hcju/isabelleps huggingface dataset:
minus_add_minus_vec: fixes u :: "'a vec"   and n :: "nat"   and v :: "'a vec"   and w :: "'a vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "u - (v + w) = u - v - w"
keys_diff: fixes a :: "'c \<Rightarrow>\<^sub>0 'd"   and b :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (a - b) \<subseteq> keys a \<union> keys b"
inverse_arrows_def: fixes f :: "'a"   and g :: "'a" shows "C.inverse_arrows f g \<equiv> C.ide (g \<cdot>\<^sub>C f) \<and> C.ide (f \<cdot>\<^sub>C g)"
inv_neg_one: shows "inv (\<ominus> \<one>) = \<ominus> \<one>"
mono_def: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f" assumes "class.order less_eq less" shows "order.mono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f x \<le> f y)"
simps: fixes a :: "trm" shows "SN a = (\<exists>t. a = t \<and> (\<forall>x. t \<mapsto> x \<longrightarrow> SN x))"
beforex_mono: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d' :: "nat"   and d :: "nat" assumes "P,C,M \<rhd> xt / I,d'"   and "d' \<le> d" shows "P,C,M \<rhd> xt / I,d"
combine_sub_sys: shows "sub_incidence_system \<V> \<B> \<V>\<^sup>+ \<B>\<^sup>+"
mssnth_arctan_series_stream_aux: fixes b :: "bool"   and n :: "real"   and m :: "nat" shows "mssnth (arctan_series_stream_aux b n) m = (- 1) ^ (if b then Suc m else m) / (2 * real m + n)"
ideD_1: fixes a :: "'a" assumes "Go\<tau>.A.ide a" shows "Go\<tau>.A.arr a"
it_to_sorted_list_triv: fixes l :: "'a list" assumes "distinct l" shows "RETURN l \<le> it_to_sorted_list (\<lambda>_ _. True) (set l)"
irsSp_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n irs-[]\<rightarrow>\<^sub>d* n"
ltln_WeakUntil_to_Release: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> W\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> R\<^sub>n (\<phi> or\<^sub>n \<psi>)"
rp: shows "rat_to_normalized_int_poly p = (a, P)"
ds_ds_subseteq_ds: fixes r :: "('a \<times> 'a) set"   and S :: "'a set" assumes "trans r" shows "r \<down>s (r \<down>s S) \<subseteq> r \<down>s S"
quarter_discrim_def: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "quarter_discrim p q \<equiv> (p \<bullet> (M *v q))\<^sup>2 - p \<bullet> (M *v p) * (q \<bullet> (M *v q))"
MapsToI: fixes f :: "'g"   and C :: "('h, 'g, 'i) Category_scheme"   and X :: "'h"   and Y :: "'h" assumes "f \<in> mor\<^bsub>C\<^esub>"   and "dom\<^bsub>C\<^esub> f = X"   and "cod\<^bsub>C\<^esub> f = Y" shows "f maps\<^bsub>C\<^esub> X to Y"
semWhileSC: fixes e :: "'exp" assumes "WhileSideCondition e" shows "\<forall>d. e \<equiv>\<^bsub>d\<^esub> e"
LIMSEQ_INF: fixes X :: "nat \<Rightarrow> 'a" assumes "decseq X" shows "X \<longlonglongrightarrow> \<Sqinter> range X"
assms_4: shows "j < card {j. j < n \<and> j \<in> J}"
zero: fixes s :: "'b"   and t :: "'b" shows "alternating_list 0 s t = []"
exhaust: fixes y :: "Ordering" obtains "y = \<bottom>"   | "y = LT"   | "y = EQ"   | "y = GT"
minimal_inD1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b" assumes "ordering less_eq less"   and "ordering.minimal_in less P x" shows "x \<in> P"
c_def: shows "c = (if l = \<infinity> then 2 else 1 + (real_of_ereal l - 1) / 2)"
PiE_Int: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'd set" shows "Pi\<^sub>E I A \<inter> Pi\<^sub>E I B = (\<Pi>\<^sub>E x\<in>I. A x \<inter> B x)"
join_lc: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
assms_1: shows "NTDom \<eta>1 = NTDom \<eta>2"
is_unit_triv: fixes a :: "'c" assumes "a \<noteq> (0::'c)" shows "is_unit [:a:]"
minus: fixes x :: "complex"   and xa :: "'d" shows "- x *\<^sub>C xa = - (x *\<^sub>C xa)"
Coplanar_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "Coplanar A B C D \<equiv> \<exists>X. Col A B X \<and> Col C D X \<or> Col A C X \<and> Col B D X \<or> Col A D X \<and> Col B C X"
get_proc_call: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "get_proc (targetnode a) = p"
\<G>\<F>_suffix: fixes \<phi> :: "'a ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'a set" shows "\<G>\<F> \<phi> (suffix i w) = \<G>\<F> \<phi> w"
assms_4: shows "\<forall>x\<in>s. ((\<lambda>k. f k x) \<longlongrightarrow> g x) at_top"
DONE_bind: fixes a :: "'i"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "DONE a \<bind> f = f a"
finfun_leD: fixes A :: "'a \<Rightarrow>f bool"   and B :: "'a \<Rightarrow>f bool"   and x :: "'a" assumes "A \<le> B"   and "A $ x" shows "B $ x"
update_convs_2: fixes trans' :: "('a \<times> 'b \<times> 'a) set \<Rightarrow> ('a \<times> 'b \<times> 'a) set"   and init :: "'a set"   and trans :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "trans_update trans' \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = \<lparr>init = init, trans = trans' trans, \<dots> = more\<rparr>"
node_ptr_kinds_M_eq: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "|h \<turnstile> object_ptr_kinds_M|\<^sub>r = |h' \<turnstile> object_ptr_kinds_M|\<^sub>r" shows "|h \<turnstile> node_ptr_kinds_M|\<^sub>r = |h' \<turnstile> node_ptr_kinds_M|\<^sub>r"
finite_map_to_set: fixes m :: "'c \<Rightarrow> 'd option" shows "finite (map_to_set m) = finite (dom m)"
Abs_cfun_eqvt: fixes f :: "'c \<Rightarrow> 'd"   and p :: "perm" assumes "cont f" shows "(p \<bullet> Abs_cfun) f = Abs_cfun f"
edges_graph': fixes V :: "'a set"   and E :: "('a \<times> 'a) set" assumes "nodes_edges_consistent V E" shows "edges (graph V E) = E"
closed_segment_image_interval: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = (\<lambda>u. (1 - u) *\<^sub>R a + u *\<^sub>R b) ` {0..1}"
repc_valid: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg" shows "repc cfg \<in> MDP.valid_cfg"
asym: fixes a :: "'a"   and b :: "'a" assumes "a < b"   and "b < a" shows "False"
mult_mv_mult_vm: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and x :: "complex \<times> complex" shows "A *\<^sub>m\<^sub>v x = x *\<^sub>v\<^sub>m mat_transpose A"
ccompatible5: fixes c :: "real"   and k :: "'d \<Rightarrow> nat"   and x :: "'d"   and X :: "'d set" assumes "c \<le> real (k x)"   and "c \<in> \<nat>"   and "x \<in> X" shows "ccompatible {region X I r |I r. valid_region X k I r} (GE x c)"
embed_defs_2: shows "embed \<equiv> \<lambda>r r' f. \<forall>a\<in>Field r. bij_betw f (under r a) (under r' (f a))"
tendsto_powr: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "a \<noteq> 0" shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
a_p: shows "f n\<^sub>1 @ a = f n\<^sub>2"
domintros_1: fixes m :: "nat"   and b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b" assumes "\<And>ma n. real m = (4 * real ma + 1) / (2 * 2 ^ n) \<Longrightarrow> dyad_rec_dom (b, l, r, (2 * real ma + 1) / 2 ^ n)"   and "\<And>ma n. real m = (4 * real ma + 3) / (2 * 2 ^ n) \<Longrightarrow> dyad_rec_dom (b, l, r, (2 * real ma + 1) / 2 ^ n)" shows "dyad_rec_dom (b, l, r, real m)"
wls_copy_of_good_item_simps_1: fixes inp :: "'g \<Rightarrow> ('g, 'h, 'varSort, 'var, 'i) qTerm set option"   and binp :: "'h \<Rightarrow> ('g, 'h, 'varSort, 'var, 'i) qAbs set option"   and inp' :: "'g \<Rightarrow> ('g, 'h, 'varSort, 'var, 'i) qTerm set option"   and binp' :: "'h \<Rightarrow> ('g, 'h, 'varSort, 'var, 'i) qAbs set option"   and delta :: "'i"   and delta' :: "'i" assumes "goodInp inp"   and "goodBinp binp"   and "goodInp inp'"   and "goodBinp binp'" shows "(Op delta inp binp = Op delta' inp' binp') = (delta = delta' \<and> inp = inp' \<and> binp = binp')"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) \<lparr>Chn = \<nu>\<pi>\<rho>.chine_assoc, Dom = Dom ((\<nu> \<star> \<pi>) \<star> \<rho>), Cod = Cod (\<nu> \<star> \<pi> \<star> \<rho>)\<rparr>"
ipath_mkTree_sdrop: fixes s :: "'state"   and rs :: "'rule stream"   and steps :: "('state \<times> 'rule) stream"   and m :: "nat" assumes "s \<in> S"   and "fair rs"   and "ipath (mkTree rs s) steps" shows "\<exists>n s'. s' \<in> S \<and> ipath (mkTree (sdrop n rs) s') (sdrop m steps)"
simps_3: fixes t :: "'c local.term"   and u :: "'c local.term" shows "Ide (t \<^bold>\<otimes> u) = (Ide t \<and> Ide u)"
red_rtrancl_\<tau>_heapD_inv: fixes t :: "'t"   and s :: "'x \<times> 'm"   and s' :: "'x \<times> 'm"   and wfs :: "'t \<Rightarrow> 'x \<times> 'm \<Rightarrow> bool" assumes "silent_moves t s s'"   and "wfs t s" shows "snd s' = snd s"
that: fixes ns :: "'node list" assumes "g \<turnstile> lookupDefNode g m v-ns\<rightarrow>m"   and "\<forall>x\<in>set (tl ns). v \<notin> defs g x" shows "thesis"
assms: shows "v \<in> dom (range_of \<Psi>)"
SubstP_graph_aux_def: shows "SubstP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SubstP_aux u p q v. x1 = (p, q) \<and> x2 = SyntaxN.Ex v (SyntaxN.Ex u (SubstFormP (Var v) (Var u) p q)) \<and> atom u \<sharp> (p, q, v) \<and> atom v \<sharp> (p, q))"
run_one_step_basic_binop_f_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x271 :: "t"   and x272 :: "binop_f"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $Binop_f x271 x272) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
runit'_simps_5: fixes f :: "'b" assumes "D.ide f" shows "D.cod \<r>\<^sub>D\<^sup>-\<^sup>1[f] = f \<star>\<^sub>D src\<^sub>D f"
heap_def: fixes s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" shows "heap s = snd (snd s)"
case_eq_if: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
monadic_nfoldli_simp_1: fixes c :: "'a \<Rightarrow> bool nres"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a nres"   and s :: "'a" shows "monadic_nfoldli [] c f s = RETURN s"
fset_fset_from_list: fixes l :: "'a list" shows "fset (fset_from_list l) = set l"
ex_inf: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "\<exists>inf. dpo.is_inf le x y inf"
rel_refl_strong: fixes x :: "('h, 'i, 'j) gpv"   and R1a :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and R2a :: "'i \<Rightarrow> 'i \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> results'_gpv x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> outs'_gpv x \<Longrightarrow> R2a z2 z2" shows "rel_gpv R1a R2a x x"
simps_4: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "fw m n k i (Suc j) = fw_upd (fw m n k i j) k i (Suc j)"
simps_2: fixes X :: "freemsg"   and Y :: "freemsg" shows "freediscrim (MPAIR X Y) = 1"
red_simps_3: fixes t :: "'a local.term" assumes "Ide t" shows "Dom (t\<^bold>\<down>) = t"
smult_eq_0_iff: fixes a :: "'a"   and p :: "'a poly" shows "(smult a p = 0) = (a = (0::'a) \<or> p = 0)"
eq_iff: fixes a :: "'b list"   and b :: "'b list" shows "(a = b) = (b \<le>p a \<and> a \<le>p b)"
simps_2: fixes n :: "nat"   and execs :: "'dom_t \<Rightarrow> 'action_t list list" shows "run (Suc n) None execs = None"
simps_1: fixes m :: "'b" shows "\<alpha>ah [] m = {}"
s0_le_s_stop: shows "ego.q \<delta> \<le> ego2.s_stop"
typeI_twoCubeImg: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" assumes "typeI_twoCube twoC" shows "\<exists>a b g1 g2. a < b \<and> (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and> cubeImage twoC = {(x, y). x \<in> {a..b} \<and> y \<in> {g2 x..g1 x}} \<and> twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b))) \<and> g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b}"
Units_inv_Units: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<in> Units G"
apps_eq_tail_conv: fixes r :: "dB"   and ts :: "dB list"   and s :: "dB" shows "(r \<degree>\<degree> ts = s \<degree>\<degree> ts) = (r = s)"
VLambda_in_vproduct: fixes I :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V \<Rightarrow> V" assumes "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f i \<in>\<^sub>\<circ> A i" shows "VLambda I f \<in>\<^sub>\<circ> vproduct I A"
bindU_stateT_StateT: fixes f :: "'b \<rightarrow> (udom \<times> 'b)\<cdot>'a"   and k :: "udom \<rightarrow> udom\<cdot>('a, 'b) stateT" shows "bindU\<cdot>(StateT\<cdot>f)\<cdot>k = StateT\<cdot>(\<Lambda> s. f\<cdot>s \<bind> (\<Lambda> (x, s'). runStateT\<cdot>(k\<cdot>x)\<cdot>s'))"
alphaAbs_qAbs_imp_all_equal_or_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X"   and "qAbs xs x X $= qAbs xs' x' X'" shows "xs = xs' \<and> (\<forall>y. (y = x \<or> qAFresh xs y X) \<and> (y = x' \<or> qAFresh xs y X') \<longrightarrow> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
lr_of_tran_s3_def: fixes ifs :: "char list list"   and ard :: "('a \<times> 32 simple_match \<times> char list \<times> simple_action) list" shows "lr_of_tran_s3 ifs ard = concat (map (\<lambda>(p, r, c, a). map (\<lambda>b. (p, b, case a of Accept \<Rightarrow> [Forward c] | Drop \<Rightarrow> [])) (simple_match_to_of_match r ifs)) ard)"
distrPUTL1: fixes x :: "'c"   and P :: "'c \<Rightarrow> 'd decision option"   and PL :: "('c \<Rightarrow> 'd decision option) list"   and PUT :: "'c \<Rightarrow> 'd decision option" assumes "x \<in> dom P"   and "list2policy PL x = P x"   and "PUTList PUT x PL" shows "PUT x = P x"
Kr: fixes x :: "'a" shows "(1::'a) + x\<^sup>\<star> \<cdot> x \<le> x\<^sup>\<star>"
cPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* P'"   and "distinct (bn \<alpha>)"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* \<Psi>\<^sub>Q"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* subject \<alpha>" shows "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<alpha> \<prec> P' \<parallel> Q"
compseriesTr3: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "compseries G n f" shows "f n = {\<one>\<^bsub>G\<^esub>}"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'c"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "k"   and x15 :: "nat"   and x16 :: "nat"   and x17 :: "nat" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 of Rreq x xa xb xc xd xe xf \<Rightarrow> f1 x xa xb xc xd xe xf | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17"
graph_union_idemp_3: fixes A :: "('a, 'b) labeled_graph"   and B :: "('a, 'b) labeled_graph" shows "graph_union A (graph_union B A) = graph_union B A"
expand: fixes id :: "'d Monomorphic_Monad.id"   and id' :: "'d Monomorphic_Monad.id" assumes "extract id = extract id'" shows "id = id'"
Ksub: fixes K :: "'M set" assumes "K \<in> \<D>" shows "\<exists>x\<in>S \<inter> K. K \<subseteq> ball x (R x)"
mini_first: fixes xs :: "'c list"   and i :: "nat"   and key :: "'c \<Rightarrow> 'd" assumes "xs \<noteq> []"   and "i < mini xs key" shows "key (xs ! mini xs key) < key (xs ! i)"
ehoare_relative_complete: fixes P :: "assn"   and e :: "expr"   and Q :: "vassn" assumes "\<Turnstile>\<^sub>e {P} e {Q}" shows "{} \<turnstile>\<^sub>e {P} e {Q}"
simps_15: fixes b :: "nat"   and \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "fvi b (Until \<phi> I \<psi>) = fvi b \<phi> \<union> fvi b \<psi>"
poly_mult_assoc: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list"   and x :: "'a" shows "poly (a *** b *** c) x = poly (a *** (b *** c)) x"
f_def: fixes k :: "'rows" shows "f k = (if b = a \<and> k = a then p else if b = a \<and> k = b then q else if b = b \<and> k = a then u else if b = b \<and> k = b then v else if b = k then 1::'a else (0::'a)) * A $ k $ j"
dd_commute: fixes r :: "('a \<times> 'a) set"   and L :: "'a \<Rightarrow> ('c \<times> 'c) set"   and R :: "'a \<Rightarrow> ('c \<times> 'c) set" assumes "wf r"   and "trans r"   and "\<And>a b s t u. \<lbrakk>(s, t) \<in> L a; (s, u) \<in> R b\<rbrakk> \<Longrightarrow> (t, u) \<in> conversion'' L R (under r a) O (R b)\<^sup>= O conversion'' L R (under r a \<union> under r b) O ((L a)\<inverse>)\<^sup>= O conversion'' L R (under r b)" shows "commute (\<Union> (range L)) (\<Union> (range R))"
single_comp_portfolio: fixes asset :: "'b"   and qty :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "portfolio (qty_single asset qty)"
distinct_7: fixes x2 :: "name"   and x3 :: "nat" shows "DBVar x2 \<noteq> DBInd x3"
measure_zero_top: fixes M :: "'b measure"   and A :: "'b set" assumes "emeasure M A = top" shows "Sigma_Algebra.measure M A = 0"
aeq: assumes "a = 0" shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. 0 < a * x\<^sup>2 + b * x + c"
assms_2: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"
cod_trg_eq_dom_trg: shows "\<epsilon>.cod.trg = \<epsilon>.dtrg"
U'_1: shows "U' \<subseteq> (+) (- x0) ` U"
inv_image_eq: shows "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = (`) p\<^sub>A ` ((\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}})"
continuous_imp_homeomorphic_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_map X Y f"   and "compact_space X"   and "Hausdorff_space Y"   and "f ` topspace X = topspace Y"   and "inj_on f (topspace X)" shows "homeomorphic_map X Y f"
simps_10: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and x :: "String.literal"   and p :: "fol fm" shows "semantics e f g (Forall x p) = (\<forall>v. semantics (e(x := v)) f g p)"
subgroup_of_group_def: fixes H :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "subgroup_of_group H G composition unit \<equiv> subgroup H G composition unit \<and> group G composition unit"
cgderiv_def: fixes f :: "'a \<Rightarrow> complex"   and x :: "'a"   and D :: "'a" shows "cGDERIV f x :> D = (f has_derivative cinner D) (at x)"
Congr: fixes A :: "'t"   and e1 :: "('t, 'f) Expression"   and e2 :: "('t, 'f) Expression"   and B :: "'t"   and f :: "'f"   and C :: "'t" assumes "Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory"   and "f \<in> Sig aS : B \<rightarrow> C" shows "Vx : A \<turnstile> f E@ e1\<equiv>f E@ e2 : C \<in> Theory"
right_connected: fixes \<mu> :: "'d"   and \<mu>' :: "'d"   and \<nu> :: "'d" assumes "D.seq \<mu> \<mu>'" shows "(\<nu> \<star>\<^sub>D \<mu> \<noteq> D.null) = (\<nu> \<star>\<^sub>D \<mu>' \<noteq> D.null)"
parallel_constructive_security: fixes real1 :: "nat \<Rightarrow> ('i + 'j, 'k + 'l) resource"   and ideal1 :: "nat \<Rightarrow> ('m + 'j, 'n + 'l) resource"   and sim1 :: "nat \<Rightarrow> ('i, 'k, 'm, 'n) converter"   and \<I>_real1 :: "nat \<Rightarrow> ('i, 'k) \<I>"   and \<I>_inner1 :: "nat \<Rightarrow> ('m, 'n) \<I>"   and \<I>_common1 :: "nat \<Rightarrow> ('j, 'l) \<I>"   and bound1 :: "nat \<Rightarrow> enat"   and lossless1 :: "bool"   and w1 :: "nat \<Rightarrow> ('m \<Rightarrow> 'i) \<times> ('k \<Rightarrow> 'n)"   and real2 :: "nat \<Rightarrow> ('o + 'p, 'q + 'r) resource"   and ideal2 :: "nat \<Rightarrow> ('s + 'p, 't + 'r) resource"   and sim2 :: "nat \<Rightarrow> ('o, 'q, 's, 't) converter"   and \<I>_real2 :: "nat \<Rightarrow> ('o, 'q) \<I>"   and \<I>_inner2 :: "nat \<Rightarrow> ('s, 't) \<I>"   and \<I>_common2 :: "nat \<Rightarrow> ('p, 'r) \<I>"   and bound2 :: "nat \<Rightarrow> enat"   and lossless2 :: "bool"   and w2 :: "nat \<Rightarrow> ('s \<Rightarrow> 'o) \<times> ('q \<Rightarrow> 't)" assumes "constructive_security real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 bound1 lossless1 w1"   and "constructive_security real2 ideal2 sim2 \<I>_real2 \<I>_inner2 \<I>_common2 bound2 lossless2 w2"   and "\<And>\<eta>. lossless2 \<Longrightarrow> lossless_resource (\<I>_real1 \<eta> \<oplus>\<^sub>\<I> \<I>_common1 \<eta>) (real1 \<eta>)"   and "\<And>\<eta>. lossless1 \<Longrightarrow> plossless_converter (\<I>_real2 \<eta>) (\<I>_inner2 \<eta>) (sim2 \<eta>)"   and "\<And>\<eta>. lossless1 \<Longrightarrow> lossless_resource (\<I>_inner2 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta>) (ideal2 \<eta>)" shows "constructive_security (\<lambda>\<eta>. parallel_wiring \<rhd> real1 \<eta> \<parallel> real2 \<eta>) (\<lambda>\<eta>. parallel_wiring \<rhd> ideal1 \<eta> \<parallel> ideal2 \<eta>) (\<lambda>\<eta>. sim1 \<eta> |\<^sub>= sim2 \<eta>) (\<lambda>\<eta>. \<I>_real1 \<eta> \<oplus>\<^sub>\<I> \<I>_real2 \<eta>) (\<lambda>\<eta>. \<I>_inner1 \<eta> \<oplus>\<^sub>\<I> \<I>_inner2 \<eta>) (\<lambda>\<eta>. \<I>_common1 \<eta> \<oplus>\<^sub>\<I> \<I>_common2 \<eta>) (\<lambda>\<eta>. min (bound1 \<eta>) (bound2 \<eta>)) (lossless1 \<or> lossless2) (\<lambda>\<eta>. w1 \<eta> |\<^sub>w w2 \<eta>)"
R_N_bound: shows "\<forall>M\<ge>N. \<bar>R M\<bar> \<le> 1 / 4"
ordinal_plus_oSuc: fixes x :: "ordinal"   and y :: "ordinal" shows "x + oSuc y = oSuc (x + y)"
OV_inv2_inductive_1: shows "init obsv_TS \<subseteq> OV_inv2"
supp_Inl: fixes x :: "'a" shows "supp (Inl x) = supp x"
applicableConflictCharacterization: fixes stateA :: "State" shows "applicableConflict stateA = (\<exists>clause. getConflictFlag stateA = False \<and> formulaEntailsClause (getF stateA) clause \<and> clauseFalse clause (elements (getM stateA)))"
step: fixes usvs :: "[:'a:] \<times> 'b" assumes "\<And>usvs'. lt\<cdot>(slength\<cdot>(cfst\<cdot>usvs'))\<cdot>(slength\<cdot>(cfst\<cdot>usvs)) = TT \<Longrightarrow> P usvs'" shows "P usvs"
assms_2: fixes x :: "'b"   and y :: "'b" assumes "r' x y" shows "r (g x) (g y)"
C2O_app: fixes xs :: "C list"   and ys :: "C list" shows "C2O (C (xs @ ys)) = addO (C2O (C ys)) (C2O (C xs))"
assms_2: shows "ring (R\<lparr>carrier := H\<rparr>)"
less_seq_def: fixes xs :: "'b seq"   and ys :: "'b seq" shows "(xs < ys) = ((xs, ys) \<in> seq_lexord {(xs, ys). xs < ys})"