hcju/isabelleps huggingface dataset:
downclosed_set_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "class.preorder less_eq less" shows "ord.downclosed_set less_eq X = (\<forall>x\<in>X. \<forall>y. less_eq y x \<longrightarrow> y \<in> X)"
rsp: shows "eq_onp (\<lambda>x. \<exists>y. x = - y) top top"
simps_1: fixes z :: "nat" shows "inside z 0 = (0 < z)"
d_gchainTr1: fixes n :: "nat" shows "0 < n \<longrightarrow> (\<forall>f. d_gchain G n f \<longrightarrow> (\<forall>l\<le>n. \<forall>j\<le>n. l < j \<longrightarrow> f j \<subseteq> f l))"
le_lt_convs_3: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Gt) = lt_of_comp comp y x"
no_step_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and aa :: "String.literal"   and ba :: "value list"   and p :: "(String.literal \<times> value list) list" assumes "step e s r aa ba = None" shows "\<not> recognises_execution e s r ((aa, ba) # p)"
slo_linorder: fixes r :: "('b \<times> 'b) set" assumes "strict_linear_order r" shows "class.linorder (\<lambda>a b. (a, b) \<in> r\<^sup>=) (\<lambda>a b. (a, b) \<in> r)"
ereal_le_add_self: fixes y :: "ereal"   and x :: "ereal" assumes "0 \<le> y" shows "x \<le> x + y"
contour_integrable_joinD2: fixes f :: "complex \<Rightarrow> complex"   and g1 :: "real \<Rightarrow> complex"   and g2 :: "real \<Rightarrow> complex" assumes "f contour_integrable_on g1 +++ g2"   and "valid_path g2" shows "f contour_integrable_on g2"
clearjunk0_def: fixes m :: "('c, 'd) fmap" shows "clearjunk0 m = fmfilter (\<lambda>k. fmlookup m k \<noteq> Some (0::'d)) m"
siso_Atm: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'c" assumes "PL_Indis indis" shows "PL_Indis.siso tval aval indis (Atm atm) = PL_Indis.compatAtm aval indis atm"
strict_trans2: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<prec> b"   and "b \<preceq> c" shows "a \<prec> c"
compute_set_of_jordan_blocks_def: fixes A :: "'a mat"   and ev :: "'a" shows "local.compute_set_of_jordan_blocks A ev \<equiv> let k = Polynomial.order ev (char_poly A); as = map (local.dim_gen_eigenspace A ev) [0..<Suc (Suc k)]; cards = map (\<lambda>k. (k, 2 * as ! k - as ! (k - 1) - as ! Suc k)) [1..<Suc k] in map (\<lambda>(k, c). (k, ev)) (filter (\<lambda>(k, c). c \<noteq> 0) cards)"
xor_compl_left: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b"   and y :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "boolean_algebra.xor conj disj compl (compl x) y = compl (boolean_algebra.xor conj disj compl x y)"
p'_2: shows "set p' \<subseteq> G.face_cycle_set a"
mod2_Suc_Suc: fixes m :: "nat" shows "Suc (Suc m) mod 2 = m mod 2"
const_def: fixes x :: "'d" shows "const x \<equiv> \<lambda>_. x"
combine_ncols_max: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'d matrix" assumes "f (0::'c) (0::'d) = (0::'b)" shows "ncols (combine_matrix f A B) \<le> max (ncols A) (ncols B)"
reflexive_axioms: shows "reflexive A (\<sqsubseteq>)"
jviewInit: shows "\<forall>a s. spr_jviewInit a (envObs a s) = spr_jview a (tInit s)"
split_impl: shows "(split_impl, split_spec) \<in> A \<rightarrow> \<langle>B \<times>\<^sub>r A\<rangle>nres_rel"
subst_preserves_beta: fixes r :: "dB"   and s :: "dB"   and t :: "dB"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "r[t/i] \<rightarrow>\<^sub>\<beta> s[t/i]"
assms_2: shows "s ! (i - 1) = s ! (j - 1)"
Diff_cancel: fixes A :: "'c set" shows "A\<setminus>A = {}"
HVZK_unfold2: assumes "\<Sigma>1.\<Sigma>_protocol" shows "\<forall>h e out. e \<in> carrier L \<longrightarrow> h \<in> valid_pub1 \<longrightarrow> out \<in> set_spmf (S1_raw h e) \<longrightarrow> check1 h (fst out) e (snd out)"
isos_compose: fixes f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "VoV.iso f"   and "VoV.iso f'"   and "VoV.seq f' f" shows "VoV.iso (VoV.comp f' f)"
lex_fun_def: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" shows "lex_fun s t \<equiv> \<forall>x. s x \<le> t x \<or> (\<exists>y<x. s y \<noteq> t y)"
integrable_std_normal_distribution_moment: fixes k :: "nat" shows "integrable std_normal_distribution (\<lambda>x. x ^ k)"
nat_pow_Suc2: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier \<G>" shows "x [^] Suc n = x \<otimes> x [^] n"
WT_obsf_resource: fixes \<I> :: "('y, 'z) \<I>"   and res :: "('y, 'z) resource" assumes "\<I> \<turnstile>res res \<surd>" shows "exception_\<I> \<I> \<turnstile>res obsf_resource res \<surd>"
subset_acq: shows "acquired_reads True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (r # sb)) \<O> \<subseteq> A"
eq0: shows "Ifm vs (x # bs) (subst0 (CP 0\<^sub>p) p) = Ifm vs ((0::'a) # bs) p"
IH: fixes x :: "nat"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v :: "exp"   and y :: "nat"   and e' :: "exp" assumes "VFun f \<in> E (ELam x e) \<rho>"   and "is_val v"   and "ELam x e = ELam x (subst y v e')"   and "\<forall>v1 v2. v2 \<in> E (subst y v e') ((x, v1) # \<rho>) \<longrightarrow> (\<exists>\<rho>' v'. v' \<in> E v [] \<and> v2 \<in> E e' \<rho>' \<and> \<rho>' \<approx> (y, v') # (x, v1) # \<rho>)"   and "x \<noteq> y" shows "\<exists>\<rho>' v''. v'' \<in> E v [] \<and> VFun f \<in> E (ELam x e') \<rho>' \<and> \<rho>' \<approx> (y, v'') # \<rho>"
Lab: fixes G :: "prog"   and s0 :: "st"   and c :: "stmt"   and n :: "nat"   and s1 :: "state"   and l :: "jump" assumes "G\<turnstile>Norm s0 \<midarrow>c\<midarrow>n\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>l\<bullet> c\<midarrow>n\<rightarrow> abupd (absorb l) s1"
constant_deriv_zero: fixes I :: "('a, 'b, 'c) interp"   and id1 :: "'a"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "is_interp I" shows "directional_derivative I ($f id1 local.empty) \<nu> = 0"
assms: shows "\<S>\<bullet>rs \<tturnstile>_ i vs;es : ts"
open_Collect_conj: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}"   and "open {x. Q x}" shows "open {x. P x \<and> Q x}"
Suc_le_D_lemma: fixes n :: "nat"   and m' :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "Suc n \<le> m'"   and "\<And>m. n \<le> m \<Longrightarrow> P (Suc m)" shows "P m'"
f3: shows "Deliver (i, Create i e2) \<notin> set (history k)"
euclid_ext_poly_dynamic_def: fixes p :: "int" shows "euclid_ext_poly_dynamic p = (if p \<le> 65535 then euclid_ext_poly_mod_main p (finite_field_ops32 (uint32_of_int p)) else if p \<le> 4294967295 then euclid_ext_poly_mod_main p (finite_field_ops64 (uint64_of_int p)) else euclid_ext_poly_mod_main p (finite_field_ops_integer (integer_of_int p)))"
intros_1: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi>' :: "nat \<Rightarrow> 'a"   and k' :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<sigma> :: "'b \<Rightarrow> 'c"   and \<sigma>' :: "'b \<Rightarrow> 'c" assumes "IFC_def.cs prob \<pi>' k' \<prec> IFC_def.cs prob \<pi> k"   and "\<pi> = IFC_def.path prob \<sigma>"   and "\<pi>' = IFC_def.path prob \<sigma>'"   and "\<pi> (Suc (IFC_def.cs_select prob \<pi> (IFC_def.cs prob \<pi>' k'))) \<noteq> \<pi>' (Suc k')" shows "IFC_def.contradicts prob (\<sigma>', k') (\<sigma>, k)"
sup_assoc: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_sup sup less_eq less" shows "sup (sup x y) z = sup x (sup y z)"
map_ugraph_trans: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat" shows "map_ugraph (g \<circ> f) = map_ugraph g \<circ> map_ugraph f"
ideD_1: fixes a :: "'a2" assumes "A2.ide a" shows "A2.arr a"
bob_def: fixes q :: "complex Matrix.mat"   and b :: "bit \<times> bit" shows "bob q b \<equiv> if (fst b, snd b) = (bit.zero, bit.zero) then q else if (fst b, snd b) = (bit.zero, bit.one) then (Quantum.Id 2 \<Otimes> X) * q else if (fst b, snd b) = (bit.one, bit.zero) then (Quantum.Id 2 \<Otimes> Z) * q else if (fst b, snd b) = (bit.one, bit.one) then (Quantum.Id 2 \<Otimes> Z * X) * q else undefined"
domSubset4: fixes x :: "(int \<times> int \<times> protocol) set set"   and y :: "(int \<times> int \<times> protocol) set set"   and dn :: "protocol \<times> int" shows "dom (Cp (DenyAllFromTo x y \<oplus> DenyAllFromTo y x \<oplus> AllowPortFromTo x y dn)) = dom (Cp (DenyAllFromTo x y \<oplus> DenyAllFromTo y x))"
Lim_def: fixes A :: "'b filter"   and f :: "'b \<Rightarrow> 'a" shows "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
WTrt_binop_i_i_i_i_oE: fixes x :: "'b prog"   and xa :: "ty"   and xb :: "bop"   and xc :: "ty"   and xd :: "ty" assumes "pred.eval (WTrt_binop_i_i_i_i_o x xa xb xc) xd" obtains "x \<turnstile> xa\<guillemotleft>xb\<guillemotright>xc : xd"
IH_1: fixes z' :: "'a"   and buf' :: "event_data option list set list list"   and nts' :: "nat list"   and i :: "nat" assumes "\<not> ([] \<in> set buf \<or> nts = [])"   and "mbufnt_take f (f (map hd buf) (hd nts) z) (map tl buf) (tl nts) = (z', buf', nts')"   and "wf_mbufn i js Ps (map tl buf)"   and "list_all2 R [i..<j] (tl nts)"   and "\<And>k. k \<in> set js \<Longrightarrow> k \<le> j" shows "wf_mbufn (Mini (i + length (tl nts)) js) js Ps buf'"
normalizability_I: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>*"   and "b \<in> NF A" shows "(a, b) \<in> A\<^sup>!"
n_S2: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and s :: "'a" assumes "Reachability_Problem K S S1 S2"   and "s \<in> S2" shows "Reachability_Problem.n K S S1 S2 s = 1"
distinct_49: fixes psi1 :: "('d, 'e, 'f) psi"   and psi2 :: "('d, 'e, 'f) psi"   and psi' :: "('d, 'e, 'f) psi" shows "psi1 \<parallel> psi2 \<noteq> !psi'"
GatewaySystem_L3aux: fixes req :: "nat \<Rightarrow> reqType list"   and stop :: "nat \<Rightarrow> 'a list"   and a :: "nat \<Rightarrow> aType list"   and lose :: "nat \<Rightarrow> bool list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and d :: "nat"   and i :: "nat \<Rightarrow> 'b list"   and dt :: "nat \<Rightarrow> 'b list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat"   and k :: "nat" assumes "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose"   and "msg (Suc 0) req \<and> msg (Suc 0) a \<and> msg (Suc 0) stop \<and> ts lose \<longrightarrow> (\<forall>t. (ack t = [init_state] \<and> req (Suc t) = [init] \<and> lose (Suc t) = [False] \<and> lose (Suc (Suc t)) = [False] \<longrightarrow> ack (Suc (Suc t)) = [connection_ok]) \<and> (ack t = [connection_ok] \<and> req (Suc t) = [send] \<and> (\<forall>k\<le>Suc d. lose (t + k) = [False]) \<longrightarrow> i (Suc (t + d)) = inf_last_ti dt t \<and> ack (Suc t) = [sending_data]) \<and> (ack (t + d) = [sending_data] \<and> a (Suc t) = [sc_ack] \<and> (\<forall>k\<le>Suc d. lose (t + k) = [False]) \<longrightarrow> vc (Suc (t + d)) = [vc_com]))" shows "ack (t + 2 + k) = [connection_ok] \<and> req (Suc (t + 2 + k)) = [send] \<and> (\<forall>j\<le>Suc d. lose (t + 2 + k + j) = [False]) \<longrightarrow> i (Suc (t + 2 + k + d)) = inf_last_ti dt (t + 2 + k)"
generatorp_generator_eq: shows "generatorp = (\<lambda>x. x \<in> generator)"
labels_pinduct: fixes a0 :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp"   and P :: "('b \<Rightarrow> ('c, 'd, 'b, 'e) seqp) \<Rightarrow> ('c, 'd, 'b, 'e) seqp \<Rightarrow> bool"   and a1 :: "('c, 'd, 'b, 'e) seqp" assumes "wellformed a0"   and "\<And>\<Gamma> l fg p. labels_dom (\<Gamma>, {l}\<langle>fg\<rangle> p) \<Longrightarrow> P \<Gamma> ({l}\<langle>fg\<rangle> p)"   and "\<And>\<Gamma> l fa p. labels_dom (\<Gamma>, {l}\<lbrakk>fa\<rbrakk> p) \<Longrightarrow> P \<Gamma> ({l}\<lbrakk>fa\<rbrakk> p)"   and "\<And>\<Gamma> p1 p2. \<lbrakk>labels_dom (\<Gamma>, p1 \<oplus> p2); P \<Gamma> p1; P \<Gamma> p2\<rbrakk> \<Longrightarrow> P \<Gamma> (p1 \<oplus> p2)"   and "\<And>\<Gamma> l fip fmsg p q. labels_dom (\<Gamma>, {l}unicast(fip, fmsg) . p \<triangleright> q) \<Longrightarrow> P \<Gamma> ({l}unicast(fip, fmsg) . p \<triangleright> q)"   and "\<And>\<Gamma> l fmsg p. labels_dom (\<Gamma>, {l}broadcast(fmsg) . p) \<Longrightarrow> P \<Gamma> ({l}broadcast(fmsg) . p)"   and "\<And>\<Gamma> l fips fmsg p. labels_dom (\<Gamma>, {l}groupcast(fips, fmsg) . p) \<Longrightarrow> P \<Gamma> ({l}groupcast(fips, fmsg) . p)"   and "\<And>\<Gamma> l fmsg p. labels_dom (\<Gamma>, {l}send(fmsg) . p) \<Longrightarrow> P \<Gamma> ({l}send(fmsg) . p)"   and "\<And>\<Gamma> l fdata p. labels_dom (\<Gamma>, {l}deliver(fdata) . p) \<Longrightarrow> P \<Gamma> ({l}deliver(fdata) . p)"   and "\<And>\<Gamma> l fmsg p. labels_dom (\<Gamma>, {l}receive(fmsg) . p) \<Longrightarrow> P \<Gamma> ({l}receive(fmsg) . p)"   and "\<And>\<Gamma> pn. \<lbrakk>labels_dom (\<Gamma>, call(pn)); P \<Gamma> (\<Gamma> pn)\<rbrakk> \<Longrightarrow> P \<Gamma> (call(pn))" shows "P a0 a1"
lift_wf: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and agents' :: "'a set"   and alts' :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "scf_lowering agents alts scf agents' alts'"   and "pref_profile_wf agents' alts' R" shows "pref_profile_wf agents alts (lift_pref_profile agents' alts' agents alts R)"
lessThan_minus_lessThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a"   and m :: "'a" assumes "class.linorder less_eq less" shows "ord.lessThan less n - ord.lessThan less m = ord.atLeastLessThan less_eq less m n"
a_def: shows "a = (\<Lambda> uu_. CFn\<cdot>(\<Lambda> uu_ uu_. CFn\<cdot>(\<Lambda> uu_ uu_. \<bottom>)))"
restrict_apply: fixes f :: "'h \<Rightarrow> 'g"   and A :: "'h set"   and x :: "'h" shows "restrict f A x = (if x \<in> A then f x else undefined)"
us1: shows "unat (2 ^ s1 * word_of_nat mq) = 2 ^ s1 * mq"
ivl_integral_zero: fixes a :: "real"   and f :: "real \<Rightarrow> 'a" shows "ivl_integral a a f = (0::'a)"
continuous_on: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and s :: "'e set"   and f :: "'e \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c" assumes "bounded_bilinear prod"   and "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. prod (f x) (g x))"
4: shows "ks \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> ms = ks \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> ms \<oplus> ks \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> ms"
smaller: shows "\<forall>n. sum (f \<circ> g) {..<n} \<le> suminf f"
Pi_I': fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
simps_2: fixes e :: "'b set \<times> 'b set"   and w :: "('b set \<times> 'b set) list"   and m :: "'b" shows "\<alpha>ah (e # w) m = (if m \<in> fst e then fst e \<union> snd e \<union> mon_pl w else \<alpha>ah w m)"
construct_basis: fixes B :: "'b set"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "vs1.independent B"   and "b \<in> B" shows "construct B f b = f b"
path_X: shows "path M (xs || trX) q2 \<and> length xs = length trX"
Conjunctive_conj: fixes HC :: "'d upred \<Rightarrow> 'd upred"   and P :: "'d upred"   and Q :: "'d upred" assumes "Conjunctive HC" shows "HC (P \<and> Q) = (HC P \<and> Q)"
eventually_nhds_within_iff_sequentially: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" shows "eventually P (inf (nhds a) (principal s)) = (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n)))"
datastate_1: fixes i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "join_ir i r \<equiv> \<lambda>x. case x of I n \<Rightarrow> input2state i $ n | R n \<Rightarrow> r $ n"
diff_delta_lnexp_cf4_def: shows "diff_delta_lnexp_cf4 \<equiv> \<lambda>x. - (x ^ 8) / (numer_cf4 (- x) * numer_cf4 x)"
simps_4: fixes \<phi> :: "'b ltl" shows "G_depth (G \<phi>) = G_depth \<phi> + 1"
nofail_SPEC_iff: fixes m :: "'c nres" shows "nofail m = (m \<le> SPEC (\<lambda>_. True))"
assms: shows "wf\<^sub>s\<^sub>t X A"
t_local_1: shows "\<S>\<bullet>(s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ map typeof vs, return := Some tls\<rparr> \<turnstile> es : [] _> tls"
zone_delay_def: fixes Z :: "('a \<Rightarrow> 'b) set" shows "Z\<^sup>\<up> = {u \<oplus> d |u d. u \<in> Z \<and> (0::'b) \<le> d}"
check_mult_l_def: fixes spec :: "(char list list \<times> int) list"   and A :: "(nat, (char list list \<times> int) list) fmap"   and \<V> :: "char list set"   and p :: "nat"   and q :: "(char list list \<times> int) list"   and i :: "nat"   and r :: "(char list list \<times> int) list" shows "check_mult_l spec A \<V> p q i r = (let b = p \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars_llist q \<subseteq> \<V> \<and> vars_llist r \<subseteq> \<V> in if \<not> b then check_mult_l_dom_err (p \<notin># dom_m A) p (i \<in># dom_m A) i \<bind> (\<lambda>c. RETURN (error_msg i c)) else ASSERT (p \<in># dom_m A) \<bind> (\<lambda>_. let p = the (fmlookup A p) in mult_poly_full p q \<bind> (\<lambda>pq. weak_equality_l pq r \<bind> (\<lambda>b. weak_equality_l r spec \<bind> (\<lambda>b'. if b then if b' then RETURN CFOUND else RETURN CSUCCESS else check_mult_l_mult_err p q pq r \<bind> (\<lambda>c. RETURN (error_msg i c)))))))"
mem_case_prodE: fixes z :: "'c"   and c :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and p :: "'d \<times> 'e" assumes "z \<in> (case p of (x, xa) \<Rightarrow> c x xa)" obtains x :: "'d" and y :: "'e" where "p = (x, y)" and "z \<in> c x y"
Interp_imp_general: fixes C :: "'a literal multiset"   and D :: "'a literal multiset"   and D' :: "'a literal multiset"   and CC :: "'a literal multiset set" assumes "C \<le> D"   and "D < D'"   and "Interp D \<Turnstile> C"   and "interp D' \<subseteq> \<Union> (production ` CC)" shows "\<Union> (production ` CC) \<Turnstile> C"
check_sat: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "Check check"   and "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<not> \<U> (check s)" shows "\<Turnstile> (check s)"
effNone: fixes pc' :: "nat"   and s' :: "(ty list \<times> ty err list) option"   and i :: "instr"   and P :: "'a prog"   and pc :: "nat"   and et :: "ex_table" assumes "(pc', s') \<in> set (eff i P pc et None)" shows "s' = None"
berlekamp_mat_closed_3: fixes u :: "'b mod_ring poly" shows "dim_col (berlekamp_mat u) = degree u"
bin_to_bl_zero: fixes n :: "nat" shows "bin_to_bl n 0 = replicate n False"
rSym: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> X" shows "(\<Psi>, Q, P) \<in> X"
pwD1: fixes S :: "'c nres"   and S' :: "'c nres" assumes "S \<le> S'"   and "nofail S'" shows "nofail S"
assms_2: fixes x :: "'a" assumes "x \<in> U" shows "(f has_derivative blinfun_apply (f' x)) (at x)"
Cnotzero_mono: fixes r :: "('c \<times> 'c) set"   and q :: "('d \<times> 'd) set" assumes "(r, czero) \<notin> ordIso \<and> Card_order r"   and "Card_order q"   and "r \<le>o q" shows "(q, czero) \<notin> ordIso \<and> Card_order q"
rs_def: shows "run_step d i (s, vs, es) = (s'', vs'', res'')"
respects_labels: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and B :: "'c set"   and \<phi> :: "'b \<Rightarrow> 'c"   and v :: "'b" assumes "ChamberComplexRetraction X f"   and "ChamberComplex.label_wrt X B \<phi>"   and "v \<in> \<Union> X" shows "\<phi> (f v) = \<phi> v"
record_simps_2: fixes f :: "nat \<Rightarrow> nat"   and x :: "'a state" shows "refs (update_clock f x) = refs x"
valid_def: fixes sys :: "('f, 'g, 'h, 'i, 'j) system"   and \<phi> :: "(nat \<Rightarrow> ('f, 'g, 'h, 'i, 'j) system_state) \<Rightarrow> bool" shows "(sys \<Turnstile> \<phi>) = (\<forall>\<sigma>. run sys \<sigma> \<longrightarrow> \<phi> \<sigma>)"
cda_ax1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<cdot> d z = x \<cdot> (y \<cdot> d z)" shows "d (x \<cdot> y \<cdot> z) = d (x \<cdot> d (y \<cdot> z))"
chine_composite_1: shows "\<mu>\<nu>\<pi>.chine = \<langle>\<mu>.chine \<cdot> Prj\<^sub>1 \<lbrakk>\<mu>.cod.leg0, \<nu>.cod.leg1 \<cdot> \<p>\<^sub>1[\<nu>.cod.leg0, \<pi>.cod.leg1]\<rbrakk> \<langle>\<nu>.chine \<cdot> \<nu>\<pi>.prj\<^sub>1 \<lbrakk>\<nu>.cod.leg0, \<pi>.cod.leg1\<rbrakk> \<pi>.chine \<cdot> \<nu>\<pi>.prj\<^sub>0\<rangle> \<cdot> \<p>\<^sub>0[\<mu>.leg0, \<nu>.leg1 \<cdot> \<nu>\<pi>.prj\<^sub>1]\<rangle>"
assms: shows "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
cINF_greatest: fixes A :: "'b set"   and m :: "'c multiset"   and f :: "'b \<Rightarrow> 'c multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "m \<subseteq># Inf (f ` A)"
SubobjsR_Rep: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Cs :: "char list list" assumes "P \<turnstile> C \<prec>\<^sub>R D"   and "Subobjs\<^sub>R P D Cs" shows "Subobjs\<^sub>R P C (C # Cs)"
fold_map_of_alt: shows "fold (\<lambda>(k, v) m. if m k \<noteq> None then m else m(k \<mapsto> v)) xs Map.empty = map_of xs ++ Map.empty"
rep_in_iso_class: fixes F :: "'a set" assumes "B.is_iso_class F" shows "B.VVV.iso_class_rep F \<in> F"
set_mono_strict_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "strict_prefix xs ys" shows "set xs \<subseteq> set (butlast ys)"
partJoin_def: fixes P :: "'b set set"   and Q :: "'b set set" shows "partJoin P Q \<equiv> partGen (P \<union> Q)"
dependent_mono: fixes B :: "'b set"   and A :: "'b set" assumes "local.dependent B"   and "B \<subseteq> A" shows "local.dependent A"
dist: shows "dist_over_sup (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
dgrad_p_set_exhaust: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" obtains m :: "nat" where "F \<subseteq> punit.dgrad_p_set d m"
Src_Dom_Cod: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Src src (bicategorical_language.Dom V src trg t) = bicategorical_language.Src src t \<and> bicategorical_language.Src src (bicategorical_language.Cod V src trg t) = bicategorical_language.Src src t"
indicator_pos_le: fixes S :: "'c set"   and x :: "'c" shows "(0::'b) \<le> indicator S x"
remdups_adj_append_two: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" shows "remdups_adj (xs @ [x, y]) = remdups_adj (xs @ [x]) @ (if x = y then [] else [y])"
r_snoc_def: shows "r_snoc \<equiv> Cn 2 r_rev [Cn 2 r_cons [recf.Id 2 1, Cn 2 r_rev [recf.Id 2 0]]]"
w2case1a: shows "sint (SCAST(32 \<rightarrow> 64) w2 + - 0x7FFFFFFF) = sint (SCAST(32 \<rightarrow> 64) w2) + sint (- 0x7FFFFFFF)"
map_option_case: fixes f :: "'d \<Rightarrow> 'c"   and y :: "'d option" shows "map_option f y = (case y of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>x\<rfloor> \<Rightarrow> \<lfloor>f x\<rfloor>)"
geodesic_subset_def: fixes S :: "'b set" shows "geodesic_subset S = (\<forall>x\<in>S. \<forall>y\<in>S. \<exists>G. geodesic_segment_between G x y \<and> G \<subseteq> S)"