hcju/isabelleps huggingface dataset:
i_extension_mono: fixes f :: "'a \<Rightarrow> 'c"   and I1 :: "'a set"   and I2 :: "'a set" assumes "f \<in> rHom R R'"   and "ideal R I1"   and "ideal R I2"   and "I1 \<subseteq> I2" shows "i_extension f R R' I1 \<subseteq> i_extension f R R' I2"
that: fixes b :: "'a"   and a :: "'a" assumes "c = to_fract b / to_fract a"   and "smult a p = smult b q"   and "coprime a b"   and "normalize a = a" shows "thesis"
berlekamp_basis_length_factorization: fixes f :: "'a mod_ring poly"   and n :: "nat"   and us :: "'a mod_ring poly list" assumes "square_free f"   and "n = length (berlekamp_basis f)"   and "monic f"   and "f = prod_list us"   and "\<And>u. u \<in> set us \<Longrightarrow> 0 < degree u" shows "length us \<le> n"
f_linear: shows "linear (FunctionFrechet I id1 (\<chi>i. sterm_sem I (local.empty i) (fst \<nu>)))"
fstI: fixes x :: "'b \<times> 'c"   and y :: "'b"   and z :: "'c" assumes "x = (y, z)" shows "fst x = y"
1: shows "cod m = cod m' \<and> local.dom e = local.dom e'"
that: assumes "\<sigma>' = Some (snd (f (the_state \<sigma>)))"   and "res = fst (f (the_state \<sigma>))" shows "thesis"
continuous_on_mult': fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on A f"   and "continuous_on A g" shows "continuous_on A (\<lambda>x. f x * g x)"
simps_3: fixes I :: "('e, 'f, 'd) interp"   and f :: "'e"   and args :: "'d \<Rightarrow> ('e, 'd) trm" shows "dterm_sem I ($f f args) = (\<lambda>v. Functions I f (\<chi>i. dterm_sem I (args i) v))"
ipresIGWls_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'sort, 'm, 'n, 'i, 'o, 'p) model_scheme" shows "ipresIGWls h termMOD MOD = ipresWls h MOD"
runit_eqI: fixes f :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "ide f"   and "\<guillemotleft>\<mu> : f \<star> src f \<Rightarrow> f\<guillemotright>"   and "\<mu> \<star> src f = (f \<star> \<i>[src f]) \<bullet> \<a>[f, src f, src f]" shows "\<mu> = \<r>[f]"
psubstt_id_2: fixes ts :: "'a FOL_Fitting.term list" shows "psubstts (\<lambda>x. x) ts = ts"
assms_3: shows "Broadcast m2 \<in> set (history i)"
get_action_def: fixes rule :: "'b rule" shows "get_action rule = (case rule of Rule x1 x2 \<Rightarrow> x2)"
reachable_edges_subset: shows "edges rg \<subseteq> edges g"
classical_operator_exists_inj: fixes \<pi> :: "'c \<Rightarrow> 'd option" assumes "inj_map \<pi>" shows "classical_operator_exists \<pi>"
Cons_replicate_eq: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat"   and y :: "'b" shows "(x # xs = replicate n y) = (x = y \<and> 0 < n \<and> xs = replicate (n - 1) x)"
l_one: fixes x :: "int" assumes "x \<in> UNIV" shows "1 * x = x"
mod_mod_trivial: fixes a :: "'b"   and b :: "'b" shows "a mod b mod b = a mod b"
is_maxchopper_def: fixes P :: "'b list \<Rightarrow> bool"   and chopper :: "'b list \<Rightarrow> 'b list list \<times> 'b list" shows "is_maxchopper P chopper = (\<forall>xs zs yss. (chopper xs = (yss, zs)) = (xs = concat yss @ zs \<and> (\<forall>ys\<in>set yss. ys \<noteq> []) \<and> (case yss of [] \<Rightarrow> is_maxpref P [] xs | us # uss \<Rightarrow> is_maxpref P us xs \<and> chopper (concat uss @ zs) = (uss, zs))))"
viewR_aux_def: fixes t :: "('c, 'd) FingerTree" shows "viewR_aux t \<equiv> Abs_viewres (FingerTreeStruc.viewR (Rep_FingerTree t))"
periodN_indeces: fixes w :: "'b list"   and n :: "nat"   and i :: "nat" assumes "periodN w n"   and "i + n < \<^bold>|w\<^bold>|" shows "w ! i = w ! (i + n)"
step_snap: shows "S t j \<turnstile> t ! j \<mapsto> S t (j + 1)"
tensor_in_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d"   and g :: "'d"   and c :: "'d"   and d :: "'d" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>f \<otimes> g : a \<otimes> c \<rightarrow> b \<otimes> d\<guillemotright>"
E2_def: shows "E2 = (\<Sum>i<2 * N. \<bar>u 1 ((T ^^ (n - (2 * N - i))) x)\<bar>)"
monom_list_degree: fixes x :: "'c"   and m :: "('d \<times> nat) list" shows "eval_monom_list (\<lambda>_. x) m = x ^ monom_list_degree m"
intros_1: fixes F :: "'b formula" assumes "F \<in> AX0" shows "F \<in> AX10"
Rep_ratfps_cases: fixes y :: "'b poly fract" assumes "y \<in> {x. constant_term (denominator x) \<noteq> (0::'b)}" obtains x :: "'b ratfps" where "y = Rep_ratfps x"
Disj_SubstAtomicP_Fls: fixes y :: "tm"   and z :: "tm"   and w :: "tm"   and v :: "tm"   and tm :: "tm"   and y' :: "tm" shows "{y EQ Q_Disj z w, SubstAtomicP v tm y y'} \<turnstile> Fls"
xs: shows "xs = concat (map (roots_of_complex_main \<circ> fst) pis)"
sum_upto_moebius_mu_integral': fixes x :: "real" assumes "1 < x" shows "(A has_integral x * A x - M x) {1..x}"
span_insertI: fixes p :: "'b"   and B :: "'b set"   and r :: "'b" assumes "p \<in> span B" shows "p \<in> span (insert r B)"
RETURN_RES_refine: fixes x :: "'c"   and R :: "('c \<times> 'a) set"   and X :: "'a set" assumes "\<exists>x'. (x, x') \<in> R \<and> x' \<in> X" shows "RETURN x \<le> \<Down> R (RES X)"
vsv_vrestriction: fixes A :: "V" shows "vsv (r\<inverse>\<^sub>\<circ> \<restriction>\<^sub>\<circ> A)"
extremum_unique: fixes a :: "'a::order_top" shows "(\<top> \<le> (a::'a::order_top)) = (a = \<top>)"
of_integer_def: fixes R :: "('b, 'c) ring_scheme"   and i :: "int" shows "\<guillemotleft>i\<guillemotright>\<^bsub>R\<^esub> = (if 0 \<le> i then \<guillemotleft>nat i\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub> else \<ominus>\<^bsub>R\<^esub> \<guillemotleft>nat (- i)\<guillemotright>\<^sub>\<nat>\<^bsub>R\<^esub>)"
l_set_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_tag_name type_wf set_tag_name set_tag_name_locs \<equiv> ((\<forall>element_ptr tag h h'. writes (set_tag_name_locs element_ptr) (set_tag_name element_ptr tag) h h') \<and> (\<forall>h element_ptr tag. type_wf h \<longrightarrow> element_ptr |\<in>| element_ptr_kinds h \<longrightarrow> h \<turnstile> ok set_tag_name element_ptr tag)) \<and> (\<forall>w element_ptr h h'. w \<in> set_tag_name_locs element_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> object_ptr_kinds h = object_ptr_kinds h') \<and> (\<forall>w element_ptr h h'. w \<in> set_tag_name_locs element_ptr \<longrightarrow> h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> type_wf h = type_wf h')"
tauCases': fixes P :: "pi"   and Rs :: "residual" assumes "\<tau>.P \<longmapsto> Rs" obtains P :: "pi" where "\<tau>.Pb__ = \<tau>.P" and "Rs = \<tau> \<prec> P"
vbrelation_vinE: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> T\<lparr>ArrVal\<rparr>" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>)" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>)"
span_insertD: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "p \<in> pmdl (insert r B)"   and "r \<in> pmdl B" shows "p \<in> pmdl B"
fmdom'_notI: fixes m :: "('b, 'a) fmap"   and x :: "'b" assumes "fmlookup m x = None" shows "x \<notin> fmdom' m"
restrict_chart_restrict_chart: fixes X :: "'c set"   and Y :: "'c set"   and c :: "('c, 'd) chart" assumes "open X"   and "open Y" shows "restrict_chart X (restrict_chart Y c) = restrict_chart (X \<inter> Y) c"
split_n_length: fixes es :: "v list"   and n :: "nat"   and es1 :: "v list"   and es2 :: "v list" assumes "split_n es n = (es1, es2)"   and "n \<le> length es" shows "length es1 = n"
bounded_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<le> min b c) = (a \<le> b \<and> a \<le> c)"
A_3: fixes a :: "'a set" assumes "a \<in> A" shows "emeasure M a \<noteq> \<infinity>"
assms_3: shows "x \<in>\<^sub>\<circ> A"
SourcesS10_L2: shows "Sources level2 sS10 = {sS9}"
E_const: fixes a :: "real" shows "E (return_pmf a) = a"
isint_neg: fixes e :: "MIR.num"   and bs :: "real list" assumes "isint e bs" shows "isint (Neg e) bs"
sel_2: fixes x1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x2 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x3 :: "('a, 'b) state \<Rightarrow> (nat, 'a \<times> 'b) mapping"   and x4 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x5 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'b option"   and x6 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x7 :: "('a, 'b) state \<Rightarrow> nat \<Rightarrow> 'a"   and x8 :: "((nat, 'a \<times> 'b) mapping \<Rightarrow> (nat, 'a \<times> 'b) mapping) \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and x9 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x10 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.LBI (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x2"
one_add_one: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k" assumes "class.numeral plus" shows "plus one one = numeral.numeral one plus (num.Bit0 num.One)"
const_of_i64: fixes cs :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context" assumes "const_list cs"   and "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_i64]" shows "\<exists>c. cs = [$C ConstInt64 c]"
poly_nate_polypoly': fixes bs :: "'b list"   and p :: "'b poly" shows "poly_nate bs p = polypoly' bs (polynate p)"
inf_continuous_sup: fixes P :: "'c \<Rightarrow> 'd"   and Q :: "'c \<Rightarrow> 'd" assumes "inf_continuous P"   and "inf_continuous Q" shows "inf_continuous (\<lambda>x. sup (P x) (Q x))"
fmapU_error_simps_3: fixes f :: "udom \<rightarrow> udom"   and x :: "udom" shows "fmapU\<cdot>f\<cdot>(Ok\<cdot>x) = Ok\<cdot>(f\<cdot>x)"
inverse_less_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse a) (inverse b) = ((less zero (times a b) \<longrightarrow> less b a) \<and> (less_eq (times a b) zero \<longrightarrow> less a b))"
valuation_of_state_variable_implies_lit_semantics_if: fixes v :: "'b"   and S :: "'b \<Rightarrow> bool option"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and k :: "nat"   and vs :: "'b list" assumes "v \<in> dom S"   and "\<A> (State k (index vs v)) = the (S v)" shows "lit_semantics \<A> (literal_formula_to_literal (encode_state_variable k (index vs v) (S v)))"
complete_bipartite_digraph_are_iso: fixes m :: "nat"   and n :: "nat"   and G :: "('c, 'd) pre_digraph"   and H :: "('e, 'f) pre_digraph" assumes "K\<^bsub>m,n\<^esub> G"   and "K\<^bsub>m,n\<^esub> H" shows "digraph_iso G H"
put_eq: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and k :: "'e"   and u :: "'e"   and \<rho> :: "'f"   and v :: "'e" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
ts_ord_anti: fixes x :: "'a"   and y :: "'a" assumes "!x \<sqsubseteq> !y" shows "t y \<sqsubseteq> t x"
uv: shows "u \<cdot> v = v \<cdot> u"
islimpt_insert: fixes x :: "'c"   and a :: "'c"   and s :: "'c set" shows "x islimpt insert a s = x islimpt s"
inorder_tree: fixes t :: "'c bro" shows "inorder (tree t) = inorder t"
bigo_real_nat_transfer: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "f \<in> O(g)" shows "(\<lambda>x. f (real x)) \<in> O(\<lambda>x. g (real x))"
prefix_FF_not_snilD: fixes xs :: "[:'r:]"   and ys :: "[:'r:]" assumes "prefix\<cdot>xs\<cdot>ys = FF" shows "xs \<noteq> [::]"
span_is_submodule: fixes S :: "'c set" assumes "S \<subseteq> carrier M" shows "LinearCombinations.submodule R (span S) M"
assms_2: shows "t \<le> v * v\<^sup>T \<sqinter> g"
power_2_ge_iff: fixes n :: "nat" shows "(2 ^ n - 1 < 2 ^ n) = (n < LENGTH('b))"
LIMSEQ_subseq_LIMSEQ: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a"   and f :: "nat \<Rightarrow> nat" assumes "X \<longlonglongrightarrow> L"   and "strict_mono f" shows "(X \<circ> f) \<longlonglongrightarrow> L"
lookup_tabulate2: fixes xs :: "'d list"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and y :: "'c" assumes "Mapping.lookup (Mapping.tabulate xs f) x = Some y" shows "y = f x"
char_poly_similar: fixes A :: "'b mat"   and B :: "'b mat" assumes "similar_mat A B" shows "char_poly A = char_poly B"
sources_determine_composability: fixes a :: "'a"   and \<tau> :: "'a"   and \<mu> :: "'a" assumes "a \<in> sources \<tau>" shows "(\<tau> \<star> \<mu> \<noteq> null) = (a \<star> \<mu> \<noteq> null)"
induct: fixes P :: "int list \<Rightarrow> nat list \<Rightarrow> bool"   and a0 :: "int list"   and a1 :: "nat list" assumes "\<And>k ks x xs. P ks xs \<Longrightarrow> P (k # ks) (x # xs)"   and "\<And>xs. P [] xs"   and "\<And>ks. P ks []" shows "P a0 a1"
that: fixes p :: "(nat \<times> nat) list" assumes "isShortestPath u p v" shows "thesis"
simps: fixes xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" shows "is_expansion (MS xs f) = is_expansion_aux xs f"
fusc'_unfold: shows "fusc' = 1 ## fusc + fusc' - 2 * (fusc mod fusc')"
inf_assoc: fixes x :: "'d upred"   and y :: "'d upred"   and z :: "'d upred" shows "((x \<and> y) \<and> z) = (x \<and> y \<and> z)"
last_take_conv_nth: fixes n :: "nat"   and l :: "'b list" assumes "0 < n"   and "n \<le> length l" shows "last (take n l) = l ! (n - 1)"
\<Delta>'''_sel_3: shows "source \<Delta>'''.\<Delta>''' = source \<Delta>'''"
inj_on_respects_range_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. inj_on f A) respects range_permutation A B"
list_update_code_2: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" shows "(x # xs)[0 := y] = y # xs"
R13_4: shows "pmf (sds R13) d = 1 / 2"
step: fixes i :: "nat"   and cfg1 :: "('p, 'v, 's) configuration"   and cfg2 :: "('p, 'v, 's) configuration" assumes "i < length exec - 1"   and "cfg1 = execConf i"   and "cfg2 = execConf (i + 1)" shows "cfg1 \<turnstile> execMsg i \<mapsto> cfg2"
budan_fourier_aux_left': fixes d1 :: "real"   and c :: "real"   and p :: "real poly" assumes "d1 < c"   and "p \<noteq> 0"   and "\<forall>x. d1 \<le> x \<and> x < c \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)" shows "int (order c p) \<le> changes_itv_der d1 c p \<and> even (changes_itv_der d1 c p - int (order c p))"
lemma_interval: fixes a :: "real"   and x :: "real"   and b :: "real" assumes "a < x"   and "x < b" shows "\<exists>d>0. \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
regexp_unfold_2: fixes A :: "'b set"   and B :: "'b set" shows "(A \<subseteq> B) = (A \<union> B = B)"
set_intros_5: fixes x7 :: "'d" shows "x7 \<in> set_seq_action (receive x7)"
Lim_def: fixes A :: "'e filter"   and f :: "'e \<Rightarrow> 'd" shows "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
P_Gauss_Jordan_def: fixes A :: "(('a, 'c) vec, 'b) vec" shows "P_Gauss_Jordan A = fst (Gauss_Jordan_PA A)"
igSubstAllIPresIGWlsAllSTR_imp_igSubstAllIPresIGWlsAll: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSubstAllIPresIGWlsAllSTR MOD" shows "igSubstAllIPresIGWlsAll MOD"
converse_rtranclpE2: fixes r :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and xa :: "'d"   and xb :: "'e"   and za :: "'d"   and zb :: "'e" assumes "r\<^sup>*\<^sup>* (xa, xb) (za, zb)" obtains "(xa, xb) = (za, zb)"   | a :: "'d" and b :: "'e" where "r (xa, xb) (a, b)" and "r\<^sup>*\<^sup>* (a, b) (za, zb)"
simps_4: fixes t :: "sterm"   and u :: "sterm" shows "consts (t $\<^sub>s u) = consts t |\<union>| consts u"
mset_sorted_list_of_multiset: fixes M :: "'b multiset" shows "mset (sorted_list_of_multiset M) = M"
subset_code_2: fixes A :: "'b set"   and ys :: "'b list" shows "(A \<subseteq> List.coset ys) = (\<forall>y\<in>set ys. y \<notin> A)"
coeffs_pCons_eq_cCons: fixes a :: "'a"   and p :: "'a poly" shows "coeffs (pCons a p) = cCons a (coeffs p)"
toplevel_summands_lang_UN: fixes s :: "'a rexp" shows "lang s = \<Union> (lang ` toplevel_summands s)"
InvIsMor: fixes f :: "'b"   and g :: "'b" assumes "cinv f g" shows "f \<in> mor \<and> g \<in> mor"
Abs_S_inverse: fixes y :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) G" assumes "y \<in> {x. P x}" shows "Rep_S (Abs_S y) = y"
lower_le_upper: fixes i :: "'b interval" shows "lower i \<le> upper i"
list_update_code_2: fixes x :: "'e"   and xs :: "'e list"   and y :: "'e" shows "(x # xs)[0 := y] = y # xs"
subspace_inter: fixes A :: "('d \<Rightarrow>\<^sub>0 'c) set"   and B :: "('d \<Rightarrow>\<^sub>0 'c) set" assumes "phull.subspace A"   and "phull.subspace B" shows "phull.subspace (A \<inter> B)"
image_image_mset: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d multiset" shows "image_mset f (image_mset g A) = {#f (g x). x \<in># A#}"
wpt_choice: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "mbt_algebra.wpt one times inf top (inf x y) = inf (mbt_algebra.wpt one times inf top x) (mbt_algebra.wpt one times inf top y)"
prems_4: fixes F1 :: "'a \<Rightarrow> 'b option"   and F2 :: "'a \<Rightarrow> 'b option"   and F3 :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "P F1 F2 F3"   and "dom F1 = dom F2"   and "dom F3 = dom F1"   and "x \<notin> dom F1" shows "P (F1(x \<mapsto> a)) (F2(x \<mapsto> b)) (F3(x \<mapsto> c))"
relcomppE: fixes r :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and c :: "'e" assumes "(r OO s) a c" obtains b :: "'d" where "r a b" and "s b c"
Nil_left_tensor: fixes xs :: "wall" shows "xs \<otimes> basic [] = xs"
that: fixes U :: "'a set"   and q :: "'a \<Rightarrow> 'a"   and V :: "'a set"   and r :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext U (n - k) q"   and "swapidseq_ext V k r"   and "p = q \<circ> r"   and "U \<union> V = S" shows "thesis"
span_mono: fixes A :: "('a \<Rightarrow> real) set"   and B :: "('a \<Rightarrow> real) set" assumes "A \<subseteq> diff_fun_space"   and "B \<subseteq> diff_fun_space"   and "A \<subseteq> B" shows "diff_fun_space.m1.span A \<subseteq> diff_fun_space.m1.span B"
3: shows "\<forall>r\<in>set rs'. \<not> contains s r"
gso_scalar_zero: fixes k :: "nat"   and i :: "nat" assumes "k < length fs"   and "i < k" shows "gso k \<bullet> fs ! i = (0::'a)"
acquired_append: fixes pending_write :: "bool"   and xs :: "'b memref list"   and ys :: "'b memref list"   and A :: "nat set" shows "acquired pending_write (xs @ ys) A = acquired (pending_write \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<noteq> {}) ys (acquired pending_write xs A)"
sort_ex_class_ex: fixes cs :: "('a \<times> 'a) set"   and S :: "'a set" shows "sort_ex cs S \<equiv> \<forall>c\<in>S. class_ex cs c"
converse_converse: fixes r :: "('a \<times> 'b) set" shows "\<smile> (\<smile> r) = r"
preserves_dom: fixes f :: "'a1" assumes "A1.arr f" shows "A2_B.dom (F f) = F (A1.dom f)"
is_lubD1: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c" assumes "class.po below"   and "po.is_lub below S x" shows "po.is_ub below S x"
hRe_hrcis: fixes r :: "real star"   and a :: "real star" shows "hRe (hrcis r a) = r * (*f* cos) a"
set_plus_rearrange2: fixes a :: "'a"   and b :: "'a"   and C :: "'a set" shows "a +o (b +o C) = (a + b) +o C"
summands_equal: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a = b"   and "c = d" shows "a + c = b + d"
NotE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<not>\<phi> in v]" shows "\<not> [\<phi> in v]"
Iff_MP_same: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IFF B"   and "H \<turnstile> A" shows "H \<turnstile> B"
trms_list\<^sub>s\<^sub>s\<^sub>t_is_trms\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('c, 'd) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t S = set (trms_list\<^sub>s\<^sub>s\<^sub>t S)"
HCSC: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula" assumes "AX10 \<union> set_mset \<Gamma> \<turnstile>\<^sub>H F" shows "\<Gamma> \<Rightarrow> F, {#}"
Array_lim_set: fixes p :: "'b array"   and xs :: "'b list"   and h :: "heap" shows "lim (Array.set p xs h) = lim h"
normalize_mult: fixes a :: "'a"   and b :: "'a" shows "normalize (a * b) = normalize a * normalize b"
wf_acyclic: fixes r :: "('g \<times> 'g) set" assumes "wf r" shows "acyclic r"
Fpow_def: fixes A :: "'h set" shows "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}"
bij_betw_def: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" shows "bij_betw f A B = (inj_on f A \<and> f ` A = B)"
assms: shows "(\<sigma>, NodeS i \<zeta> R) \<in> oreachable (\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o) (otherwith (\<lambda>_ _. True) {ii} (oarrivemsg I)) (other (\<lambda>_ _. True) {ii})"
a2: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "sinvar G nP = (\<forall>(e1, e2)\<in>edges G. P (nP e1) (nP e2))"
lossless_exec_gpv: fixes gpv :: "('d, 'b, 'c) gpv"   and callee :: "'e \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'e) spmf"   and s :: "'e" assumes "lossless_gpv \<I>_full gpv"   and "\<And>s out. \<lbrakk>out \<in> outs_\<I> \<I>_full; True\<rbrakk> \<Longrightarrow> lossless_spmf (callee s out)"   and "\<I>_full \<turnstile>g gpv \<surd>"   and "True" shows "lossless_spmf (exec_gpv callee gpv s)"
select_convs_4: fixes Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e" shows "In \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = In"
pcr_cr_eq: shows "pcr_dnelist (=) = cr_dnelist"
simps_3: fixes n :: "nat"   and p :: "pf"   and q :: "pf" shows "dfa_of_pf' n (And p q) = and_dfa (dfa_of_pf' n p) (dfa_of_pf' n q)"
valid_unconc_1: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and c :: "'a list multiset"   and c' :: "'a list multiset" shows "valid fg (c + c') = (valid fg c \<and> valid fg c' \<and> mon_c fg c \<inter> mon_c fg c' = {})"
cblinfun_compose_1_right: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "x o\<^sub>C\<^sub>L 1 = x"
1: shows "\<forall>PROB x. P PROB \<and> finite PROB \<longrightarrow> x \<in> MPLS PROB \<longrightarrow> x < f PROB"
is_prefix_cancel: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" shows "is_prefix (a @ b) (a @ c) = is_prefix b c"
abs_eq: fixes xc :: "('a, 'b) mapping"   and xb :: "'b"   and xa :: "'b \<Rightarrow> 'b"   and x :: "'a set" shows "upd_set' xc xb xa x = Mapping (\<lambda>a. if a \<in> x then case Mapping.lookup xc a of None \<Rightarrow> Some xb | Some b \<Rightarrow> Some (xa b) else Mapping.lookup xc a)"
ground_not_none: shows "\<G>_I_L \<iota> \<noteq> None"
lprefix_conv_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "xs \<sqsubseteq> ys = (\<exists>zs. ys = lappend xs zs)"
elims_3: fixes x :: "nat"   and xa :: "'b btree" assumes "\<not> invar_inorder x xa" obtains k :: "nat" and t :: "'b btree" where "x = k" and "xa = t" and "\<not> (bal t \<and> root_order k t)"
last_sts: shows "llast Sts = (N', P', Q', n')"