemma: fixes x :: "ordinal" assumes "0 < x" shows "x - \<omega> ** oLog \<omega> x < x"
inv\<^sub>2E_1: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "inv\<^sub>2 T A j" shows "lb T A j"
new_alive_alloc: fixes s :: "Store"   and t :: "New" shows "alive (new s t) (s\<langle>t\<rangle>)"
return_bind: fixes x :: "'a multiset"   and f :: "'a multiset \<Rightarrow> 'm" shows "bind (return x) f = f x"
times_commute_eqpoll: fixes A :: "'c set"   and B :: "'d set" shows "A \<times> B \<approx> B \<times> A"
PiF_def: fixes I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" shows "Pi\<^sub>F I M \<equiv> sigma (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j)) {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
cong: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and h :: "'b \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "sum g A = sum h B"
rotate2_rotate2: fixes x :: "(real, 2) vec" shows "rotate2 (rotate2 x) = - x"
Class_Union: shows "\<Union> (left.transformation.orbit.Class ` G) = G"
image_subsetI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
product_prob_space_def: fixes M :: "'c \<Rightarrow> 'd measure" shows "product_prob_space M \<equiv> product_sigma_finite M \<and> product_prob_space_axioms M"
surjective: fixes r :: "('b, 'c) Vector_scheme" shows "r = \<lparr>tdir = tdir r, xdir = xdir r, ydir = ydir r, zdir = zdir r, \<dots> = Vector.more r\<rparr>"
finite_proj_diag: fixes d :: "'f \<Rightarrow> 'g \<Rightarrow> 'd"   and P :: "'e \<Rightarrow>\<^sub>F 'f"   and Q :: "'e \<Rightarrow>\<^sub>F 'g"   and S :: "'e set" shows "finite ((\<lambda>i. d ((P)\<^sub>F i) ((Q)\<^sub>F i)) ` S)"
CHAR_eq0_iff: shows "(semiring_char TYPE('a) = 0) = (\<forall>n>0. of_nat n \<noteq> (0::'a))"
Suc_mult_less_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(Suc k * m < Suc k * n) = (m < n)"
subst_eql: fixes t :: "'trm"   and t1 :: "'trm"   and t2 :: "'trm"   and x :: "'var" assumes "t \<in> trm"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x \<in> var" shows "subst (eql t1 t2) t x = eql (substT t1 t x) (substT t2 t x)"
digraph_isomorphism_subgraphI: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and H :: "('a, 'b) pre_digraph" assumes "H.digraph_isomorphism hom"   and "subgraph H H" shows "pre_digraph.digraph_isomorphism H hom"
dvd_0_left_iff: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and a :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times zero a = (a = zero)"
list_assn_aux_append: fixes l1 :: "'a list"   and l1' :: "'b list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l2 :: "'a list"   and l2' :: "'b list" assumes "length l1 = length l1'" shows "list_assn P (l1 @ l2) (l1' @ l2') = list_assn P l1 l1' * list_assn P l2 l2'"
inj_imp_Ker0: assumes "inj_on T (carrier V)" shows "carrier (V.vs kerT) = {\<zero>\<^bsub>V\<^esub>}"
NatTransCompDefined_1: fixes f :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and g :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and h :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "f \<approx>>\<bullet> g"   and "g \<approx>>\<bullet> h" shows "f \<bullet> g \<approx>>\<bullet> h"
FG_advice_prop_entailment_2: fixes Y :: "'b ltln set"   and \<A> :: "'b ltln set"   and \<phi> :: "'b ltln" assumes "{\<psi>. \<psi>[Y]\<^sub>\<mu> \<in> \<A>} \<Turnstile>\<^sub>P \<phi>" shows "\<A> \<Turnstile>\<^sub>P \<phi>[Y]\<^sub>\<mu>"
uint32_simps_2: fixes x :: "uint32"   and y :: "uint32" shows "int_of_uint32 (x + y) = (int_of_uint32 x + int_of_uint32 y) mod 4294967296"
i_Exec_Stream_nth_Suc: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a"   and n :: "nat" shows "i_Exec_Comp_Stream trans_fun input c (Suc n) = trans_fun (input (Suc n)) (i_Exec_Comp_Stream trans_fun input c n)"
induct: fixes x1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and x2 :: "'e"   and x3 :: "'f \<Rightarrow> 'g"   and x4 :: "'e"   and x5 :: "'f \<Rightarrow> 'g"   and P :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint) \<Rightarrow> 'e \<Rightarrow> ('f \<Rightarrow> 'g) \<Rightarrow> 'e \<Rightarrow> ('f \<Rightarrow> 'g) \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<rightarrow> \<langle>x4,x5\<rangle>"   and "\<And>A l u a l' u'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle> \<Longrightarrow> P A l u l' u'"   and "\<And>A l u d l' u'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsup>d\<^esup> \<langle>l', u'\<rangle> \<Longrightarrow> P A l u l' u'" shows "P x1 x2 x3 x4 x5"
brec_\<alpha>_def: shows "brec_\<alpha> \<equiv> \<lambda>(Q, W, rcm, f). (hm.\<alpha> Q, set (fifo.\<alpha> W), hm.\<alpha> rcm)"
ask_option_def: fixes ask :: "('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and f :: "'i \<Rightarrow> ('g, 'h) optionT" shows "ask_option ask f = OptionT (ask (\<lambda>r. run_option (f r)))"
insert_Diff1: fixes x :: "'c"   and B :: "'c set"   and A :: "'c set" assumes "x \<in> B" shows "insert x A - B = A - B"
mat_def: fixes k :: "'a" shows "mat k = (\<chi>i j. if i = j then k else (0::'a))"
max_enat_simps_1: fixes m :: "nat"   and n :: "nat" shows "max (enat m) (enat n) = enat (max m n)"
psubstEnv_preserves_wls: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsEnv rho"   and "wlsEnv rho'" shows "wlsEnv rho &[rho']"
ide_dom: fixes f :: "'a" assumes "hom.arr f" shows "hom.ide (hom.dom f)"
open_fmap_def: shows "open = generate_topology {Pi' a b |a b. \<forall>i\<in>a. open (b i)}"
BCh_simps_3: fixes h :: "'c"   and X :: "'a set" assumes "h \<notin> {Xh d1h1, Xh d1h3}" shows "BCh h X = {}"
naturality_hom_induced: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and T :: "'c set"   and q :: "int" assumes "continuous_map X Y f"   and "f ` S \<subseteq> T" shows "hom_boundary q Y T \<circ> hom_induced q X S Y T f = hom_induced (q - 1) (subtopology X S) {} (subtopology Y T) {} f \<circ> hom_boundary q X S"
u: fixes x :: "real" assumes "u \<le> x" shows "f x < y"
mcont_emeasure_spmf: shows "mcont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. emeasure (measure_spmf p))"
sub3_det: fixes mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and A :: "'a mat"   and d :: "nat"   and l :: "nat"   and r :: "'a"   and r'' :: "'a"   and A'' :: "'a mat" assumes "mute_fun mf"   and "det_selection_fun sel_fun"   and "A \<in> carrier_mat d d"   and "sub3 sel_fun mf d l (r, A) = (r'', A'')"   and "r \<noteq> (0::'a)"   and "l \<le> d" shows "r * det A'' = r'' * det A"
cases_scheme: fixes r :: "('e, 'f) graph_rec_scheme" obtains g_V :: "'e set" and g_E :: "('e \<times> 'e) set" and g_V0 :: "'e set" and more :: "'f" where "r = \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr>"
bij_betw_same_card: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "bij_betw f A B" shows "card A = card B"
compact_frontier_bounded: fixes S :: "'b set" assumes "bounded S" shows "compact (frontier S)"
order_iff: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (a = inf a b)"
assms: shows "f = Polynomial.smult c (\<Prod>a\<leftarrow>as. [:- a, 1::'a:])"
pred_inject_3: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'b" shows "pred_xstate P1 P2 (Fault a) = P2 a"
minus_divide_left: fixes a :: "'a"   and b :: "'a" shows "- (a / b) = - a / b"
sqrtl_2_eq: fixes x :: "real" shows "sqrtl x 2 = 8 * x * (x + 1) * (x\<^sup>2 + 6 * x + 1) / (x ^ 4 + 28 * x ^ 3 + 70 * x\<^sup>2 + 28 * x + 1)"
not_reset_dur': fixes i :: "nat"   and c :: "'c"   and xs :: "('c \<Rightarrow> real) stream"   and j :: "nat" assumes "\<forall>j\<ge>i. \<not> zero c ([xs !! j]\<^sub>\<R>)"   and "i \<le> j"   and "c \<in> \<X>"   and "stream_trans xs"   and "\<forall>i. eq_elapsed (xs !! i) (xs !! Suc i)"   and "\<forall>j. xs !! j \<in> V" shows "dur xs j - dur xs i = (xs !! j) c - (xs !! i) c"
take_Suc: fixes xs :: "'b list"   and n :: "nat" assumes "xs \<noteq> []" shows "take (Suc n) xs = hd xs # take n (tl xs)"
assms_2: shows "v - p \<in> orthogonal_complement S"
frac_le: fixes y :: "'b"   and x :: "'b"   and w :: "'b"   and z :: "'b" assumes "(0::'b) \<le> y"   and "x \<le> y"   and "(0::'b) < w"   and "w \<le> z" shows "x / z \<le> y / w"
singletonD: fixes b :: "'k"   and a :: "'k" assumes "b \<in> {a}" shows "b = a"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "comm_ring_hom hom" shows "map_poly_comm_ring_hom hom"
initAborts_def: fixes s :: "('a, 'b, 'c) SLin_state" shows "initAborts s \<equiv> {d. dstate s \<preceq> d \<and> ((\<exists>rs\<in>pendingSeqs s. d = dstate s \<star> rs) \<or> (\<exists>ivs\<in>initSets s. dstate s \<preceq> \<Sqinter> ivs \<and> (\<exists>rs\<in>pendingSeqs s. d = \<Sqinter> ivs \<star> rs)))}"
delta_def: shows "delta = (\<lambda>l. if prev_N l \<le> 1 then real K else eps (prev_N l))"
subseq_Cons': fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "subseq (x # xs) ys" shows "subseq xs ys"
reachable_step: fixes s :: "'d"   and A :: "('d, 'e) automaton"   and I :: "'e \<Rightarrow> bool"   and a :: "'e"   and s' :: "'d" assumes "s \<in> reachable A I"   and "(s, a, s') \<in> automaton.trans A"   and "I a" shows "s' \<in> reachable A I"
identify_blocks: fixes i_start :: "nat"   and i_end :: "nat"   and B :: "'b mat"   and k :: "nat" assumes "(i_start, i_end) \<in> set (??.Jordan_Normal_Form_Existence.identify_blocks B k)" shows "i_start \<le> i_end \<and> i_end < k"
restrict_partition_merge: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and I :: "nat set"   and J :: "nat set"   and V :: "nat set"   and n :: "nat"   and z :: "'b option list"   and xx :: "'b option list"   and t :: "'b option list" assumes "getIJ getIJ"   and "I \<union> J = V"   and "wf_tuple n V z"   and "xx = restrict J z"   and "t = restrict I z"   and "Set.is_empty (I \<inter> J)" shows "z = New_max.merge xx t"
ab_diff_conv_add_uminus: fixes a :: "'d"   and b :: "'d" shows "a - b = a + - b"
surjective: fixes r :: "('b, 'c) sparc_state_scheme" shows "r = \<lparr>cpu_reg = cpu_reg r, user_reg = user_reg r, sys_reg = sys_reg r, mem = mem r, mmu = mmu r, cache = cache r, dwrite = dwrite r, state_var = state_var r, traps = traps r, undef = undef r, \<dots> = sparc_state.more r\<rparr>"
pm_of_idx_pm_def: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd" shows "pm_of_idx_pm xs f = Abs_poly_mapping (\<lambda>x. lookup f (Min {i. i < length xs \<and> xs ! i = x}) when x \<in> set xs)"
case_prodE': fixes c :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'b \<times> 'c"   and z :: "'d" assumes "(case p of (a, b) \<Rightarrow> c a b) z" obtains x :: "'b" and y :: "'c" where "p = (x, y)" and "c x y z"
exact_seq_imp_triviality: fixes E :: "'b monoid"   and D :: "'b monoid"   and C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and k :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "exact_seq ([E, D, C, B, A], [k, h, g, f])"   and "f \<in> Group.iso A B"   and "k \<in> Group.iso D E" shows "trivial_group C"
hta_reduce_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme"   and Q :: "('a, unit) HashMap.hashmap" shows "hta_reduce H Q \<equiv> init_hta (hs.inter Q (hta_Qi H)) (ll_set_xy.g_image_filter (\<lambda>r. if hs.memb (lhs r) Q \<and> list_all (\<lambda>q. hs.memb q Q) (rhsq r) then Some r else None) (hta_\<delta> H))"
\<phi>_def: fixes y :: "'b"   and h :: "'a" shows "Adj'.\<eta>\<epsilon>.\<phi> y h = G' h \<cdot>\<^sub>D Adj'.\<eta> y"
c_hd_c_cons: fixes x :: "nat"   and y :: "nat" shows "c_hd (c_cons x y) = x"
is_partial_heap_smaller_back: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and m :: "int"   and r :: "int"   and r' :: "int" assumes "is_partial_heap cmp heap l m r"   and "r' \<le> r" shows "is_partial_heap cmp heap l m r'"
consistentDP_def: shows "state_dp_consistency.consistentDP \<equiv> state_dp_consistency.rel_fun_lifted (=) (=) dp"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) (J_C.Map d) (J_C.Map d) (J_C.Map d)"
OclInt6_def: shows "\<six> = (\<lambda>_. \<lfloor>\<lfloor>6\<rfloor>\<rfloor>)"
simps_1: fixes less_eq0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and uu :: "'a option" assumes "option_order less_eq0 less0" shows "option_order.less_eq_option less_eq0 None uu = True"
fully_faithful_functor_axioms: shows "fully_faithful_functor (\<cdot>\<^sub>B) (\<cdot>) UP"
that: fixes u :: "'a list"   and a :: "'a"   and v :: "nat \<Rightarrow> 'a" assumes "w = u \<frown> [a] \<frown> v"   and "A \<inter> set u = {}"   and "a \<in> A" shows "thesis"
length_n_lists: fixes n :: "nat"   and xs :: "'a list" shows "length (List.n_lists n xs) = length xs ^ n"
sum_over_degree_insertion: fixes var :: "nat"   and L :: "real list"   and p :: "real mpoly"   and d :: "nat"   and x :: "real" assumes "var < length L"   and "MPoly_Type.degree p var = d" shows "(\<Sum>i = 0..<d + 1. insertion (nth_default 0 (L[var := x])) (isolate_variable_sparse p var i) * x ^ i) = insertion (nth_default 0 (L[var := x])) p"
dX_range': fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> ManiPref'.dX X d" shows "x \<in> X \<and> Xd x = d"
bd_cts_trI: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>M. \<lbrakk>\<And>i. le_trans (M i) (M (Suc i)); \<And>i. feasible (M i)\<rbrakk> \<Longrightarrow> equiv_trans (T (Sup_trans (range M))) (Sup_trans (range (T \<circ> M)))" shows "bd_cts_tr T"
maintain_prim_invar_impl_2: fixes Qi :: "'q"   and \<pi>i :: "'m"   and u :: "'v"   and d :: "nat"   and Qi' :: "'q"   and \<pi>i' :: "'m" assumes "prim_invar_impl Qi \<pi>i"   and "\<not> Q_is_empty Qi"   and "Q_getmin Qi = (u, d)"   and "foreach_impl Qi \<pi>i u (G_adj g u) = (Qi', \<pi>i')" shows "T_measure_impl (Q_delete u Qi', \<pi>i') < T_measure_impl (Qi, \<pi>i)"
F'_start: shows "(XD_smallest ds X, dual (XH_largest ds X)) \<le> F' (XD_smallest ds X, dual (XH_largest ds X))"
strict_prefernce_iff_strict_utility: fixes x :: "'a pmf"   and y :: "'a pmf" assumes "x \<in> \<P> outcomes"   and "y \<in> \<P> outcomes" shows "(x \<succeq> y \<and> (y, x) \<notin> relation) = (measure_pmf.expectation y u < measure_pmf.expectation x u)"
finfun_ext: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" assumes "\<And>a. f $ a = g $ a" shows "f = g"
sort_is_sortedQ: fixes p :: "('a, 'b) Combinators list"   and l :: "'a set list" assumes "all_in_list p l"   and "singleCombinators p" shows "FWNormalisationCore.sorted (qsort p l) l"
dom_mult_closed: fixes x :: "'b"   and y :: "'b" shows "ad (ad (ad (ad x) \<cdot> ad (ad y))) = ad (ad x) \<cdot> ad (ad y)"