hcju/isabelleps huggingface dataset:
hyps_4: shows "Abs bT t = subst_bv1 ta (length (T # Ts)) x"
GEN_OP_tag_def: fixes P :: "'c" shows "GEN_OP_tag P \<equiv> P"
no_abs_simps_2: fixes name :: "name" shows "no_abs (free name) = True"
h_divide: shows "\<forall>w\<in>cball z r. (w \<noteq> z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0"
axiom_eqrefl_def: fixes t :: "tm" shows "axiom_eqrefl t \<equiv> Thm (Atom (Rl STR ''='' [t, t]))"
seqsim_notstutstep: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<not> stutstep s n" shows "s |\<^sub>s Suc n \<approx> nextsuffix (s |\<^sub>s n)"
trivial_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and defs :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and "uses" :: "'a \<Rightarrow> 'b \<Rightarrow> 'd set"   and phis :: "'a \<Rightarrow> 'b \<times> 'd \<Rightarrow> 'd list option"   and g :: "'a"   and v :: "'d" shows "CFG_SSA_wf_base.trivial \<alpha>n inEdges' defs uses phis g v \<equiv> \<exists>v'\<in>CFG_SSA_base.allVars \<alpha>n inEdges' defs uses phis g. CFG_SSA_wf_base.isTrivialPhi \<alpha>n defs phis g v v'"
set_finsert: fixes x :: "'m"   and A :: "'m fset" assumes "x |\<in>| A" obtains B :: "'m fset" where "A = finsert x B" and "x |\<notin>| B"
simps_9: fixes v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp One (Plus v va) = True"
bound_main_lemma: fixes PROB :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" assumes "finite PROB" shows "problem_plan_bound PROB \<le> 2 ^ card (prob_dom PROB) - 1"
s_def: shows "s = (\<lambda>n. t + T * real n)"
Un_upper2: fixes B :: "'c set"   and A :: "'c set" shows "B \<subseteq> A \<union> B"
at_conj_distr: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v' \<and> \<psi> ts v') = ((\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v') \<and> (\<forall>v'. v = c > v' \<longrightarrow> \<psi> ts v'))"
atLeastatMost_empty_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.preorder less_eq less" shows "(ord.atLeastAtMost less_eq a b = {}) = (\<not> less_eq a b)"
exhaust_sel: fixes rbt :: "('a, 'b) rbt" obtains "rbt = Leaf"   | "rbt = Node (lsub rbt) (cl rbt) (key rbt) (val rbt) (rsub rbt)"
get_shadow_root_pure: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap" shows "pure (get_shadow_root element_ptr) h"
simps: fixes A :: "'a mat"   and B :: "'a mat" shows "two_block_non_interfering A B = (let z1 = 0\<^sub>m (dim_row A) (dim_col B); z2 = 0\<^sub>m (dim_row B) (dim_col A) in four_block_mat A z1 z2 B)"
preserves_dom: fixes f :: "('a, 'a) CC.arr" assumes "CC.arr f" shows "local.dom (if CC.arr f then CC.Map f else null) = (if CC.arr (CC.dom f) then CC.Map (CC.dom f) else null)"
from_into_WW: fixes k :: "nat"   and ns :: "nat list" shows "from_WW k (into_WW k ns) = ns"
bit_minus_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (uminus a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> \<not> bit (minus a one) n)"
prod_mOp_func: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_mOp I A \<in> carr_prodag I A \<rightarrow> carr_prodag I A"
HFail_HInv4c_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p"   and "HInv4c s p" shows "HInv4c s' p"
rules: shows "R' \<subseteq> upRules \<and> R = Ax \<union> R'"
\<R>\<^sub>2_elim': fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma> :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds\<^sub>2), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>" obtains \<Gamma>\<^sub>1 :: "'b \<Rightarrow> Sec option" and \<Gamma>\<^sub>2 :: "'b \<Rightarrow> Sec option" where "mds\<^sub>2 = mds" and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_security.mm_equiv dma (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B)" and "\<forall>x\<in>dom \<Gamma>. \<Gamma> x = Some High" and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>\<^sub>1 c\<^sub>1 \<Gamma>" and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>\<^sub>2 c\<^sub>2 \<Gamma>" and "sifum_types.mds_consistent dma mds \<Gamma>\<^sub>1" and "sifum_types.mds_consistent dma mds \<Gamma>\<^sub>2"
multiplicity_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times (power.power one times p (factorial_semiring.multiplicity times one p x)) x"
coherent_Runit_Ide: fixes t :: "'c local.term" assumes "Ide t" shows "coherent \<^bold>\<r>\<^bold>[t\<^bold>]"
group_axioms: shows "Group.group \<lparr>carrier = UNIV, monoid.mult = (+), one = 0::'b, \<dots> = undefined\<rparr>"
less_trans: fixes v1 :: "val"   and v2 :: "val"   and v3 :: "val" assumes "v1 \<sqsubseteq> v2"   and "v2 \<sqsubseteq> v3" shows "v1 \<sqsubseteq> v3"
flattening_lemma_ii: fixes \<psi> :: "('c, 'd) sas_plus_operator list list"   and \<Psi> :: "('c, 'd) sas_plus_problem"   and G :: "'c \<Rightarrow> 'd option"   and I :: "'c \<Rightarrow> 'd option" assumes "\<forall>ops\<in>set \<psi>. \<exists>op. ops = [op] \<and> is_valid_operator_sas_plus \<Psi> op"   and "G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I \<psi>" shows "G \<subseteq>\<^sub>m execute_serial_plan_sas_plus I (concat \<psi>)"
diff_conv_add_uminus: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "minus a b = plus a (uminus b)"
content_ff_to_fract_coeffs_to_fract: fixes p :: "'b fract poly" assumes "content_ff p \<in> range to_fract" shows "set (coeffs p) \<subseteq> range to_fract"
exec_step_ind_Putstatic_Init: fixes D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and D :: "char list"   and F :: "char list"   and C :: "char list"   and sh :: "sheap"   and ics :: "init_call_status"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "(D', b, t) = field P D F"   and "P \<turnstile> C has F,Static:t in D"   and "\<forall>sfs. sh D' \<noteq> \<lfloor>(sfs, Done)\<rfloor>"   and "\<forall>Cs. ics \<noteq> Called Cs" shows "exec_step_ind (StepI (Putstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Calling D' []) # frs, sh)"
simps: fixes g :: "('b, 'c) generator"   and s :: "'c" shows "list.unfoldr g s = (if list.has_next g s then let (a, s') = list.next g s in a # list.unfoldr g s' else [])"
accept\<^sub>G\<^sub>R_LTS_I: fixes \<delta> :: "('d \<times> 'e \<times> 'd) set"   and q\<^sub>0 :: "'d"   and F :: "('d \<times> 'e \<times> 'd) set"   and \<I> :: "('d \<times> 'e \<times> 'd) set set"   and w :: "nat \<Rightarrow> 'e"   and \<alpha> :: "(('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set set) set" assumes "accepting_pair\<^sub>G\<^sub>R_LTS \<delta> q\<^sub>0 (F, \<I>) w"   and "(F, \<I>) \<in> \<alpha>" shows "accept\<^sub>G\<^sub>R_LTS (\<delta>, q\<^sub>0, \<alpha>) w"
assms: shows "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = x"
ref_two_step: fixes A :: "'b nres"   and R :: "('b \<times> 'c) set"   and B :: "'c nres"   and C :: "'c nres" assumes "A \<le> \<Down> R B"   and "B \<le> C" shows "A \<le> \<Down> R C"
monofun_cfun: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "f \<sqsubseteq> g"   and "x \<sqsubseteq> y" shows "f\<cdot>x \<sqsubseteq> g\<cdot>y"
flow_initial_time: fixes T :: "real set"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and X :: "'d set"   and t0 :: "real"   and x0 :: "'d" assumes "ll_on_open_it T f X"   and "t0 \<in> T"   and "x0 \<in> X" shows "ll_on_open.flow T f X t0 x0 t0 = x0"
mult_single: fixes k :: "'c"   and a :: "'d"   and l :: "'c"   and b :: "'d" shows "monomial a k * monomial b l = monomial (a * b) (k + l)"
assms_2: fixes x :: "complex" assumes "x \<in> A" shows "conv_abscissa f < ereal (Re (g x))"
less_Assertion_def: fixes p :: "'c Assertion"   and q :: "'c Assertion" shows "(p < q) = ({\<cdot> p } < {\<cdot> q })"
topspace_def: fixes T :: "'d topology" shows "topspace T = \<Union> {S. openin T S}"
Re_Arctan_bounds: fixes z :: "complex" assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1" shows "\<bar>Re (Arctan z)\<bar> < pi / 2"
t_stop_pos: assumes "v \<noteq> 0" shows "0 < t_stop"
\<pi>0': shows "(\<pi> @\<^bsup>l\<^esup> \<pi>l) 0 = \<pi>' 0"
W2_nneg: fixes I :: "nat set" assumes "I \<in> P" shows "0 \<le> W cf2 (F I)"
real_pair_basis: shows "Basis = {(1, 0), (0, 1)}"
ide_char: fixes a :: "'c" shows "C.ide a = (C.arr a \<and> C.dom a = a \<and> C.cod a = a)"
diff_Suc_diff_eq2: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "Suc (j - k) - i = Suc j - (k + i)"
Compl_iff: fixes c :: "'k"   and A :: "'k set" shows "(c \<in> - A) = (c \<notin> A)"
continuous_on_If: fixes s :: "'b set"   and t :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t g"   and "\<And>x. \<lbrakk>x \<in> s; \<not> P x\<rbrakk> \<Longrightarrow> f x = g x"   and "\<And>x. \<lbrakk>x \<in> t; P x\<rbrakk> \<Longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
IH: assumes "i < card {a. a < m \<and> a \<in> I}" shows "pick I i = pick {a. a < m \<and> a \<in> I} i"
\<eta>o_def: fixes y :: "'b" shows "Adj.\<eta>o y = \<phi> y (F y)"
listrel_trans: fixes r :: "('a \<times> 'a) set" assumes "trans r" shows "trans (listrel r)"
AssertionStatImp_def: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b" shows "AssertionStatImp \<Psi> \<Psi>' \<equiv> \<forall>\<Phi>. \<Psi> \<turnstile> \<Phi> \<longrightarrow> \<Psi>' \<turnstile> \<Phi>"
A: shows "\<forall>r>c. \<forall>\<^sub>F x in F. ereal r < f x"
A_def: shows "A = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {0<..(int p - 1) div 2} \<and> x mod int q \<in> {(int q - 1) div 2<..int q - 1}}"
order_gt_0_iff_finite: shows "(0 < order \<G>) = finite (carrier \<G>)"
set_fold_cfi_def: shows "set_fold_cfi \<equiv> map_fun comp_fun_idem_apply id Finite_Set.fold"
impossible_Cons: fixes xs :: "'e list"   and ys :: "'e list"   and x :: "'e" assumes "length xs \<le> length ys" shows "(xs = x # ys) = False"
simps_3: fixes x2 :: "'b" shows "Some x2 \<noteq> None"
completeness: fixes e :: "exp"   and v :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "e \<longrightarrow>* v"   and "is_val v" shows "\<exists>v'. v' \<in> E e \<rho> \<and> v' \<in> E v []"
prod_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'e" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq zero (f i) \<and> less_eq (f i) (g i)" shows "less_eq (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
decisionD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and X :: "final_decision" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "s = Decision X" shows "t = Decision X"
non_atomic_Pair: fixes x :: "msg"   and y :: "msg" shows "composed \<langle>x, y\<rangle>"
eval_Call: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and a' :: "val"   and s1 :: "state"   and ps :: "expr list"   and pvs :: "val list"   and s2 :: "state"   and D :: "qtname"   and mode :: "inv_mode"   and statT :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list"   and s3 :: "state"   and s3' :: "state"   and accC :: "qtname"   and v :: "val"   and s4 :: "state"   and s4' :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>ps\<doteq>\<succ>pvs\<rightarrow> s2"   and "D = invocation_declclass G mode (snd s2) a' statT \<lparr>name = mn, parTs = pTs\<rparr>"   and "s3 = init_lvars G D \<lparr>name = mn, parTs = pTs\<rparr> mode a' pvs s2"   and "s3' = check_method_access G accC statT mode \<lparr>name = mn, parTs = pTs\<rparr> a' s3"   and "G\<turnstile>s3' \<midarrow>Methd D \<lparr>name = mn, parTs = pTs\<rparr>-\<succ>v\<rightarrow> s4"   and "s4' = set_lvars (locals (snd s2)) s4" shows "G\<turnstile>Norm s0 \<midarrow>{accC,statT,mode}e\<cdot>mn( {pTs}ps)-\<succ>v\<rightarrow> s4'"
adjoint_pair_antipar_4: fixes f :: "('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr" assumes "S.adjoint_pair f g" shows "S.src g = S.trg f"
le_enorm: fixes i :: "nat"   and x :: "nat \<Rightarrow> real" assumes "i < n" shows "\<bar>x i\<bar> \<le> enorm x"
Equality: fixes S :: "('fun, 'var) strand_step list"   and \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and uu :: "poscheckvariant"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "simple S"   and "Some \<delta> = mgu t t'" shows "(S @ \<langle>uu: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S', \<theta>) \<leadsto> (S @ S' \<cdot>\<^sub>s\<^sub>t \<delta>, \<theta> \<circ>\<^sub>s \<delta>)"
continuous_at_split: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f x = (continuous (at_left x) f \<and> continuous (at_right x) f)"
preserves_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>F f : F a \<rightarrow>\<^sub>C F b\<guillemotright>"
ModPon_imp_Pf: fixes e :: "(name, hf) finfun"   and x :: "tm"   and y :: "tm" assumes "Pf \<lbrakk>Q_Imp x y\<rbrakk>e"   and "Pf \<lbrakk>x\<rbrakk>e" shows "Pf \<lbrakk>y\<rbrakk>e"
frameImpE: fixes A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b"   and \<Phi> :: "'c" assumes "(\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) \<turnstile>\<^sub>F \<Phi>"   and "A\<^sub>F \<sharp>* \<Phi>" shows "\<Psi>\<^sub>F \<turnstile> \<Phi>"
r_zero: fixes x :: "'a list" assumes "x \<in> carrier (poly_ring R)" shows "x \<oplus>\<^bsub>poly_ring R\<^esub> \<zero>\<^bsub>poly_ring R\<^esub> = x"
insertI1: fixes a :: "'a"   and B :: "'a set" shows "a \<in> a \<triangleright> B"
aezfun_setspan_def: fixes A :: "'d set" shows "aezfun_setspan A = {x. supp (aezfun x) \<subseteq> A}"
strict_boundedE: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a <s signed.min b c" obtains "a <s b" and "a <s c"
to_nat_eq: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'c \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "(mod_type.to_nat Rep x = mod_type.to_nat Rep y) = (x = y)"
distr_reorder: fixes t :: "'b \<Rightarrow> 'i"   and J :: "'b set"   and K :: "'i set" assumes "inj_on t J"   and "t \<in> J \<rightarrow> K"   and "finite K" shows "distr (Pi\<^sub>M K M) (Pi\<^sub>M J (\<lambda>x. M (t x))) (\<lambda>\<omega>. \<lambda>n\<in>J. \<omega> (t n)) = Pi\<^sub>M J (\<lambda>x. M (t x))"
sup_mono: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b" shows "c \<sqinter> d \<le> a \<sqinter> b"
waux2: fixes xs :: "('a, 'b) Combinators list" assumes "DenyAll \<notin> set xs" shows "wellformed_policy1 xs"
linear: fixes x :: "'g"   and y :: "'g" shows "x \<le> y \<or> y \<le> x"
assms_1: fixes i :: "'a" assumes "i \<in> Basis - {j}" shows "(\<lambda>x. f x \<bullet> i) C1_differentiable_on s"
domain_subset_carrier: fixes c :: "('a, 'b) chart" assumes "c \<in> charts" shows "domain c \<subseteq> carrier"
l'_in_v: shows "l' \<in> LID\<^sub>V v"
comp_reduce: fixes f :: "'c \<times> 'c"   and g :: "'c \<times> 'c"   and k :: "'c \<times> 'c"   and h :: "'c \<times> 'c" assumes "C.VV.comp f g = k"   and "C.VV.seq f g"   and "C.VV.seq g h" shows "C.VV.comp f (C.VV.comp g h) = C.VV.comp k h"
mult_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d"   and "bot \<le> b"   and "bot \<le> c" shows "a * c \<le> b * d"
star_zero_def: shows "0 \<equiv> star_of (0::'b)"
berlekamp_mat_i: shows "(poly_rel ===> mat_rel R) (berlekamp_mat_i p ff_ops) berlekamp_mat"
sep_disj_fun_def: fixes m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b" shows "m1 ## m2 \<equiv> domain m1 \<inter> domain m2 = {}"
rep_eq: fixes x :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> 'f"   and xa :: "('g, 'e) uexpr"   and xb :: "('h, 'e) uexpr"   and xc :: "('i, 'e) uexpr" shows "\<lbrakk>trop x xa xb xc\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b) (\<lbrakk>xc\<rbrakk>\<^sub>e b))"
Simulation_Invariants_axioms: shows "Simulation_Invariants A B (\<sim>) PA QA PB QB"
euclidean_domainI: fixes \<phi> :: "'a list \<Rightarrow> nat" assumes "\<And>a b. \<lbrakk>a \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}; b \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}\<rbrakk> \<Longrightarrow> \<exists>q r. q \<in> carrier (K [X]) \<and> r \<in> carrier (K [X]) \<and> a = b \<otimes>\<^bsub>K [X]\<^esub> q \<oplus>\<^bsub>K [X]\<^esub> r \<and> (r = \<zero>\<^bsub>K [X]\<^esub> \<or> \<phi> r < \<phi> b)" shows "euclidean_domain (K [X]) \<phi>"
theorem3: shows "All admits_triangular_reduction = (\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'b})"
cobounded2: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "semilattice_order f less_eq less" shows "less_eq (f a b) b"
intro: fixes Inf :: "'a set \<Rightarrow> 'a"   and U :: "'a set" assumes "Inf ` Pow U \<subseteq> U" shows "Inf_ow U Inf"
AF_def: fixes p :: "'b set" shows "\<^bold>A\<^bold>F p = - \<^bold>E\<^bold>G - p"
M: shows "\<forall>(u', v')\<in>edges rg \<inter> S (A Q \<pi>) \<times> - S (A Q \<pi>). w {v, u} \<le> w {u', v'}"
reindex_product_bij: fixes a :: "nat"   and k :: "nat"   and m :: "nat"   and b :: "nat" assumes "coprime a k" shows "bij_betw (\<lambda>(d1, d2). d1 * d2) {(d1, d2). d1 dvd gcd a m \<and> d2 dvd gcd k b} {d. d dvd gcd a m * gcd k b}"
finite_char: fixes C :: "'a set set" shows "finite_char (mk_finite_char C)"
prems: fixes xs :: "'a list" assumes "rtrancl_path r x_ xs z_" shows "thesis"
less_mreal_def: fixes x :: "mreal"   and y :: "mreal" shows "x < y \<equiv> x \<le> y \<and> x \<noteq> y"
root'': shows "root \<midarrow>x\<rightarrow> root''"
1: shows "[\<lbrace>a\<^sup>P,\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P\<rparr>\<rbrace> in v]"
map_1: fixes f :: "'c \<Rightarrow> 'b" shows "map f [] = []"
less_numeral_extra_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
refine: fixes n :: "nat"   and ai :: "'b list list"   and a :: "'c set set" assumes "DIM_precond TYPE('c) n"   and "(ai, a) \<in> \<langle>appr_rel\<rangle>list_wset_rel" shows "(RETURN (ivl_rep_of_sets_impl n ai), op_ivl_rep_of_sets a) \<in> \<langle>lv_rel \<times>\<^sub>r lv_rel\<rangle>nres_rel"
perm_comp: fixes f :: "atom \<Rightarrow> atom"   and g :: "atom \<Rightarrow> atom" assumes "f \<in> perm"   and "g \<in> perm" shows "f \<circ> g \<in> perm"
setsum_valid_index_split: fixes f :: "nat list \<Rightarrow> 'a"   and ds1 :: "nat list"   and ds2 :: "nat list" shows "(\<Sum>is | is \<lhd> ds1 @ ds2. f is) = (\<Sum>is1 | is1 \<lhd> ds1. \<Sum>is2 | is2 \<lhd> ds2. f (is1 @ is2))"
modulo_gauss_int_def: fixes a :: "gauss_int"   and b :: "gauss_int" shows "a mod b = a - a div b * b"
bex_disj_distrib: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>A. P x \<or> Q x) = ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>A. Q x))"
assms: shows "d0 = lderiv0 \<and> lformula \<phi> \<or> d0 = rderiv0"
a_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus> y = y \<oplus> x"
normalize_canceled: fixes l :: "(bool \<times> 'd) list"   and l' :: "(bool \<times> 'd) list" assumes "cancels_to l l'" shows "Cancelation.normalize l = Cancelation.normalize l'"
horizontal_composition_axioms: shows "horizontal_composition (\<cdot>\<^sub>B) (\<star>\<^sub>B) src\<^sub>B trg\<^sub>B"
member_remove: fixes x :: "'b"   and xs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "DList_Set.member (DList_Set.remove x xs) = (DList_Set.member xs)(x := False)"
refl_term: fixes s :: "trm" shows "s \<longlonglongrightarrow>\<^sup>* s"