hcju/isabelleps huggingface dataset:
assms_4: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "xs \<noteq> []"   and "last xs \<noteq> (0::'a)"   and "x + y + sum_list xs = (0::'a)"   and "(0::'a) \<le> x" shows "P x y xs"
out2__bet: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Out B C"   and "C Out A B" shows "Bet A B C"
sup_ge2: fixes y :: "'b multiset"   and x :: "'b multiset" shows "y \<subseteq># x \<union># y"
to_fun_X_plus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_fun (X_poly_plus R a) b = b \<oplus> a"
coprime_poly_0: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" assumes "algebraic_semidom_class.coprime p q" shows "poly p x \<noteq> (0::'a) \<or> poly q x \<noteq> (0::'a)"
circ_rtc_2: fixes x :: "'a" shows "Sf x (x\<^sup>\<star>) = x\<^sup>\<star>"
iT_Plus_mem_iff: fixes x :: "nat"   and I :: "nat set"   and k :: "nat" shows "(x \<in> I \<oplus> k) = (k \<le> x \<and> x - k \<in> I)"
snd_proj_def: fixes M :: "('b \<times> 'a) set" shows "snd_proj M \<equiv> {B. \<exists>A. (A, B) \<in> M}"
termi: shows "\<Gamma>\<turnstile>dynCall init p return c \<down> Normal s"
assms_4: shows "q \<in> dgrad_p_set d m"
p_eq_lfp_F_sup: shows "p = lfp F_sup"
lens_prism_put_inverse: fixes X :: "'c \<Longrightarrow> 'd" assumes "pbij_lens X" shows "put\<^bsub>prism_lens (lens_prism X)\<^esub> = put\<^bsub>X\<^esub>"
wd_loop: fixes body :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and G :: "'b \<Rightarrow> bool" assumes "well_def body" shows "well_def (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)"
simps_7: fixes x3 :: "nat"   and x1 :: "nat" shows "Greater x3 \<noteq> Const x1"
c_K5_list_def: shows "c_K5_list \<equiv> ([0..4], concat (map (\<lambda>x. concat (map (\<lambda>y. if x \<noteq> y then [(x, y)] else []) [0..4])) [0..4]))"
prod_mono_strict: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "finite A"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq zero (f i) \<and> less (f i) (g i)"   and "A \<noteq> {}" shows "less (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
Pbij_composeI: fixes l1 :: "Location"   and l2 :: "Location"   and \<beta> :: "(Location \<times> Location) set"   and l3 :: "Location"   and \<gamma> :: "(Location \<times> Location) set" assumes "(l1, l2) \<in> \<beta>"   and "(l2, l3) \<in> \<gamma>" shows "(l1, l3) \<in> Pbij_compose \<beta> \<gamma>"
succ_def: fixes \<R> :: "('a \<Rightarrow> 'b) set set"   and R :: "('a \<Rightarrow> 'b) set" shows "succ \<R> R \<equiv> if \<forall>t\<ge>0::'b. \<forall>u\<in>R. u \<oplus> t \<in> R then R else THE R'. R' \<noteq> R \<and> R' \<in> Succ \<R> R \<and> (\<forall>u\<in>R. \<forall>t\<ge>0::'b. u \<oplus> t \<notin> R \<longrightarrow> (\<exists>t'\<le>t. u \<oplus> t' \<in> R' \<and> (0::'b) \<le> t'))"
span_eq: fixes S :: "('b \<Rightarrow>\<^sub>0 'c) set"   and T :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "(phull S = phull T) = (S \<subseteq> phull T \<and> T \<subseteq> phull S)"
sin_i_eq: fixes z :: "complex" shows "(\<lambda>n. \<i> * complex_of_real (sin_coeff n) * z ^ n) sums (\<i> * sin z)"
remove_vertex_V: fixes G :: "('c, 'd) Graph_scheme"   and x :: "'c" assumes "Digraph G" shows "V\<^bsub>Digraph.remove_vertex G x\<^esub> = V\<^bsub>G\<^esub> - {x}"
poly_mapping_adds_iff: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "(s adds t) = (lookup s adds lookup t)"
let2: fixes l :: "nat"   and binds :: "((nat \<times> char list) \<times> lambda) list"   and c' :: "call"   and x :: "lambda" assumes "call.Let l binds c' \<in> calls x" shows "c' \<in> calls x"
simps_40: fixes x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt"   and x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "Await x71 x72 \<noteq> x31@[x32]"
add_diff_eq: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a + (b - c) = a + b - c"
assms: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j"   and "j < k"   and "k < m"   and "A $$ (k, k) = A $$ (i, i)" shows "A $$ (j, j) = A $$ (i, i)"
sprefD: fixes u :: "'b list"   and v :: "'b list" assumes "u <p v" shows "u \<le>p v \<and> u \<noteq> v"
easy: assumes "norm x \<le> 1" shows "norm x \<le> 1 + max_list_non_empty (map (\<lambda>i. norm (coeff p i)) [0..<degree p]) / norm (lead_coeff p)"
secure_implies_next_event_consistency: fixes P :: "'a process"   and I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "secure P I D"   and "(xs, ys) \<in> rel_ipurge P I D (D x)"   and "x \<in> next_events P xs" shows "x \<in> next_events P ys"
less_append: fixes s :: "'b list"   and t :: "'b list"   and a :: "'b list" assumes "s < t" shows "a @ s < a @ t"
distinct_zipI1: fixes xs :: "'c list"   and ys :: "'d list" assumes "distinct xs" shows "distinct (zip xs ys)"
antisym: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b"   and "b \<le> a" shows "a = b"
sublist_cons: fixes l1 :: "'b list"   and l2 :: "'b list"   and h :: "'b" assumes "subseq l1 l2" shows "subseq l1 (h # l2)"
Sup_upper2: fixes u :: "'d \<times> 'e"   and A :: "('d \<times> 'e) set"   and v :: "'d \<times> 'e" assumes "u \<in> A"   and "less_eq_prod v u" shows "less_eq_prod v (Sup_prod A)"
lnull_def: fixes llist :: "'a llist" shows "lnull llist = (llist = LNil)"
\<phi>\<psi>: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "(\<phi> \<circ> \<psi>) T = T"
label_tree_def: shows "label_tree = fold_tree (\<lambda>_. State_Monad.ap (State_Monad.return Tree_Relabelling.tree.Leaf) fresh) (\<lambda>l. State_Monad.ap (State_Monad.ap (State_Monad.return Tree_Relabelling.tree.Node) l))"
preserves_reflects_arr: fixes f :: "'b" shows "arr (\<ll> f) = arr f"
lens_comp_def: fixes Y :: "'c \<Longrightarrow> 'e"   and X :: "'e \<Longrightarrow> 'd" shows "Y ;\<^sub>L X = \<lparr>lens_get = get\<^bsub>Y\<^esub> \<circ> get\<^bsub>X\<^esub>, lens_put = \<lambda>\<sigma> v. put\<^bsub>X\<^esub> \<sigma> (put\<^bsub>Y\<^esub> (get\<^bsub>X\<^esub> \<sigma>) v)\<rparr>"
a_comm': fixes y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" shows "rel_ar y O rel_ar x = rel_ar x O rel_ar y"
J2: shows "finite {j. j < length a \<and> is_nonterminal (a ! j)}"
Quotient3_int: shows "Quotient3 Int.intrel Abs_Integ Rep_Integ"
ij_le1_2: shows "j < card {i. i < dim_col A \<and> i \<in> J}"
1: shows "\<forall>PROB s as. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> Inf {n. n \<in> PLS_charles s as PROB} < f PROB"
splitAt_dist_ram22: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(c, []) = splitAt ram1 (snd (splitAt ram2 vs))"
progress_LN1: fixes j :: "nat"   and lholed :: "Lholed"   and k :: "nat"   and es :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "Lfilled j lholed [$Br (j + k)] es"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'" shows "k < length (label \<C>)"
TS_inv_def: fixes c :: "('a list \<times> 'a list) pmf"   and x :: "'a"   and i :: "'a list" shows "TS_inv c x i \<equiv> (\<exists>hs. c = return_pmf (if x = hd i then i else rev i, [x, x] @ hs)) \<or> c = return_pmf (if x = hd i then i else rev i, [])"
iWeakUntil_conj_iUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 I. P t2 \<and> Q t2) = (\<not> (\<not> Q t1. t1 \<U> t2 I. \<not> P t2))"
find_closest_bf_set: fixes ps :: "(int \<times> int) list"   and p :: "int \<times> int" assumes "0 < length ps" shows "find_closest_bf p ps \<in> set ps"
and_dfa_accepts: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and N :: "nat bdd list \<times> bool list"   and bs :: "bool list list" assumes "wf_dfa M n"   and "wf_dfa N n"   and "list_all (is_alph n) bs" shows "dfa_accepts (and_dfa M N) bs = (dfa_accepts M bs \<and> dfa_accepts N bs)"
in_domain_in_carrier: fixes c :: "('c, 'd) chart"   and charts :: "('c, 'd) chart set"   and x :: "'c" assumes "c \<in> charts"   and "x \<in> domain c" shows "x \<in> manifold.carrier charts"
that: shows "x \<notin> fst ` set_spmf p"
abs_eq: fixes x :: "int extended \<times> int extended" shows "- abs_ivl x = abs_ivl (uminus_rep x)"
prems: shows " App (Lam [x].t1) s1 \<longrightarrow>\<^sub>1 t2"
prod_unions_def: fixes A :: "'b set \<times> 'c set"   and B :: "'b set \<times> 'c set" shows "prod_unions A B \<equiv> case A of (a, b) \<Rightarrow> case B of (c, d) \<Rightarrow> (a \<union> c, b \<union> d)"
FAddSA_Root_def: fixes S :: "'d"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and SA :: "('d, 'e, 'f) seqauto"   and F :: "('d, 'e, 'f) seqauto set" assumes "S \<in> dom G"   and "States SA \<inter> dom G = {}" shows "Root F (G [f+] (S, SA)) = (SOME A. A \<in> F \<and> A \<notin> insert SA (\<Union> (ran G)))"
trans_def: fixes r :: "('e \<times> 'e) set" shows "trans r = (\<forall>x y z. (x, y) \<in> r \<longrightarrow> (y, z) \<in> r \<longrightarrow> (x, z) \<in> r)"
cell_complex_subdivision_exists: fixes e :: "real"   and \<F> :: "'b set set"   and d :: "int" assumes "0 < e"   and "finite \<F>"   and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"   and "\<And>X. X \<in> \<F> \<Longrightarrow> aff_dim X \<le> d"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X" obtains \<F>' :: "'b set set" where "finite \<F>'" and "\<Union> \<F>' = \<Union> \<F>" and "\<And>X. X \<in> \<F>' \<Longrightarrow> diameter X < e" and "\<And>X. X \<in> \<F>' \<Longrightarrow> polytope X" and "\<And>X. X \<in> \<F>' \<Longrightarrow> aff_dim X \<le> d" and "\<And>X Y. \<lbrakk>X \<in> \<F>'; Y \<in> \<F>'\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X" and "\<And>C. C \<in> \<F>' \<Longrightarrow> \<exists>D. D \<in> \<F> \<and> C \<subseteq> D" and "\<And>C x. C \<in> \<F> \<and> x \<in> C \<Longrightarrow> \<exists>D. D \<in> \<F>' \<and> x \<in> D \<and> D \<subseteq> C"
induct: fixes x1 :: "'a option list"   and x2 :: "'a option set"   and P :: "'a option process"   and Q :: "'a option process"   and Pa :: "'a option list \<Rightarrow> 'a option set \<Rightarrow> bool" assumes "(x1, x2) \<in> seq_comp_failures P Q"   and "\<And>xs X. \<lbrakk>xs \<notin> sentences P; (xs, X) \<in> failures P; None \<notin> set xs\<rbrakk> \<Longrightarrow> Pa xs X"   and "\<And>xs X Y. \<lbrakk>xs \<in> sentences P; (xs, X) \<in> failures P; ([], Y) \<in> failures Q\<rbrakk> \<Longrightarrow> Pa xs (insert None X \<inter> Y)"   and "\<And>xs ys Y. \<lbrakk>xs \<in> sentences P; (ys, Y) \<in> failures Q; ys \<noteq> []\<rbrakk> \<Longrightarrow> Pa (xs @ ys) Y"   and "\<And>xs X Y. \<lbrakk>(xs, X) \<in> seq_comp_failures P Q; Pa xs X; (xs, Y) \<in> seq_comp_failures P Q; Pa xs Y\<rbrakk> \<Longrightarrow> Pa xs (X \<union> Y)" shows "Pa x1 x2"
cross_ratio_in_terms_of_cosh_dist: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "are_endpoints_in_S p q a b"   and "B\<^sub>\<real> cart2_pt a cart2_pt b cart2_pt p" shows "cross_ratio p q a b = 2 * (cosh_dist a b)\<^sup>2 + 2 * cosh_dist a b * sqrt ((cosh_dist a b)\<^sup>2 - 1) - 1"
induct: fixes x :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set"   and P :: "(('a, 'b, 'c) prot_fun, 'd) Term.term \<Rightarrow> bool" assumes "x \<in> timpl_closure t TI"   and "P t"   and "\<And>u a b s. \<lbrakk>u \<in> timpl_closure t TI; P u; (a, b) \<in> TI; term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) u s\<rbrakk> \<Longrightarrow> P s" shows "P x"
ord_of_nat_strict_mono_iff: fixes i :: "nat"   and j :: "nat" shows "(ord_of_nat i < ord_of_nat j) = (i < j)"
one_nuc: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and cl_op :: "'b \<Rightarrow> 'b" assumes "class.unital_quantale_with_nuc one times Inf Sup inf less_eq less sup bot top cl_op" shows "less_eq one (cl_op one)"
gcd_red_int: fixes x :: "int"   and y :: "int" shows "gcd x y = gcd y (x mod y)"
pre_on_finish_def: fixes u :: "'v"   and s' :: "'v tarjan_state" shows "pre_on_finish u s' \<equiv> \<exists>s. DFS_invar G tarjan_params s \<and> cond s \<and> stack s \<noteq> [] \<and> u = hd (stack s) \<and> pending s `` {u} = {} \<and> s' = finish u s"
dfree_Const: fixes r :: "real" shows "dfree (Const r)"
eq_lemma: fixes x :: "nat"   and y :: "nat" shows "rec_exec rec_eq [x, y] = (if x = y then 1 else 0)"
svar_case: fixes I :: "('b, 'c, 'a) interp"   and x :: "'a" shows "sterm_sem I (Var x) = (\<lambda>v. v $ x)"
assms_2: shows "(\<box>\<^sub>c\<gamma>' \<longrightarrow>\<^sup>c \<gamma>'') t n"
ide_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "ide (trg \<mu>)"
t: shows "\<Gamma> \<turnstile> t : U \<rightarrow> T"
dg_def: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "\<CC> = [\<CC>\<lparr>Obj\<rparr>, \<CC>\<lparr>Arr\<rparr>, \<CC>\<lparr>Dom\<rparr>, \<CC>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
Fn: fixes f :: "'fsym"   and Tl :: "'fsym trm list" assumes "wtFsym f"   and "list_all wt Tl"   and "arOf f = length Tl"   and "list_all \<phi> Tl" shows "\<phi> (Fn f Tl)"
dens': shows "has_parametrized_subprob_density (state_measure (set vs \<union> set vs') \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure ({}, set vs \<union> set vs', \<Gamma>, \<lambda>_. 1) \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> b)) (stock_measure BOOL) (\<lambda>\<sigma> v. ennreal (eval_cexpr f \<sigma> v))"
rexp_DA_no_post_def: fixes \<sigma> :: "nat \<Rightarrow> 'c list"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and init :: "'d rexp \<Rightarrow> 'e"   and delta :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and final :: "'e \<Rightarrow> bool"   and wf_state :: "'e \<Rightarrow> bool"   and L :: "'e \<Rightarrow> 'c list set"   and n :: "nat" shows "rexp_DA_no_post \<sigma> wf_atom project lookup init delta final wf_state L n \<equiv> rexp_DA \<sigma> wf_atom project lookup init delta final wf_state id L n"
success_not_cond_effect: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref"   and h :: "heap" assumes "success (shiftr_p a key i) h"   and "\<not> cond (shiftr_p a key i) h" shows "effect (shiftr_p a key i) h h False"
assms_2: fixes conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "X conv"   and "a \<in> outs_\<I> \<I>" shows "plossless_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). X conv' \<or> plossless_converter \<I> \<I>' conv')"
u_galois_connection: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice"   and y :: "'b::refinement_lattice" assumes "dist_over_sup (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice)" shows "((F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (x::'b::refinement_lattice) \<sqsubseteq> (y::'b::refinement_lattice)) = (x \<sqsubseteq> (F\<^sup>#) y)"
U_par_p_id: shows "1\<^sub>\<pi> \<cdot> 1\<^sub>\<pi> \<parallel> U = U"
fempty_fsubsetI: fixes x :: "'a fset" shows "{||} |\<subseteq>| x"
exhaust_disc: fixes vertex :: "'b vertex" obtains "vertex = SOURCE"   | "vertex = vertex.SINK"   | "is_Inner vertex"
is_lub_singleton: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.po below" shows "po.is_lub below {x} x"
smult_r_distr: fixes a :: "'a"   and x :: "'c"   and y :: "'c" assumes "a \<in> carrier R"   and "x \<in> carrier M"   and "y \<in> carrier M" shows "a \<odot>\<^bsub>M\<^esub> (x \<oplus>\<^bsub>M\<^esub> y) = a \<odot>\<^bsub>M\<^esub> x \<oplus>\<^bsub>M\<^esub> a \<odot>\<^bsub>M\<^esub> y"
simps: fixes a1 :: "'e"   and a2 :: "'e" shows "hull_relp a1 a2 = (\<exists>p x. a1 = p \<bullet> x \<and> a2 = x)"
local_ring_axioms: shows "local_ring stx.carrier_stalk stx.add_stalk stx.mult_stalk (stx.zero_stalk (identity S \<^sup>\<inverse> S V)) (stx.one_stalk (identity S \<^sup>\<inverse> S V))"
nth_nths_card: fixes j :: "nat"   and xs :: "'a list"   and J :: "nat set" assumes "j < length xs"   and "j \<in> J" shows "nths xs J ! card {j0. j0 < j \<and> j0 \<in> J} = xs ! j"
mod_subset_empty: fixes a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and p :: "nat"   and X :: "'c topology" shows "((a, b) \<in> mod_subset p (subtopology X {})) = (a = b)"
IH_1: assumes "matches \<gamma> m p"   and "all_chains (no_call_to c) \<Gamma> rs" shows "\<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> [Rule m (Call c)] \<Rightarrow>\<^sub>z Undecided"
hyps: shows "g\<cdot>(iterate i\<cdot>f\<cdot>\<bottom>) = iterate i\<cdot>h\<cdot>\<bottom>"
distinct_29: fixes x3 :: "fresh_t"   and x71 :: "key"   and x72 :: "msg" shows "Nonce x3 \<noteq> Crypt x71 x72"
stepxy: fixes q :: "'a"   and x :: "'a"   and y :: "'a"   and A :: "'a list"   and a :: "nat \<times> nat list" assumes "q \<in> {x, y}"   and "A \<in> {[x, y], [y, x]}" shows "step A q a \<in> {[x, y], [y, x]}"
map_eq_zero_iff: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(Poly_Mapping.map f p = 0) = (\<forall>k\<in>Poly_Mapping.keys p. f (poly_mapping.lookup p k) = (0::'b))"
max_Var_floatarith_inner_floatariths: fixes f :: "floatarith list"   and g :: "floatarith list" assumes "length f = length g" shows "max_Var_floatarith (inner_floatariths f g) = max (max_Var_floatariths f) (max_Var_floatariths g)"
cproper_interval_bool_def: shows "cproper_interval = proper_interval"
InvariantsClAfterApplyConflict: fixes state :: "State"   and F0 :: "Literal list list" assumes "getConflictFlag state"   and "InvariantUniq (getM state)"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantEquivalentZL (getF state) (getM state) F0"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)"   and "0 < currentLevel (getM state)" shows "let state' = applyConflict state in InvariantCFalse (getConflictFlag state') (getM state') (getC state') \<and> InvariantCEntailed (getConflictFlag state') F0 (getC state') \<and> InvariantClCharacterization (getCl state') (getC state') (getM state') \<and> InvariantClCurrentLevel (getCl state') (getM state') \<and> InvariantCnCharacterization (getCn state') (getC state') (getM state') \<and> InvariantUniqC (getC state')"
rGamma_complex_def: fixes z :: "complex" shows "rGamma z = lim (rGamma_series z)"
subst_fv: fixes e :: "exp"   and x :: "nat"   and v :: "exp" assumes "FV e \<subseteq> {x}"   and "FV v = {}" shows "FV (subst x v e) = {}"
lebesgue_integral_def: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'c" shows "integral\<^sup>L M f = (if \<exists>x. has_bochner_integral M f x then THE x. has_bochner_integral M f x else (0::'c))"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(pure x2 = pure y2) = (x2 = y2)"
bmod_comp_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'd"   and x :: "'e" shows "\<otimes> R f g x = \<Squnion>{f y \<cdot> g z |y z. R x y z}"
in_hom_def: fixes f :: "'a1 \<times> 'a2"   and a :: "'a1 \<times> 'a2"   and b :: "'a1 \<times> 'a2" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 b\<guillemotright> \<equiv> a \<in> A1xA2.domains f \<and> b \<in> A1xA2.codomains f"
collision_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and time_set :: "real set" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o time_set \<equiv> \<exists>t\<in>time_set. movement.s a\<^sub>e v\<^sub>e s\<^sub>e t = movement.s a\<^sub>o v\<^sub>o s\<^sub>o t"
matchId: fixes a :: "name"   and P :: "pi" shows "[a\<frown>a]P \<sim>\<^sub>e P"
mrbst_insert_altdef: fixes x :: "'a"   and t :: "'a tree" shows "mrbst_insert x t = mrbst_union \<langle>\<langle>\<rangle>, x, \<langle>\<rangle>\<rangle> t"
1: shows "w\<^sub>1 =\<^sub>F w\<^sub>2"
image_uminus_greaterThan: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus ` ord.greaterThan less x = ord.lessThan less (uminus x)"
tendsto_ln: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> 0" shows "((\<lambda>x. ln (f x)) \<longlongrightarrow> ln a) F"
ref_union_closed_def: fixes P :: "'a process" shows "ref_union_closed P \<equiv> \<forall>xs A. (\<exists>X. X \<in> A) \<longrightarrow> (\<forall>X\<in>A. (xs, X) \<in> failures P) \<longrightarrow> (xs, \<Union>X\<in>A. X) \<in> failures P"
fst_divmod: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "fst (divmod m n) = divide (numeral.numeral one plus m) (numeral.numeral one plus n)"
minim_Under_under: fixes A :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> Field r" shows "r.Under A = r.under (r.minim A)"
subspace_is_vs: fixes K :: "('e, 'f) ring_scheme"   and V :: "('e, 'g, 'h) module_scheme"   and W :: "'g set" assumes "vectorspace K V"   and "subspace K W V" shows "vectorspace K (V\<lparr>carrier := W\<rparr>)"
less_add_iff1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and e :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.ordered_ring plus minus zero less_eq less times uminus" shows "less (plus (times a e) c) (plus (times b e) d) = less (plus (times (minus a b) e) c) d"
H_values: fixes i :: "nat"   and j :: "nat" assumes "i < dim_row H"   and "j < dim_col H"   and "i \<noteq> 1 \<or> j \<noteq> 1" shows "H $$ (i, j) = complex_of_real (1 / sqrt 2)"