hcju/isabelleps huggingface dataset:
resolvable_design_num_res_classes: shows "int (size \<P>) = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
assms_3: shows "AE x\<in>A in M. 0 < f x"
clamp_empty_interval: fixes i :: "'b"   and b :: "'b"   and a :: "'b" assumes "i \<in> Basis"   and "b \<bullet> i < a \<bullet> i" shows "clamp a b = (\<lambda>_. a)"
less_imp_neq: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "ko.lt ko x y" shows "x \<noteq> y"
bounded_uminus: fixes X :: "'b set" shows "bounded (uminus ` X) = bounded X"
simps_1: fixes G :: "('c \<times> ('c, 'd) RHS) list"   and k :: "nat" shows "init G [] k = None"
Or\<^sub>n_And\<^sub>n_image_prop_semantics: fixes \<A> :: "'b ltln set set"   and \<I> :: "'b ltln set" assumes "finite \<A>"   and "\<And>\<Phi>. \<Phi> \<in> \<A> \<Longrightarrow> finite \<Phi>" shows "\<I> \<Turnstile>\<^sub>P Or\<^sub>n (And\<^sub>n ` \<A>) = (\<exists>\<Phi>\<in>\<A>. \<forall>\<phi>\<in>\<Phi>. \<I> \<Turnstile>\<^sub>P \<phi>)"
inv_ordermap_mono_less: fixes M :: "'a set"   and r :: "('a \<times> 'a) set"   and \<alpha> :: "V"   and \<beta> :: "V" assumes "(inv_into M (ordermap M r) \<alpha>, inv_into M (ordermap M r) \<beta>) \<in> r"   and "small M"   and "\<alpha> \<in> elts (ordertype M r)"   and "\<beta> \<in> elts (ordertype M r)"   and "wf r"   and "trans r" shows "\<alpha> < \<beta>"
Schwarz1: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "continuous_on (closure S) f"   and "open S"   and "connected S"   and "bounded S"   and "S \<noteq> {}" obtains w :: "complex" where "w \<in> frontier S" and "\<And>z. z \<in> closure S \<Longrightarrow> cmod (f z) \<le> cmod (f w)"
diff_zero: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and a :: "'g" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
C_def: shows "C = (\<lambda>t. {x \<in> B. f x \<le> t})"
nat_pow_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier (mult_of R)" shows "x [^] n \<in> carrier (mult_of R)"
that: fixes x :: "'a" assumes "x \<in> carrier" shows "\<exists>c1\<in>atlas. x \<in> domain c1 \<and> k-smooth_on (codomain c1) (f \<circ> inv_chart c1)"
swap_rel_trans: fixes a :: "atom"   and b :: "atom"   and c :: "atom"   and x :: "'a" assumes "sort_of a = sort_of b"   and "sort_of b = sort_of c"   and "(a \<rightleftharpoons> c) \<bullet> x = x"   and "(b \<rightleftharpoons> c) \<bullet> x = x" shows "(a \<rightleftharpoons> b) \<bullet> x = x"
singular_chain_sum: fixes I :: "'b set"   and p :: "nat"   and X :: "'c topology"   and f :: "'b \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "\<And>i. i \<in> I \<Longrightarrow> singular_chain p X (f i)" shows "singular_chain p X (sum f I)"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "list_distinct (x # xs) = (\<not> list_member xs x \<and> list_distinct xs)"
coprime_0_2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "comm_monoid_mult.coprime times one zero p = dvd.dvd times p one"
times_hf_def: fixes A :: "hf"   and B :: "hf" shows "A * B = (\<Sum>x\<^bold>\<in>A. B)"
Suc_2: fixes i :: "nat" assumes "i \<le> Suc n" shows "ideal (I i) R"
fmul1fadd2fold: fixes x :: "'b"   and s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "fmul1 x (foldseq fadd2 s n) = foldseq fadd2 (\<lambda>k. fmul1 x (s k)) n"
measure_spmf_scale_spmf: fixes r :: "real"   and p :: "'c spmf" shows "measure_spmf (scale_spmf r p) = scale_measure (ennreal (min (inverse (weight_spmf p)) r)) (measure_spmf p)"
CondRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
sum_list_sum_nth: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and xs :: "'d list" assumes "class.monoid_add plus zero" shows "sum_list xs = sum ((!) xs) {0..<length xs}"
wfp_on_iff_inductive_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wfp_on P A = inductive_on P A"
pdec_zero_simps_2: shows "pdec2 0 = 0"
mem_sphere: fixes y :: "'b"   and x :: "'b"   and e :: "real" shows "(y \<in> sphere x e) = (dist x y = e)"
simps: fixes x :: "nat"   and n :: "nat" shows "newton_sqrt_aux x n = (let y = (x + n div x) div 2 in if y < x then newton_sqrt_aux y n else x)"
pos_noteq: fixes a :: "'b"   and l :: "'b list"   and b :: "'b"   and c :: "'b" assumes "a \<in> set l"   and "b \<in> set l"   and "c \<in> set l"   and "a \<noteq> b"   and "position a l \<le> position b l"   and "position b l \<le> position c l" shows "a \<noteq> c"
normalize_length_le: fixes p :: "'a list list" shows "length (UP.normalize p) \<le> length p"
comp_inv_arr: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "C g f = C.dom f"
valid_def: fixes stack :: "(recf \<times> nat list \<times> nat list) list" shows "valid stack \<equiv> \<forall>s\<in>set stack. recfn (length (fst (snd s))) (fst s)"
weak_ranking_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "weak_ranking R = (SOME xs. is_weak_ranking xs \<and> R = of_weak_ranking xs)"
U_3: assumes "Fu f \<noteq> g" shows "Abs a = g \<and> Fu f = Abs b"
l9_2: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B TS P Q" shows "A B TS Q P"
assms_2: fixes X :: "'a \<times> 'a pdevs" assumes "X \<in> set XS" shows "degree_aform X \<le> d"
A_events: fixes M :: "'a measure"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'a set"   and i :: "'b" assumes "prob_space_with_indicators M I A"   and "i \<in> I" shows "A i \<in> sets M"
PG: shows "prv (imp (PP \<langle>\<phi>G\<rangle>) (PP \<langle>fls\<rangle>))"
TBase_def: shows "TBase \<equiv> Abs_ty TBase_Rep"
ics_SDG_path_Append: fixes n'' :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n :: "'node SDG_node"   and ns :: "'node SDG_node list" assumes "n'' ics-ns'\<rightarrow>\<^sub>d* n'"   and "n ics-ns\<rightarrow>\<^sub>d* n''" shows "n ics-ns @ ns'\<rightarrow>\<^sub>d* n'"
simps_1: fixes t :: "'b tree" shows "subtract Tip t = Some t"
measurable_bind': fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e measure"   and R :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "(\<lambda>(x, y). g x y) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g x) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
accD_pos: fixes x :: "'a"   and y :: "'a" assumes "(x, y) \<in> K1.acc" shows "\<exists>n. 0 < K1.p x y n"
dim_factorization_lattice: fixes x :: "nat"   and u :: "'b poly"   and d :: "nat"   and m :: "'b" assumes "x < degree u + d" shows "dim_vec (factorization_lattice u d m ! x) = degree u + d"
INF_apply: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and A :: "'d set"   and x :: "'c" shows "Inf (f ` A) x = (INF y\<in>A. f y x)"
invertible_left_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "inverse u \<cdot> u = \<one>"
rel_gpv_Pause: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and out :: "'f"   and c :: "('d, 'f, 'h) rpv"   and out' :: "'g"   and c' :: "('e, 'g, 'h) rpv" shows "rel_gpv A C (Generative_Probabilistic_Value.Pause out c) (Generative_Probabilistic_Value.Pause out' c') = (C out out' \<and> (\<forall>x. rel_gpv A C (c x) (c' x)))"
kills_e2: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>"
simps_4: shows "eval_ternary_And TernaryFalse TernaryFalse = TernaryFalse"
subset_card_intvl_is_intvl: fixes A :: "nat set"   and k :: "nat" assumes "A \<subseteq> {k..<k + card A}" shows "A = {k..<k + card A}"
lunit_simps_4: fixes f :: "'b" assumes "D.ide f" shows "D.dom \<l>\<^sub>D[f] = trg\<^sub>D f \<star>\<^sub>D f"
sup_ge1_type: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" shows "\<tau> \<le> \<tau> \<squnion> \<sigma>"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "C.arr (C.map f)"
assms_1: fixes A :: "'a tensor" assumes "dims A = []" shows "P A"
le_supE: fixes a :: "'b multiset"   and b :: "'b multiset"   and x :: "'b multiset" assumes "a \<union># b \<subseteq># x" obtains "a \<subseteq># x" and "b \<subseteq># x"
simps_1: shows "dual\<^sub>s\<^sub>t [] = []"
is_category_with_terminal_object: shows "category_with_terminal_object C"
failingBranchExistence: fixes subs :: "'a \<Rightarrow> 'a set"   and P :: "(nat \<times> 'a) set \<Rightarrow> bool"   and gamma :: "'a" assumes "inherited subs P"   and "fans subs"   and "\<not> P (tree subs gamma)" shows "\<exists>f. branch subs gamma f \<and> (\<forall>n. \<not> P (tree subs (f n)))"
dvd_div_eq_0_iff: fixes b :: "'a"   and a :: "'a" assumes "b dvd a" shows "(a div b = (0::'a)) = (a = (0::'a))"
closed_sum_left_subset: fixes B :: "'d set"   and A :: "'d set" assumes "(0::'d) \<in> B" shows "A \<subseteq> A +\<^sub>M B"
equivalence_preserved_by_iso_left: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a"   and \<phi> :: "'a"   and f' :: "'a" assumes "internal_equivalence f g \<eta> \<epsilon>"   and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"   and "local.iso \<phi>" shows "internal_equivalence f' g ((g \<star> \<phi>) \<cdot> \<eta>) (\<epsilon> \<cdot> (local.inv \<phi> \<star> g))"
reads_writes_separate_backwards: fixes SR :: "('a \<Rightarrow> 'a \<Rightarrow> bool) set"   and getter :: "('a, 'b, 'c) prog"   and h :: "'a"   and h' :: "'a"   and SW :: "('a, 'd, 'e) prog set"   and setter :: "('a, 'd, 'f) prog"   and x :: "'c" assumes "reads SR getter h h'"   and "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> getter \<rightarrow>\<^sub>r x"   and "\<And>h h'. \<forall>w\<in>SW. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>SR. r h h')" shows "h \<turnstile> getter \<rightarrow>\<^sub>r x"
subsequence_length: fixes w :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" shows "length (w [i \<rightarrow> j]) = j - i"
const_lookup_zero: fixes p :: "real"   and x :: "nat" shows "isolate_variable_sparse (Const p) x 0 = Const p"
uintNeq2: shows "uint (~~ w) \<noteq> 2147483647"
times_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and y :: "'b"   and x :: "'b" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero z"   and "less_eq y x" shows "less_eq (times y z) (times x z)"
proots_within_pCons_1_iff_2: fixes a :: "'b"   and s :: "'b set" shows "proots_within [:a, - (1::'b):] s = (if a \<in> s then {a} else {})"
has_field_derivative_cong_ev': fixes x :: "'a"   and y :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and u :: "'a"   and v :: "'a"   and t :: "'a set" assumes "x = y"   and "\<forall>\<^sub>F x in nhds x. x \<in> s \<longrightarrow> f x = g x"   and "u = v"   and "s = t"   and "f x = g y" shows "(f has_field_derivative u) (at x within s) = (g has_field_derivative v) (at y within t)"
rewrite_negated_primitives_unchanged_if_not_has_disc_negated: fixes m :: "'b match_expr"   and disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and C :: "'c \<Rightarrow> 'b"   and negate_f :: "('c \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc_negated disc False m" shows "rewrite_negated_primitives (disc, sel) C negate_f m = m"
\<alpha>_\<beta>: shows "LeftDerivation \<alpha> (take u D) \<beta>"
unity_root_pow: fixes k :: "nat"   and n :: "int"   and m :: "nat" shows "unity_root k n ^ m = unity_root k (n * int m)"
thetaParZOL2_def: shows "thetaParZOL2 \<equiv> {(c1 | c2, d) |c1 c2 d. discr c1 \<and> c2 \<approx>01 d}"
Ssafe_def: fixes S :: "('a, 'b, 'c) formula list \<times> ('a, 'b, 'c) formula list" shows "Ssafe S = ((\<forall>i\<ge>0. i < length (fst S) \<longrightarrow> fsafe (fst S ! i)) \<and> (\<forall>i\<ge>0. i < length (snd S) \<longrightarrow> fsafe (snd S ! i)))"
subset_smash_signs: fixes a1 :: "rat list list"   and a2 :: "rat list list"   and b1 :: "rat list list"   and b2 :: "rat list list" assumes "set a1 \<subseteq> set a2"   and "set b1 \<subseteq> set b2" shows "set (signs_smash a1 b1) \<subseteq> set (signs_smash a2 b2)"
disjnt_empty2: fixes A :: "'a set" shows "disjnt A {}"
nth_coeffs_coeff: fixes i :: "nat"   and f :: "'b poly" assumes "i < length (coeffs f)" shows "coeffs f ! i = coeff f i"
timpls_transformable_to_pred_finite: fixes A :: "'sets set set"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term" assumes "finite A"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "finite {s. timpls_transformable_to_pred A t s}"
prefix_eq_prefix_take_ex: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix xs ys = (\<exists>n. ys \<down> n = xs)"
simps: fixes uu :: "'b ext" shows "uu\<^sup>\<star> = ext.Top"
degree_polyadd: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1" shows "Reflected_Multivariate_Polynomial.degree (p +\<^sub>p q) \<le> max (Reflected_Multivariate_Polynomial.degree p) (Reflected_Multivariate_Polynomial.degree q)"
inject: fixes x1 :: "'a list"   and x2 :: "nat"   and y1 :: "'a list"   and y2 :: "nat" shows "(Fundef x1 x2 = Fundef y1 y2) = (x1 = y1 \<and> x2 = y2)"
csum_def: fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set" shows "r1 +c r2 \<equiv> |Field r1 <+> Field r2|"
or_lta2_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "D \<noteq> E"   and "F \<noteq> E" shows "A B C LtA D E F \<or> D E F LtA A B C \<or> A B C CongA D E F"
is_SDG_path_valid_SDG_node_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n"
literalElListIffOppositeLiteralElOppositeLiteralList: fixes literal :: "Literal"   and literalList :: "Literal list" shows "literal el literalList = opposite literal el oppositeLiteralList literalList"
append_self_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs @ ys = xs) = (ys = [])"
decycle_def: fixes E :: "'a uprod set"   and u :: "'a"   and p :: "'a uprod list" shows "decycle E u p \<equiv> epath E u p u \<and> 2 < length p \<and> distinct p"
SAT_Sup_upper: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and Y :: "('k \<times> 'k \<Rightarrow> ennreal) set"   and \<Gamma> :: "('k, 'l) web_scheme" assumes "f \<in> Y" shows "SAT \<Gamma> f \<subseteq> SAT \<Gamma> (\<Squnion> Y)"
add_le_cancel_right: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus a c) (plus b c) = less_eq a b"
subfrmlsn_id: fixes \<phi> :: "'a ltln" shows "\<phi> \<in> subfrmlsn \<phi>"
cases: fixes x :: "H1 \<times> X4 set" obtains uu_ :: "H1" where "x = (uu_, {X4.Xd1})"   | uv_ :: "H1" where "x = (uv_, {X4.Xd1'})"   | uw_ :: "H1" where "x = (uw_, {X4.Xd2})"   | ux_ :: "H1" where "x = (ux_, {Xd2'})"   | uy_ :: "H1" where "x = (uy_, {X4.Xd1, X4.Xd1'})"   | uz_ :: "H1" where "x = (uz_, {X4.Xd1, X4.Xd2})"   | va_ :: "H1" where "x = (va_, {X4.Xd1, Xd2'})"   | vb_ :: "H1" where "x = (vb_, {X4.Xd1', X4.Xd2})"   | vc_ :: "H1" where "x = (vc_, {X4.Xd1', Xd2'})"   | vd_ :: "H1" where "x = (vd_, {X4.Xd2, Xd2'})"   | ve_ :: "H1" where "x = (ve_, {X4.Xd1, X4.Xd1', X4.Xd2})"   | vf_ :: "H1" where "x = (vf_, {X4.Xd1, X4.Xd1', Xd2'})"   | vg_ :: "H1" where "x = (vg_, {X4.Xd1, X4.Xd2, Xd2'})"   | vh_ :: "H1" where "x = (vh_, {X4.Xd1', X4.Xd2, Xd2'})"   | vi_ :: "H1" where "x = (vi_, {X4.Xd1, X4.Xd1', X4.Xd2, Xd2'})"   | vj_ :: "H1" where "x = (vj_, {})"
scale_zero_left: fixes x :: "'b" shows "(0::'a) *a x = (0::'b)"
iso_is_arr: fixes f :: "'a" assumes "hom.iso f" shows "hom.arr f"
get_lower_bound_1: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and U :: "'c set"   and V :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "U \<in> I"   and "V \<in> I" shows "direct_lim.get_lower_bound I U V \<in> I"
map_poly_comm_semiring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_comm_semiring_hom hom \<equiv> comm_semiring_hom hom"
zero_less_diff: fixes n :: "nat"   and m :: "nat" shows "order.greater (n - m) 0 = order.greater n m"
im_zero: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b" assumes "GroupHom G T" shows "T (0::'a) = (0::'b)"
tails_strict: shows "tails\<cdot>\<bottom> = \<bottom>"
div_gt_0: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "0 < m" shows "0 < n div m"
double_negation_dist_implies: fixes y :: "'a"   and x :: "'a" shows "- - (y - x) = - - y - - - x"
Orbit_Relation_def: shows "Orbit_Relation = {(x, y). x \<in> S \<and> y \<in> S \<and> (\<exists>\<alpha>\<in>G. y = \<alpha> x)}"
intro: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open"   and "class.t2_space_axioms open" shows "class.t2_space open"
scaleRat_leq2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq a b"   and "k < 0" shows "less_eq (scaleRat k b) (scaleRat k a)"
assms_1: shows "pdevs_apply x d \<noteq> (0::'a)"
S_eq: fixes a :: "real"   and i :: "'b" shows "S a i = f -` F (a, i) \<inter> space M"
assms_3: shows "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"
top_eq_inf_iff: fixes x :: "'b upred"   and y :: "'b upred" shows "(true = (x \<and> y)) = (x = true \<and> y = true)"
supp_Conj: fixes A :: "fm"   and B :: "fm" shows "supp (A AND B) = supp A \<union> supp B"
rqrm_prop_def: fixes \<delta> :: "('a, 'b) ta_rule set"   and rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap" shows "rqrm_prop \<delta> rqrm \<equiv> \<forall>q. ls.\<alpha> (rqrm_lookup rqrm q) = {r \<in> \<delta>. q \<in> set (rhsq r)}"
assms_1: fixes x :: "'b"   and s :: "'a" assumes "x \<in> S s" shows "healthy (wp (prog x))"
trms\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('c, 'd) stateful_strand_step list"   and B :: "('c, 'd) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t (A @ B) = trms\<^sub>s\<^sub>s\<^sub>t A \<union> trms\<^sub>s\<^sub>s\<^sub>t B"
simps_5: fixes K :: "clock"   and n :: "nat" shows "\<lbrakk> K \<not>\<Up> \<ge> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = {\<rho>. \<forall>i\<ge>n. \<not> hamlet (Rep_run \<rho> i K)}"
assms_1: shows "\<turnstile>l \<oplus>\<^sub>\<or>\<^bsup>(s, s')\<^esup>"
y_eq_new_last: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" assumes "ProofStepInduct_y_eq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post" shows "y = DisjointPathsPlusOne.new_last P_new"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "'a list"   and y :: "'a list" assumes "swap_list x xa xb = y" obtains i :: "nat" and j :: "nat" and l :: "'a list" where "x = i" and "xa = j" and "xb = l" and "y = l[j := l ! i, i := l ! j]"
converse_iff: fixes a :: "'e"   and b :: "'f"   and r :: "('f \<times> 'e) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
subtypings_length: fixes CT :: "nat \<Rightarrow> classDef option"   and As :: "nat list"   and Bs :: "nat list" assumes "CT \<turnstile>+ As <: Bs" shows "length As = length Bs"
simps_2: fixes x :: "'a"   and e :: "'b \<Rightarrow> 'c"   and v :: "'d"   and xs :: "('a \<times> ('b \<Rightarrow> 'c) \<times> 'd) list"   and l :: "'a \<Rightarrow> 'c"   and s :: "'b" shows "postList ((x, e, v) # xs) l s = (l x = e s \<and> postList xs l s)"
pickEff_def: fixes r :: "'rule"   and s :: "'state" shows "pickEff r s \<equiv> if enabled r s then SOME sl. eff r s sl else the None"
simps_5: fixes S :: "'av Abs_State.st option"   and Inv :: "'av Abs_State.st option"   and b :: "bexp"   and c :: "'av Abs_State.st option acom"   and P :: "'av Abs_State.st option" shows "step' S ({Inv} WHILE b DO c {P}) = {S \<squnion> post c} WHILE b DO step' Inv c {Inv}"
space_empty_eq_bot: fixes a :: "'b measure" shows "(space a = {}) = (a = bot)"
intros_1: fixes out :: "'b"   and c :: "('a, 'b, 'c) rpv"   and gpv :: "('a, 'b, 'c) gpv"   and input :: "'c"   and \<I> :: "('b, 'c) \<I>"   and x :: "('a, 'b, 'c) gpv" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "x = c input" shows "x \<in> sub_gpvs \<I> gpv"
assms: shows "\<forall>x. SMT.trigger (SMT.Symb_Cons (SMT.Symb_Cons (SMT.pat (f x)) SMT.Symb_Nil) SMT.Symb_Nil) (f x = x)"
unsigned_eq_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and w :: "'b word" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.unsigned one plus zero w = zero) = (w = 0)"