hcju/isabelleps huggingface dataset:
n_1: shows "ltake (enat x) (lfilter P lst) = lfilter P (ltake (enat n) lst)"
absorb2: fixes b :: "'g"   and a :: "'g" assumes "b \<le> a" shows "min a b = b"
Rs_def: shows "Rs \<equiv> {R. linorder_on (set xs) R}"
step: fixes Y :: "'a \<Rightarrow> 'b set"   and x :: "'a" assumes "\<And>x. countable (Y x)" shows "countable (F Y x)"
fold_graph_AND: fixes \<Phi> :: "'b pltl set" assumes "finite \<Phi>" shows "fold_graph And_ltlp true\<^sub>p \<Phi> (AND \<Phi>)"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_S: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>t S)"
min_def: fixes a :: "'c list"   and b :: "'c list" shows "suffix_order.min a b = (if suffix a b then a else b)"
converse_Times: fixes A :: "'e set"   and B :: "'d set" shows "(A \<times> B)\<inverse> = B \<times> A"
prj_simps_4: shows "C.arr \<mu>\<nu>_\<pi>.prj\<^sub>0"
fls_divide_X_intpow_conv_times: fixes f :: "'a fls"   and i :: "int" shows "f / fls_X_intpow i = f * fls_X_intpow (- i)"
fst_sel_fun_def: fixes x :: "(nat \<times> 'b) list" shows "fst_sel_fun x = fst (hd x)"
elims: fixes x :: "'a extT"   and xa :: "'a extT"   and y :: "'a extT" assumes "x + xa = y" obtains x :: "'a" and y :: "'a" where "xa__ = extT.Val x" and "xa = extT.Val y" and "ya__ = extT.Val (x + y)"   | uu_ :: "'a" where "xa__ = extT.Val uu_" and "xa = extT.Top" and "ya__ = extT.Top"   | uv_ :: "'a extT" where "xa__ = extT.Top" and "xa = uv_" and "ya__ = extT.Top"
rep_eq: fixes x :: "uint32" shows "Rep_uint32 (double x) = Rep_uint32 x + Rep_uint32 x"
kE_int_toFull: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'fsym trm" assumes "I.wtE (kE \<xi>)"   and "wt T" shows "toFull (tpOf T) (I.int (kE \<xi>) T) = F.int \<xi> T"
inv_imps_work_sum_def: fixes c :: "('loc, 't) configuration" shows "inv_imps_work_sum c \<equiv> \<forall>loc. c_imp c loc + c_work c loc = zmset_of (mset_set (set_antichain (frontier (c_pts c loc)))) + (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c loc'))))) (summary loc' loc))"
Fun_mkArr'_1: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B F)" shows "\<guillemotleft>S'.mkArr A B F : S'.mkIde A \<rightarrow>' S'.mkIde B\<guillemotright>"
obs: shows "\<forall>x. \<parallel>x\<parallel> = 1 \<longrightarrow> \<parallel>A *v x\<parallel> \<le> sqrt (\<parallel>Finite_Cartesian_Product.transpose A ** A\<parallel>\<^sub>o\<^sub>p) * (\<parallel>x\<parallel>)"
generate_valid_stateful_policy_IFSACS_2_select_simps_1: fixes G :: "'b graph"   and M :: "'b SecurityInvariant_configured list"   and edgesList :: "('b \<times> 'b) list" shows "hosts (generate_valid_stateful_policy_IFSACS_2 G M edgesList) = nodes G"
submoduleE_3: fixes H :: "(nat \<Rightarrow> 'a) set"   and aa :: "nat \<Rightarrow> 'a" assumes "submodule H R P"   and "aa \<in> H" shows "\<ominus>\<^bsub>P\<^esub> aa \<in> H"
reindex_cong: fixes l :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "bij l"   and "g \<circ> l = h" shows "Sum_any g = Sum_any h"
nonTickFree_n_frontTickFree: fixes s :: "'a event list" assumes "\<not> tickFree s"   and "front_tickFree s" shows "\<exists>t. s = t @ [tick]"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b AND a AND c = a AND b AND c"
cl_in_def: fixes \<E> :: "'a set"   and X :: "'a set" shows "\<E>.cl_in \<E> X = matroid.cl \<E> (\<E>.indep_in \<E>) X"
0: shows "(a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p)) = monom P ((a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p))) (deg R (a \<odot>\<^bsub>P\<^esub> p)) (deg R (monom P ((a \<odot>\<^bsub>P\<^esub> p) (deg R (a \<odot>\<^bsub>P\<^esub> p))) (deg R (a \<odot>\<^bsub>P\<^esub> p))))"
ty_program_def: fixes prog :: "program" shows "ty_program prog \<equiv> let \<Gamma>g = \<lambda>x. if x \<in> set (global_vars prog) then Some () else None in \<forall>pd\<in>set (program.processes prog). ty_proc_decl \<Gamma>g pd"
induced_automorphism_morphism: shows "ChamberComplexEndomorphism X \
"
LocalSecretsComposition_ine1_k: fixes k :: "Keys"   and PQ :: "specID"   and P :: "specID"   and Q :: "specID" assumes "kKS k \<in> LocalSecrets PQ"   and "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "\<not> ine Q (kE k)"   and "kKS k \<notin> LocalSecrets P"   and "kKS k \<notin> LocalSecrets Q" shows "ine P (kE k)"
RemoveMax_axioms: shows "RemoveMax empty is_empty of_list multiset remove_max inv"
kernel_is_closed_csubspace: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "closed_csubspace (f -` {0::'c})"
\<theta>_def: fixes xb :: "('d, 'e) F \<times> 'f" shows "\<theta> xb = F id \<langle>id, \<lambda>a. snd xb\<rangle> (fst xb)"
Tree\<^sub>\<alpha>_free_2: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['a]"   and f :: "'d"   and \<phi> :: "'b" shows "Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<noteq> Pred\<^sub>\<alpha> f \<phi>"
arr_dom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "arr (local.dom f)"
induct: fixes P :: "'b set list \<Rightarrow> 'b set list \<Rightarrow> bool"   and a0 :: "'b set list"   and a1 :: "'b set list" assumes "\<And>ns. P [] ns"   and "\<And>x xs ns. (find_sub_in x ns False = None \<Longrightarrow> P xs ns) \<Longrightarrow> P (x # xs) ns" shows "P a0 a1"
ty_dom_existence: fixes X :: "tyvrs"   and \<Gamma> :: "binding list" assumes "X \<in> ty_dom \<Gamma>" shows "\<exists>T. TVarB X T \<in> set \<Gamma>"
adjoint_functors_def: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd" shows "adjoint_functors C D F G = (\<exists>\<phi> \<psi>. meta_adjunction C D F G \<phi> \<psi>)"
i_times_eq_iff: fixes w :: "complex"   and z :: "complex" shows "(\<i> * w = z) = (w = - (\<i> * z))"
countable_basis: fixes x :: "'a" obtains A :: "nat \<Rightarrow> 'a set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> x"
less_2: shows "set \<sigma> \<subseteq> {x, y}"
seq_step_invariant_sterms_TT: fixes A :: "('a \<times> ('a, 'b, 'c, 'd) seqp, 'b seq_action) automaton"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and P :: "('a \<times> 'd) \<times> 'b seq_action \<times> 'a \<times> 'd \<Rightarrow> bool"   and l' :: "'d"   and q :: "('a, 'b, 'c, 'd) seqp"   and \<xi> :: "'a"   and p :: "('a, 'b, 'c, 'd) seqp"   and I :: "'b seq_action \<Rightarrow> bool"   and p' :: "('a, 'b, 'c, 'd) seqp"   and a :: "'b seq_action"   and \<xi>' :: "'a" assumes "A \<TTurnstile>\<^sub>A onll \<Gamma> P"   and "wellformed \<Gamma>"   and "automaton.trans A = seqp_sos \<Gamma>"   and "l' \<in> labels \<Gamma> q"   and "(\<xi>, p) \<in> reachable A I"   and "((\<xi>, p'), a, \<xi>', q) \<in> automaton.trans A"   and "I a"   and "p' \<in> sterms \<Gamma> p" shows "\<forall>l\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
closure_SUPSEQ: fixes A :: "'b list set" shows "regular (SUPSEQ A)"
hcard_def: fixes x :: "hf" shows "hcard x = card (hfset x)"
sandwitch_real: fixes l :: "nat \<Rightarrow> real"   and a :: "real"   and r :: "nat \<Rightarrow> real"   and m :: "nat \<Rightarrow> real" assumes "l \<longlonglongrightarrow> a"   and "r \<longlonglongrightarrow> a"   and "\<And>i. l i \<le> m i"   and "\<And>i. m i \<le> r i" shows "m \<longlonglongrightarrow> a"
subset_Imagep: fixes B :: "'c set"   and r :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'d set" shows "(B \<subseteq> r ``` A) = (\<forall>b\<in>B. \<exists>a\<in>A. r a b)"
simps_16: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case D of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f4"
continuous_on_artanh: fixes A :: "real set" assumes "A \<subseteq> {- 1<..<1}" shows "continuous_on A artanh"
ipurge_ref_eq: fixes D :: "'f \<Rightarrow> 'e"   and x :: "'f"   and I :: "('e \<times> 'e) set"   and u :: "'e"   and xs :: "'f list"   and X :: "'f set" assumes "D x \<in> sinks I D u (xs @ [x])" shows "ipurge_ref I D u (xs @ [x]) X = ipurge_ref I D u xs {x' \<in> X. (D x, D x') \<notin> I}"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "\<alpha> \<in> S \<rightarrow>\<^sub>E T" shows "map \<alpha> S T"
ide_eval_Can: fixes t :: "'c local.term" assumes "Can t" shows "D.ide \<lbrace>t\<rbrace>"
polymul_head_polyeq: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "p \<noteq> 0\<^sub>p"   and "q \<noteq> 0\<^sub>p" shows "head (p *\<^sub>p q) = head p *\<^sub>p head q"
conga_diff56: 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"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "C' \<noteq> B'"
intros_1: fixes \<sigma>' :: "nat \<Rightarrow> 'e"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'e"   and l :: "'h"   and s\<^sub>m\<^sub>s\<^sub>g :: "'e \<Rightarrow> 'f"   and p :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, {l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p), broadcast (s\<^sub>m\<^sub>s\<^sub>g (\<sigma> i)), \<sigma>', p) \<in> oseqp_sos \<Gamma> i"
body_def: fixes T :: "Unary_PCF.type"   and t :: "expr" shows "body T t \<equiv> \<exists>X. \<forall>x. (x, T) |\<notin>| X \<longrightarrow> lc (open0_Var (x, T) t)"
disj_move_sparse_vec_mat: fixes j :: "nat"   and a :: "nat"   and c :: "(nat \<times> 'a) list"   and as :: "(nat \<times> (nat \<times> 'a) list) list"   and b :: "(nat \<times> 'a) list"   and i :: "int" assumes "j \<le> a"   and "sorted_spvec ((a, c) # as)" shows "disj_matrices (sparse_row_matrix as) (move_matrix (sparse_row_vector b) (int j) i)"
f_def: shows "f \<equiv> \<lambda>k. k mod m"
fv_subterms_set: fixes M :: "('b, 'a) Term.term set" shows "fv\<^sub>s\<^sub>e\<^sub>t (subterms\<^sub>s\<^sub>e\<^sub>t M) = fv\<^sub>s\<^sub>e\<^sub>t M"
fps_sum_nth: fixes f :: "'b \<Rightarrow> 'a fps"   and S :: "'b set"   and n :: "nat" shows "fps_nth (sum f S) n = (\<Sum>k\<in>S. fps_nth (f k) n)"
coeffs_eq_iff: fixes p :: "'c poly"   and q :: "'c poly" shows "(p = q) = (Polynomial.coeffs p = Polynomial.coeffs q)"
quotient_compose_list_g: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b" assumes "Quotient3 R Abs Rep"   and "equivp R" shows "Quotient3 (list_all2 R OOO (\<approx>)) (abs_fset \<circ> map Abs) (map Rep \<circ> rep_fset)"
pull_up_senior_older_seniors_less: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "senior x n < senior y n" shows "older_seniors x n \<subseteq> older_seniors y n"
subset_code_1: fixes xs :: "'e list"   and B :: "'e set" shows "(set xs \<subseteq> B) = (\<forall>x\<in>set xs. x \<in> B)"
R11_defs_1: shows "R11 \<equiv> {(s, s'). s = med11 s'}"
fdia_add2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x + y\<rangle> z =  |x\<rangle> z +  |y\<rangle> z"
simps_6: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "subformulas\<^sub>\<mu> (\<phi> W\<^sub>n \<psi>) = subformulas\<^sub>\<mu> \<phi> \<union> subformulas\<^sub>\<mu> \<psi>"
lookup_empty_empty: fixes t :: "('a, 'b) RBT.rbt" shows "(RBT.lookup t = Map.empty) = (t = RBT.empty)"
14: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e * (e\<^sup>T * e \<sqinter> g) \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * e"
int_of_fbint_cases: fixes y :: "int" assumes "y \<in> UNIV" obtains x :: "fbint" where "y = int_of_fbint x"
construct_lhs_vector_clean: fixes p :: "real poly"   and i :: "nat"   and signs :: "rat list list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "i < length signs" shows "of_rat (construct_lhs_vector p qs signs $ i) = of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = signs ! i})"
take_Suc1: fixes x :: "nat" shows "SList.take [] (Suc x) = []"
bot_\<I>_def: shows "\<bottom> = \<I>_uniform {} UNIV"
stateful_policy_compliance_def: fixes \<T> :: "'b stateful_policy"   and G :: "'b graph"   and M :: "'b SecurityInvariant_configured list" shows "stateful_policy_compliance \<T> G M \<equiv> ((wf_graph G \<and> valid_reqs M) \<and> all_security_requirements_fulfilled M G \<and> wf_stateful_policy \<T>) \<and> (hosts \<T> = nodes G \<and> flows_fix \<T> \<subseteq> edges G) \<and> all_security_requirements_fulfilled (get_IFS M) (stateful_policy_to_network_graph \<T>) \<and> (\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>))"
that: fixes A :: "nat \<Rightarrow> 'a set" assumes "\<And>i. open (A i)"   and "\<And>i. x \<in> A i"   and "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> x" shows "thesis"
partition_on_eq_quotient: fixes A :: "'a set"   and P :: "'a set set" assumes "partition_on A P" shows "A // {(x, y). \<exists>p\<in>P. x \<in> p \<and> y \<in> p} = P"
simps_2: fixes e :: "'a \<times> 'b"   and es :: "('a \<times> 'b) list" shows "single_valued_code (e # es) = (foldr (\<lambda>x. (\<and>) (fst x \<noteq> fst e \<or> snd x = snd e)) es True \<and> single_valued_code es)"
lm024: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c" shows "snd \<circ> summedBid bids = sum bids \<circ> omega"
prod_mwb_lens: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'e" assumes "mwb_lens X"   and "mwb_lens Y" shows "mwb_lens (X \<times>\<^sub>L Y)"
set_error_locally_respects: shows "\<forall>s u a. \<not> ifp (current s) u \<longrightarrow> vpeq u s (set_error_code s a)"
inf_absorb1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y" shows "inf x y = x"
prems: shows "\<forall>l<Suc k. v \<notin> local.writes (path \<sigma> l)"
assms_5: shows "(r, Expunge e2 mo r) \<in> set (node_deliver_messages xs)"
r_coset_eq_set_mult: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and x :: "'c" shows "H #>\<^bsub>G\<^esub> x = H <#>\<^bsub>G\<^esub> {x}"
word_upto_upt: fixes a :: "'a word"   and b :: "'a word" shows "word_upto a b = (if a \<le> b then map word_of_nat [unat a..<Suc (unat b)] else word_upto a b)"
iMODb_mod_0_icard: fixes r :: "nat"   and c :: "nat" shows "icard [ r, mod 0, c ] = eSuc 0"
terminal_tllist_of_llist_lnull: fixes xs :: "'a llist"   and b :: "'b" assumes "lnull xs" shows "terminal (tllist_of_llist b xs) = b"
seqI': fixes f :: "'c"   and a :: "'c"   and b :: "'c"   and g :: "'c"   and c :: "'c" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "C.seq g f"
cm_preserves_inv_implications_nonneg: fixes c0 :: "('loc, 't) Propagate.configuration"   and c1 :: "('loc, 't) Propagate.configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "next_change_multiplicity' c0 c1 loc t n"   and "inv_implications_nonneg c0" shows "inv_implications_nonneg c1"
set_img: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "local.set (img f) = Img f"
fcomp_comp: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" shows "fcomp f g = g \<circ> f"
simps: fixes f :: "nat \<Rightarrow> nat"   and a :: "nat" shows "f5 f (Suc a) = f a"
DEG_def: fixes to :: "'b nat_term_order" shows "DEG to = Abs_nat_term_order (deg_comp (nat_term_compare to))"
triv_measurable_cst: fixes N :: "'b measure"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "space N = space M"   and "space M \<noteq> {}"   and "sets N = {{}, space M}"   and "f \<in> borel_measurable N" shows "\<exists>c. \<forall>x\<in>space N. f x = c"
deg_fun_superset: fixes s :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "supp_fun s \<subseteq> A"   and "finite A" shows "deg_fun s = sum s A"
hcomp_iso_in_hom: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d"   and \<nu> :: "'d"   and h :: "'d"   and k :: "'d" assumes "\<guillemotleft>\<mu> : f \<cong>\<^sub>D g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<cong>\<^sub>D k\<guillemotright>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<mu>" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : h \<star>\<^sub>D f \<cong>\<^sub>D k \<star>\<^sub>D g\<guillemotright>"
offs_mono_aux: fixes i :: "nat"   and j :: "nat"   and ns :: "nat list"   and k :: "nat" assumes "i \<le> j"   and "j < length ns" shows "offs ns k ! i \<le> offs ns k ! (i + (j - i))"
abs_eq: fixes xa :: "'c"   and x :: "'c" assumes "eq_onp dense xa xa"   and "eq_onp dense x x" shows "Abs_dense xa \<sqinter> Abs_dense x = Abs_dense (xa \<sqinter> x)"
midpoint_distinct: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and X :: "'p"   and C' :: "'p" assumes "\<not> Col A B C"   and "Col A B X"   and "X Midpoint C C'" shows "C \<noteq> C'"
rcos_const: fixes h :: "'a" assumes "Group.group G"   and "h \<in> A" shows "A #> h = A"
apartment_gallery: fixes Y :: "'a set set"   and Cs :: "'a set list" assumes "Y \<in> \<A>"   and "SimplicialComplex.maxsimpchain Y Cs" shows "gallery Cs"
refinesI: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "\<And>P. sound P \<Longrightarrow> wp prog P \<tturnstile> wp prog' P" shows "prog \<sqsubseteq> prog'"
nat_pow_0: fixes x :: "int" shows "[0] \<cdot>\<^bsub>INTEG\<^esub> x = \<zero>\<^bsub>INTEG\<^esub>"
last_append: fixes xs :: "'c list"   and ys :: "'c list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
mojmir_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set" shows "mojmir \<Sigma> \<delta> q\<^sub>0 w F \<equiv> semi_mojmir \<Sigma> \<delta> q\<^sub>0 w \<and> mojmir_axioms \<delta> F"
d2_def: shows "d.d2 = prod_list d.dims2"
simps_1: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and s1 :: "'c stream"   and s2 :: "'d stream" shows "shd (smap2 f s1 s2) = f (shd s1) (shd s2)"
lim_le: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b" assumes "convergent f"   and "\<And>n. f n \<le> x" shows "lim f \<le> x"
simps_1: fixes f :: "recf" shows "wrap_Id f 0 = f"
add_strict_increasing2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_cancel_comm_monoid_add plus minus zero less_eq less"   and "less_eq zero a"   and "less b c" shows "less b (plus a c)"
mr_cos_sprod_distrib2: fixes H :: "'a set"   and a :: "'b"   and X :: "'a set"   and Y :: "'a set" assumes "submodule R M H"   and "a \<in> carrier R"   and "X \<in> set_mr_cos M H"   and "Y \<in> set_mr_cos M H" shows "mr_cos_sprod M H a (mr_cospOp M H X Y) = mr_cospOp M H (mr_cos_sprod M H a X) (mr_cos_sprod M H a Y)"
support_single: fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and a :: "char list" shows "support (\<lambda>l. P (l a)) \<subseteq> {a}"
A: fixes x :: "'a" assumes "cont f" shows "cont (\<lambda>y. y [+] f x)"
simps: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a nres"   and s :: "'a"   and l :: "'b list" shows "rfoldl f s l = (case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> f s x \<bind> (\<lambda>s. rfoldl f s ls))"
rebase_eq_0: fixes x :: "'a mod_ring" shows "(@x = 0) = (x = 0)"
dom_weakly_local: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y = (zero_class.zero::'a)) = (x \<cdot> ad (ad y) = (zero_class.zero::'a))"
gcd_pos_int: fixes m :: "int"   and n :: "int" shows "(0 < gcd m n) = (m \<noteq> 0 \<or> n \<noteq> 0)"
product_closure: fixes x :: "i"   and y :: "i" assumes "O x y" shows "\<exists>z. \<forall>w. P w z = (P w x \<and> P w y)"
Lyndon_conj_greater: fixes u :: "'a list"   and v :: "'a list" assumes "Lyndon (u \<cdot> v)"   and "u \<noteq> \<epsilon>"   and "v \<noteq> \<epsilon>" shows "u \<cdot> v <lex v \<cdot> u"
Pbij_extends_inverse: fixes \<gamma> :: "(Location \<times> Location) set"   and \<beta> :: "(Location \<times> Location) set" shows "Pbij_extends \<gamma> (Pbij_inverse \<beta>) = Pbij_extends (Pbij_inverse \<gamma>) \<beta>"
closureI: fixes a :: "'a set"   and x :: "'a" assumes "\<And>c. \<lbrakk>c closed; a \<subseteq> c\<rbrakk> \<Longrightarrow> x \<in> c" shows "x \<in> closure a"
AE_one_D_integrals_eq: shows "AE x in lborel. LBINT y. f (y, x) = integral UNIV (\<lambda>y. f (y, x))"
fls_regpart_to_fls_trivial: fixes f :: "'b fls" assumes "0 \<le> fls_subdegree f" shows "fls_regpart_as_fls f = f"
vors: shows "i < |ls| \<and> j < |ls| \<and> ls ! i = x \<and> ls ! j = y \<and> Suc i mod |ls| = j"
imply_mem: fixes p :: "('a, 'b) fm"   and ps :: "('a, 'b) fm list" assumes "p \<in> set ps" shows "ps \<turnstile> p"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := generate G {a}\<rparr>)" shows "\<exists>y\<in>carrier (G\<lparr>carrier := generate G {a}\<rparr>). y \<otimes>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub> x = \<one>\<^bsub>G\<lparr>carrier := generate G {a}\<rparr>\<^esub>"
abs_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b set \<Rightarrow> 'e"   and Rep :: "'e \<Rightarrow> 'b set"   and x :: "'b" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
contour_integral_diff: fixes f1 :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex"   and f2 :: "complex \<Rightarrow> complex" assumes "f1 contour_integrable_on g"   and "f2 contour_integrable_on g" shows "contour_integral g (\<lambda>x. f1 x - f2 x) = contour_integral g f1 - contour_integral g f2"
lift_set_mono: fixes A :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and B :: "('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and i :: "nat" assumes "A \<subseteq> B" shows "lift_set i A \<subseteq> lift_set i B"
radical_idealI_extend_indets: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and f :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "finite X"   and "F \<subseteq> P[X]"   and "\<V> (insert (1 - punit.monom_mult (1::'d) (monomial 1 None) (extend_indets f)) (extend_indets ` F)) = {}" shows "f \<in> \<surd>ideal F"
WHILEIT_def: shows "WHILEIT \<equiv> \<lambda>I b f. REC\<^sub>T (WHILEI_body (\<bind>) RETURN I b f)"
compare_double_trans: fixes x :: "double"   and y :: "double"   and z :: "double" assumes "compare_double x y \<le> 0"   and "compare_double y z \<le> 0" shows "compare_double x z \<le> 0"
NDVDJ_NDVD: fixes x :: "real"   and n :: "int"   and bs :: "real list"   and i :: "int"   and s :: "MIR.num" assumes "0 \<le> x"   and "x < 1"   and "0 < real_of_int n" shows "Ifm (x # bs) (NDVDJ i n s) = Ifm (x # bs) (NDvd i (CN 0 n s))"
inverse_nonnegative_iff_nonnegative: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (inverse a) = less_eq zero a"
leg_in_hom_1: shows "\<guillemotleft>r0 : ra \<rightarrow>\<^sub>C r.dsrc\<guillemotright>"