hcju/isabelleps huggingface dataset:
relpowp_commute: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and n :: "nat" shows "P OO P ^^ n = P ^^ n OO P"
cmult_in_bigo_iff: fixes c :: "'b"   and f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. c * f x) \<in> O[F](g)) = (c = (0::'b) \<or> f \<in> O[F](g))"
min_range_antimono_def: fixes p :: "nat"   and F :: "floatarith \<Rightarrow> floatarith"   and DF :: "floatarith \<Rightarrow> floatarith"   and l :: "float"   and u :: "float"   and X :: "(real \<times> real pdevs) \<times> real" shows "min_range_antimono p F DF l u X = (let L = floatarith.Num l; U = floatarith.Num u in approx p (floatarith.Max (DF L) (DF U)) [] \<bind> (\<lambda>aivl. let a = upper aivl; A = floatarith.Num a in approx p (Half (F L + F U - A * (L + U))) [] \<bind> (\<lambda>bivl. let (b, be) = mid_err bivl; (B, Be) = (floatarith.Num (float_of b), floatarith.Num (float_of be)) in approx p (floatarith.Add (Half (F L - F U + A * (U - L))) Be) [] \<bind> (\<lambda>divl. Some (affine_unop p (real_of_float a) b (real_of_float (upper divl)) X)))))"
flow_tendsto: fixes t :: "real"   and x0 :: "'a"   and ts :: "'c \<Rightarrow> real"   and F :: "'c filter" assumes "t \<in> existence_ivl0 x0"   and "(ts \<longlongrightarrow> t) F" shows "((\<lambda>s. flow0 x0 (ts s)) \<longlongrightarrow> flow0 x0 t) F"
taut_conj: fixes A :: "'a upred"   and B :: "'a upred" shows "`A \<and> B` = (`A` \<and> `B`)"
submonoid_inverse_closed: fixes u :: "'b" assumes "JB.additive.sub.invertible u"   and "u \<in> J" shows "JB.additive.inverse u \<in> J"
fps_of_ratfps_add: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "fps_of_ratfps (x + y) = fps_of_ratfps x + fps_of_ratfps y"
sep_reorder_dollar_aux_2: fixes X :: "nat"   and Y :: "nat" shows "($ X \<and>* $ Y) = $ (X + Y)"
altinsertb: shows "\<forall>xa. insertion (nth_default 0 ((xs' @ x # xs)[var := xa])) b = B"
similar_mat_wit_pow_id: fixes A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat"   and k :: "nat" assumes "similar_mat_wit A B P Q" shows "A ^\<^sub>m k = P * B ^\<^sub>m k * Q"
residualInject'_2: fixes \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi"   and M :: "'d"   and B :: "('d, 'e, 'f) boundOutput" shows "(\<alpha> \<prec> P = ROut M B) = (\<exists>xvec N. \<alpha> = M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<and> B = \<lparr>\<nu>*xvec\<rparr>N \<prec>' P)"
field_lbound_gt_zero: fixes d1 :: "'d"   and d2 :: "'d" assumes "(0::'d) < d1"   and "(0::'d) < d2" shows "\<exists>e>0::'d. e < d1 \<and> e < d2"
invertible: fixes u :: "'a" assumes "u \<in> Units" shows "units.sub.invertible u"
pr1_simps_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "cod (pr1 a b) = a"
cA_index: fixes i :: "'n"   and j :: "'n" shows "cA $h i $h j = complex_of_real (A $h i $h j)"
chainsD2: fixes c :: "'b set set"   and S :: "'b set set" assumes "c \<in> chains S" shows "c \<subseteq> S"
length_generate_DB: fixes PS :: "bit list" shows "length (generate_DB PS) = length PS + 8 + sLen"
degree_pCons_eq_if: fixes a :: "'a"   and p :: "'a poly" shows "degree (pCons a p) = (if p = 0 then 0 else Suc (degree p))"
subset_singletonD: fixes A :: "'g set"   and x :: "'g" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
simps_6: fixes t :: "'c local.term" shows "Arr \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Arr t"
abs_eq: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> 'd" shows "Square_Matrix.row (of_fun x) = (\<lambda>i. vec_lambda (x i))"
has_finite_products': fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and I :: "'c set" assumes "category_with_finite_products C"   and "I \<noteq> UNIV"   and "finite I" shows "category.has_products C I"
vmc_path_lnull_ltl_no_deadend: assumes "\<not> lnull (ltl P)" shows "vmc_path_no_deadend G P v p** \<sigma>'"
elims_2: fixes x :: "ivl" assumes "is_empty x" obtains l :: "int" and h :: "int" where "x = {l\<dots>h}" and "h < l"
cr_generator_def: shows "cr_generator \<equiv> \<lambda>x y. x = generator y"
eq: shows "set (map (consistent_sign_vec qs) (characterize_root_list_p p)) = set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 1} \<union> set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = - 1} \<union> set (map (consistent_sign_vec qs) (characterize_root_list_p p)) \<inter> {s. z_R I s = 0}"
err_step_defs_1: fixes n :: "nat"   and app :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and p :: "nat"   and t :: "'b err" shows "err_step n app step p t = (case t of Err \<Rightarrow> error n | OK \<tau> \<Rightarrow> if app p \<tau> then map_snd OK (step p \<tau>) else error n)"
C: shows "cond xh \<longrightarrow> step xh = RES X' \<and> x \<in> X'"
eq_upred_sym: fixes x :: "('e, 'd) uexpr"   and y :: "('e, 'd) uexpr" shows "(x =\<^sub>u y) = (y =\<^sub>u x)"
that: fixes tr :: "'a list" assumes "target (io || tr) q = p \<and> path M (io || tr) q \<and> length io = length tr" shows "thesis"
extend_2: fixes K :: "'a set" assumes "K \<in> \<D>" shows "extend K \<subseteq> cbox a b"
CH_range': fixes c :: "'a"   and B :: "'a set" assumes "c \<in> CH B" shows "c \<in> B"
vars1'_def: shows "ps2_P.vars1' = ind_in_set ps2_P.vars0 ` vars1"
boffa: fixes x :: "'a" assumes "x \<cdot> x = x" shows "x\<^sup>\<star> = (1::'a) + x"
vconn_implies_hpar_1: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>S g\<guillemotright>" shows "S.src \<mu> = S.src f"
mult_le_cancel_right1: fixes c :: "'b"   and b :: "'b" shows "(c \<le> b * c) = (((0::'b) < c \<longrightarrow> (1::'b) \<le> b) \<and> (c < (0::'b) \<longrightarrow> b \<le> (1::'b)))"
total_preorder_on_def: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "total_preorder_on carrier le \<equiv> Order_Predicates.preorder_on carrier le \<and> total_preorder_on_axioms carrier le"
eigen_vector_norm_sr: fixes v :: "(complex, 'n) Finite_Cartesian_Product.vec"   and \<alpha> :: "complex" assumes "eigen_vector cA v \<alpha>" shows "cmod \<alpha> \<le> sr"
one_Characters: fixes G :: "('c, 'd) monoid_scheme" shows "\<one>\<^bsub>Characters G\<^esub> = principal_char G"
mono_SEQI2a: assumes "P \<sqsubseteq> Q" shows "\<forall>s. s \<notin> D (P `;` S) \<longrightarrow> Ra (Q `;` S) s \<subseteq> Ra (P `;` S) s"
subst_closed_except_preserved: fixes t :: "'a"   and S :: "name fset"   and env :: "(name, 'a) fmap" assumes "closed_except t S"   and "fdisjnt (fmdom env) S" shows "closed_except (subst t env) S"
ensures_eq: fixes A :: "'d set"   and B :: "'d set" shows "A ensures B = (A unless B) \<inter> transient (A - B)"
poly_as_sum_of_monoms: fixes p :: "'a poly" shows "(\<Sum>i\<le>Polynomial.degree p. Polynomial.monom (poly.coeff p i) i) = p"
i_set0_mult_mod_0_div_i_set0: fixes I :: "nat set"   and k :: "nat"   and d :: "nat" assumes "I \<in> i_set0_mult k"   and "k mod d = 0" shows "I \<oslash> d \<in> i_set0"
seq: fixes c\<^sub>1 :: "(Eg1.var, aexp, bexp) Stmt"   and mds :: "Mode \<Rightarrow> Eg1.var set"   and mem :: "Eg1.var \<Rightarrow> nat"   and c\<^sub>1' :: "(Eg1.var, aexp, bexp) Stmt"   and mds' :: "Mode \<Rightarrow> Eg1.var set"   and mem' :: "Eg1.var \<Rightarrow> nat"   and c\<^sub>2 :: "(Eg1.var, aexp, bexp) Stmt" assumes "eval_abv\<^sub>A \<langle>c\<^sub>1, mds, mem\<rangle>\<^sub>A \<langle>c\<^sub>1', mds', mem'\<rangle>\<^sub>A" shows "eval_abv\<^sub>A \<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle>\<^sub>A \<langle>c\<^sub>1' ;; c\<^sub>2, mds', mem'\<rangle>\<^sub>A"
cs_def: shows "\<exists>c1 c2. cs = [$C ConstFloat64 c1, $C ConstFloat64 c2]"
sample_set_excl_def: fixes Q :: "nat set"   and P :: "nat set" shows "sample_set_excl Q P = spmf_of_set (Q - P)"
rk_\<alpha>\<beta>\<gamma>: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {\<alpha>, \<beta>, \<gamma>} \<le> 2"
VSigmaD2: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "\<langle>a, b\<rangle> \<in> elts (VSigma A B)" shows "b \<in> elts (B a)"
inj_comm_ring_hom_axioms: shows "inj_comm_ring_hom of_int"
Rep_plus_one_le_card: fixes a :: "'a" assumes "a + (1::'a) \<noteq> (0::'a)" shows "Rep a + 1 < int CARD('a)"
gauss_jordan_single_i_def: fixes ops :: "'b arith_ops_record"   and A :: "'b mat" shows "gauss_jordan_single_i ops A \<equiv> gauss_jordan_main_i ops (dim_row A) (dim_col A) A 0 0"
edge_rev_in_arcs: fixes a :: "'b" shows "(edge_rev M a \<in> arcs G) = (a \<in> arcs G)"
fully_connected_wf: fixes G :: "'b graph" assumes "wf_graph G" shows "wf_graph (fully_connected G)"
Diag_TensorE_2: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag (t \<^bold>\<otimes> u)" shows "C.arr (un_Prim t)"
bij_betw_inv_into_left: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and A' :: "'e set"   and a :: "'d" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A f (f a) = a"
simps_2: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and xs :: "'f list"   and x :: "'f" shows "sinks I D u (xs @ [x]) = (if (u, D x) \<in> I \<or> (\<exists>v\<in>sinks I D u xs. (v, D x) \<in> I) then insert (D x) (sinks I D u xs) else sinks I D u xs)"
univ_poly_subfield_of_consts: fixes K :: "'a list set set" assumes "subfield K (Rupt K p)" shows "subfield (Hom.poly_of_const ` K) (K [X]\<^bsub>Rupt K p\<^esub>)"
word_2p_lem: fixes n :: "nat"   and w :: "'a word" assumes "n < size w" shows "(w < 2 ^ n) = (uint w < 2 ^ n)"
ih_17: shows "InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
shift2: fixes I :: "nat set"   and cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and ii :: "nat" assumes "I \<subseteq> {..<brn (cl ! n)}"   and "ii \<in> local.shift cl n I" shows "brnL cl n \<le> ii \<and> ii < brnL cl n + brn (cl ! n)"
tObsC_tStep: fixes t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" shows "tObsC (t \<leadsto> s) = tObsC t \<leadsto> (envObsC (es s), pubActs s)"
max2_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<in> Field s"   and "b \<in> Field s" shows "((s.max2 a b, c) \<in> s) = ((a, c) \<in> s \<and> (b, c) \<in> s)"
moebius_rotation_uminus: fixes \<phi> :: "real" shows "- moebius_rotation \<phi> = moebius_rotation (- \<phi>)"
wf_RESTRICT: fixes idx :: "'n"   and \<phi> :: "('a, 'k) aformula" assumes "local.wf idx \<phi>" shows "local.wf idx (RESTRICT \<phi>)"
app_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "\<not> pre_term.is_app unapp (const name)"
esqrt_square: fixes x :: "ereal" assumes "0 \<le> x" shows "esqrt x * esqrt x = x"
inf_fun_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
in_reals_code: fixes x :: "complex" shows "(x \<in> \<real>) = (Im x = 0)"
mk_weak_until_def: fixes x :: "'a ltln"   and y :: "'a ltln" shows "mk_weak_until x y \<equiv> case y of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> mk_globally x | _ \<Rightarrow> case x of true\<^sub>n \<Rightarrow> true\<^sub>n | false\<^sub>n \<Rightarrow> y | _ \<Rightarrow> x W\<^sub>n y"
char_poly_four_block_zeros_col: fixes A1 :: "'c mat"   and A2 :: "'c mat"   and n :: "nat"   and A3 :: "'c mat" assumes "A1 \<in> carrier_mat 1 1"   and "A2 \<in> carrier_mat 1 n"   and "A3 \<in> carrier_mat n n" shows "char_poly (four_block_mat A1 A2 (0\<^sub>m n 1) A3) = char_poly A1 * char_poly A3"
blinfun_elim: fixes g :: "'b \<Rightarrow> 'a \<Rightarrow>\<^sub>L real" shows "blinfun_apply (blinfun_vec g) = (\<lambda>x. \<chi>i. blinfun_apply (g i) x)"
cmod_unit_one: fixes a :: "real" shows "cmod (complex_of_real (cos a) + \<i> * complex_of_real (sin a)) = 1"
moduleI: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme" assumes "cring R"   and "abelian_group M"   and "\<And>a x. \<lbrakk>a \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> x \<in> carrier M"   and "\<And>a b x. \<lbrakk>a \<in> carrier R; b \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> (a \<oplus>\<^bsub>R\<^esub> b) \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> b \<odot>\<^bsub>M\<^esub> x"   and "\<And>a x y. \<lbrakk>a \<in> carrier R; x \<in> carrier M; y \<in> carrier M\<rbrakk> \<Longrightarrow> a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y"   and "\<And>a b x. \<lbrakk>a \<in> carrier R; b \<in> carrier R; x \<in> carrier M\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> b \<odot>\<^bsub>M\<^esub> x = a \<odot>\<^bsub>M\<^esub> (b \<odot>\<^bsub>M\<^esub> x)"   and "\<And>x. x \<in> carrier M \<Longrightarrow> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> x = x" shows "module R M"
equivalent_on_cl: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and C :: "'b literal set"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" assumes "equivalent_on \<sigma> \<eta> (vars_of_cl C) I"   and "fo_interpretation I" shows "validate_ground_clause I (subst_cl C \<sigma>) = validate_ground_clause I (subst_cl C \<eta>)"
iSPEC_rule: fixes \<Phi> :: "'f \<Rightarrow> bool"   and \<Psi> :: "'f \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "SPEC \<Phi> \<le> SPEC \<Psi>"
pair_in_Id_conv: fixes a :: "'c"   and b :: "'c" shows "((a, b) \<in> Id) = (a = b)"
eq: shows "[n..<n + length (x # xs)] = n # [Suc n..<Suc (n + length xs)]"
x1_gt_1: fixes as :: "real list"   and bs :: "real list"   and hs :: "(real \<Rightarrow> real) list"   and k :: "nat"   and x\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and hb :: "real"   and e :: "real"   and p :: "real" assumes "akra_bazzi_real_recursion as bs hs k x\<^sub>0 x\<^sub>1 hb e p" shows "1 < x\<^sub>1"
\<alpha>_ide: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<a>[fst (f, g, h), fst (snd (f, g, h)), snd (snd (f, g, h))] = \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc (\<cdot>) prj0 prj1 f g h, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr>"
is_swap_sum_mod_2: assumes "is_swap" shows "(f 0 + f 1) mod 2 = 1"
prems_11: shows "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r p)) k'"
the_elem_set: fixes x :: "'c" shows "the_elem (set [x]) = x"
on_circline_def: shows "on_circline \<equiv> map_fun rep_circline (map_fun rep_complex_homo id) on_circline_clmat_hcoords"
eventually_at_Pair_within_TimesI1: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and X :: "'a set"   and y :: "'b"   and Y :: "'b set" assumes "eventually P (at x within X)"   and "P x" shows "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P x'"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "semiring_hom hom" shows "comm_semiring_hom hom"
bdd_above_norm_cont_comp: fixes T :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_on T f"   and "compact T" shows "bdd_above {\<parallel>f t\<parallel> |t. t \<in> T}"
simps: fixes a1 :: "cld"   and a2 :: "char list" shows "class_name a1 a2 = (\<exists>dcl cl fds meth_defs. a1 = cld_def dcl cl fds meth_defs \<and> a2 = dcl)"
seqsimilar_def: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" shows "\<sigma> \<approx> \<tau> \<equiv> \<natural> \<sigma> = \<natural> \<tau>"
execute_operator_def: fixes s :: "'c \<Rightarrow> bool option"   and op :: "'c strips_operator" shows "s \<then> op \<equiv> s ++ map_of (effect_to_assignments op)"
compactin_subset_topspace: fixes X :: "'c topology"   and S :: "'c set" assumes "compactin X S" shows "S \<subseteq> topspace X"
from_bool_0: fixes x :: "bool" shows "(from_bool x = 0) = (\<not> x)"
order_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "order r A = ((\<forall>x\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> x) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> x \<longrightarrow> x = y) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z \<longrightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> z))"
arc_simple_path: fixes g :: "real \<Rightarrow> 'b" shows "arc g = (simple_path g \<and> pathfinish g \<noteq> pathstart g)"
le: fixes x :: "'n" assumes "x \<in> S" shows "norm (f x) \<le> g x"
lub: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<sqsubseteq>\<^bsub>r\<^esub> z"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z"
elims: fixes x :: "nat"   and y :: "recf" assumes "r_const x = y" obtains "x = 0" and "y = Z"   | c :: "nat" where "x = Suc c" and "y = Cn 1 S [r_const c]"
divideR_right: fixes r :: "real"   and y :: "'d"   and x :: "'d" assumes "r \<noteq> 0" shows "(y = x /\<^sub>R r) = (r *\<^sub>R y = x)"
induce_subgraph_verts: fixes G :: "('e, 'f) pre_digraph"   and vs :: "'e set" shows "verts (G \<restriction> vs) = vs"
apsnd_def': fixes f :: "'e \<Rightarrow> 'f" shows "apsnd f = (\<lambda>(k, v). (k, f v))"
sinh_real_pos_iff: fixes x :: "real" shows "(0 < sinh x) = (0 < x)"
ordLeq_transitive: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set"   and r'' :: "('h \<times> 'h) set" assumes "r \<le>o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
null_eqI: fixes n :: "'a" assumes "\<And>f. n \<cdot> f = n \<and> f \<cdot> n = n" shows "n = C.null"
simps: fixes r1 :: "le_rel"   and p1 :: "linear_poly"   and c1 :: "'a"   and r2 :: "le_rel"   and p2 :: "linear_poly"   and c2 :: "'a" shows "Le_Constraint r1 p1 c1 + Le_Constraint r2 p2 c2 = Le_Constraint (r1 + r2) (p1 + p2) (c1 + c2)"
assms_1: shows "- z \<le> - x \<squnion> - y"
show_l_zero: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier V"   and "a \<in> carrier V" shows "(a \<oplus>\<^bsub>V\<^esub> x = x) = (a = \<zero>\<^bsub>V\<^esub>)"
2: shows "map getGMUserCom (tl sl) = ucl1"
keys_hom_component: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" shows "keys (hom_component p n) = {t \<in> keys p. deg_pm t = n}"
if_elim: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and b :: "'d"   and p :: "('a, 'c, 'd) Stmt"   and q :: "('a, 'c, 'd) Stmt"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "(((Stmt.If b p q, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B" obtains "c' = p" and "mem' = mem" and "mds' = mds" and "eval\<^sub>B mem b"   | "c' = q" and "mem' = mem" and "mds' = mds" and "\<not> eval\<^sub>B mem b"
ns_1: shows "g \<turnstile> Entry g-ns\<rightarrow>defNode g v"
head_last_cases: fixes xs :: "bool list" obtains "xs = []"   | ys :: "bool list" where "xs = True # ys__"   | ys :: "bool list" where "xs = ysa__ @ [False]"   | ys :: "bool list" where "xs = False # ys @ [True]"
mult_Suc_right: fixes m :: "nat"   and n :: "nat" shows "m * Suc n = m + m * n"
f: shows "f \<in> M \<rightarrow>\<^sub>M prob_algebra N"
st_proj1_inv: fixes x :: "'b sphere" assumes "x \<noteq> top_sphere" shows "st_proj1_inv (st_proj1 x) = x"
pcancel1_cancel_confluent: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "pcancel1 xs ys"   and "cancel zs ys" shows "\<exists>us. cancel us xs \<and> pcancel1 us zs"
RedBlock: fixes V :: "char list"   and T :: "ty"   and u :: "val"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>{V:T; Val u},s,b\<rangle> \<rightarrow> \<langle>Val u,s,b\<rangle>"
minimum_def: fixes x :: "'c"   and y :: "'c" shows "minimum x y = (if x \<sqsubseteq> y then x else y)"
intro: fixes n :: "nat"   and fs :: "'a vec list" assumes "\<And>i j. \<lbrakk>i < n; j < length fs\<rbrakk> \<Longrightarrow> fs ! j $ i \<in> \<int>" shows "gram_schmidt_fs_int_axioms n fs"
intersect_tsD2: fixes c :: "'a"   and A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and k :: "'b" assumes "c \<in> (A \<Rightarrow>\<inter>  B) k" shows "c \<in> B k"
zero: fixes a :: "complex" shows "a *\<^sub>C (0::'d) = (0::'d)"
map_of_foldr: fixes xs :: "('b \<times> 'c) list" shows "map_of xs = foldr (\<lambda>(k, v) m. m(k \<mapsto> v)) xs Map.empty"
trans_SeqT: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>t s'" shows "(c1 ;; c2, s) \<rightarrow>c (c2, s')"
simps_2: fixes f :: "'i \<Rightarrow> 'g"   and t :: "'f"   and b :: "bool" shows "convert_new_thread_action f (ThreadExists t b) = ThreadExists t b"
scaleC_nonneg_nonpos: fixes a :: "complex"   and x :: "'b" assumes "0 \<le> a"   and "x \<le> (0::'b)" shows "a *\<^sub>C x \<le> (0::'b)"
rep_eq: fixes x :: "linear_poly" shows "vars_list x = fun_vars_list (Rep_linear_poly x)"
IdE: fixes p :: "'b \<times> 'b" assumes "p \<in> Id" obtains x :: "'b" where "p = (x, x)"
compact_eq_Heine_Borel: fixes S :: "'b set" shows "compact S = (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union> C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union> D))"
step_command: fixes c :: "cmd"   and d :: "cmd"   and e :: "cmd" assumes "c \<^sub>C\<longlonglongrightarrow> d"   and "d \<^sub>C\<longlonglongrightarrow>\<^sup>* e" shows "c \<^sub>C\<longlonglongrightarrow>\<^sup>* e"
trace_rest'_def: fixes rest :: "('l, 'o, 'm, 'p, 'n, 'q, 'r) rest_scheme" shows "trace_rest' rest \<equiv> \<lambda>uu uua. rec_list (\<lambda>S. (\<lambda>ia. S \<bind> (\<lambda>s. map_spmf fst (rfunc_adv rest s ia)), \<lambda>iu. S \<bind> (\<lambda>s. map_spmf fst (rfunc_usr rest s iu)))) (\<lambda>obs tr tra S. case obs of Inl (ia, oa) \<Rightarrow> tra (cond_spmf_fst (S \<bind> (\<lambda>s. rfunc_adv rest s ia)) oa) | Inr (iu, ou) \<Rightarrow> tra (cond_spmf_fst (S \<bind> (\<lambda>s. rfunc_usr rest s iu)) ou)) uua uu"