hcju/isabelleps huggingface dataset:
sees_method_mono: fixes C' :: "char list"   and C :: "char list"   and P :: "'a prog"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M, b :  Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> subtype P T' T"
coincidence_dterm': fixes \<theta> :: "('d, 'e) trm"   and \<nu> :: "(real, 'e) vec \<times> (real, 'e) vec"   and \<nu>' :: "(real, 'e) vec \<times> (real, 'e) vec"   and I :: "('d, 'f, 'e) interp"   and J :: "('d, 'f, 'e) interp" assumes "dsafe \<theta>"   and "Vagree \<nu> \<nu>' (FVT \<theta>)"   and "Iagree I J {Inl x |x. x \<in> SIGT \<theta>}" shows "dterm_sem I \<theta> \<nu> = dterm_sem J \<theta> \<nu>'"
split_list_last_propE: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'b list" and x :: "'b" and zs :: "'b list" where "xs = ys @ x # zs" and "P x" and "\<forall>z\<in>set zs. \<not> P z"
neg_0_less_iff_less: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (uminus a) = less a zero"
intro: fixes dom :: "'c set"   and carrier :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "dom \<noteq> {}"   and "\<And>i. i \<in> dom \<Longrightarrow> Order_Predicates.preorder_on carrier (R i)"   and "\<And>i x y. i \<notin> dom \<Longrightarrow> \<not> R i x y" shows "preorder_family dom carrier R"
score_1_def: fixes e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and strat :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat" shows "score_1 e strat = (let states = Inference.S e; pairs_to_score = ffilter (\<lambda>(x, y). x < y) (states |\<times>| states) in Let ((\<lambda>(s1, s2). \<lparr>Score = score_state_pair strat e s1 s2, S1 = s1, S2 = s2\<rparr>) |`| pairs_to_score) (ffilter (\<lambda>x. 0 < Score x)))"
fnn: fixes x :: "'a" assumes "x \<in> M" shows "0 \<le> f x"
list_insert_def: fixes x :: "'a"   and xs :: "'a list" shows "list_insert x xs = (if list_member xs x then xs else x # xs)"
o_def: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "f \<circ> g = (\<lambda>x. f (g x))"
open_nat_def: shows "open = generate_topology (range lessThan \<union> range greaterThan)"
conseq: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com" assumes "\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> e s \<le> k * e' s \<and> (\<forall>t. \<exists>l'. P l' s \<and> (Q l' t \<longrightarrow> Q' l t))"   and "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}" shows "\<turnstile>\<^sub>1 {P'} c { e' \<Down> Q'}"
in_abs_v_sec_length: fixes v :: "nat" shows "(v \<in> set abs_ast_variable_section) = (v < length astDom)"
bounds_lg_6: fixes c :: "'a"   and b :: "'a option"   and x :: "'a" assumes "c \<le>\<^sub>l\<^sub>b b"   and "x \<ge>\<^sub>l\<^sub>b b" shows "c \<le> x"
res: shows "(if g = 0 then (0, []) else if n = 0 then (d, hs) else (d, (monom 1 1, n - 1) # hs)) = (c, fs)"
DomF': shows "A_B.Dom (fst Fg') = A_B.Map (A_B.dom (fst Fg'))"
subspace_subset: fixes U :: "'b set"   and V :: "'b set" assumes "Subspace.subspace U V" shows "U \<subseteq> V"
P: shows "P \<subseteq> {x. a *\<^sub>R i \<bullet> x = 0}"
pconj_idem: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" shows "\<guillemotleft> P \<guillemotright> s .& \<guillemotleft> P \<guillemotright> s = \<guillemotleft> P \<guillemotright> s"
assms_3: shows "nprv (insert \<psi> F) \<chi>"
Diamondo_undef: fixes \<alpha> :: "game option"   and \<phi> :: "fml option" shows "(Diamondo \<alpha> \<phi> = undeff) = (\<alpha> = undefg \<or> \<phi> = undeff)"
tendsto_imp_eventually_ne: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and c :: "'c"   and F :: "'d filter"   and c' :: "'c" assumes "class.t1_space open"   and "filterlim f (topological_space.nhds open c) F"   and "c \<noteq> c'" shows "\<forall>\<^sub>F z in F. f z \<noteq> c'"
int_of_uint32_mult: fixes x :: "uint32"   and y :: "uint32" shows "int_of_uint32 (x * y) = int_of_uint32 x * int_of_uint32 y mod 4294967296"
add_strict_increasing: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "{#} \<subset># a"   and "b \<subseteq># c" shows "b \<subset># a + c"
fps_conv_radius_fps_X_power: fixes n :: "nat" shows "fps_conv_radius (fps_X ^ n) = \<infinity>"
length_concat_negGorbit_list: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" shows "length (concat (negHorbit_list gs T as)) = length gs * length as"
eq_add_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.ab_group_add plus zero minus uminus" shows "(x = plus x y) = (y = zero)"
map_upd_cut_irrelevant: fixes tab :: "'b \<Rightarrow> 'a option"   and x :: "'b"   and y :: "'a"   and vn :: "'b"   and el :: "'a"   and tab' :: "'b \<Rightarrow> 'a option" assumes "(tab(x \<mapsto> y)) vn = Some el"   and "(tab'(x \<mapsto> y)) vn = None" shows "tab vn = Some el"
eta_reds_abs: fixes s :: "Core.term"   and t :: "Core.term"   and T :: "typ" assumes "s \<rightarrow>\<^sub>\<eta>\<^sup>* t" shows "Abs T s \<rightarrow>\<^sub>\<eta>\<^sup>* Abs T t"
eucl_of_list_replicate_zero: fixes E :: "nat" shows "eucl_of_list (replicate E 0) = (0::'a)"
nlists_def: fixes A :: "'b set"   and n :: "nat" shows "nlists A n \<equiv> {l. set l \<subseteq> A \<and> length l = n}"
le_trans_top: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy t" shows "le_trans t (\<lambda>P s. bound_of P)"
ab_semigroup_add_ow_def: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ab_semigroup_add_ow S pls \<equiv> semigroup_add_ow S pls \<and> ab_semigroup_add_ow_axioms S pls"
pathfinish_in_path_image: fixes g :: "real \<Rightarrow> 'c" shows "pathfinish g \<in> path_image g"
inf_top_left: fixes x :: "'b upred" shows "(true \<and> x) = x"
fst_row_add_iterate_PA_eq_fst_Gauss_Jordan_in_ij_PA: fixes P :: "(('b, 'c) vec, 'c) vec"   and i :: "'c"   and A :: "(('b, 'd) vec, 'c) vec"   and j :: "'d" shows "fst (row_add_iterate_PA (mult_row (interchange_rows P i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n)) i ((1::'b) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n) $ i $ j), mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n)) i ((1::'b) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n) $ i $ j)) (nrows A - 1) i j) = fst (Gauss_Jordan_in_ij_PA (P, A) i j)"
strong_postdominate_path_branch: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and n :: "'b"   and as :: "'a list"   and n'' :: "'b"   and n' :: "'b" assumes "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFG.path sourcenode targetnode valid_edge n as n''"   and "StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n' n''"   and "\<not> StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n' n" obtains a :: "'a" and as' :: "'a list" and as'' :: "'a list" where "as = as' @ a # as''" and "valid_edge a" and "\<not> StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n' (sourcenode a)" and "StrongPostdomination.strong_postdominate sourcenode targetnode valid_edge Exit n' (targetnode a)"
alphaEnv_def: shows "alphaEnv = {(rho, rho'). \<forall>xs. sameDom (rho xs) (rho' xs) \<and> liftAll2 (#=) (rho xs) (rho' xs)}"
partial_net_preserves_subnets: fixes s :: "'c net_state"   and t :: "'c net_state"   and a :: "'d node_action"   and st' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree" assumes "(SubnetS s t, a, st') \<in> pnet_sos (automaton.trans (pnet np p1)) (automaton.trans (pnet np p2))" shows "\<exists>s' t'. st' = SubnetS s' t'"
linmap_im_of_lincombTr: fixes A :: "'b set"   and N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "R module N"   and "f \<in> mHom R M N"   and "H \<subseteq> carrier M" shows "s \<in> {j. j \<le> n} \<rightarrow> A \<and> g \<in> {j. j \<le> n} \<rightarrow> H \<longrightarrow> f (l_comb R M n s g) = l_comb R N n s (cmp f g)"
f_Streams_Connected_def: fixes outS :: "'d message_af list"   and inS :: "'d message_af list" shows "f_Streams_Connected outS inS \<equiv> inS = NoMsg # outS"
word_mult_le_mono1: fixes i :: "'b word"   and j :: "'b word"   and k :: "'b word" assumes "i \<le> j"   and "0 < k"   and "unat j * unat k < 2 ^ LENGTH('b)" shows "i * k \<le> j * k"
rec_dummy_pi_def: shows "rec_dummy_pi = Pr 1 (constn 2) (Cn 3 rec_np [recf.id 3 2])"
Nats_diff_nonpos_Ints: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<nat>"   and "y \<in> \<int>\<^sub>\<le>\<^sub>0" shows "x - y \<in> \<nat>"
non_empty: fixes V :: "'b set" assumes "vectorspace V" shows "V \<noteq> {}"
filterlim_at_top_div_const_nat: fixes c :: "nat" assumes "0 < c" shows "LIM x sequentially. x div c :> sequentially"
monom_of_nat: fixes n :: "nat" shows "MPoly_Type.monom 0 (of_nat n) = of_nat n"
fun_upd_same: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'c" shows "(f(x := y)) x = y"
getNonWatchedUnfalsifiedLiteralSomeCharacterization_2: fixes clause :: "Literal list"   and w1 :: "Literal"   and w2 :: "Literal"   and M :: "(Literal \<times> bool) list"   and l :: "Literal" assumes "getNonWatchedUnfalsifiedLiteral clause w1 w2 M = Some l" shows "l \<noteq> w1"
emeasure_exponential_Ioi_cutoff: fixes l :: "real"   and x :: "real" assumes "0 < l" shows "emeasure (exponential l) {x<..} = ennreal (exp (- max 0 x * l))"
Outside_def: fixes R :: "('c \<times> 'd) set"   and X :: "'c set" shows "R outside X = R - X \<times> Range R"
can_step_def: fixes l :: "'l"   and s :: "'c" shows "C.can_step l s \<equiv> \<exists>s'. clstep l s s'"
preserves_dom: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "local.dom (src f) = src (local.dom f)"
IH: assumes "inj_on f (set xs)"   and "x \<in> set xs" shows "foldr (\<lambda>x \<A>. \<A>(f x := g x)) xs \<A> (f x) = g x"
strict_weak_trans: fixes a :: "'alt pmf"   and b :: "'alt pmf"   and c :: "'alt pmf" assumes "a \<prec>[Pareto (SD \<circ> R)] b"   and "Pareto (SD \<circ> R) b c" shows "a \<prec>[Pareto (SD \<circ> R)] c"
Plus_ZeroR: fixes r :: "'a language" shows "Plus r Zero = r"
Polys_mono: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<subseteq> Y" shows "P[X] \<subseteq> P[Y]"
div_mod_imp_dvdm: fixes b :: "int poly"   and a :: "int poly" assumes "\<exists>q r. b = q * a + smult p r" shows "a dvdm b"
ModuleHom_def: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" shows "ModuleHom smult M smult' T \<equiv> RModuleHom UNIV smult M smult' T"
LIMSEQ_abs_realpow_zero: fixes c :: "real" assumes "\<bar>c\<bar> < 1" shows "(^) \<bar>c\<bar> \<longlonglongrightarrow> 0"
eventually_zero: fixes k :: "nat"   and m :: "nat" assumes "f \<bullet> ns (k + m) = \<zero>" shows "f \<bullet> ns (k + Suc m) = \<zero>"
iso_inv_iso: fixes f :: "'a" assumes "S.iso f" shows "S.iso (S.inv f)"
axioms_1: fixes inner_cols :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> 'a"   and inner_rows :: "('a, 'c) vec \<Rightarrow> ('a, 'c) vec \<Rightarrow> 'a" assumes "Generalizations2.matrix inner_cols inner_rows" shows "vec_real_inner (\<lambda>x y. x * cnj_class.cnj y) inner_cols"
extremum_uniqueI: fixes a :: "'a" assumes "a \<le> bot" shows "a = bot"
not_neutral_contains_not_neutral: fixes g :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "prod g A \<noteq> (1::'a)" obtains a :: "'b" where "a \<in> A" and "g a \<noteq> (1::'a)"
simps_3: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F> (X\<^sub>n \<phi>) w = \<F> \<phi> w"
component_in_hom: fixes j :: "'b" assumes "J1.arr j" shows "\<guillemotleft>\<pi>1 j : a1 \<rightarrow> D1 (J1.cod j)\<guillemotright>"
trd_def: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "trd fs p = trd_aux fs p 0"
arrE: fixes f :: "'a \<times> 'a" assumes "VxV.arr f" obtains "arr (fst f) \<and> arr (snd f)"
assms_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" shows "P f [x]"
\<zeta>: fixes p :: "fm"   and a :: "real"   and bs :: "real list" assumes "iszlfm p (a # bs)" shows "0 < \<zeta> p \<and> d_\<beta> p (\<zeta> p)"
ocircline_eq_cmat_def: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "ocircline_eq_cmat A B = (\<exists>k>0. B = cor k *\<^sub>s\<^sub>m A)"
pos_imp_zdiv_neg_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(a div b < 0) = (a < 0)"
Collect_bex_eq: fixes A :: "'d set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "{x. \<exists>y\<in>A. P x y} = (\<Union>y\<in>A. {x. P x y})"
ilist_Previous_gr0: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd"   and init :: "'d" assumes "0 < n" shows "f\<^bsup>\<leftarrow> init\<^esup> n = f (n - Suc 0)"
semigroup_axioms: shows "semigroup (\<^bold>\<sqinter>)"
Polygamma_of_real: fixes x :: "real"   and n :: "nat" assumes "x \<noteq> 0" shows "Polygamma n (of_real x) = of_real (Polygamma n x)"
countable_enum_cases: fixes S :: "'b set" assumes "countable S" obtains f :: "'b \<Rightarrow> nat" where "finite S" and "bij_betw fa__ S {..<card S}"   | f :: "'b \<Rightarrow> nat" where "infinite S" and "bij_betw f S UNIV"
inner_node_Entry_edge: fixes n :: "'node" assumes "inner_node n" obtains a :: "'edge" where "valid_edge a" and "intra_kind (kind a)" and "inner_node (targetnode a)" and "sourcenode a = (_Entry_)"
continuous_at_imp_continuous_at_within: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'e"   and s :: "'d set" assumes "isCont f x" shows "continuous (at x within s) f"
ssufI1: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<cdot> v = w"   and "u \<noteq> \<epsilon>" shows "v <s w"
sg_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "G \<guillemotright> H  = (H \<noteq> {} \<and> H \<subseteq> carrier G \<and> (\<forall>a\<in>H. \<forall>b\<in>H. a \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> b \<in> H))"
flow_unlift: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a node \<times> 'a node \<Rightarrow> ennreal" assumes "flow_attainability \<Delta>"   and "flow (flow_attainability.\<Delta>' \<Delta>) f" shows "flow \<Delta> (\<lambda>(x, y). f (Inner x, Inner y))"
NCol_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C \<or> \<not> Tarski_neutral_dimensionless.Col Bet A C B \<or> \<not> Tarski_neutral_dimensionless.Col Bet B A C \<or> \<not> Tarski_neutral_dimensionless.Col Bet B C A \<or> \<not> Tarski_neutral_dimensionless.Col Bet C A B \<or> \<not> Tarski_neutral_dimensionless.Col Bet C B A" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C"
fminus_iff: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" shows "(c |\<in>| A |-| B) = (c |\<in>| A \<and> c |\<notin>| B)"
wfI_min: fixes R :: "('e \<times> 'e) set" assumes "\<And>x Q. x \<in> Q \<Longrightarrow> \<exists>z\<in>Q. \<forall>y. (y, z) \<in> R \<longrightarrow> y \<notin> Q" shows "wf R"
inverse_pow_pow: fixes G :: "('c, 'd) cyclic_group_scheme"   and a :: "'c"   and r :: "nat" assumes "cyclic_group G"   and "a \<in> carrier G" shows "inv\<^bsub>G\<^esub> (a [^]\<^bsub>G\<^esub> r) = inv\<^bsub>G\<^esub> a [^]\<^bsub>G\<^esub> r"
tendsto_of_real: fixes g :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. of_real (g x)) \<longlongrightarrow> of_real a) F"
assms_3: fixes p :: "real poly" assumes "p \<in> set fss" shows "p \<noteq> 0"
cod_mkArr: fixes A :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and B :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr set"   and F :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr \<Rightarrow> (('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "S.arr (S.mkArr A B F)" shows "S.cod (S.mkArr A B F) = S.mkIde B"
pcompose_mult: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "(p * q) \<circ>\<^sub>p r = p \<circ>\<^sub>p r * q \<circ>\<^sub>p r"
alphaAbs_qAbs_iff_alphaAbs_all_equal_or_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_all_equal_or_qAFresh xs x X xs' x' X'"
minus_less_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (uminus a) b = less (uminus b) a"
w_simps_6: shows "trg\<^sub>B w = a"
Extend_def: fixes S :: "('c, 'd) fm set"   and f :: "nat \<Rightarrow> ('c, 'd) fm" shows "Extend S f \<equiv> \<Union> (range (extend S f))"
split_list_last: fixes x :: "'a"   and xs :: "'a list" assumes "x mem xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
distinct_1: fixes x1 :: "'a"   and x2 :: "'a" shows "LOC x1 \<noteq> ENV x2"
ltrm_deg_0: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "deg R p = 0" shows "monom P (p (deg R p)) (deg R p) = p"
distinct_fst: fixes A :: "('a \<times> 'b) list" assumes "distinct (map fst A)" shows "distinct A"
ccw_convex': fixes u :: "real"   and v :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real"   and d :: "real \<times> real" assumes "0 \<le> u"   and "0 \<le> v"   and "u + v = 1"   and "ccw a b c"   and "coll a b c"   and "ccw a b d"   and "det3 a b d \<noteq> 0" shows "ccw a b (u *\<^sub>R c + v *\<^sub>R d)"
success_nthI: fixes i :: "nat"   and h :: "heap"   and a :: "'b array" assumes "i < Array.length h a" shows "success (a.(i)) h"
R0: fixes out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g" shows "([], {(x, p). p \<noteq> out s\<^sub>0 x}) \<in> c_failures step out s\<^sub>0"
i_cut_singleton_1: fixes a :: "'a"   and t :: "'a" shows "{a} \<down>\<le> t = (if a \<le> t then {a} else {})"
T_def: shows "T = \<lparr>verts = {x}, arcs = {}\<rparr>"
s\<^sub>1_eq_s\<^sub>1': shows "s\<^sub>1 \<approx> s\<^sub>1'"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "ring_isom hom" shows "comm_ring_isom hom"
monomial_adds_nat_iff: fixes i :: "'a"   and p :: "nat"   and c :: "'a \<Rightarrow>\<^sub>0 nat" shows "(monomial p i adds c) = (p \<le> lookup c i)"
arev_in_arcs: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev GM a \<in> arcs G"
p_in_ns: shows "p \<in> set prx \<or> p \<in> set sfx"
cnf_to_bool_append: fixes \<gamma> :: "'b \<Rightarrow> bool"   and a1 :: "'b negation_type list"   and a2 :: "'b negation_type list" shows "cnf_to_bool \<gamma> (a1 @ a2) = (cnf_to_bool \<gamma> a1 \<and> cnf_to_bool \<gamma> a2)"
np_1: fixes a :: "'a" assumes "a \<in> carrier R" shows "a^\<^bsup>R Suc 0\<^esup> = a"
comp_by_index_is_pr: shows "comp_by_index \<in> PrimRec2"
lift_def: fixes h :: "'d \<Rightarrow> 'c"   and inp :: "'b \<Rightarrow> 'd option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
sup_singleton: fixes a :: "'a" shows "a \<in> supremum {a}"
eager_def: fixes p :: "'v \<Rightarrow> int pmf"   and e :: "'v exp" shows "eager p e = sample_vars p (vars e) (eval lookup e)"
prime_imp_power_coprime: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and p :: "'c"   and a :: "'c"   and m :: "nat" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "\<not> dvd.dvd times p a" shows "algebraic_semidom.coprime times one a (power.power one times p m)"
simps: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and s :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a1 :: "'g"   and a2 :: "'i" shows "(r OO s) a1 a2 = (\<exists>a b c. a1 = a \<and> a2 = c \<and> r a b \<and> s b c)"
ccr_flush_def: fixes state :: "'b sparc_state" shows "ccr_flush state \<equiv> let ccr_val = sys_reg_val CCR state; fi_val = ccr_val AND 2097152 >> 21; fd_val = ccr_val AND 4194304 >> 22; state1 = if fi_val = 1 then flush_instr_cache state else state in if fd_val = 1 then flush_data_cache state1 else state1"
simps_3: fixes f :: "'c \<Rightarrow> 'a"   and b :: "'b"   and ts :: "'c FOL_Fitting.term list" shows "psubst f (Pred b ts) = Pred b (psubstts f ts)"
representation_sum: fixes basis :: "'b set"   and I :: "'c set"   and v :: "'c \<Rightarrow> 'b" assumes "independent basis"   and "\<And>i. i \<in> I \<Longrightarrow> v i \<in> span basis" shows "representation basis (sum v I) = (\<lambda>b. \<Sum>i\<in>I. representation basis (v i) b)"
counit_simps_4: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (counit a) = G.map\<^sub>0 a"
cases: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "a \<in> least_map f S" obtains x :: "'b" where "a = x" and "x \<in> S" and "\<forall>x'\<in>S. f x \<le> f x'"
pred_of_set_def: shows "pred_of_set = pred.Pred \<circ> (\<lambda>A x. x \<in> A)"
6: shows "(\<sqinter>xa\<in>range left \<union> range right \<rightarrow>  \<sqinter>xa\<in>range left \<union> range right \<rightarrow>  P) \<le> (left`?`x \<rightarrow> right`!`x \<rightarrow> P)"
SUCC_subst: fixes i :: "name"   and t :: "tm"   and k :: "tm" shows "subst i t (SUCC k) = SUCC (subst i t k)"
sdiv_neg_pos: fixes a :: "int"   and b :: "int" assumes "a < 0"   and "0 \<le> b" shows "a sdiv b = - (- a div b)"