hcju/isabelleps huggingface dataset:
pair_relator_halflist_append: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b" assumes "class.group_add minus plus zero uminus" shows "alternating_list (monoid_add.add_order plus zero (plus s t)) s t @ (if gcd_nat.greater_eq (monoid_add.add_order plus zero (plus s t)) 2 then alternating_list (monoid_add.add_order plus zero (plus s t)) s t else alternating_list (monoid_add.add_order plus zero (plus t s)) t s) = alternating_list (2 * monoid_add.add_order plus zero (plus s t)) s t"
M_def: fixes \<pi> :: "'b prefix" shows "M \<pi> = {(i, v). i < progress \<pi> \<and> wf_tuple nfv fv v \<and> (\<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> sat \<sigma> (map the v) i)}"
InitMsgs: shows "msgs cJ -# <pJ, inM False> = msgs cJ1 -# <pJ, inM True>"
add_assocD: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "(D y z \<and> D x (y \<oplus> z)) = (D x y \<and> D (x \<oplus> y) z)"
times_le_abs: fixes err :: "real"   and B :: "real" assumes "- 1 \<le> err"   and "err \<le> 1" shows "err * B \<le> \<bar>B\<bar>"
span_clauses_1: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "a \<in> cspan S"
dg_Rel_components_1: fixes \<alpha> :: "V" shows "dg_Rel \<alpha>\<lparr>Obj\<rparr> = Vset \<alpha>"
f: shows "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright> \<and> C.ide f \<and> D.isomorphic (F f) f'"
redT_updWs_total: fixes t :: "'a"   and ws :: "'a \<Rightarrow> 'b wait_set_status option"   and was :: "('a, 'b) wait_set_action list" shows "\<exists>ws'. redT_updWs t ws was ws'"
a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes document_ptr :: "(_) document_ptr"   and uu :: "unit" shows "local.a_get_owner_document\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr uu = Heap_Error_Monad.bind document_ptr_kinds_M (\<lambda>document_ptrs. if document_ptr \<in> set document_ptrs then return document_ptr else error SegmentationFault)"
borel_eq_halfspace_greater: shows "borel = sigma UNIV ((\<lambda>(a, i). {x. a < x \<bullet> i}) ` (UNIV \<times> Basis))"
right_idem: fixes a :: "'b"   and b :: "'b" shows "min (min a b) b = min a b"
\<F>\<G>_af: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set"   and i :: "nat" shows "\<F>\<G> (af \<phi> (prefix i w)) (suffix i w) = \<F>\<G> \<phi> (suffix i w)"
ord_eq_le_trans: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a = b"   and "b \<subseteq># c" shows "a \<subseteq># c"
merge_env_def: fixes merge :: "'f cset \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and A :: "'f cset"   and f :: "'f \<Rightarrow> ('d, 'e) envT" shows "merge_env merge A f = EnvT (\<lambda>r. merge A (\<lambda>a. run_env (f a) r))"
dghm_ObjMap_vdomain: shows "\<D>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
Spy_not_see_NA: fixes A :: "agent"   and B :: "agent"   and NA :: "nat"   and evs :: "event list" assumes "Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_public" shows "Nonce NA \<notin> analz (knows Spy evs)"
rep_eq: fixes x :: "mini_alg_unique"   and xa :: "mini_alg_unique" shows "Rep_mini_alg_unique (mau_times x xa) = ma_times (Rep_mini_alg_unique x) (Rep_mini_alg_unique xa)"
step: fixes a' :: "'a"   and bs :: "'b list"   and a'' :: "'a"   and a :: "'a"   and b :: "'b" assumes "rtrancl3p r a' bs a''"   and "r a b a'"   and "P a' bs a''" shows "P a (b # bs) a''"
prefix_to_carriers: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "set xs \<subseteq> set (history i)"
PREFER_id_D: fixes R :: "('a \<times> 'a) set" assumes "PREFER_id R" shows "R = Id"
streams_iff_sset: fixes s :: "'c stream"   and A :: "'c set" shows "(s \<in> streams A) = (sset s \<subseteq> A)"
i_maps_\<iota>_to_\<iota>\<^sub>1: shows "i \<cdot> \<iota> = \<iota>\<^sub>1 \<cdot> (i \<otimes>\<^sub>1 i)"
subst_poly_scaleRat: fixes \<sigma> :: "nat \<Rightarrow> linear_poly"   and r :: "rat"   and p :: "linear_poly" shows "subst_poly \<sigma> (r *R p) = r *R subst_poly \<sigma> p"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_def: fixes n :: "nat"   and \<phi> :: "'a formula" shows "lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi> = \<Union> {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) \<phi> \<and> (w, I) \<Turnstile> \<phi>}"
less_1: fixes y :: "'a list" assumes "length y < length xs"   and "\<And>xs'. \<lbrakk>rtrancl_path r x xs' y; distinct (x # xs'); set xs' \<subseteq> set y\<rbrakk> \<Longrightarrow> thesis"   and "rtrancl_path r x y y" shows "thesis"
T_sync: fixes P :: "'b process"   and A :: "'b set"   and Q :: "'b process" shows "T (P \<lbrakk>A\<rbrakk> Q) = {s. \<exists>t u. t \<in> T P \<and> u \<in> T Q \<and> s setinterleaves ((t, u), ev ` A \<union> {tick})} \<union> {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
twoCisTypeII_3: shows "cubeImage twoC = {(y, x). x \<in> cbox a b \<and> y \<in> cbox (g2 x) (g1 x)}"
A_def: shows "A = {0<..(int (nat p) - 1) div 2}"
IH1: fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "Vagree \<nu> \<omega> (\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE1}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"   and "osafe ODE1" shows "ODE_sem (local.adjoint I \<sigma> \<nu>) ODE1 = ODE_sem (local.adjoint I \<sigma> \<omega>) ODE1"
new_asms_NoReadOrWrite_only_def: fixes mdss\<^sub>C :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "new_asms_NoReadOrWrite_only mdss\<^sub>C \<equiv> \<forall>i<length cms. (mdss\<^sub>C ! i) AsmNoWrite \<inter> - range var\<^sub>C_of = {}"
supt_imp_supteq: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<rhd> t" shows "s \<unrhd> t"
iteration_sep2: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> x \<cdot> y\<^sup>\<star>" shows "(x + y)\<^sup>\<infinity> = x\<^sup>\<infinity> \<cdot> y\<^sup>\<infinity>"
qp: shows "is_quantum_predicate (adjoint (tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K)) * P * tensor_P (hadamard_on_i (Suc k)) (1\<^sub>m K))"
plus_of_nat: fixes x :: "'a"   and n :: "nat" shows "cos (x + of_nat n * ((2::'a) * of_real pi)) = cos x"
cases: fixes x :: "('a \<Rightarrow> bool) \<times> 'a queue" obtains f :: "'a \<Rightarrow> bool" and q :: "'a queue" where "x = (f, q)"
CSP1_Inf: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` A} is CSP1 healthy"
odd_n0: fixes m :: "'a" assumes "odd m" shows "m \<noteq> (0::'a)"
simps_1: fixes uu :: "Arity list" shows "transform_alts uu [] = []"
WHILE_unfold: fixes b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'm"   and s :: "'a" shows "WHILE b f s = (if b s then bind (f s) (WHILE b f) else return s)"
ccw_system1235_axioms: shows "ccw_system1235 ccw S"
intro: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "H \<lhd> add_monoid G" shows "abelian_subgroup_axioms H G"
wqo_on_imp_reflp_on: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "reflp_on P A"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom (f \<star> src f))"
distinct_5: fixes x1 :: "'b"   and x41 :: "'a"   and x42 :: "('a, 'b) com" shows "Atm x1 \<noteq> while x41 do x42"
content_cbox_pair_eq0_D: fixes a :: "'a"   and c :: "'b"   and b :: "'a"   and d :: "'b" assumes "content (cbox (a, c) (b, d)) = 0" shows "content (cbox a b) = 0 \<or> content (cbox c d) = 0"
assms_1: fixes k :: "'a"   and x :: "'b" shows "f k x (0::'b) = x"
adds_term_pprod_def: shows "adds_term_pprod \<equiv> pprod.adds_term"
abs_def: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e set \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e set"   and x :: "'e" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
assms: shows "\<exists>c. \<forall>x\<in>space M. f x = c"
step_k_2: fixes y :: "V"   and l :: "V" assumes "y \<in> elts k"   and "Ord l" shows "y \<in> elts l \<or> y = l \<or> l \<in> elts y"
Wk: shows "tts (s (Suc i)) t = WAIT k"
simps_2: fixes x :: "nat" assumes "gx1 \<le> x" shows "f2' x = g' (real x) + (\<Sum>i<k. as ! i * f2' ((ts ! i) x))"
f'_def: fixes n :: "nat" shows "f' n = (if n = 0 then 0 else f n)"
isPrefixPrefixToLevel_auxLowerLevel: fixes i :: "nat"   and j :: "nat"   and M :: "('b \<times> bool) list"   and k :: "nat" assumes "i \<le> j" shows "isPrefix (prefixToLevel_aux M i k) (prefixToLevel_aux M j k)"
extr_insert: fixes At :: "atom"   and T :: "atom set"   and H :: "chmsg set"   and C :: "chmsg" assumes "At \<in> extr T H" shows "At \<in> extr T (insert C H)"
aux: fixes a :: "int"   and c :: "int"   and b :: "int" assumes "a \<in> {1, - 1, 0}"   and "c \<noteq> 0"   and "a * b = c" shows "b = c * a"
mem_diff: fixes c' :: "'Com\<^sub>A"   and mds' :: "Mode \<Rightarrow> 'Var\<^sub>A set"   and mem' :: "'Var\<^sub>A \<Rightarrow> 'Val"   and lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and mem'' :: "'Var\<^sub>A \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle>\<^sub>A \<in> abs.loc_reach lc"   and "\<forall>x. var_asm_not_written mds' x \<longrightarrow> mem' x = mem'' x \<and> dma\<^sub>A mem' x = dma\<^sub>A mem'' x" shows "\<langle>c', mds', mem''\<rangle>\<^sub>A \<in> abs.loc_reach lc"
keys_monom_mult: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "c \<noteq> (0::'c)" shows "keys (monom_mult c t p) = (\<oplus>) t ` keys p"
FINITE_valid_states: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set" assumes "finite PROB" shows "finite (valid_states PROB)"
nths_vars2': shows "nths dims0 (- vars1') = dims2"
ltake_0: fixes xs :: "'a llist" shows "ltake 0 xs = []\<^sub>l"
chine_assoc_props_3: shows "Prj\<^sub>1\<^sub>0 \<cdot> chine_assoc = Prj\<^sub>0\<^sub>1"
mask_or_not_mask: fixes x :: "'b word"   and n :: "nat" shows "x AND mask n OR x AND NOT (mask n) = x"
integral_cong: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L N g"
simps_1: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'b"   and ys :: "'b list"   and z :: "'c"   and zs :: "'c list" shows "list_all3 P (x # xs) (y # ys) (z # zs) = (P x y z \<and> list_all3 P xs ys zs)"
spanning_tree_def: fixes t :: "'a"   and g :: "'a"   and r :: "'a" shows "spanning_tree t g r \<equiv> forest t \<and> t \<le> component g r\<^sup>T * component g r \<sqinter> - - g \<and> component g r \<le> r\<^sup>T * t\<^sup>\<star> \<and> regular t"
mr_charn: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "a \<in> set p"   and "x \<in> dom (Cp a)"   and "\<And>r. r \<in> set p \<and> x \<in> dom (Cp r) \<Longrightarrow> r = a" shows "applied_rule_rev Cp x p = \<lfloor>a\<rfloor>"
process_ret_Decision_sound: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and X :: "final_decision" assumes "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), Undecided\<rangle> \<Rightarrow> Decision X" shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>[Rule m (Call chain)], Undecided\<rangle> \<Rightarrow> Decision X"
cases: fixes trsys :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a"   and a2 :: "'b llist" assumes "trsys.inf_step trsys a1 a2" obtains a :: "'a" and b :: "'b" and a' :: "'a" and bs :: "'b llist" where "a1 = a" and "a2 = LCons b bs" and "trsys a b a'" and "trsys.inf_step trsys a' bs"
rev_nth_append: fixes xs :: "'c list"   and v :: "nat"   and y :: "'c"   and ys :: "'c list" assumes "xs !. v = Some y" shows "(ys @ xs) !. v = Some y"
Nr: fixes n :: "nat" assumes "Nr \<le> n" shows "ereal (K + deltaG TYPE('a) + 1) \<le> extended_Gromov_product_at basepoint (to_Gromov_completion ((inv f ^^ n) basepoint)) (repelling_fixed_point f)"
happ_mapsto_1: fixes F :: "hf"   and B :: "hf"   and C :: "hf"   and Y :: "hf" assumes "F \<^bold>\<in> hexp B C"   and "Y \<^bold>\<in> B" shows "happ F Y \<^bold>\<in> C"
dist_triangle_third: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b"   and e :: "real"   and x3 :: "'b"   and x4 :: "'b" assumes "class.metric_space dist uniformity open"   and "dist x1 x2 < e / 3"   and "dist x2 x3 < e / 3"   and "dist x3 x4 < e / 3" shows "dist x1 x4 < e"
Gromov_extension_isometry_on_composition: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'e" assumes "isometry_on UNIV f"   and "isometry_on UNIV g" shows "Gromov_extension (g \<circ> f) = Gromov_extension g \<circ> Gromov_extension f"
q_1: shows "integrable M (\<lambda>x. q (X x))"
nontrivial_integer_group: shows "\<not> trivial_group integer_group"
assms_2: shows "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X"
assum: shows "elem \<in> set (signs_smash signs1 signs2)"
Int_space_eq1: fixes x :: "'b set"   and M :: "'b measure" assumes "x \<in> sets M" shows "space M \<inter> x = x"
unitary11_orientation: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" assumes "unitary11_gen M"   and "M = (A, B, C, D)" shows "\<exists>k'. sgn k' = sgn (if B = 0 then 1 else sgn (Re (A * D / (B * C)) - 1)) \<and> congruence M (1, 0, 0, - 1) = cor k' *\<^sub>s\<^sub>m (1, 0, 0, - 1)"
assms_2: shows "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>1, l\<^sub>1, sh\<^sub>1),False\<rangle>"
CESem_simps_no_tick_1: fixes x :: "Vars.var"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and r :: "C" shows "(\<N>\<lbrakk> Lam [x]. e \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<sqsubseteq> CFn\<cdot>(\<Lambda> v. (\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<rho>(x := v)\<^esub>)|\<^bsub>r\<^esub>)"
VE: shows "G = \<lparr>nodes = V, edges = E\<rparr>"
intro: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a"   and v :: "char list \<Longrightarrow> 'a" assumes "myss2 x y a z"   and "myss3_axioms x y a z v" shows "myss3 x y a z v"
less_le_not_le: fixes x :: "'a list"   and y :: "'a list" shows "strict_prefix x y = (prefix x y \<and> \<not> prefix y x)"
Var: fixes env :: "nat \<Rightarrow> LambdaType.type"   and x :: "nat"   and T :: "LambdaType.type" assumes "env x = T" shows "env \<turnstile> dB.Var x : T"
asympartpE: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "asympartp r x y" obtains "r x y" and "\<not> r y x"
MkArr_Map: fixes f :: "('a, 'a) CC.arr" assumes "CC.arr f" shows "CC.MkArr (CC.Dom f) (CC.Cod f) (CC.Map f) = f"
proj2_line_rep_dependent: fixes i :: "real"   and l :: "proj2_line"   and j :: "real"   and m :: "proj2_line" assumes "i *\<^sub>R proj2_line_rep l + j *\<^sub>R proj2_line_rep m = 0"   and "i \<noteq> 0 \<or> j \<noteq> 0" shows "l = m"
character_with_value_exists: fixes x :: "'a"   and z :: "complex" assumes "x \<in> carrier G"   and "x \<noteq> \<one>"   and "z ^ ord x = 1" obtains \<chi> :: "'a \<Rightarrow> complex" where "character G \<chi>" and "\<chi> x = z"
partially_vec_diff: shows "\<forall>a\<in>Dy_pair. partially_vector_differentiable (\<lambda>a. F a \<bullet> i) j a"
prod_cases: fixes P :: "'a \<times> 'b \<Rightarrow> bool"   and p :: "'a \<times> 'b" assumes "\<And>a b. P (a, b)" shows "P p"
conductor_eq_1_iff_principal: shows "(\<Phi>.conductor = 1) = (\<Phi> = principal_dchar d)"
card_le_nth_prime: fixes n :: "nat" shows "card {q. prime q \<and> q \<le> nth_prime n} = Suc n"
ChiRel_HAStates: fixes a :: "'a"   and b :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(a, b) \<in> ChiRel A" shows "a \<in> HAStates A"
plus_le_conv: fixes A :: "'a set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "Semilat A r f"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "(x \<squnion>\<^bsub>f\<^esub> y \<sqsubseteq>\<^bsub>r\<^esub> z) = (x \<sqsubseteq>\<^bsub>r\<^esub> z \<and> y \<sqsubseteq>\<^bsub>r\<^esub> z)"
dim_row_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'b Matrix.vec" shows "dim_row (mat\<^sub>r nr nc g) = nr"
simps_5: fixes g :: "char list \<Rightarrow> bool"   and h :: "'b fm \<Rightarrow> bool"   and p :: "'b fm"   and q :: "'b fm" shows "eval g h (p \<^bold>\<longrightarrow> q) = (eval g h p \<longrightarrow> eval g h q)"
that: fixes zs :: "'a list" assumes "ys = zs @ xs" shows "thesis"
bres_galois: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "less_eq_prod (times_prod' x y) z = less_eq_prod y (prod.bres x z)"
lexmin: fixes C :: "(nat \<Rightarrow> 'a) set"   and i :: "nat" shows "lexmin C i = min_elt (ith (eq_upto C (lexmin C) i) i)"
simps_2: fixes x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool"   and y2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool" shows "((x2)\<^sub>\<surd> = (y2)\<^sub>\<surd>) = (x2 = y2)"
col_trivial_3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Col Bet A B A"
concat_eq_append_conv: fixes xss :: "'d list list"   and ys :: "'d list"   and zs :: "'d list" shows "(concat xss = ys @ zs) = (if xss = [] then ys = [] \<and> zs = [] else \<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2)"
bits_mod_0: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'b" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo zero a = zero"
member_of_member_of_declC: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_of C" shows "G \<turnstile> m member_of declclass m"
hyps: fixes bs :: "'a list"   and n :: "nat"   and t :: "tm" assumes "boundslt (length bs) p"   and "n \<le> length bs" shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n := Itm vs bs t]) p"
mat_empty_column_length: fixes nr :: "nat"   and nc :: "nat"   and M :: "'c list list" assumes "mat nr nc M"   and "M = []" shows "nc = 0"
per_init_Domain: fixes D :: "'a set" shows "Domain (per_init D) = D"
cadjoint_def: fixes G :: "'c \<Rightarrow> 'b" shows "G\<^sup>\<dagger> = (SOME F. is_cadjoint F G)"
reindex_nontrivial: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; h x = h y\<rbrakk> \<Longrightarrow> g (h x) = (1::'c)" shows "prod g (h ` A) = prod (g \<circ> h) A"
WT_gpv_right_gpv: fixes \<I>2 :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and \<I>1 :: "('k, 'l) \<I>" assumes "\<I>2 \<turnstile>g gpv \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>2 \<turnstile>g right_gpv gpv \<surd>"
greaterThanAtMost_iff: fixes i :: "'a \<times> ('a, 'b) operation"   and l :: "'a \<times> ('a, 'b) operation"   and u :: "'a \<times> ('a, 'b) operation" shows "(i \<in> hb.greaterThanAtMost l u) = (hb l i \<and> weak_hb i u)"
VPow_in_Vset: shows "VPow \<omega> \<in>\<^sub>\<circ> Vset (ZFC_in_HOL.succ (ZFC_in_HOL.succ \<omega>))"
carrierI: fixes t :: "'a llist set"   and x :: "'a llist" assumes "t open"   and "x \<in> t" shows "x \<in> carrier"
nres_monad_laws_2: fixes M :: "'b nres" shows "M \<bind> RETURN = M"
bounded_antilinear_cinner_left_comp: fixes g :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "bounded_clinear g" shows "bounded_antilinear (\<lambda>x. g x \<bullet>\<^sub>C b)"
fst_lens_plus: fixes y :: "'b \<Longrightarrow> 'c"   and x :: "'d \<Longrightarrow> 'c" assumes "wb_lens y" shows "fst\<^sub>L ;\<^sub>L (x +\<^sub>L y) = x"
gfp_const: fixes t :: "'d" shows "gfp (\<lambda>x. t) = t"
unit_simps_4: shows "src\<^sub>D (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) = src\<^sub>D (F f)"
stable_pair_on_CD_on: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH" shows "fst XD_XH \<inter> snd XD_XH = Contracts.CD_on Pd ds (fst XD_XH)"
cases: fixes a :: "color list" assumes "valid a" obtains "a = []"   | xs :: "color list" where "a = B # xs__" and "valid xs__"   | xs :: "color list" and n :: "nat" where "a = replicate n R @ xs" and "valid xs" and "3 \<le> n"
IH: shows "dmu_array_row_main (IArray fs) (IArray fs !! mm) mm (IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (if i = mm then Suc (Suc j) else Suc i)) (Suc mm)) (Suc j) = IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (Suc i)) (Suc mm)"
segment_sub_sub: fixes S :: "'a set"   and d :: "'a" assumes "S \<subseteq> carrier D"   and "d \<in> S" shows "segment (Iod D S) d \<subseteq> segment D d"
INF_UNION: fixes f :: "'i \<Rightarrow> 'h"   and B :: "'j \<Rightarrow> 'i set"   and A :: "'j set" shows "\<Sqinter> (f ` \<Union> (B ` A)) = (\<Sqinter>x\<in>A. \<Sqinter> (f ` B x))"
pdevs_val_def: fixes e :: "nat \<Rightarrow> real"   and x :: "'b pdevs" shows "pdevs_val e x = (\<Sum>i. e i *\<^sub>R pdevs_apply x i)"
insert_tries_def: fixes ks :: "'c list"   and v :: "'d" shows "insert_tries ks v = update_with_trie ks (\<lambda>vo. case vo of None \<Rightarrow> [v] | Some vs \<Rightarrow> v # vs)"
supE: fixes A :: "'a pred"   and B :: "'a pred"   and x :: "'a" assumes "eval (A \<squnion> B) x" obtains "eval A x"   | "eval B x"
select_convs_2: fixes mag :: "'d"   and dim :: "(int, 'c) dimvec"   and more :: "'e" shows "Quantity.dim \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = dim"
sortedConc: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "FWNormalisationCore.sorted (a # p) l" shows "FWNormalisationCore.sorted p l"
MINIM: fixes a' :: "'a" assumes "a' \<in> local.Above B" shows "(a, a') \<in> r"
sign_simps_12: fixes ac :: "'a"   and bc :: "'a"   and cc :: "'a" shows "ac - bc + cc = ac + cc - bc"
AMin_z: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<exists>a. AMin {x \<in> v ` carrier K. 0 < x} = ant a"
mset_subset_eq_multiset_union_diff_commute: fixes B :: "'b multiset"   and A :: "'b multiset"   and C :: "'b multiset" assumes "B \<subseteq># A" shows "A - B + C = A + C - B"
uminus: shows "(R ===> R) (arith_ops_record.uminus ops) uminus_class.uminus"