hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "('b, 'c) oalist_ntm" shows "lookup (MP_oalist x) = OAlist_lookup_ntm x"
relabel_preserve_other: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "Labeling c s t f l"   and "u \<noteq> v" shows "Network.relabel_effect c f l u v = l v"
tr_leaking_prefix_exists: fixes A' :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and A :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list"   and B :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and n :: "'lbl"   and t :: "('fun, 'var) Term.term" assumes "A' \<in> set (tr\<^sub>p\<^sub>c A [])"   and "prefix B A'"   and "ik\<^sub>s\<^sub>t (proj_unl n B) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>" shows "\<exists>C D. prefix C B \<and> prefix D A \<and> C \<in> set (tr\<^sub>p\<^sub>c D []) \<and> ik\<^sub>s\<^sub>t (proj_unl n C) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>"
pending_reachable: shows "pending s \<subseteq> reachable \<times> reachable"
oppositeLiteralList_def: fixes clause :: "Literal list" shows "oppositeLiteralList clause \<equiv> map opposite clause"
R: fixes l :: "'a llist"   and a :: "'a" assumes "l \<in> A\<^sup>\<omega>"   and "a \<in> A"   and "s = a ## l" shows "R"
lt_plus_eqI: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.lt p \<prec> punit.lt q" shows "punit.lt (p + q) = punit.lt q"
less_imp_not_eq2: fixes y :: "'a"   and x :: "'a" assumes "y < x" shows "(y = x) = False"
\<Z>_csgrpI: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_sgrp \<alpha> \<SS>"   and "\<And>a b. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b = b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a" shows "\<Z>_csgrp \<alpha> \<SS>"
fact4: fixes i :: "'c" assumes "i \<in> - ODE_vars I ODE" shows "(\<lambda>t. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol t) $ i else 0) = (\<lambda>t. 0)"
inf_bot_right: fixes 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 x :: "'b" assumes "class.bounded_lattice_bot inf less_eq less sup bot" shows "inf x bot = bot"
inf_right_zero: fixes x :: "'a" shows "- x \<sqinter> top = - x"
bottomI: fixes x :: "'a" assumes "x \<sqsubseteq> \<bottom>" shows "x = \<bottom>"
cong_cong_prod_coprime: fixes A :: "'b set"   and x :: "'c"   and y :: "'c"   and m :: "'b \<Rightarrow> 'c" assumes "\<forall>i\<in>A. [x = y] (mod m i)"   and "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)" shows "[x = y] (mod prod m A)"
field_cont_on_gamma3_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g1 x)} (\<lambda>a. F a \<bullet> i)"
trans_greek_less: fixes r :: "('a \<times> 'a) set" shows "trans (greek_less r)"
pivotandupdate_tableau_valuated: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<nabla> (pivot_and_update x\<^sub>i x\<^sub>j c s)"
stc_eq_approx_iff: fixes x :: "complex star"   and y :: "complex star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "(x \<approx> y) = (stc x = stc y)"
addition_inv_init: fixes m :: "nat"   and n :: "nat"   and p :: "nat"   and lm :: "nat list" assumes "m \<noteq> n"   and "max m n < p"   and "p < length lm"   and "lm ! p = 0" shows "addition_inv (0, lm) m n p lm"
reduction_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and L1 :: "'b literal"   and C :: "'b eclause"   and \<sigma>' :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and s :: "'b trm"   and polarity :: "sign"   and L2 :: "'b literal"   and u :: "'b trm"   and u' :: "'b trm"   and p :: "indices list"   and v :: "'b trm"   and D :: "'b eclause"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.reduction trm_ord sel pos_ord L1 C \<sigma>' t s polarity L2 u u' p v D I S \<sigma> = (D \<in> S \<and> basic_superposition.eligible_literal trm_ord sel L1 C \<sigma>' \<and> basic_superposition.eligible_literal trm_ord sel L2 D \<sigma>' \<and> ground_clause (subst_cl (cl_ecl D) \<sigma>') \<and> basic_superposition.minimal_redex trm_ord sel pos_ord p (t \<lhd> \<sigma>) C S t \<and> coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C)) \<and> allowed_redex u' C \<sigma> \<and> \<not> is_a_variable u' \<and> L1 \<in> cl_ecl C \<and> L2 \<in> cl_ecl D \<and> basic_superposition.orient_lit_inst trm_ord L1 t s polarity \<sigma>' \<and> basic_superposition.orient_lit_inst trm_ord L2 u v pos \<sigma>' \<and> u \<lhd> \<sigma>' \<noteq> v \<lhd> \<sigma>' \<and> u' \<lhd> \<sigma>' = u \<lhd> \<sigma>' \<and> \<not> validate_ground_clause I (subst_cl (cl_ecl D - {L2}) \<sigma>') \<and> (subst_lit L2 \<sigma>', subst_lit L1 \<sigma>') \<in> basic_superposition.lit_ord trm_ord \<and> (\<forall>x\<in>cl_ecl D - {L2}. (subst_lit x \<sigma>', subst_lit L2 \<sigma>') \<in> basic_superposition.lit_ord trm_ord) \<and> all_trms_irreducible (subst_set (trms_ecl D) \<sigma>') (\<lambda>t. basic_superposition.trm_rep trm_ord sel t S) \<and> I (u \<lhd> \<sigma>') (v \<lhd> \<sigma>') \<and> subterm t p u')"
loop: fixes st :: "'s \<times> ('c \<Rightarrow> real) set" shows "\<K>p st (return_pmf st)"
While_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e :: "'addr expr1" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (while (e) e') h (stk, loc, Suc (length (compE2 e) + pc), xcp) (stk', loc', Suc (length (compE2 e) + pc'), xcp')"
simps_2: fixes e :: "'a expr1"   and es :: "'a expr1 list" shows "compEs2 (e # es) = compE2 e @ compEs2 es"
stutter_equiv_suffixes_right: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b"   and m :: "nat" assumes "\<sigma> \<approx> \<tau>" obtains n :: "nat" where "\<sigma> [m..] \<approx> \<tau> [n..]" and "\<forall>j<n. \<exists>k<m. \<sigma> [k..] \<approx> \<tau> [j..]"
in_span_listI: fixes v :: "rat vec"   and c :: "nat \<Rightarrow> rat"   and vs :: "rat vec list" assumes "True"   and "v = gs.lincomb_list c vs" shows "v \<in> gs.span_list vs"
ACI_trans: fixes r :: "'a rexp"   and s :: "'a rexp"   and t :: "'a rexp" assumes "ACI r s"   and "ACI s t" shows "ACI r t"
simps_13: fixes v :: "tm" shows "qfree (NEq v) = True"
finite: fixes k :: "nat" shows "finite {p. p partitions n \<and> sum p {..n} = k}"
partitions_of_eq: shows "partitions_of A B F = (\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}"
fimageI: fixes x :: "'b"   and A :: "'b fset"   and f :: "'b \<Rightarrow> 'c" assumes "x |\<in>| A" shows "f x |\<in>| f |`| A"
subdg_objD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
size_4: fixes x21 :: "'d"   and x22 :: "'d list" shows "length (x21 # x22) = length x22 + Suc 0"
pdiv_def: fixes p :: "'a list list"   and q :: "'a list list" shows "UP.pdiv p q = (if q = [] then [] else fst (UP.long_division p q))"
eq_paths: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and P :: "'b set"   and Q :: "'b set"   and a :: "'b"   and b :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "P \<in> \<P>"   and "Q \<in> \<P>"   and "a \<in> P"   and "b \<in> P"   and "a \<in> Q"   and "b \<in> Q"   and "a \<noteq> b" shows "P = Q"
assms_6: shows "mat_of_rows n gs = U2 * mat_of_rows n fs_init"
gt_and_basic_gt_le_basic_gt: shows "(\<lambda>t s. local.kbo_std.gt t s \<and> t >\<^sub>t s) \<le> (>\<^sub>t)"
app_inject: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b"   and u\<^sub>3 :: "'b"   and u\<^sub>4 :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "app u\<^sub>1 u\<^sub>2 = app u\<^sub>3 u\<^sub>4" shows "u\<^sub>1 = u\<^sub>3 \<and> u\<^sub>2 = u\<^sub>4"
sat\<^sub>1: shows "eval\<^sub>c F G C\<^sub>1"
authK_not_AKcryptSK: fixes A :: "agent"   and authK :: "nat"   and tk :: "msg"   and evs :: "event list"   and K :: "nat" assumes "Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, tk\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> kerbV" shows "\<not> AKcryptSK K authK evs"
fBallE: fixes A :: "'b fset"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "fBall A P" obtains "P x"   | "x |\<notin>| A"
assms: shows "I \<le> |x\<acute>=f & \<lambda>s. True on U S @ t\<^sub>0] I"
simps: fixes K :: "'a set"   and Us :: "'a list" shows "R.Span K Us = foldr (R.line_extension K) Us {\<zero>}"
lm08: fixes X :: "'b set" assumes "Pow X \<subseteq> {{}, X}" shows "trivial X"
add_spairs_def: fixes ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" shows "add_spairs ps bs p = merge_wrt pair_ord (new_spairs bs p) ps"
multi_member_this: fixes x :: "'b"   and XS :: "'b multiset" shows "x \<in># (x, {#}) + XS"
Rep_eqD: fixes x :: "'a"   and y :: "'a" assumes "Rep x = Rep y" shows "x = y"
pre_zeta_aux_def: fixes N :: "nat"   and a :: "real"   and s :: "complex" shows "pre_zeta_aux N a s = complex_of_real a powr - s / 2 + (\<Sum>i = 1..N. (bernoulli (2 * i) / fact (2 * i)) *\<^sub>R (pochhammer s (2 * i - 1) * complex_of_real a powr (- s - of_nat (2 * i - 1)))) + EM_remainder (Suc (2 * N)) (\<lambda>x. - (pochhammer s (Suc (2 * N)) * complex_of_real (x + a) powr (- 1 - 2 * of_nat N - s))) 0"
rank_link: fixes t :: "('c, 'd) SkewBinomialTree"   and t' :: "('c, 'd) SkewBinomialTree" assumes "rank t = rank t'" shows "rank (link t t') = rank t + 1"
2135: fixes x :: "'a"   and y :: "'a" shows "- - ((top \<squnion> x) \<sqinter> y) \<sqinter> y = y"
mod_nonneg_pos_int_def: fixes x :: "int"   and y :: "int" shows "mod_nonneg_pos_int x y = int_of_integer (mod_nonneg_pos (integer_of_int x) (integer_of_int y))"
fmax: shows "\<forall>E\<in>Filters. F \<subseteq> E \<longrightarrow> F = E"
simps_2: fixes v :: "'a"   and uu :: "'a ifex"   and uv :: "'a ifex"   and r :: "'a ifex list" shows "lowest_tops (IF v uu uv # r) = Some (case lowest_tops r of None \<Rightarrow> v | Some u \<Rightarrow> min u v)"
is_Read\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Read\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
linear_sum: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and g :: "'d \<Rightarrow> ('a, 'b) vec"   and S :: "'d set" assumes "linear (*s) (*s) f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
b_def: shows "b = 13355545553549848714922837267299490903143206628621657811747118592 / 23792006023392488526789546722992491355941103837356113731091180925"
subformulas\<^sub>\<nu>_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "subformulas\<^sub>\<nu> \<psi> \<subseteq> subformulas\<^sub>\<nu> \<phi>"
select_fold_exec\<^sub>S\<^sub>e\<^sub>t: fixes \<tau> :: "'d state \<times> 'd state"   and l :: "('d state \<times> 'd state \<Rightarrow> 'e) list" assumes "list_all (\<lambda>f. \<tau> \<Turnstile> \<upsilon> f) l" shows "\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (foldl UML_Set.OclIncluding Set{} l \<tau>)\<rceil>\<rceil> = set (map (\<lambda>f. f \<tau>) l)"
distinct_169: fixes x9 :: "Term.var" shows "Super \<noteq> Acc x9"
bconf_While: fixes P :: "'a prog"   and sh :: "sheap"   and b :: "'b exp"   and e :: "'b exp"   and b' :: "bool" shows "(P,sh \<turnstile>\<^sub>b (while (b) e,b') \<surd>) = (\<not> b')"
theFTOne_def: fixes cl :: "('g, 'h, 'i) cmd list"   and dl :: "('j, 'k, 'l) cmd list" shows "theFTOne cl dl \<equiv> theFT cl \<union> theFT dl"
sorted_list_of_set_2: fixes A :: "'a word set" shows "signed.sorted (signed.sorted_list_of_set A)"
assms: shows "\<langle>c \<and> t\<rangle> < n"
simps_4: fixes \<psi>\<^sub>1 :: "'b ltln"   and \<psi>\<^sub>2 :: "'b ltln"   and Y :: "'b ltln set" shows "(\<psi>\<^sub>1 U\<^sub>n \<psi>\<^sub>2)[Y]\<^sub>\<mu> = \<psi>\<^sub>1[Y]\<^sub>\<mu> U\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>"
prob_1_bell_snd: fixes v :: "complex mat" assumes "v = |\<beta>\<^sub>0\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>0\<^sub>1\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>0\<rangle> \<or> v = |\<beta>\<^sub>1\<^sub>1\<rangle>" shows "prob1 2 v 1 = 1 / 2"
pcompose_diff: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "(p - q) \<circ>\<^sub>p r = p \<circ>\<^sub>p r - q \<circ>\<^sub>p r"
trm_subst_ident_subterms: fixes t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t \<inter> subst_domain \<theta> = {}" shows "subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = subterms t"
wcode_double_case_first_correctness: fixes ires :: "cell list"   and rs :: "nat"   and m :: "nat"   and n :: "nat" shows "let P = \<lambda>(st, l, r). st = 13; Q = \<lambda>(st, l, r). wcode_double_case_inv st ires rs (l, r); f = steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Oc # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main in \<exists>n. P (f n) \<and> Q (f n)"
moebius_pt_comp_inv_image_left: fixes M :: "moebius"   and A :: "complex_homo set" shows "moebius_pt (moebius_inv M) ` moebius_pt M ` A = A"
WTrt_binop_simps_11: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>Mod\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
fps_inv_def: fixes a :: "'a fps" shows "fps_inv a = Abs_fps (compinv a)"
itypeI: fixes c :: "'a"   and I :: "'b itself" shows "c ::\<^sub>i I"
Int_lower2: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq B (A \<inter> B)"
Cantor_1: shows "list.set \<alpha>s \<subseteq> ON"
subtopology_eq_discrete_topology_eq: fixes X :: "'c topology"   and U :: "'c set" shows "(subtopology X U = discrete_topology U) = (U \<subseteq> topspace X \<and> U \<inter> X derived_set_of U = {})"
cltn2_left_id: fixes A :: "cltn2" shows "cltn2_compose cltn2_id A = A"
process_rule_3_futures: fixes ys :: "'e list"   and Y :: "'e set"   and P :: "'e process"   and xs :: "'e list"   and Y' :: "'e set" assumes "(ys, Y) \<in> futures P xs"   and "Y' \<subseteq> Y" shows "(ys, Y') \<in> futures P xs"
fun_left_idem: fixes x :: "'a"   and z :: "'a" shows "x \<^bold>* (x \<^bold>* z) = x \<^bold>* z"
utp_theory_mono_unital_def: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and utp_unit :: "'a hrel" shows "utp_theory_mono_unital hcond utp_unit \<equiv> utp_theory_unital hcond utp_unit \<and> utp_theory_mono hcond"
assms_2: shows "appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB \<or> appliedBackjump stateA stateB"
assms_2: shows "x2 \<in> {x1<--<b}"
vwalk_join_arcs: fixes p :: "'a list"   and q :: "'a list" assumes "joinable p q" shows "vwalk_arcs (p \<oplus> q) = vwalk_arcs p @ vwalk_arcs q"
radical_superset: fixes F :: "'c set" shows "F \<subseteq> \<surd>F"
valid_path_ends_on_deadend: fixes P :: "'a llist"   and i :: "nat" assumes "valid_path P"   and "enat i < llength P"   and "deadend (P $ i)" shows "enat (Suc i) = llength P"
leg_simps_4: shows "C.dom Dom_\<rho>.leg1 = Dom_\<rho>.apex"
distinct_indices_state_def: fixes s :: "('a, 'b) state" shows "distinct_indices_state s = (\<forall>i x b x' b'. Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x' = Some (i, b') \<or> Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x' = Some (i, b') \<longrightarrow> x = x' \<and> b = b')"
ordIso_imp_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r =o r'" shows "r \<le>o r'"
inf_sup_aci_1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
order_trans: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y"   and "less_eq_prod y z" shows "less_eq_prod x z"
rep_eq: fixes x :: "'c" shows "Rep_monom (var_monom x) = [(x, 1)]"
prv_eql_subst_trm_id_rev: fixes y :: "'var"   and \<phi> :: "'fmla"   and n :: "'trm" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "n \<in> num" shows "bprv (imp (eql n (Var y)) (imp \<phi> (subst \<phi> n y)))"
downshift_def: fixes m :: "nat" shows "downshift m \<equiv> m div 2"
diff_numeral_special_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "minus (numeral.numeral one plus m) one = neg_numeral.sub minus plus one m num.One"
partial_equivalence_def: fixes A :: "'b set"   and equiv :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "partial_equivalence A equiv \<equiv> symmetric A equiv \<and> partial_equivalence_axioms A equiv"
binary_endofunctor_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b" shows "binary_endofunctor C T \<equiv> (category C \<and> product_category C C) \<and> product_category C (product_category.comp C C) \<and> binary_functor C C C T"
lead_coeff_mult: fixes p :: "'d poly"   and q :: "'d poly" shows "lead_coeff (p * q) = lead_coeff p * lead_coeff q"
assm_4: fixes K :: "'a set" assumes "K \<in> d" shows "\<exists>a b. K = cbox a b"
implies_mp_below: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<leadsto> y) \<le> y"
poly2_def: fixes p :: "'b poly poly"   and x :: "'b"   and y :: "'b" shows "poly2 p x y = poly (poly p [:y:]) x"
less_eq_imp_interp_subseteq_interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<le> D" shows "interp C \<subseteq> interp D"
index_upt: fixes m :: "nat"   and i :: "nat"   and n :: "nat" assumes "m \<le> i"   and "i < n" shows "index [m..<n] i = i - m"
unitary_intros_3: fixes P :: "'a \<Rightarrow> bool" shows "unitary \<guillemotleft> P \<guillemotright>"
less_add_one: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less a (plus a one)"
locally_path_connected_open_path_component: fixes S :: "'b set" shows "locally path_connected S = (\<forall>t x. openin (top_of_set S) t \<and> x \<in> t \<longrightarrow> openin (top_of_set S) (path_component_set t x))"
mgp_next_plane0_if_next_plane: fixes g :: "graph"   and g' :: "graph"   and p :: "nat" assumes "minGraphProps g"   and "g' \<in> set (next_plane\<^bsub>p\<^esub> g)" shows "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"
simps: fixes a1 :: "'b \<Rightarrow> 'b ttree"   and a2 :: "'b set"   and a3 :: "'b list"   and a4 :: "'b list" shows "substitute'' a1 a2 a3 a4 = ((\<exists>f T. a1 = f \<and> a2 = T \<and> a3 = [] \<and> a4 = []) \<or> (\<exists>zs f x xs' xs T ys. a1 = f \<and> a2 = T \<and> a3 = x # xs \<and> a4 = x # ys \<and> zs \<in> paths (f x) \<and> xs' \<in> xs \<otimes> zs \<and> substitute'' (f_nxt f T x) T xs' ys))"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p. (\<And>x. P p) \<Longrightarrow> P (E p)"   and "\<And>p. (\<And>x. P (NOT p)) \<Longrightarrow> P (A p)"   and "\<And>p. (\<And>x. P p) \<Longrightarrow> P (NOT p)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>\<And>x. P (NOT p); \<And>x. P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>\<And>x. P p; \<And>x. P q\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "P T"   and "P F"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Le v)"   and "\<And>v. P (Gt v)"   and "\<And>v. P (Ge v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v va. P (Dvd v va)"   and "\<And>v va. P (NDvd v va)" shows "P a0"
g_nonneg: shows "AE x\<in>einterval a b in lborel. 0 \<le> g x"
in_set_conv_decomp_first: fixes x :: "'c"   and xs :: "'c list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set ys)"
twofin: shows "finite (atoms F \<union> (\<Union>I\<in>set (subformulae F). atoms (tseytin_toatom F I)))"
assms_1: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "norm (f x) \<le> norm (f y)"
higher_pderiv_pcompose_linear: fixes n :: "nat"   and p :: "'a poly"   and c :: "'a" shows "(pderiv ^^ n) (p \<circ>\<^sub>p [:0::'a, c:]) = Polynomial.smult (c ^ n) ((pderiv ^^ n) p \<circ>\<^sub>p [:0::'a, c:])"
type_less_x_Sequence: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> < Sequence \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Sequence \<rho>" and "\<rho> < \<sigma>"
simps_1: fixes uu :: "'c list" shows "List.product [] uu = []"
elims_2: fixes x :: "'a extT"   and xa :: "'a extT" assumes "x \<le> xa" obtains x :: "'a" and y :: "'a" where "xa__ = extT.Val x" and "xa = extT.Val y" and "x \<le> y"   | uv_ :: "'a extT" where "xa__ = uv_" and "xa = extT.Top"
rel_generat_is_PureD: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Out :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat" assumes "rel_generat A Out R x y" shows "is_Pure x = is_Pure y"
Write_Sim: fixes P :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and d :: "'e \<Rightarrow> 'f"   and c :: "'d \<Rightarrow> 'f"   and a :: "'f \<Rightarrow> 'c" assumes "P \<preceq>S Q"   and "\<And>A. d A = c (S A)" shows "a`!`c \<rightarrow> P \<preceq>S a`!`d \<rightarrow> Q"
pt_eqvt_fun2a: fixes f :: "'a \<Rightarrow> 'c" assumes "pt TYPE('a) TYPE('b)"   and "pt TYPE('c) TYPE('b)"   and "at TYPE('b)"   and "supp f = {}" shows "\<forall>pi x. pi \<bullet> f x = f (pi \<bullet> x)"
le_approx1: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq> Q" shows "D Q \<subseteq> D P"
Pd_singletons_for_ds_range: fixes X :: "'a set"   and ds :: "'b set"   and d :: "'b" shows "Field (TruePref.Pd_singletons_for_ds X ds d) \<subseteq> {x. Xd x = d}"