hcju/isabelleps huggingface dataset:
nAct_mono_back: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and n :: "nat"   and t :: "nat \<Rightarrow> cnf"   and n' :: "nat" assumes "dynamic_component.nAct tCMP active c (enat n) (inf_llist t) \<le> dynamic_component.nAct tCMP active c (enat n') (inf_llist t)" shows "dynamic_component.lNAct active c t n \<le> n'"
distinct_3: shows "X4.Xd1 \<noteq> X4.Xd2"
l2_inv3_step4: fixes Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and gnx :: "msg" shows "{l2_inv3} l2_step4 Rb A B Ni gnx {> l2_inv3}"
eSuc_ile_mono: fixes n :: "eint"   and m :: "eint" shows "(n + 1 \<le> m + 1) = (n \<le> m)"
Union_least: fixes A :: "'d set set"   and C :: "'d set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> C" shows "\<Union> A \<subseteq> C"
simps_2: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'a list"   and x :: "'a" shows "ipurge_tr I D u (xs @ [x]) = (if D x \<in> sinks I D u (xs @ [x]) then ipurge_tr I D u xs else ipurge_tr I D u xs @ [x])"
hash_rt_F\<^sub>m_alt_def: fixes h :: "'c \<Rightarrow> 'e"   and rhm :: "'d \<Rightarrow> 'f" shows "hash_rt_F\<^sub>m h rhm = hash_blindable (map_prod h (map rhm))"
MatchOr_MatchNot: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (MatchNot (MatchOr m1 m2)) a p = (matches \<gamma> (MatchNot m1) a p \<and> matches \<gamma> (MatchNot m2) a p)"
sum_2: fixes f :: "2 \<Rightarrow> 'b" shows "sum f UNIV = f 1 + f 2"
semilattice_set_axioms: shows "semilattice_set (\<^bold>*)"
B_def: shows "B = sum_list (ss @ [s, t]) `\<rightarrow> C0"
absolutely_integrable_mult_Fejer_kernel_reflected2: fixes f :: "real \<Rightarrow> real"   and n :: "nat"   and t :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>x. Fejer_kernel n x * f (t - x)) absolutely_integrable_on {- pi..pi}"
val_def: fixes qd :: "QDelta"   and \<delta> :: "rat" shows "val qd \<delta> = qdfst qd + \<delta> * qdsnd qd"
space_return: fixes M :: "'a measure"   and x :: "'a" shows "space (return M x) = space M"
one: shows "x ; y \<cdot> - (x ; z) \<le> x ; (y \<cdot> - z)"
trans: fixes b :: "'a word"   and a :: "'a word"   and c :: "'a word" assumes "(b::'a word) \<le>s (a::'a word)"   and "(c::'a word) \<le>s (b::'a word)" shows "(c::'a word) \<le>s (a::'a word)"
monotone_image: fixes f :: "'b \<Rightarrow> 'c" shows "monotone (\<subseteq>) (\<subseteq>) ((`) f)"
ccSUP_upper: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<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 bot :: "'c"   and top :: "'c"   and A :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "i \<in> A" shows "less_eq (f i) (Sup (f ` A))"
face_cycle_succ_pred: fixes a :: "'b" shows "face_cycle_succ (face_cycle_pred a) = a"
gcd_eq_0_iff: 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 a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(gcd a b = zero) = (a = zero \<and> b = zero)"
advantage3_def: fixes \<G> :: "'b cyclic_group"   and x :: "nat"   and \<A> :: "'b \<times> 'b \<Rightarrow> nat spmf" assumes "dis_log_alt \<G>" shows "dis_log_alt.advantage3 \<G> x \<A> \<equiv> spmf (dis_log_alt.dis_log3 \<G> x \<A>) True"
quasi_inverses_def: fixes f :: "'a"   and g :: "'a" shows "quasi_inverses f g \<equiv> \<exists>\<phi> \<psi>. internal_equivalence f g \<phi> \<psi>"
f_def: shows "bij_betw f (range to_nat) (range to_nat)"
assms: fixes v :: "'v"   and u :: "'v" assumes "v \<in> V"   and "u \<in> V"   and "v \<noteq> u" shows "\<exists>!n. adjacent v n \<and> adjacent u n"
condensation_inequality_1: fixes n :: "nat" assumes "\<And>m n. \<lbrakk>0 < m; m \<le> n\<rbrakk> \<Longrightarrow> f n \<le> f m" shows "(\<Sum>k = 1..<n. f (2 * 2 ^ Discrete.log k)) \<le> sum f {1..<n}"
sum_closed: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ('f, 'g) aezfun" assumes "finite A"   and "f ` A \<subseteq> FG" shows "sum f A \<in> FG"
start_stepsize_impl_def: shows "start_stepsize_impl = (if 0 < start_stepsize optns then start_stepsize optns else 1)"
r_ifeq: fixes a :: "nat"   and b :: "nat"   and v\<^sub>0 :: "nat"   and v\<^sub>1 :: "nat" shows "eval r_ifeq [a, b, v\<^sub>0, v\<^sub>1] \<down>= (if a = b then v\<^sub>0 else v\<^sub>1)"
assms_3: shows "sKS secret \<notin> LocalSecrets PQ"
nprv_subst: fixes x :: "'var"   and t :: "'trm"   and \<psi> :: "'fmla"   and F :: "'fmla set" assumes "x \<in> var"   and "t \<in> trm"   and "\<psi> \<in> fmla"   and "finite F"   and "F \<subseteq> fmla"   and "nprv F \<psi>" shows "nprv ((\<lambda>\<phi>. subst \<phi> t x) ` F) (subst \<psi> t x)"
inject_3: fixes x31 :: "('a, 'b) MWLsCom"   and x32 :: "('a, 'b) MWLsCom"   and y31 :: "('a, 'b) MWLsCom"   and y32 :: "('a, 'b) MWLsCom" shows "(x31;x32 = y31;y32) = (x31 = y31 \<and> x32 = y32)"
wfP_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "wfP r = wf {(x, y). r x y}"
union_is_single: fixes M :: "'c multiset"   and N :: "'c multiset"   and a :: "'c" shows "(M + N = {#a#}) = (M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#})"
Nil_less_Cons: fixes a :: "'b"   and x :: "'b list" shows "[] < a # x"
that: fixes p :: "'item list"   and r :: "'item list"   and s :: "'item list"   and q :: "'item list" assumes "t =\<^sub>F p @ r"   and "u =\<^sub>F s @ q"   and "v =\<^sub>F p @ s"   and "w =\<^sub>F r @ q"   and "Ind (set r) (set s)" shows "thesis"
2: shows "(v, u) \<in> \<E> f\<^sub>1 \<and> (u, v) \<in> \<E> f\<^sub>2 \<and> vs = [] \<or> (\<exists>v\<in>\<V> f\<^sub>1 \<inter> \<V> f\<^sub>2. v \<notin> \<V> g)"
pred_True: shows "pred_option (\<lambda>_. True) = (\<lambda>_. True)"
lex_ord_raw_def: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and ko :: "'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> order option"   and xs :: "('b \<times> 'c) list \<times> 'a"   and ys :: "('b \<times> 'd) list \<times> 'a" shows "oalist_raw.lex_ord_raw rep_key_order ko f xs ys = comparator.lex_ord_pair (key_compare (rep_key_order ko)) f (oalist_raw.sort_oalist_aux rep_key_order ko xs) (oalist_raw.sort_oalist_aux rep_key_order ko ys)"
some_H'h't: fixes M :: "('b \<times> real) set set"   and E :: "'b set"   and p :: "'b \<Rightarrow> real"   and F :: "'b set"   and f :: "'b \<Rightarrow> real"   and c :: "('b \<times> real) set set"   and H :: "'b set"   and h :: "'b \<Rightarrow> real"   and x :: "'b" assumes "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "graph H h = \<Union> c"   and "x \<in> H" shows "\<exists>H' h'. graph H' h' \<in> c \<and> (x, h x) \<in> graph H' h' \<and> linearform H' h' \<and> Subspace.subspace H' E \<and> Subspace.subspace F H' \<and> graph F f \<subseteq> graph H' h' \<and> (\<forall>x\<in>H'. h' x \<le> p x)"
add_order_least: fixes n :: "nat"   and a :: "'a" assumes "order.greater n 0"   and "a +^ n = (0::'a)" shows "order.greater_eq n (add_order a)"
meetI: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'a"   and x :: "'a"   and y :: "'a" assumes "dlat le"   and "le i x"   and "le i y"   and "\<And>z. \<lbrakk>le z x; le z y\<rbrakk> \<Longrightarrow> le z i" shows "dlat.meet le x y = i"
alphas_1: fixes bs :: "atom set"   and x :: "'e"   and R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> atom set"   and p :: "perm"   and cs :: "atom set"   and y :: "'e" shows "(bs, x) \<approx>set R f p (cs, y) = (f x - bs = f y - cs \<and> (f x - bs) \<sharp>* p \<and> R (p \<bullet> x) y \<and> p \<bullet> bs = cs)"
assms: shows "A \<subseteq>\<^sub>\<circ> vinsert b B"
cA: fixes c :: "'a" assumes "extreme_bound A (\<sqsubseteq>) C c" shows "c \<in> A"
axioms_1: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "ring R addition multiplication zero unit" shows "abelian_group R addition zero"
pivotandupdate_rvars: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "PivotAndUpdate eq_idx_for_lvar pivot_and_update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (EqForLVar.eq_for_lvar eq_idx_for_lvar (\<T> s) x\<^sub>i)" shows "rvars (\<T> (pivot_and_update x\<^sub>i x\<^sub>j c s)) = rvars (\<T> s) - {x\<^sub>j} \<union> {x\<^sub>i}"
cases: fixes x :: "symbol list" obtains "xa__ = []"   | x :: "symbol" and xs :: "symbol list" where "xa__ = x # xs"
big_def: fixes P :: "'prog"   and \<sigma> :: "'state" shows "big P \<sigma> \<equiv> {\<sigma>'. \<exists>n. \<sigma>' \<in> small_nstep P \<sigma> n \<and> \<sigma>' \<in> endset}"
simps: fixes a1 :: "(C, BD) sequent list"   and a2 :: "(C, BD) sequent" shows "g3mod2p a1 a2 = ((\<exists>A. a1 = [ \<Empt> \<Rightarrow>* \<LM> A  \<RM>] \<and> a2 = ( \<Empt> \<Rightarrow>* \<LM> \<diamond> A  \<RM>)) \<or> (\<exists>A. a1 = [ \<LM> A  \<RM> \<Rightarrow>* \<Empt>] \<and> a2 = ( \<LM> \<box> A  \<RM> \<Rightarrow>* \<Empt>)))"
generate_minimal: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "generate G H = \<Inter> {H'. subgroup H' G \<and> H \<subseteq> H'}"
pred_entailment_def: fixes P :: "'BExp set"   and P' :: "'BExp set" shows "P \<turnstile> P' \<equiv> \<forall>mem. pred P mem \<longrightarrow> pred P' mem"
Nml_HcompD_3: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "Nml t"
hyps: fixes env :: "('a, 'b, 'c) env"   and e :: "('a, 'b, 'c) env" assumes "lookup env xs = Some e" shows "lookup env (xs @ ys) = lookup e ys"
pcr_ennreal_enn2ereal: fixes x :: "ennreal" shows "pcr_ennreal (enn2ereal x) x"
pochhammer_product': fixes z :: "'b"   and n :: "nat"   and m :: "nat" shows "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
scale_right_diff_distrib: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and x :: "'b"   and y :: "'b" assumes "Modules.module scale" shows "scale a (x - y) = scale a x - scale a y"
prems: shows "k < length (trace_parallel_plan_strips I (a # \<pi>))"
lexord_cancel_right_linorder: fixes u :: "'a list"   and z :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "u \<cdot> z <lex v \<cdot> w"   and "\<not> u \<bowtie> v" shows "u <lex v"
uedge_in_set_eq: fixes u :: "'b"   and v :: "'b"   and S :: "('b \<times> 'b) set" shows "(uedge (u, v) \<in> uedge ` S) = ((u, v) \<in> S \<or> (v, u) \<in> S)"
image_to_nat_on: fixes A :: "'a set" assumes "countable A"   and "infinite A" shows "to_nat_on A ` A = UNIV"
xs_def: shows "xs = Time_Monad.val (mergesort_tm fst (x # y # zs))"
cInf_singleton: fixes x :: "'b multiset" shows "Inf {x} = x"
interpretation_grounds_all': fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and M :: "('b, 'a) Term.term set" assumes "interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>" shows "ground (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>)"
resTrans_2: fixes x :: "name"   and y :: "name"   and P :: "pi"   and Rs :: "residual" assumes "<\<nu>x>x<y>.P \<longmapsto> Rs" shows "False"
Max_in_mset: fixes M :: "'b multiset" assumes "M \<noteq> {#}" shows "Max_mset M \<in># M"
polyfun_submatrix_deep_model: fixes rs :: "nat list"   and y :: "nat"   and i :: "nat"   and j :: "nat"   and shared_weights :: "bool" assumes "deep_model_correct_params_y rs y"   and "i < deep_model_correct_params.r rs ^ deep_model_correct_params.N_half rs"   and "j < deep_model_correct_params.r rs ^ deep_model_correct_params.N_half rs" shows "polyfun {..<deep_model_correct_params.weight_space_dim shared_weights rs} (\<lambda>f. deep_model_correct_params_y.witness_submatrix shared_weights rs y f $$ (i, j))"
pin: fixes P :: "'a multiset" assumes "P \<in># {#mset_set (bl \<inter> blv). bl \<in># filter_mset ((\<in>) p) (remove1_mset blv \<B>)#}" shows "count P p = 1"
simps_2: fixes red :: "bool list \<Rightarrow> bool"   and ds :: "bool list" shows "cutoff red ds Leaf = Leaf"
ccpo_gfp_simp: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "ccpo.fixp Inf (\<lambda>x y. y \<le> x) f = gfp f"
l11_30: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p" assumes "A B C LeA D E F"   and "A B C CongA A' B' C'"   and "D E F CongA D' E' F'" shows "A' B' C' LeA D' E' F'"
ab_group_add_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" shows "class.ab_group_add plus zero minus uminus \<equiv> class.comm_monoid_add plus zero \<and> class.ab_group_add_axioms plus zero minus uminus"
has_integral_componentwise_iff: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and A :: "'c set" shows "(f has_integral y) A = (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y \<bullet> b) A)"
r: shows "row_echelon_form (1\<^sub>m 0)"
apply_cm_def: fixes c :: "('c, 't, 'd) Propagate.configuration_scheme"   and loc :: "'c"   and t :: "'t"   and n :: "int" shows "apply_cm c loc t n = (let new_pointstamps = \<lambda>loc'. if loc' = loc then update_zmultiset (c_pts c loc') t n else c_pts c loc' in c\<lparr>c_pts := new_pointstamps, c_work := \<lambda>loc'. c_work c loc' + frontier_changes (new_pointstamps loc') (c_pts c loc')\<rparr>)"
\<xi>: fixes x :: "'a" assumes "x \<in> K" shows "\<exists>C. C \<in> components (T - S) \<and> x \<in> C \<and> \<xi> x \<in> C \<and> \<xi> x \<in> L"
atlas_is_atlas: fixes a1 :: "('a, 'b) chart"   and a2 :: "('a, 'b) chart" assumes "a1 \<in> sub.atlas"   and "a2 \<in> sub.atlas" shows "k-smooth_compat a1 a2"
lemma2_6_5_a': fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset"   and N :: "'a multiset"   and S :: "'a set" assumes "trans r"   and "(M, N) \<in> mul_eq r" shows "(M -s r \<down>s S, N -s r \<down>s S) \<in> mul_eq r"
analytic_on_compose_gen: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and g :: "complex \<Rightarrow> complex"   and T :: "complex set" assumes "f analytic_on S"   and "g analytic_on T"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<in> T" shows "g \<circ> f analytic_on S"
deg_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c" shows "deg R p = (LEAST n. bound \<zero>\<^bsub>R\<^esub> n (coeff (UP R) p))"
i': shows "atom i' \<sharp> (t, u, z, X0, v1, v2, X3)"
cod_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.cod (B.VVV.dom f) = B.VVV.dom f"
bind_alt1: fixes m :: "'m"   and m' :: "'m"   and f :: "'a option \<Rightarrow> 'm" shows "bind (alt m m') f = alt (bind m f) (bind m' f)"
fix_indices: fixes xs :: "nat list"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'d \<Rightarrow> nat" assumes "set xs \<subseteq> {0..n}"   and "distinct xs"   and "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "canonical M n" shows "\<exists>M'. ((\<exists>u. u \<turnstile>\<^bsub>v,n\<^esub> M') \<longrightarrow> (\<exists>u. u \<turnstile>\<^bsub>v,n\<^esub> M)) \<and> (\<forall>i\<in>set xs. i \<noteq> 0 \<longrightarrow> M' 0 i + M' i 0 = \<one>) \<and> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M' i i xs) \<and> (\<forall>i\<le>n. i \<notin> set xs \<and> M 0 i + M i 0 = \<one> \<longrightarrow> M' 0 i + M' i 0 = \<one>)"
total_preorder_on_restrict_subset: fixes A :: "'alt set" assumes "A \<subseteq> \<Union> (set (x # xs))" shows "total_preorder_on A (restrict_relation A (of_weak_ranking (x # xs)))"
assms_1: shows "invariantsHoldInState state F0 decisionVars"
mkVarChannel_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and v :: "varDecl"   and upd :: "((String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list) \<Rightarrow> 'a gState_scheme \<times> pState \<Rightarrow> 'a gState_scheme \<times> pState" assumes "gState_inv prog g"   and "\<And>k v' cs. toVariable g p v = (k, v', cs) \<Longrightarrow> (g, fst (upd (lm.update k v') (g, p))) \<in> gState_progress_rel prog" shows "(g, fst (mkVarChannel v upd g p)) \<in> gState_progress_rel prog"
swap_pair_fun''': fixes C :: "'b \<times> 'a \<Rightarrow> real \<times> real" shows "(\<lambda>y. prod.swap (C (1::'b, y))) = (\<lambda>x. (C (1::'b, x) \<bullet> j, C (1::'b, x) \<bullet> i))"
mult_vec_mat_def: fixes v :: "'c vec"   and A :: "'c mat" shows "v \<^sub>v* A \<equiv> vec (dim_col A) (\<lambda>j. v \<bullet> col A j)"
H_cond: fixes p :: "'a"   and r :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a" assumes "H (\<tt>\<tt> p \<cdot> \<tt>\<tt> r) x q"   and "H (\<tt>\<tt> p \<cdot> !r) y q" shows "H p (if r then x else y) q"
finite_comm_group_Characters: shows "finite_comm_group (Characters (Characters (Characters G)))"
inf2E: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "inf A B x y" obtains "A x y" and "B x y"
mop_set_pick_def: shows "mop_set_pick \<equiv> mop pre_set_pick IICF_Set.op_set_pick"
seq_comp_prop_3: fixes xs :: "'c option list"   and Y :: "'c option set"   and P :: "'c option process"   and Q :: "'c option process"   and X :: "'c option set" assumes "(xs, Y) \<in> seq_comp_failures P Q"   and "X \<subseteq> Y" shows "(xs, X) \<in> seq_comp_failures P Q"
nataction_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.monoid_add plus zero" shows "power.power zero plus a 2 = plus a a"
IH_2: fixes d :: "nat"   and t' :: "'a tree" assumes "ins n d x r = Bal t'"   and "bal_i n (height r + d)" shows "bal_i (n + 1) (height t' + d)"
polyfun_linear_factor: fixes c :: "nat \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" shows "\<exists>b. \<forall>z. (\<Sum>i\<le>n. c i * z ^ i) = (z - a) * (\<Sum>i<n. b i * z ^ i) + (\<Sum>i\<le>n. c i * a ^ i)"
Poincare_recurrence_thm_2: fixes A :: "'a set" assumes "A \<in> sets M" shows "A - recurrent_subset_infty A \<in> null_sets M"
hext_new: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and x :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set" assumes "h a = None" shows "h \<unlhd> h(a \<mapsto> x)"
nodes_restrict_edges: fixes g :: "'b ugraph"   and E :: "('b \<times> 'b) set" shows "nodes (restrict_edges g E) = nodes g"
minus_eq_nat_subst: fixes A :: "nat"   and B :: "nat"   and C :: "nat"   and D :: "nat"   and E :: "nat" assumes "A = B - C - D"   and "- int E = - int C - int D" shows "A = B - E"
unfold_while: fixes 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 b :: "expr"   and c :: "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 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)" shows "P,E \<turnstile> \<langle>while (b) c,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = P,E \<turnstile> \<langle>if (b) (c;; while (b) c) else unit,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
intros: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and H :: "'henv"   and \<Sigma> :: "'dyn list" assumes "Finca_get F f = Some fd"   and "pc = length (body fd)" shows "Sinca.final (State F H [Frame f pc \<Sigma>])"
interior_rel_interior_gen: fixes S :: "'b set" shows "interior S = (if aff_dim S = int DIM('b) then rel_interior S else {})"
simps_3: fixes n :: "nat" assumes "2 \<le> n" shows "karatsuba_cost n = 3 * karatsuba_cost (nat \<lceil>real n / 2\<rceil>) + real n"
assms_2: fixes r :: "'a" assumes "r \<in> carrier R" shows "f r = g r"
id_blinfun_def: shows "id_blinfun \<equiv> Blinfun (\<lambda>x. x)"
gcd_add_mult: fixes m :: "'b"   and k :: "'b"   and n :: "'b" shows "gcd m (k * m + n) = gcd m n"
b_def: shows "b = dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<lparr>i\<rparr>"
distinct_map_fst_snd_eqD: fixes xs :: "('d \<times> 'e) list"   and i :: "'d"   and a :: "'e"   and b :: "'e" assumes "distinct (map fst xs)"   and "(i, a) \<in> set xs"   and "(i, b) \<in> set xs" shows "a = b"
IH: fixes \<sigma> :: "nat \<Rightarrow> val"   and f :: "nat \<Rightarrow> nat" shows "cexpr_sem \<sigma> (map_vars f e) = cexpr_sem (\<sigma> \<circ> f) e"
in_set_least_squares_approximation_eq_full_rank_QR2: fixes A :: "((real, 'a) vec, 'b) vec"   and x :: "(real, 'a) vec"   and b :: "(real, 'b) vec" assumes "rank A = ncols A" shows "(x \<in> set_least_squares_approximation A b) = (x = matrix_inv (snd (QR_decomposition A)) ** Finite_Cartesian_Product.transpose (fst (QR_decomposition A)) *v b)"
compat_list: fixes y :: "'a"   and x :: "'a"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and xs' :: "'a list" assumes "y \<noteq> x"   and "gt y x" shows "ext gt (xs @ y # xs') (xs @ x # xs')"
closed_segment_same_Re: fixes a :: "complex"   and b :: "complex" assumes "Re a = Re b" shows "closed_segment a b = {z. Re z = Re a \<and> Im z \<in> closed_segment (Im a) (Im b)}"
abs_eq_iff: fixes x :: "'a rexp"   and y :: "'a rexp" shows "(ACI_class x = ACI_class y) = ACI x y"
arcs_ends_wf: fixes G :: "('a, 'b) pre_digraph"   and x :: "'a"   and y :: "'a" assumes "graph G"   and "x \<rightarrow>\<^bsub>G\<^esub> y" shows "x \<in> verts G \<and> y \<in> verts G"
fact: shows "\<forall>i. (Decreasing_Diagrams.measure r (s i), Decreasing_Diagrams.measure r (s (Suc i))) \<in> (mul r)\<inverse>"
card_Suc_eq: fixes A :: "'d set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
ipresCons_imp_ipresSubstAbs: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.ipresCons Delta h hA MOD"   and "FixSyn.igSubstCls Delta MOD"   and "FixSyn.igConsIPresIGWls Delta MOD"   and "FixSyn.igFreshCls Delta MOD" shows "FixSyn.ipresSubstAbs Delta h hA MOD"
card1_eI: fixes S :: "'a set" assumes "1 \<le> card S" shows "\<exists>y S'. S = {y} \<union> S' \<and> y \<notin> S'"
Max_in: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Max less_eq A \<in> A"