hcju/isabelleps huggingface dataset:
i_Exec_Stream_Acc_LocalState__State_Idle_nth: fixes l :: "nat"   and k :: "nat"   and localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and i :: "nat"   and output_fun :: "'d \<Rightarrow> 'g message_af"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'d" assumes "0 < l"   and "l \<le> k"   and "Exec_Equal_State localState trans_fun"   and "\<And>n. n \<le> i \<Longrightarrow> State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState l localState trans_fun input c n)" shows "i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c i = i_Exec_Comp_Stream_Acc_LocalState l localState trans_fun input c i"
zero_induct: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat" assumes "P k"   and "\<And>n. P (Suc n) \<Longrightarrow> P n" shows "P 0"
Proj_on_own_range: fixes P :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'a" assumes "is_Proj P" shows "Proj (P *\<^sub>S \<top>) = P"
wf_Fields_Ex: fixes wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and C :: "char list" assumes "wf_prog wf_md P"   and "is_class P C" shows "\<exists>FDTs. P \<turnstile> C has_fields FDTs"
inf_sup_distrib1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
W'_4: shows "diff_fun k (charts_submanifold W') f'"
assms_3: fixes i :: "'b" assumes "i \<in> T - S" shows "h i = \<^bold>1"
double_compl: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L" shows "\<^bold>-\<^bsub>L\<^esub> (\<^bold>-\<^bsub>L\<^esub> x) = x"
IBuffer_def: fixes ic :: "state \<Rightarrow> 'a"   and q :: "state \<Rightarrow> 'a list"   and oc :: "state \<Rightarrow> 'a" shows "IBuffer ic q oc \<equiv> Init BInit ic q oc \<and> \<box>[Next ic q oc]_(ic, q, oc) \<and> WF(Deq ic q oc)_(ic, q, oc)"
my_U_is_defined_2: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and p :: "'a pmf" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>"   and "p \<in> \<P> outcomes" shows "p \<approx>[\<R>] mix_pmf (my_U outcomes \<R> p) (some_best outcomes \<R>) (some_worst outcomes \<R>)"
set_simps_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = insert x21 (set x22)"
adm_def: fixes P :: "'b \<Rightarrow> bool" shows "adm P = (\<forall>Y. chain Y \<longrightarrow> (\<forall>i. P (Y i)) \<longrightarrow> P (\<Squnion>i. Y i))"
star_rtc_least_eq: fixes x :: "'a ppath set"   and y :: "'a ppath set" assumes "pp_one \<union> x \<union> pp_prod y y = y" shows "pp_star x \<subseteq> y"
scale_sum_right: fixes a :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "a * sum f A = (\<Sum>x\<in>A. a * f x)"
abs_eq: shows "x_axis = abs_circline x_axis_clmat"
degree_frame: fixes x :: "'a"   and v :: "'a"   and v' :: "'a"   and w :: "'b" assumes "finite G.E"   and "x \<notin> {v, v'}" shows "degree x (del_unEdge v w v' G) = degree x G"
euler_witness_context_def: fixes p :: "nat" shows "euler_witness_context p \<equiv> 1 < p \<and> odd p"
restrict_def: fixes r :: "('b \<times> 'b) set"   and S :: "'b set" shows "restrict r S = {(x, y). x \<in> S \<and> y \<in> S \<and> (x, y) \<in> r}"
minus_minus: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "- (- x) = x"
slp_vp_Append: fixes n :: "'node"   and as :: "'edge list"   and n'' :: "'node"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^bsub>sl\<^esub>* n''"   and "n'' -as'\<rightarrow>\<^sub>\<surd>* n'" shows "n -as @ as'\<rightarrow>\<^sub>\<surd>* n'"
fps_neg_nth: fixes f :: "'b fps"   and n :: "nat" shows "- f $ n = - (f $ n)"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "norm (A $$ (i, j)) \<le> b"
sectionE: fixes f :: "'b" assumes "D.section f" obtains g :: "'b" where "D.ide (g \<cdot>\<^sub>D f)"
lookup_RBT: fixes t :: "('b, 'c) rbt" assumes "ord.is_rbt cless t" shows "RBT_Mapping2.lookup (Mapping_RBT t) = rbt_comp_lookup ccomp t"
elims: fixes x :: "'a ifex"   and y :: "'a list" assumes "ifex_vars x = y" obtains v :: "'a" and t :: "'a ifex" and e :: "'a ifex" where "x = IF v t e" and "y = v # ifex_vars t @ ifex_vars e"   | "x = Trueif" and "y = []"   | "x = Falseif" and "y = []"
spn_props_2: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "is_left_adjoint (spn \<mu>)"
Join_project_increasing: fixes F :: "'a program"   and G :: "'c program"   and func :: "'a \<Rightarrow> 'e" shows "(F \<squnion> project h UNIV G \<in> increasing func) = (extend h F \<squnion> G \<in> increasing (func \<circ> f))"
analytic_onE_box: fixes f :: "complex \<Rightarrow> complex"   and A :: "complex set"   and s :: "complex" assumes "f analytic_on A"   and "s \<in> A" obtains a :: "complex" and b :: "complex" where "Re a < Re b" and "Im a < Im b" and "s \<in> box a b" and "f analytic_on box a b"
assms_2: shows "orb \<in> orbits G E \<phi>"
effect_returnE: fixes x :: "'b"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "effect (return x) h h' r" obtains "r = x" and "h' = h"
matrix_vector_mul_linear: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "linear ((*v) A)"
conf_widen: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and P :: "'d prog"   and h :: "'b"   and v :: "'c val"   and T :: "ty"   and T' :: "ty" assumes "heap_base.conf typeof_addr P h v T"   and "P \<turnstile> T \<le> T'" shows "heap_base.conf typeof_addr P h v T'"
QR_def: fixes p :: "nat"   and q :: "nat" shows "QR p q \<equiv> (prime p \<and> 2 < p) \<and> prime q \<and> 2 < q \<and> p \<noteq> q"
reparam_eq_line_integrals_2: fixes \<gamma>1 :: "real \<Rightarrow> 'a"   and \<gamma>2 :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a"   and b :: "'a" assumes "reparam \<gamma>1 \<gamma>2"   and "\<gamma>2 piecewise_C1_differentiable_on {0..1}"   and "continuous_on (path_image \<gamma>2) (\<lambda>x. F x \<bullet> b)"   and "line_integral_exists F {b} \<gamma>2" shows "line_integral_exists F {b} \<gamma>1"
smult_axis: fixes x :: "'c"   and i :: "'d"   and y :: "'c" shows "x *s axis i y = axis i (x * y)"
identity_implies_part: fixes x :: "i"   and y :: "i" assumes "PM"   and "x = y" shows "P x y"
assms_5: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> dom (\<alpha> m)"   and "it \<noteq> dom (\<alpha> m)"   and "\<not> c \<sigma>"   and "I it \<sigma>" shows "P \<sigma>"
cut_le_def: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<le> t \<equiv> {x \<in> I. x \<le> t}"
notStuck_GuardD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and m :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>Guard m g c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "s \<in> g" shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
insertI: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and y :: "'c" assumes "x \<notin> A"   and "fold_graph f z A y" shows "fold_graph f z (insert x A) (f x y)"
distinct_2: fixes x21 :: "'e"   and x22 :: "'e list" shows "x21 # x22 \<noteq> []"
obtain_pos_sum: fixes r :: "rat" assumes "0 < r" obtains s :: "rat" and t :: "rat" where "0 < s" and "0 < t" and "r = s + t"
bubble_minD_size: fixes xs :: "'a list"   and ys :: "'a list" assumes "bubble_min xs = ys" shows "length xs = length ys"
b: shows "abc_steps_l (0, lm) A na = (length A, lma)"
intros_3: fixes c :: "f32"   and fop :: "unop_f" shows "\<lparr>[$C ConstFloat32 c, $Unop_f T_f32 fop]\<rparr> \<leadsto> \<lparr>[$C ConstFloat32 (app_unop_f fop c)]\<rparr>"
Quotient_alt_def3: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> (\<forall>x y. R x y = (\<exists>z. T x z \<and> T y z)))"
UN_extend_simps_5: fixes A :: "'a set"   and B :: "'b \<Rightarrow> 'a set"   and C :: "'b set" shows "A \<inter> \<Union> (B ` C) = (\<Union>x\<in>C. A \<inter> B x)"
not_eq_extremum: fixes a :: "'c" shows "(a \<noteq> top) = (a < top)"
successively_altdef: shows "successively = (\<lambda>P. rec_list True (\<lambda>x xs b. case xs of [] \<Rightarrow> True | y # xa \<Rightarrow> P x y \<and> b))"
prod_mono: 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 zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq zero (f i) \<and> less_eq (f i) (g i)" shows "less_eq (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
interpreted: shows "Interruptible_Separation_Kernel kstep output_f s0 current cswitch kprecondition realistic_execution CISK_control kinvolved ifp vpeq AS_set invariant AS_precondition aborting waiting"
traverse_tree_def: fixes path :: "dir list" shows "traverse_tree path \<equiv> foldr (\<lambda>d f. f \<circ> (case d of L \<Rightarrow> left | R \<Rightarrow> right)) path id"
divideC_field_simps_2: fixes c :: "complex"   and b :: "'d"   and a :: "'d" assumes "c \<noteq> 0" shows "(b /\<^sub>C c = a) = (b = c *\<^sub>C a)"
III: fixes xa :: "'a" shows "- (- (- xa)) = - (- (- xa \<squnion> - (- (- xa))))"
glb_eqI: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c set"   and l :: "'c" assumes "class.po below"   and "po.is_glb below M l" shows "po.glb below M = l"
ucast_down_ucast_id: fixes d :: "'b word \<Rightarrow> 'c word"   and w :: "'c word" assumes "is_down d" shows "ucast (ucast w) = w"
assms_4: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
sheaf_spec_is_sheaf: shows "sheaf_of_rings Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
insertion_sub: fixes f :: "nat \<Rightarrow> real"   and A :: "real mpoly"   and B :: "real mpoly" shows "insertion f (A - B) = insertion f A - insertion f B"
index_smult_vec_1: fixes i :: "nat"   and v :: "'b Matrix.vec"   and a :: "'b" assumes "i < dim_vec v" shows "(a \<cdot>\<^sub>v v) $v i = a * v $v i"
index_middle_element: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "MinkowskiBetweenness.fin_long_chain betw f a b c X" shows "\<exists>n>0. n < card X - 1 \<and> f n = b"
vars_mset_add_mset: fixes x :: "('d, 'c) Term.term \<times> ('e, 'c) Term.term"   and M :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) multiset" shows "vars_mset (add_mset x M) = vars_term (fst x) \<union> vars_term (snd x) \<union> vars_mset M"
reductio_aa_2: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] \<Longrightarrow> [\<^bold>\<not>\<psi> in v]"   and "[\<phi> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<^bold>\<not>\<phi> in v]"
oldest_token_monotonic_Suc: fixes x :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "x \<le> n"   and "oldest_token (token_run x n) n = Some i"   and "oldest_token (token_run x (Suc n)) (Suc n) = Some j" shows "j \<le> i"
Ar: fixes y :: "'a list set"   and x :: "'a list set"   and z :: "'a list set" assumes "\<not> l_ewp y"   and "x = x \<cdot> y + z" shows "x = z \<cdot> y\<^sup>\<star>"
uniformity_ell2_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). norm (x - y) < e})"
set_filterQuery: fixes QQ :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set" assumes "QQ = New_max.filterQuery I Q"   and "non_empty_query Q" shows "\<forall>X\<in>Q. (1 \<le> card (fst X \<inter> I)) = (X \<in> QQ)"
verit_comp_simplify1_1: fixes a :: "'a" shows "(a < a) = False"
bet_out: fixes B :: "'p"   and A :: "'p"   and C :: "'p" assumes "B \<noteq> A"   and "Bet A B C" shows "A Out B C"
I_minWeightBasisD_1: fixes carrier :: "'c set"   and indep :: "'c set \<Rightarrow> bool"   and weight :: "'c \<Rightarrow> 'd"   and T :: "'c set"   and E :: "'c set" assumes "weighted_matroid carrier indep"   and "weighted_matroid.I_minWeightBasis carrier indep weight (T, E)" shows "indep T"
getFrN_def: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and n :: "nat" shows "getFrN xs ts \<phi>s n = getFreshN (set xs \<union> \<Union> (FvarsT ` set ts) \<union> \<Union> (Fvars ` set \<phi>s)) n"
entails_trans: fixes N1 :: "'a literal multiset set"   and N2 :: "'a literal multiset set"   and N3 :: "'a literal multiset set" assumes "N1 \<TTurnstile>\<G>e N2"   and "N2 \<TTurnstile>\<G>e N3" shows "N1 \<TTurnstile>\<G>e N3"
insert_map_inject1: fixes i :: "nat"   and x :: "'c"   and f :: "nat \<Rightarrow> 'c"   and y :: "'c"   and g :: "nat \<Rightarrow> 'c" assumes "insert_map i x f = insert_map i y g" shows "x = y"
RECT_refine: fixes body :: "('b \<Rightarrow> 'c nres) \<Rightarrow> 'b \<Rightarrow> 'c nres"   and x :: "'b"   and x' :: "'d"   and R :: "('b \<times> 'd) set"   and S :: "('c \<times> 'e) set"   and body' :: "('d \<Rightarrow> 'e nres) \<Rightarrow> 'd \<Rightarrow> 'e nres" assumes "trimono body"   and "(x, x') \<in> R"   and "\<And>f f' x x'. \<lbrakk>\<And>x x'. (x, x') \<in> R \<Longrightarrow> f x \<le> \<Down> S (f' x'); (x, x') \<in> R\<rbrakk> \<Longrightarrow> body f x \<le> \<Down> S (body' f' x')" shows "REC\<^sub>T body x \<le> \<Down> S (REC\<^sub>T body' x')"
assms_2: shows "\<forall>\<^sub>F x in at_top. g x = g' x"
inj_comm_semiring_hom_axioms: shows "inj_comm_semiring_hom (map_poly (\<lambda>a. [:a:]))"
r_code_const1_prim: shows "prim_recfn 1 r_code_const1"
OclAsInteger\<^sub>R\<^sub>e\<^sub>a\<^sub>l_def: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>real\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "X->oclAsType\<^sub>R\<^sub>e\<^sub>a\<^sub>l(Integer) = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> then \<lfloor>\<lfloor>floor \<lceil>\<lceil>X \<tau>\<rceil>\<rceil>\<rfloor>\<rfloor> else invalid \<tau>)"
cost_eq_Sum_freq_mult_depth: fixes t :: "'a tree" assumes "consistent t" shows "cost t = (\<Sum>a\<in>alphabet t. freq t a * depth t a)"
float_plus_down: fixes prec :: "nat"   and x :: "float"   and y :: "float" shows "real_of_float (float_plus_down prec x y) \<le> real_of_float x + real_of_float y"
neg_equal_0_iff_equal: fixes minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q" assumes "class.group_add minus plus zero uminus" shows "(uminus a = zero) = (a = zero)"
less_character_data_ptr_def: fixes x :: "(_) character_data_ptr"   and y :: "(_) character_data_ptr" shows "x < y \<equiv> x \<le> y \<and> \<not> y \<le> x"
is_pref_profile_update: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent" assumes "lowered.is_pref_profile R"   and "total_preorder_on alts' Ri'"   and "i \<in> agents'" shows "lowered.is_pref_profile (R(i := Ri'))"
shunting_1_pp: fixes x :: "'a"   and y :: "'a" shows "(x \<le> - - y) = (x \<sqinter> - y = bot)"
i_shrink_last_def: fixes f :: "nat \<Rightarrow> 'd"   and k :: "nat" shows "f \<div>\<^sub>l k \<equiv> i_aggregate f k last"
segment_unique: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Worder D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "ord_equiv (Iod D (segment D a)) (Iod D (segment D b))" shows "a = b"
strict_mono_suc: fixes f :: "nat \<Rightarrow> nat"   and sn :: "nat"   and n :: "nat" assumes "strict_mono f"   and "f sn = Suc (f n)" shows "sn = Suc n"
infsetsum_mono_neutral: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set"   and g :: "'c \<Rightarrow> real"   and B :: "'c set" assumes "f abs_summable_on A"   and "g abs_summable_on B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"   and "\<And>x. x \<in> A - B \<Longrightarrow> f x \<le> 0"   and "\<And>x. x \<in> B - A \<Longrightarrow> 0 \<le> g x" shows "infsetsum f A \<le> infsetsum g B"
bap: shows "eval (Cn 1 r_snoc [r_pdec1, Z]) [arg] \<down>= e_snoc b 0"
assms_3: shows "e1 \<in> out_arcs G v"
nest: fixes n :: "nat" shows "S n \<subseteq> S (Suc n)"
decr0: fixes p :: "fm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "bound0 p" shows "Ifm vs (x # bs) p = Ifm vs bs (decr0 p)"
minus_plus': fixes s :: "'a"   and t :: "'a"   and u :: "'a" assumes "s adds t" shows "u + (t - s) = u + t - s"
assms_2: shows "GE.satPB \<xi> (gC ` \<Phi>)"
lessThan_iff: fixes i :: "'b word"   and k :: "'b word" shows "(i \<in> signed.lessThan k) = (i <s k)"
res: shows "set_iterator (set_iterator_filter P it) {x \<in> S. P x}"
rank_link: fixes t\<^sub>1 :: "'b tree"   and t\<^sub>2 :: "'b tree" shows "rank (link t\<^sub>1 t\<^sub>2) = rank t\<^sub>1 + 1"
nf_invar_axioms: shows "nf_invar c src dst PRED C N d"
LIM_cong: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and l :: "'c"   and m :: "'c" assumes "a = b"   and "\<And>x. x \<noteq> b \<Longrightarrow> f x = g x"   and "l = m" shows "f \<midarrow>a\<rightarrow> l = g \<midarrow>b\<rightarrow> m"
continuous_vnm_def: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" shows "continuous_vnm C P = (\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. p \<succeq>[P] q \<and> q \<succeq>[P] r \<longrightarrow> (\<exists>\<alpha>\<in>{0..1}. mix_pmf \<alpha> p r \<approx>[P] q))"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "B.dom (G f) = G (A.dom f)"
irewrite_step_def: fixes name :: "name"   and pats :: "Term_Class.term list"   and rhs :: "pterm"   and t :: "pterm" shows "irewrite_step name pats rhs t = map_option (subst rhs) (match (name $$ pats) t)"
sum_splice: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum f {0..<2 * n} = (\<Sum>i = 0..<n. f (2 * i)) + (\<Sum>i = 0..<n. f (2 * i + 1))"
axioms_1: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "reducedly_statically_complete_calculus Bot Inf entails Red_I Red_F" shows "calculus Bot Inf entails Red_I Red_F"
swapped_preserves_good: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "good X"   and "(X, Y) \<in> swapped" shows "good Y"
v1: shows "v = lt (q g1 \<odot> g1)"
intro: fixes V\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>C :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>C :: "'r \<Rightarrow> 'r"   and src\<^sub>C :: "'r \<Rightarrow> 'r"   and trg\<^sub>C :: "'r \<Rightarrow> 'r"   and V\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>D :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>D :: "'s \<Rightarrow> 's"   and src\<^sub>D :: "'s \<Rightarrow> 's"   and trg\<^sub>D :: "'s \<Rightarrow> 's"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>G :: "'r \<times> 'r \<Rightarrow> 's"   and \<tau>\<^sub>0 :: "'r \<Rightarrow> 's"   and \<tau>\<^sub>1 :: "'r \<Rightarrow> 's" assumes "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"   and "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"   and "equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F"   and "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G \<Phi>\<^sub>G"   and "pseudonatural_equivalence V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudofunctor_pseudonaturally_equivalent_to_equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<tau>\<^sub>0 \<tau>\<^sub>1"
dickson_less_v_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'t"   and u :: "'t" shows "dickson_less_v d m v u = (d (pp_of_term v) \<le> m \<and> d (pp_of_term u) \<le> m \<and> v \<prec>\<^sub>t u)"
fds_abs_converges_imp_converges: fixes f :: "'b fds"   and s :: "'b" assumes "fds_abs_converges f s" shows "fds_converges f s"
finite_insert_lepoll: fixes A :: "'a set"   and a :: "'a" assumes "finite A" shows "(insert a A \<lesssim> A) = (a \<in> A)"
empty_fsp: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N" shows "({}, \<lambda>x\<in>{\<zero>}. \<zero>\<^bsub>N\<^esub>) \<in> fsps R M N f H"
Src_Src: fixes t :: "'c local.term" assumes "Arr t" shows "Src (Src t) = Src t"
of_int_poly_add: fixes p :: "int poly"   and q :: "int poly" shows "of_int_poly (p + q) = of_int_poly p + of_int_poly q"
wagner_1: fixes x :: "'a" shows "(x \<cdot> x\<^sup>\<star>)\<^sup>\<omega> = x\<^sup>\<omega>"
mtx_invertibleI: fixes A :: "'b sq_mtx"   and B :: "'b sq_mtx" assumes "A * B = 1"   and "B * A = 1" shows "mtx_invertible A"
prod_nonzeroI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> (0::'b)" shows "prod f A \<noteq> (0::'b)"
Set_filter_unchanged: fixes X :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> X \<Longrightarrow> P x" shows "Set.filter P X = X"
cong_exp_trans_2: fixes c :: "'a"   and a :: "'a"   and b :: "nat"   and n :: "'a"   and d :: "'a" assumes "[c = a ^ b] (mod n)"   and "[a = d] (mod n)" shows "[c = d ^ b] (mod n)"
on_curvep_infinity: fixes a :: "'b"   and b :: "'b"   and x :: "'b"   and y :: "'b" shows "on_curvep a b (x, y, 0::'b)"
np_specD4: fixes np :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "np_spec np"   and "(True, p, q) \<in> set (np gs bs hs data)" shows "q \<in> set gs \<union> set bs"
path_is_step: fixes astep :: "'b \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and p :: "'d list"   and c' :: "'b" assumes "LTS.path astep c p c'" shows "(c, c') \<in> (asystem.step astep)\<^sup>*"