hcju/isabelleps huggingface dataset:
max_diff_distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "minus (ord.max less_eq x y) z = ord.max less_eq (minus x z) (minus y z)"
F1a: shows "\<forall>x\<^sub>2 x\<^sub>1. x\<^sub>1 \<subseteq> x\<^sub>2 \<longrightarrow> x\<^sub>2 = x\<^sub>2 \<union> x\<^sub>1"
simps_6: fixes x61 :: "nat"   and x62 :: "('a, 'b) MWLsCom list"   and y61 :: "nat"   and y62 :: "('a, 'b) MWLsCom list" shows "(spawn\<^bsub>x61\<^esub> x62 = spawn\<^bsub>y61\<^esub> y62) = (x61 = y61 \<and> x62 = y62)"
map_add_find_right: fixes n :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and xx :: "'b"   and m :: "'c \<Rightarrow> 'b option" assumes "n k = Some xx" shows "(m ++ n) k = Some xx"
injective_inv: fixes t :: "'a"   and e :: "'a" assumes "t \<sqinter> id t \<le> top"   and "e \<sqinter> id t = bot"   and "(e \<sqinter> top \<sqinter> id (e \<sqinter> top) \<le> top \<and> top \<sqinter> (e \<sqinter> top) = top) \<and> id e \<sqinter> top \<sqinter> id (id e \<sqinter> top) \<le> top \<and> top \<sqinter> (id e \<sqinter> top) = top" shows "(t \<squnion> e) \<sqinter> id (t \<squnion> e) \<le> top"
calc_perpendicular_to_x_axis: fixes z :: "complex_homo" assumes "z \<noteq> of_complex 1"   and "z \<noteq> of_complex (- 1)" shows "z \<in> circline_set (calc_perpendicular_to_x_axis z) \<and> is_poincare_line (calc_perpendicular_to_x_axis z) \<and> perpendicular_to_x_axis (calc_perpendicular_to_x_axis z)"
weak_tau_modality_eqvt: fixes p :: "perm"   and x :: "('idx, 'pred, 'act) formula" shows "p \<bullet> weak_tau_modality x = weak_tau_modality (p \<bullet> x)"
iT_Minus_Minus_eq2: fixes I :: "nat set"   and k :: "nat" assumes "I \<subseteq> [\<dots>k]" shows "k \<ominus> (k \<ominus> I) = I"
more_arith_simps_6: fixes a :: "'c" shows "a * (1::'c) = a"
simps_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Gt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f3"
a_star_preserves_SNa: fixes M :: "trm"   and M' :: "trm" assumes "SNa M"   and "M \<longrightarrow>\<^sub>a* M'" shows "SNa M'"
poly_finite_fourier_poly: fixes m :: "int"   and ws :: "complex list" assumes "m \<in> {0..<int (length ws)}"   and "m < int (length ws)" shows "poly (finite_fourier_poly ws) (unity_root (length ws) m) = ws ! nat m"
elims: fixes x :: "'b equation"   and y :: "'b set" assumes "vars_of_eq x = y" obtains t :: "'b trm" and s :: "'b trm" where "x = Eq t s" and "y = vars_of t \<union> vars_of s"
lm004: fixes R :: "('a \<times> 'b) set" assumes "runiq R" shows "R = graph (Domain R) (toFunction R)"
arctan_lower_41_def: shows "arctan_lower_41 \<equiv> \<lambda>x. - (pi / 2) - (15159 + 147455 * x\<^sup>2 + 345345 * x ^ 4 + 225225 * x ^ 6) * x / (35 * (35 + 1260 * x\<^sup>2 + 6930 * x ^ 4 + 12012 * x ^ 6 + 6435 * x ^ 8))"
DBDeq_def: shows "DBDeq \<equiv> Deq mid q2 out \<and> unchanged (inp, q1)"
WT_callee: fixes x :: "'a"   and \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and s :: "('a, 'b, 'c, 'd) converter" assumes "x \<in> outs_\<I> \<I>"   and "\<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "\<I>' \<turnstile>g run_converter s x \<surd>"
contradictD: fixes as :: "atom list"   and cs :: "real list"   and xs :: "real list" assumes "contradict as cs" shows "\<exists>a\<in>set as. \<not> I\<^sub>R a xs"
T'_def: shows "T' \<equiv> {y. \<forall>x\<in>T. orthogonal x y}"
lang_rderiv_lderiv: fixes n :: "nat"   and a :: "'a"   and b :: "'a"   and r :: "'b rexp" shows "lang n (rderiv a (lderiv b r)) = lang n (lderiv b (rderiv a r))"
uniformity_point_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
piecewise_continuous_on_congI: fixes a :: "'c"   and b :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "piecewise_continuous_on a b I f"   and "\<And>x. x \<in> {a..b} - I \<Longrightarrow> g x = f x" shows "piecewise_continuous_on a b I g"
strong_fermat_witness_def: fixes a :: "nat"   and n :: "nat" shows "strong_fermat_witness a n = (\<not> strong_fermat_liar a n)"
m_closed: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and y :: "'e" assumes "subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
thetaSPar_Sretr: shows "thetaSPar \<subseteq> Sretr thetaSPar"
seg_sym: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.segment \<P> betw a b = MinkowskiBetweenness.segment \<P> betw b a"
comp_associative: fixes CC :: "('b, 'c, 'd) category_scheme"   and f :: "'c"   and g :: "'c"   and h :: "'c" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>"   and "g \<in> Ar\<^bsub>CC\<^esub>"   and "h \<in> Ar\<^bsub>CC\<^esub>"   and "Cod\<^bsub>CC\<^esub> h = Dom\<^bsub>CC\<^esub> g"   and "Cod\<^bsub>CC\<^esub> g = Dom\<^bsub>CC\<^esub> f" shows "f \<bullet>\<^bsub>CC\<^esub> (g \<bullet>\<^bsub>CC\<^esub> h) = f \<bullet>\<^bsub>CC\<^esub> g \<bullet>\<^bsub>CC\<^esub> h"
word_sless_eq: fixes x :: "'b word"   and y :: "'b word" shows "(x <s y) = (x \<le>s y \<and> x \<noteq> y)"
suntil_implies_until: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "(\<phi> suntil \<psi>) \<omega>" shows "(\<phi> until \<psi>) \<omega>"
add_0_right: fixes a :: "'a" shows "a + (0::'a) = a"
arr_dom: fixes f :: "'a \<times> 'a" assumes "CC.arr f" shows "CC.arr (CC.dom f)"
locally_sound_mode_use_def: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "locally_sound_mode_use lc = (\<forall>c' mds' mem'. \<langle>c', mds', mem'\<rangle> \<in> loc_reach lc \<longrightarrow> (\<forall>x. (x \<in> mds' GuarNoReadOrWrite \<longrightarrow> doesnt_read_or_modify c' x) \<and> (x \<in> mds' GuarNoWrite \<longrightarrow> doesnt_modify c' x)))"
take_bit_push_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)"
NN_def: fixes e :: "real" shows "NN e = 1 + nat \<lceil>max (ln e / ln (real k * \<delta> / 2)) (- ln e / ln (real k * \<delta>))\<rceil>"
poly_prod: fixes p :: "'d \<Rightarrow> 'c poly"   and A :: "'d set"   and x :: "'c" shows "poly (prod p A) x = (\<Prod>k\<in>A. poly (p k) x)"
n0'_1: shows "\<forall>n<n\<^sub>0'. s (w \<triangleright> n) \<down>= 0"
iso_dual: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq' :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less' :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "OrderingSetIso less_eq less less_eq' less' P f" shows "OrderingSetIso (\<lambda>a b. less_eq b a) (\<lambda>a b. less b a) (\<lambda>a b. less_eq' b a) (\<lambda>a b. less' b a) P f"
diff_delta_arctan_upper_13_def: shows "diff_delta_arctan_upper_13 \<equiv> \<lambda>x. x\<^sup>2 / (1 + x\<^sup>2)"
K_S_closed: fixes \<mu> :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and s :: "'s \<times> ('c \<Rightarrow> real)"   and s' :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<mu> \<in> K s"   and "s' \<in> set_pmf \<mu>"   and "s \<in> S" shows "s' \<in> S"
dom_def: fixes f :: "'a" shows "H.dom f = (if H.domains f \<noteq> {} then SOME a. a \<in> H.domains f else H.null)"
PhiWhileP_def: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "PhiWhileP A b \<Phi> = FIX (PhiWhilePOp A b \<Phi>)"
disjnt_insert2: fixes Y :: "'d set"   and a :: "'d"   and X :: "'d set" shows "disjnt Y (insert a X) = (a \<notin> Y \<and> disjnt Y X)"
exists_vector_in_Berlekamp_basis_dvd: fixes u :: "'b mod_ring poly"   and B :: "'b mod_ring poly set"   and P :: "'b mod_ring poly set"   and p_i :: "'b mod_ring poly"   and p_j :: "'b mod_ring poly"   and w :: "'b mod_ring poly" assumes "0 < degree u"   and "vectorspace.basis class_ring (W u) B"   and "finite B"   and "finite P"   and "u = \<Prod>P"   and "P \<subseteq> {q. irreducible q \<and> monic q}"   and "p_i \<in> P"   and "p_j \<in> P"   and "p_i \<noteq> p_j"   and "monic u"   and "square_free u"   and "\<not> irreducible w"   and "w dvd u"   and "monic w"   and "p_i dvd w"   and "p_j dvd w" shows "\<exists>v\<in>B. v mod p_i \<noteq> v mod p_j \<and> degree (v mod p_i) = 0 \<and> degree (v mod p_j) = 0 \<and> (\<exists>s. gcd w (v - [:s:]) \<noteq> w \<and> \<not> comm_monoid_mult_class.coprime w (v - [:s:]))"
Schottky_lemma3: fixes z :: "complex" assumes "z \<in> (\<Union>m\<in>\<int>. \<Union>n\<in>{0<..}. {Complex m (ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)}) \<union> (\<Union>m\<in>\<int>. \<Union>n\<in>{0<..}. {Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)})" shows "cos (complex_of_real pi * cos (complex_of_real pi * z)) = 1 \<or> cos (complex_of_real pi * cos (complex_of_real pi * z)) = - 1"
fbrelation_vintersectionI: fixes s :: "V" shows "fbrelation (r\<^sub>2 \<inter>\<^sub>\<circ> s)"
hull_minimal: fixes s :: "'b set"   and t :: "'b set"   and S :: "'b set \<Rightarrow> bool" assumes "s \<subseteq> t"   and "S t" shows "S hull s \<subseteq> t"
fls_shift_subdegree: fixes f :: "'b fls"   and n :: "int" assumes "f \<noteq> 0" shows "fls_subdegree (fls_shift n f) = fls_subdegree f - n"
add_diff_inverse_ennreal: fixes x :: "ennreal"   and y :: "ennreal" assumes "x \<le> y" shows "x + (y - x) = y"
tl_list_minus_butlast_empty: fixes s :: "'b list"   and t :: "'b list" assumes "strict_prefix (butlast s) t"   and "0 < length s"   and "length t = length s" shows "tl (t - butlast s) = []"
merge_code: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "merge t1 t2 = (case t1 of \<langle>\<rangle> \<Rightarrow> t2 | \<langle>l1, a1, r1\<rangle> \<Rightarrow> case t2 of \<langle>\<rangle> \<Rightarrow> t1 | \<langle>l2, a2, r2\<rangle> \<Rightarrow> if a1 \<le> a2 then \<langle>merge t2 r1, a1, l1\<rangle> else \<langle>merge t1 r2, a2, l2\<rangle>)"
weakTransitiveCoinduct': fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "eqvt X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>[{uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}] Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> {uu_. \<exists>\<Psi> P P' Q' Q. uu_ = (\<Psi>, P, Q) \<and> \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}" shows "\<Psi> \<rhd> P \<sim> Q"
vsv_vcomp_at: fixes r :: "V"   and s :: "V"   and a :: "V" assumes "vsv r"   and "vsv s"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" shows "(s \<circ>\<^sub>\<circ> r)\<lparr>a\<rparr> = s\<lparr>r\<lparr>a\<rparr>\<rparr>"
y_point: fixes y :: "'a" assumes "y \<in># mset_set (\<V> - {x})" shows "y \<in> \<V>"
H2: fixes n :: "nat" shows "u n \<in> bcontfun"
subst_word_append: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list"   and ys :: "'b list" shows "subst_word f (xs @ ys) = subst_word f xs @ subst_word f ys"
pivot_def: fixes x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and s :: "('b, 'a) state" shows "pivot x\<^sub>i x\<^sub>j s \<equiv> \<T>_update (pivot_tableau x\<^sub>i x\<^sub>j (\<T> s)) s"
xz: shows "infdist w Gxz \<le> 4 * deltaG TYPE('a)"
cs_not_nil: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "IFC prob" shows "IFC_def.cs prob \<pi> n \<noteq> []"
trans: fixes b :: "('c, 'd) Term.term"   and a :: "('c, 'd) Term.term"   and c :: "('c, 'd) Term.term" assumes "b \<sqsubseteq> a"   and "c \<sqsubseteq> b" shows "c \<sqsubseteq> a"
diffh: shows "h differentiable_on sphere (0::'a) 1 \<inter> S"
add1: fixes A :: "'f"   and k :: "'f"   and a :: "'f"   and b :: "'f" assumes "A \<equiv> k + a" shows "A + b \<equiv> k + (a + b)"
valid2: shows "digit_encode (nths ds (- rmodes)) c \<lhd> nths ds (- rmodes)"
member_inI: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and provC :: "qtname"   and C :: "qtname" assumes "G \<turnstile> m member_of provC"   and "G\<turnstile>C\<preceq>\<^sub>C provC" shows "G \<turnstile> m member_in C"
interior_of_closedin_union_empty_interior_of: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "closedin X S"   and "X interior_of T = {}" shows "X interior_of (S \<union> T) = X interior_of S"
is_OK_returns_heap_E: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog" assumes "h \<turnstile> ok f" obtains h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>h h'"
not_gr_zero: fixes n :: "'k multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
mem_upD: fixes f :: "nat \<Rightarrow> 'a"   and R :: "('a, 'b) ring_scheme"   and n :: "nat" assumes "f \<in> up R" shows "f n \<in> carrier R"
derg: fixes z :: "complex" assumes "z \<in> S" shows "deriv f z * deriv g (f z) = 1"
suffix_singleton_suffix: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "w i ## w [Suc i..] = w [i..]"
elims_2: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and x :: "('b \<times> 'c) list \<times> 'a"   and xa :: "('b \<times> 'c) list \<times> 'a" assumes "oalist_raw.oalist_eq_raw rep_key_order x xa" obtains xs :: "('b \<times> 'c) list" and ox :: "'a" and ys :: "('b \<times> 'c) list \<times> 'a" where "x = (xs, ox)" and "xa = ys" and "xs = oalist_raw.sort_oalist_aux rep_key_order ox ys"
basic_subtype_x_String: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B String" obtains "\<tau> = OclVoid"
pwalk_awalk: fixes v :: "'a" assumes "v \<in> verts G"   and "dist v \<noteq> \<infinity>" shows "awalk s (pwalk v) v"
bfs_refines_shortest_augmenting_spec: fixes cf :: "nat \<times> nat \<Rightarrow> 'capacity" shows "Graph.bfs cf s t \<le> find_shortest_augmenting_spec_cf cf"
\<sigma>1_2: shows "\<sigma>' \<in> small_nstep P \<sigma>1 n1"
simps_1: fixes ys :: "'b list" shows "subtract [] ys = ys"
quotient_rel_set_disjoint: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "equivp R"   and "C \<in> UNIV // {(x, y). R x y}"   and "rel_set R A B" shows "(A \<inter> C = {}) = (B \<inter> C = {})"
INFUV_def: shows "INFUV = (INF i. U *\<^sub>S V i)"
insert_is_Un: fixes a :: "'e"   and A :: "'e set" shows "insert a A = {a} \<union> A"
not_less_zero: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "\<not> less n zero"
ideal_def: fixes I :: "'a set" shows "ideal I = (left_ideal I \<and> right_ideal I)"
assms: shows "set \<Gamma>\<^sub>1 \<subseteq> set \<Gamma>\<^sub>2"
scale_right_distrib: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'c"   and x :: "'d"   and y :: "'d" assumes "module scale" shows "scale a (x + y) = scale a x + scale a y"
el_in_restriction_clm_singleton: fixes n :: "nat"   and v :: "view"   and ts :: "traffic"   and c :: "cars" shows "n \<^bold>\<in> restrict v (clm ts) c \<longrightarrow> clm ts c = Abs_nat_int {n}"
set_split_bst2: fixes t :: "'a tree"   and x :: "'a" assumes "bst t" shows "set_tree (snd (split_bst x t)) = {y \<in> set_tree t. x < y}"
ltlr_to_ltln_semantics: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr" shows "w \<Turnstile>\<^sub>n ltlr_to_ltln \<phi> = w \<Turnstile>\<^sub>r \<phi>"
step_redex: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and r :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and r' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and c :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')" shows "\<Gamma>\<turnstile> (subst_redex c r, s) \<rightarrow> (subst_redex c r', s')"
get_host_def: shows "get_host = (\<lambda>shadow_root_ptr. Heap_Error_Monad.bind (Heap_Error_Monad.bind element_ptr_kinds_M (filter_M (\<lambda>element_ptr. Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>shadow_root_opt. return (shadow_root_opt = Some shadow_root_ptr))))) (case_list (error HierarchyRequestError) (\<lambda>host_ptr. case_list (return host_ptr) (\<lambda>aa lista. error HierarchyRequestError))))"
eij_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" shows "Linear_Diophantine_Equations.eij a b i j = lcm (a ! i) (b ! j) div b ! j"
run_step_break_imp_lfilled_prop_def: fixes s' :: "s"   and vs' :: "v list"   and n :: "nat"   and res :: "v list" shows "run_step_break_imp_lfilled_prop s' vs' n res = (\<lambda>d i (s, vs, es). run_step d i (s, vs, es) = (s', vs', RSBreak n res) \<longrightarrow> s = s' \<and> vs = vs' \<and> (\<exists>n' lfilled es_c. n \<le> n' \<and> Lfilled_exact (n' - n) lfilled (vs_to_es res @ [$Br n'] @ es_c) es))"
cart2_append1_apply_cltn2: fixes p :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "\<exists>k. k \<noteq> 0 \<and> cart2_append1 p v* cltn2_rep J = k *\<^sub>R cart2_append1 (apply_cltn2 p J)"
inf_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "x \<sqinter> y = (if x = a\<^sub>1 \<or> y = a\<^sub>1 then a\<^sub>1 else a\<^sub>2)"
pmf_prob_pmf_set_integral: fixes p :: "'b pmf"   and A :: "'b set" shows "pmf_prob p A = pmf_set_integral p (\<lambda>_. 1) A"
gauge_ball_dependent: fixes e :: "'a \<Rightarrow> real" assumes "\<forall>x. 0 < e x" shows "gauge (\<lambda>x. ball x (e x))"
add_mset_inter_add_mset: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(a, A) \<inter># (a, B) = a, A \<inter># B"
seq_pr_tuple: fixes f :: "'a"   and g :: "'a" assumes "span f g" shows "seq \<pp>\<^sub>0[cod f, cod g] \<langle>f, g\<rangle>"
BlockRedSome: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   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 v :: "val" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e,(h, l(V := None))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l' V = \<lfloor>v\<rfloor>"   and "\<not> assigned V e" shows "P,E \<turnstile> \<langle>{V:T; e},(h, l)\<rangle> \<rightarrow> \<langle>{V:T; V:=Val v;; e'},(h', l'(V := l V))\<rangle>"
N: fixes n :: "nat" assumes "N \<le> n" shows "norm (f n) \<le> g n"
weak_lower_semilattice_axioms: shows "weak_lower_semilattice L"
simps_2: fixes t :: "'b Trace"   and s :: "'b" shows "tFirst (t \<leadsto> s) = tFirst t"
lsc_hull_lsc: fixes f :: "'b \<Rightarrow> ereal" shows "lsc f = (f = lsc_hull f)"
inj_on_one_iff: shows "inj_on (\<lambda>x. finprod G x I) (carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)) = (\<forall>x. x \<in> carrier (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) \<longrightarrow> finprod G x I = \<one>\<^bsub>G\<lparr>carrier := J\<rparr>\<^esub> \<longrightarrow> x = \<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub>)"
s2r_union_hom: fixes A :: "'a set"   and B :: "'a set" shows "s2r (A \<union> B) = s2r A \<union> s2r B"
other_claim': fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat"   and j :: "nat" assumes "is_path \<pi>"   and "\<pi> i = \<pi> j"   and "\<pi> i \<noteq> local.return"   and "\<forall>k. \<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"   and "\<forall>k. \<not> j cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "i = j"
inj: fixes f :: "'k \<Rightarrow> 'k" assumes "bijection f" shows "inj f"
second_addend: fixes e\<^sub>2 :: "expr"   and x :: "nat"   and y :: "nat" assumes "e\<^sub>2 = Double (Add (Var ''x'') (Var ''y''))" shows "eval (\<E>\<^sub>x\<^sub>y x y) e\<^sub>2 = Some (2 * x + 2 * y) \<and> wfe \<Gamma>\<^sub>x\<^sub>y e\<^sub>2 \<and> coste e\<^sub>2 = 2"
T_opt_def: fixes s :: "'state"   and rs :: "'request list" shows "T_opt s rs = Inf {T s rs as |as. length as = length rs}"
sepnMT: fixes p :: "('b, 'c) Combinators list" assumes "p \<noteq> []" shows "separate p \<noteq> []"
eq_fin_indoc: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a list"   and v :: "'a list"   and c :: "'a" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* u v"   and "traces.indoc ind c u" shows "traces.indoc ind c v"
not_rt_fresher_single: fixes ip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "\<not> rt1 \<sqsubseteq>\<^bsub>ip\<^esub> rt2" shows "\<not> the (rt1 ip) \<sqsubseteq> the (rt2 ip)"
g: shows "retraction_maps (nsphere p) (nsphere p) f g"
aprimedivisor_prime_power: fixes p :: "'b"   and k :: "nat" assumes "prime p"   and "0 < k" shows "aprimedivisor (p ^ k) = p"
Trg_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Trg (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Trg t"
intros_2: fixes n :: "nat"   and l :: "'b tree234"   and r :: "'b tree234"   and p :: "'b" assumes "full n l"   and "full n r" shows "full (Suc n) \<langle>l, p, r\<rangle>"
simps_2: fixes B :: "'c mat"   and i :: "nat"   and ai'j :: "'c"   and i' :: "nat"   and "is" :: "('c \<times> nat) list" shows "eliminate_entries_rec B i ((ai'j, i') # is) = eliminate_entries_rec (addrow ai'j i' i B) i is"
aall_def: fixes F :: "(state \<Rightarrow> 'a) \<Rightarrow> behavior \<Rightarrow> bool" shows "\<forall>\<forall> x. F x \<equiv> \<not> (\<exists>\<exists> x. \<not> F x)"