hcju/isabelleps huggingface dataset:
cod_dom: fixes f :: "'a" assumes "A.arr f" shows "A.cod (A.dom f) = A.dom f"
echelon_form_of_upt_k_det_unit: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and A :: "(('b, 'c) vec, 'c) vec"   and k :: "nat" shows "is_unit (fst (echelon_form_of_upt_k_det bezout (1::'b, A) k))"
pr_conv_3_to_1_def: shows "pr_conv_3_to_1 = (\<lambda>f x. f (c_fst (c_fst x)) (c_snd (c_fst x)) (c_snd x))"
MWLfSteps_det_cases_1: fixes m :: "'id \<Rightarrow> 'val"   and W :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr skip m W m'" obtains "W = []" and "m' = m"
ServiceCenter_L2: fixes x :: "nat"   and i :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list" assumes "\<forall>t2<x. i t2 = []"   and "ServiceCenter i a" shows "\<forall>t3\<le>x. a t3 = []"
rep_eq: fixes x :: "'a"   and xa :: "'b" shows "pp.mapping_of (single_pp x xa) = monomial xa x"
remove_vertex_path_from_to: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a"   and x :: "'a" assumes "v \<leadsto>xs\<leadsto> w"   and "x \<in> V"   and "x \<notin> set xs" shows "v \<leadsto>xs\<leadsto>\<^bsub>remove_vertex x\<^esub> w"
perm_swap_2: fixes a :: "atom"   and b :: "atom"   and x :: "'a" shows "[(a, b)] \<bullet> [(b, a)] \<bullet> x = x"
dfull_del: fixes n :: "nat"   and t :: "'a tree23"   and k :: "int option" assumes "full (Suc n) t" shows "dfull n (del k t)"
Mismatch_def: fixes x3 :: "name"   and x2 :: "name"   and x1 :: "pi" shows "[x3\<noteq>x2]x1 \<equiv> Abs_pi (Mismatch_Rep x3 x2 (Rep_pi x1))"
maximal_prime_Tr0: fixes mx :: "'a set"   and x :: "'a" assumes "maximal_ideal R mx"   and "x \<in> carrier R"   and "x \<notin> mx" shows "mx \<minusplus> R \<diamondsuit>\<^sub>p x = carrier R"
inj_rename_imp_inj: fixes h :: "'c \<Rightarrow> 'd" assumes "inj (rename h)" shows "inj h"
fold_coeffs_0_eq: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "fold_coeffs f 0 = id"
converse_inject: fixes r :: "('c \<times> 'b) set"   and s :: "('c \<times> 'b) set" shows "(r\<inverse> = s\<inverse>) = (r = s)"
index_not_zero_impl_diff_group: fixes ps :: "'a set"   and G :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "int (\<B> index ps) = \<Lambda>"   and "G \<in> \<G>" shows "\<not> ps \<subseteq> G"
cont2cont_case_prod: fixes f :: "'e \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'c \<times> 'd" assumes "\<And>a b. cont (\<lambda>x. f x a b)"   and "\<And>x b. cont (\<lambda>a. f x a b)"   and "\<And>x a. cont (f x a)"   and "cont g" shows "cont (\<lambda>x. case g x of (a, b) \<Rightarrow> f x a b)"
ex_post_efficient_sds_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "ex_post_efficient_sds_axioms agents alts sds \<equiv> \<forall>R. pref_profile_wf agents alts R \<longrightarrow> set_pmf (sds R) \<inter> pareto_losers R = {}"
cart_basis'_def: shows "cart_basis' = (\<chi>i. axis i (1::'b))"
length_negGorbit_list: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and gs :: "'c list"   and T :: "'d \<Rightarrow> 'a"   and as :: "'d list" shows "length (aezfun_scalar_mult.negGorbit_list smult gs T as) = length gs"
assert_bound_index_valid: fixes s :: "('i, 'a) state"   and as :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "index_valid as s" shows "index_valid (insert a as) (assert_bound a s)"
STSteps_steps_2: fixes T :: "'procT"   and P' :: "('procS, 'procT) Proc" shows "TargetTerm T \<longmapsto>ST* P' = (\<exists>T'. T' \<in>T P' \<and> T \<longmapsto>Target* T')"
nz: shows "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'a)"
add_mono_thms_linordered_semiring_3: fixes i :: "'k"   and j :: "'k"   and k :: "'k"   and l :: "'k" assumes "i \<le> j \<and> k = l" shows "i + k \<le> j + l"
mono: fixes g :: "'c \<Rightarrow> ('d, 'c) step"   and x :: "'c" shows "mono_llist (\<lambda>lunstream. case g x of Done \<Rightarrow> LNil | Skip s' \<Rightarrow> lunstream s' | Yield x s' \<Rightarrow> LCons x (lunstream s'))"
numeral_eq_one_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "Num.num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus n = one) = (n = num.One)"
Fst_def: fixes q :: "ZF" shows "Fst q \<equiv> SOME x. \<exists>y. q = Opair x y"
ideD_1: fixes a :: "'a" assumes "Left_a.ide a" shows "Left_a.arr a"
inner_node_Entry_Exit_path: fixes l :: "nat"   and prog :: "cmd" assumes "l < #:prog" shows "(\<exists>as. While_CFG.path prog (_ l _) as (_Exit_)) \<and> (\<exists>as. While_CFG.path prog (_Entry_) as (_ l _))"
standard_redundancy_criterion_reductive_axioms: shows "standard_redundancy_criterion_reductive gr.ord_\<Gamma>"
SUCC_inject_E: fixes x :: "tm"   and y :: "tm"   and H :: "fm set"   and A :: "fm" assumes "insert (x EQ y) H \<turnstile> A" shows "insert (SUCC x EQ SUCC y) H \<turnstile> A"
power_divide: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and n :: "nat" assumes "class.field divide inverse times one plus zero minus uminus" shows "power.power one times (divide a b) n = divide (power.power one times a n) (power.power one times b n)"
tdghm_NTDGDom: shows "\<NN>\<lparr>NTDGDom\<rparr> = \<AA>"
IH: shows "t (\<lambda>x. \<Sum>y\<in>T. P y x) = (\<lambda>x. \<Sum>y\<in>T. t (P y) x)"
H3_4: fixes x :: "'a" shows "C x - f2 x = g3 x + u3 x - u3 (T x)"
lim_set: fixes r :: "'b ref"   and v :: "'b"   and h :: "heap" shows "lim (Ref.set r v h) = lim h"
conditional_subset: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "{x \<in> A. P x} \<subseteq> A"
globally_sound_mode_use_def: fixes gc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "globally_sound_mode_use gc \<equiv> \<forall>mdss. mdss \<in> reachable_mode_states gc \<longrightarrow> compatible_modes mdss"
convex_convex_hull: fixes n :: "nat"   and S :: "'b vec set" assumes "gram_schmidt.convex n S" shows "gram_schmidt.convex_hull n S = S"
abs_mult_less: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (abs a) c"   and "less (abs b) d" shows "less (times (abs a) (abs b)) (times c d)"
nth'_list_augment: fixes xs :: "'a list"   and k :: "nat"   and v :: "'a" shows "nth' (list_augment xs k v) k = v"
Qcomplete_def: shows "Qcomplete \<equiv> insert None (Some ` Q)"
disj_ac_2: fixes a :: "'a"   and b :: "'a" shows "a OR b = b OR a"
simps_2: fixes s :: "nat \<Rightarrow> 'a list"   and i :: "nat" shows "inf_last_ti s (Suc i) = (if s (Suc i) \<noteq> [] then s (Suc i) else inf_last_ti s i)"
is_interval_convex: fixes S :: "'a set" assumes "is_interval S" shows "convex S"
axioms_1: fixes R :: "('a, 'b) ring_scheme" assumes "cring R" shows "ring R"
LIM_Im: fixes f :: "'a \<Rightarrow> complex"   and L :: "complex"   and a :: "'a" assumes "f \<midarrow>a\<rightarrow> L" shows "(\<lambda>x. Im (f x)) \<midarrow>a\<rightarrow> Im L"
lp_monom_mult: fixes c :: "'b"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'b)"   and "p \<noteq> 0" shows "lpp (local.punit.monom_mult c t p) = t + lpp p"
sqrt_int_ceiling_pos: fixes x :: "int" assumes "0 \<le> x" shows "sqrt_int_ceiling_pos x = \<lceil>sqrt (real_of_int x)\<rceil>"
singletonI: fixes a :: "'b" shows "a \<in> {a}"
sin_bound_lemma: fixes x :: "real"   and y :: "real"   and u :: "real"   and v :: "real" assumes "x = y"   and "\<bar>u\<bar> \<le> v" shows "\<bar>x + u - y\<bar> \<le> v"
T_closed: fixes n :: "nat"   and k :: "nat" shows "liftn n \<T> k = \<T>"
dir_image_ordIso: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "inj_on f (Field r)" shows "r =o dir_image r f"
\<Delta>'''_sel_4: shows "sink \<Delta>''' = sink \<Delta>"
is_quantum_predicate_def: fixes P :: "complex mat" shows "is_quantum_predicate P = (P \<in> carrier_mat d d \<and> positive P \<and> P \<le>\<^sub>L 1\<^sub>m d)"
bij_betw_linorder_rank: fixes B :: "'a set"   and R :: "('a \<times> 'a) set"   and A :: "'a set" assumes "linorder_on B R"   and "finite A"   and "A \<subseteq> B" shows "bij_betw (linorder_rank R A) A {..<card A}"
tendsto_set_lebesgue_integral_at_right: fixes a :: "real"   and b :: "real"   and M :: "real measure"   and f :: "real \<Rightarrow> 'a" assumes "a < b"   and "\<And>a'. a' \<in> {a<..b} \<Longrightarrow> {a'..b} \<in> sets M"   and "set_integrable M {a<..b} f" shows "((\<lambda>a'. set_lebesgue_integral M {a'..b} f) \<longlongrightarrow> set_lebesgue_integral M {a<..b} f) (at_right a)"
CD_on_simps: fixes ds :: "'b set" shows "Bossy''.CD_on ds {} = {}"
wf\<^sub>s\<^sub>t_induct: fixes P :: "'c set \<Rightarrow> ('d, 'c) strand_step list \<Rightarrow> bool"   and a0 :: "'c set"   and a1 :: "('d, 'c) strand_step list" assumes "\<And>V. P V []"   and "\<And>V t S. P V S \<Longrightarrow> P V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V t S. P (V \<union> fv t) S \<Longrightarrow> P V (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V s t S. P (V \<union> fv s) S \<Longrightarrow> P V (\<langle>assign: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V s t S. P V S \<Longrightarrow> P V (\<langle>check: s \<doteq> t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>V uu_ uv_ S. P V S \<Longrightarrow> P V (\<forall>uu_\<langle>\<or>\<noteq>: uv_\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0 a1"
card_pderivs_lang_UNIV_le_awidth: fixes r :: "'a rexp" shows "card (pderivs_lang UNIV r) \<le> awidth r + 1"
additive_isotone: fixes f :: "'a \<Rightarrow> 'a" assumes "additive f" shows "isotone f"
inter: fixes x :: "'a"   and y :: "'a"   and alg :: "interpolation_algorithm" assumes "(x, y) \<in> set xs_ys" shows "poly (interpolation_poly alg xs_ys) x = y"
Init_simps1_1: fixes F :: "'a \<Rightarrow> bool" shows "Init \<not> F \<equiv> \<not> Init F"
coef_hd_cong: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and \<zeta> :: "('b, 'a) hd"   and A :: "'a pvar \<Rightarrow> zhmultiset"   and B :: "'a pvar \<Rightarrow> zhmultiset"   and i :: "nat" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "\<forall>x\<in>vars_hd \<zeta>. \<forall>i. A (PCoef x i) = B (PCoef x i)" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.coef_hd coef_sym \<zeta> i)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (tkbo_coefs.coef_hd coef_sym \<zeta> i))"
convex_linear_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f"   and "convex S" shows "convex (f ` S)"
NSLIM_inverse: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and L :: "'d" assumes "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"   and "L \<noteq> (0::'d)" shows "(\<lambda>x. inverse (f x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S inverse L"
erfc_real_ge_0_iff: fixes x :: "real" shows "(1 \<le> erfc x) = (x \<le> 0)"
VInter_antimono: fixes B :: "V"   and A :: "V" assumes "B \<noteq> 0"   and "B \<subseteq>\<^sub>\<circ> A" shows "\<Inter>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> \<Inter>\<^sub>\<circ> B"
axioms_4: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and create_element :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptrs type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name set_tag_name_locs set_disconnected_nodes set_disconnected_nodes_locs create_element" shows "l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_tag_name set_tag_name_locs type_wf create_element known_ptr"
select_convs_2: fixes g_V :: "'b set"   and g_E :: "('b \<times> 'b) set"   and g_V0 :: "'b set"   and more :: "'c" shows "g_E \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_E"
map_of_list_nil: shows "map_of_list [] = Map.empty"
inst_output_consistency: fixes s :: "rstate_t"   and t :: "rstate_t"   and a :: "int_point_t" assumes "rvpeq (rcurrent s) s t"   and "rcurrent s = rcurrent t" shows "routput_f s a = routput_f t a"
eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "injective f" shows "(f x = f y) = (x = y)"
fi: shows "(f has_contour_integral i) (part_circlepath z r s t)"
step: shows "\<forall>b. (x, b) \<in> r\<^sup>+ \<longrightarrow> (\<exists>c. (b, c) \<in> r)"
l1_init_def: shows "l1_init \<equiv> {\<lparr>ik = {}, secret = {}, progress = Map.empty, signalsInit = \<lambda>x. 0, signalsResp = \<lambda>x. 0\<rparr>}"
cofinal_def: fixes A :: "'c set"   and r :: "('c \<times> 'c) set" shows "cofinal A r \<equiv> \<forall>a\<in>Field r. \<exists>b\<in>A. a \<noteq> b \<and> (a, b) \<in> r"
seqE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq g f" obtains "VoV.arr f" and "VoV.arr g" and "VoV.dom g = VoV.cod f"
convex_imp_path_connected: fixes S :: "'b set" assumes "convex S" shows "path_connected S"
simps_2: fixes n :: "nat"   and l :: "nat" shows "T Del (n, l) = (if f1 * real l \<le> real (n - 1) then 1 else if int l0 \<le> \<lfloor>real l / c\<rfloor> then real n else 1)"
Sups_least: fixes X :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" shows "z \<le> \<Sqinter> X"
nonfinal_deterministic_RP_step: fixes St :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat"   and St' :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" assumes "\<not> is_final_dstate St"   and "St' = deterministic_RP_step St" shows "wstate_of_dstate St \<leadsto>\<^sub>w\<^sup>+ wstate_of_dstate St'"
attach_converter_of_resource_conv_parallel_resource2: fixes res :: "('j, 'l) resource"   and res' :: "('i, 'k) resource" shows "1\<^sub>C |\<^sub>\<propto> converter_of_resource res \<rhd> res' = res' \<parallel> res"
L1: fixes v :: "letter list"   and ws :: "letter list list"   and w :: "letter list" assumes "L v ws" shows "L v (w # ws)"
matrix_to_iarray_snd_Gauss_Jordan_PA: fixes A :: "(('b, 'c) vec, 'd) vec" shows "matrix_to_iarray (snd (Gauss_Jordan_PA A)) = snd (Gauss_Jordan_iarrays_PA (matrix_to_iarray A))"
mult_zero_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" shows "class.mult_zero times zero \<equiv> (\<forall>a. times zero a = zero) \<and> (\<forall>a. times a zero = zero)"
S: shows "set p \<subseteq> set yvec \<times> set zvec"
Rep_ivl: fixes x :: "'b ivl" shows "Rep_ivl x \<in> {p. fst p \<le> snd p}"
of_int_less_1_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus z) one = (z < 1)"
3: shows "summable (\<lambda>x. LINT xa|lebesgue. cmod (indicat_real {0<..} xa *\<^sub>R (complex_of_real xa powr (z - 1) / complex_of_real (exp ((real x + a) * xa)))))"
matching_Goto: assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"   and "\<not> no_matching_Goto \<gamma> p rs\<^sub>1" shows "thesis"
sincos_total_pi: fixes y :: "real"   and x :: "real" assumes "0 \<le> y"   and "x\<^sup>2 + y\<^sup>2 = 1" shows "\<exists>t\<ge>0. t \<le> pi \<and> x = cos t \<and> y = sin t"
iprev_neq_Max_not_singleton: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "\<nexists>a. I = {a}" shows "iprev n I \<noteq> Max I"
lookup_pair_inj: fixes xs :: "('a \<times> 'c) list"   and ys :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "lookup_pair xs = lookup_pair ys" shows "xs = ys"
compatible_cs_plus: fixes c1 :: "rat le_constraint"   and d1 :: "QDelta le_constraint"   and c2 :: "rat le_constraint"   and d2 :: "QDelta le_constraint" assumes "compatible_cs c1 d1"   and "compatible_cs c2 d2" shows "compatible_cs (c1 + c2) (d1 + d2)"
distinct_zipI2: fixes ys :: "'a list"   and xs :: "'b list" assumes "distinct ys" shows "distinct (zip xs ys)"
qPsubstAll_preserves_qFreshAll: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and zs :: "'varSort"   and z :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodEnv rho" shows "(qFresh zs z X \<longrightarrow> qGood X \<and> qFreshEnv zs z rho \<longrightarrow> qFresh zs z X #[[rho]]) \<and> (qFreshAbs zs z A \<longrightarrow> qGoodAbs A \<and> qFreshEnv zs z rho \<longrightarrow> qFreshAbs zs z A $[[rho]])"
HUnion_hempty: shows "\<Squnion>0 = 0"
OK_iff_ok: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c program" shows "OK I F = (\<forall>i\<in>I. \<forall>j\<in>I - {i}. F i ok F j)"
is_extensional: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "functor A B F"   and "\<not> partial_magma.arr A f" shows "F f = partial_magma.null B"
simps_1: fixes S :: "nat set" shows "pick S 0 = (LEAST a. a \<in> S)"
inf_left_idem: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<inter># (x \<inter># y) = x \<inter># y"
monom_to_monomial: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "to_fun (monom P a n) = monomial_function R a n"
trancl_into_trancl: fixes a :: "'e"   and b :: "'e"   and r :: "('e \<times> 'e) set"   and c :: "'e" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
or: shows "(\<exists>v\<in>W. c v \<noteq> (0::'a)) \<or> (\<exists>v\<in>X. c v \<noteq> (0::'a))"
Conseq: fixes A :: "triple set"   and P' :: "state \<Rightarrow> assn"   and c :: "stmt"   and Q' :: "state \<Rightarrow> assn"   and P :: "assn"   and Q :: "assn" assumes "\<forall>Z. A \<turnstile> {P' Z} c {Q' Z}"   and "\<forall>s t. (\<forall>Z. P' Z s \<longrightarrow> Q' Z t) \<longrightarrow> P s \<longrightarrow> Q t" shows "A \<turnstile> {P} c {Q}"
minus_add_cancel: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "- x + (x + y) = y"
assms: shows "ASSUME_\<Theta> \<pi> f\<^sub>0 s\<^sub>0 R (insert (f, P, c, Q) \<Theta>)"
HEndPhase2_HInv5_q2: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "p \<noteq> q"   and "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq) \<and> \<not> hasRead s q d qq" shows "\<exists>D\<in>MajoritySet. \<exists>qq. \<forall>d\<in>D. bal (dblock s' q) < mbal (disk s' d qq) \<and> \<not> hasRead s' q d qq"
lincomb_closed: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "rs \<bullet>\<cdot> ms \<in> V"
assoc_simps_4: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.dom \<a>\<^sub>B[f, g, h] = (B.dom f \<star>\<^sub>B B.dom g) \<star>\<^sub>B B.dom h"
uv_1: shows "\<forall>i\<in>Basis. u \<bullet> i \<le> v \<bullet> i"
Range_Id_on: fixes A :: "'b set" shows "Range (s2r A) = A"
sums_unique2: fixes f :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "f sums a"   and "f sums b" shows "a = b"
eq_uptoE: fixes g :: "nat \<Rightarrow> 'b"   and B :: "(nat \<Rightarrow> 'b) set"   and f :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "g \<in> eq_upto B f i" obtains "g \<in> B" and "\<And>j. j < i \<Longrightarrow> f j = g j"
map_spmf_conv_bind_spmf: fixes f :: "'f \<Rightarrow> 'e"   and p :: "'f spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
fdomain_fid_on: fixes A :: "V" shows "\<D>\<^sub>\<bullet> (fid_on A) = A"
strict_implies_order: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subset># a" shows "b \<subseteq># a"
Cantor_sum_Cons: fixes \<alpha> :: "V"   and \<alpha>s :: "V list"   and m :: "nat"   and ms :: "nat list" shows "Cantor_sum (\<alpha> # \<alpha>s) (m # ms) = \<omega> \<up> \<alpha> * ord_of_nat m + Cantor_sum \<alpha>s ms"
r_prenum: fixes e :: "nat"   and x :: "nat" shows "eval r_prenum [e, x] \<down>= (if x < e_length e then e_nth e x else 0)"
connected_equivalence_relation: fixes S :: "'b set"   and a :: "'b"   and b :: "'b"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "connected S"   and "a \<in> S"   and "b \<in> S"   and "\<And>x y. \<lbrakk>R x y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> R y x"   and "\<And>x y z. \<lbrakk>R x y; R y z; x \<in> S; y \<in> S; z \<in> S\<rbrakk> \<Longrightarrow> R x z"   and "\<And>a. a \<in> S \<Longrightarrow> \<exists>T. openin (top_of_set S) T \<and> a \<in> T \<and> (\<forall>x\<in>T. R a x)" shows "R a b"
assms_3: shows "x [^]\<^bsub>R\<^esub> CARD('a) ^ m' = x"
elims_3: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat" assumes "\<not> bn_valid x xa xb" obtains n :: "nat" and p :: "nat" and q :: "nat" where "x = Suc (Suc n)" and "xa = p" and "xb = q" and "q \<notin> {0<..p}"