hcju/isabelleps huggingface dataset:
cases: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and a1 :: "('a, 'i, 'j) state"   and a2 :: "('f, 'i, 'j) state" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"   and "std_inca_simulation.match Fstd_get Finca_get \<DD>\<ee>\<II>\<nn>\<ll> a1 a2" obtains F1 :: "'a" and F2 :: "'f" and H :: "'i" and st :: "'j list" where "a1 = State F1 H st" and "a2 = State F2 H st" and "std_inca_simulation.rel_fundefs \<DD>\<ee>\<II>\<nn>\<ll> (Fstd_get F1) (Finca_get F2)"
min_simps_2: fixes xs :: "('c, 'd) bintree option list" shows "Binomial_Queue.min (None # xs) = Binomial_Queue.min xs"
hyps_3: assumes "P \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   and "iconfs (shp s2) es2"   and "P,shp s2 \<turnstile>\<^sub>b (es2,b2) \<surd>" shows "P \<turnstile> \<langle>es2,s2\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
Scheduler_L2: fixes C :: "Config"   and activation :: "nat \<Rightarrow> nat list"   and t :: "nat" assumes "Scheduler C activation"   and "\<not> t mod cycleLength C mem schedule C" shows "activation t = []"
a_set_child_nodes_def: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "a_set_child_nodes ptr children = invoke a_set_child_nodes_tups ptr children"
val_Zp_ultrametric: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "min (val_Zp x) (val_Zp y) \<le> val_Zp (x \<oplus> y)"
keys_MP_oalist: fixes xs :: "('a, 'b) oalist_ntm" shows "keys (MP_oalist xs) = set (map fst (fst (list_of_oalist_ntm xs)))"
CFG_call_edge_params: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and i :: "nat"   and ins :: "'var list"   and outs :: "'var list"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "i < length ins"   and "(p, ins, outs) \<in> set procs"   and "pred (kind a) s"   and "pred (kind a) s'"   and "\<forall>V\<in>ParamUses (sourcenode a) ! i. state_val s V = state_val s' V" shows "params fs (state_val s) ! i = params fs (state_val s') ! i"
\<G>\<F>_\<F>_subset: fixes \<phi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set" shows "\<G>\<F> \<phi> w \<subseteq> \<F> \<phi> w"
Sum_any_left_distrib: fixes g :: "'b \<Rightarrow> 'c"   and r :: "'c" assumes "finite {a. g a \<noteq> (0::'c)}" shows "Sum_any g * r = (\<Sum>n. g n * r)"
comm_add_exp: fixes u :: "'b list"   and v :: "'b list"   and n :: "nat" assumes "u \<cdot> v = v \<cdot> u" shows "u \<^sup>@ n \<cdot> v = v \<cdot> u \<^sup>@ n"
better_LNot_intro: fixes y :: "name"   and N :: "trm"   and a :: "coname"   and M :: "trm"   and x :: "name"   and b :: "coname" assumes "y \<sharp> N"   and "a \<sharp> M" shows "Cut <a>.NotR x.M a y.NotL <b>.N y \<longrightarrow>\<^sub>l Cut <b>.N x.M"
ideal_decomp_aux_def: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "ideal_decomp_aux F f = (let J = ideal F; L = (J \<div> f) \<inter> P[X]; L' = lpp ` punit.reduced_GB L in ((*) f ` normal_form L ` P[X], map (apfst ((*) f)) (snd (split 0 X L'))))"
linorder_neq_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<noteq> y) = (x < y \<or> y < x)"
get_min: fixes t :: "('b \<times> nat) tree" assumes "Leftist_Heap.linorder_class.heap t"   and "t \<noteq> \<langle>\<rangle>" shows "get_min t = Min (Tree2.set_tree t)"
standard_decomp_snd_split: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "S \<subseteq> .[X]"   and "t \<in> .[X]" shows "standard_decomp (deg_pm t) (snd (split t U S))"
choice_iff: fixes Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<forall>x. \<exists>y. Q x y) = (\<exists>f. \<forall>x. Q x (f x))"
c_unfold_len: fixes k :: "nat" shows "\<forall>u. length (c_unfold k u) = k"
semiring_normalization_rules_24: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "plus a c = plus c a"
step_preserves_le: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "(char list \<Rightarrow> 'av) option"   and c :: "(char list \<Rightarrow> int) set acom"   and c' :: "(char list \<Rightarrow> 'av) option acom" assumes "S \<subseteq> \<gamma>\<^sub>o S'"   and "c \<le> \<gamma>\<^sub>c c'" shows "step S c \<le> \<gamma>\<^sub>c (step' S' c')"
lexord_Nil_left: fixes y :: "'g list"   and r :: "('g \<times> 'g) set" shows "(([], y) \<in> lexord r) = (\<exists>a x. y = a # x)"
lexext_simps_2: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "\<not> lexext gt [] xs"
lookup: fixes x :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup x \<in> {f. finite {x. f x \<noteq> (0::'b)}}"
l_set_disconnected_nodes_get_tag_name_axioms: shows "l_set_disconnected_nodes_get_tag_name ShadowRootClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs"
set_disconnected_nodes_is_l_set_disconnected_nodes: shows "l_set_disconnected_nodes ShadowRootClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs"
isomorphicI': fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.iso_in_hom f a b" shows "VV.isomorphic a b"
interleave_comm: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<in> ys \<otimes> zs" shows "xs \<in> zs \<otimes> ys"
snd_splitI: fixes U :: "'x set"   and S :: "('x \<Rightarrow>\<^sub>0 nat) set"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "finite X"   and "U \<subseteq> X"   and "finite S"   and "0 \<notin> S" obtains V :: "'x set" where "V \<subseteq> U" and "(monomial (1::'b) t, V) \<in> set (snd (split t U S))"
vars_s: shows "vars_mset s \<subseteq># vars_mset t"
RootState_notmem_Range_ChiRel: fixes ST :: "('d, 'e, 'f) status" shows "RootState ST \<notin> Range (ChiRel (HA ST))"
lead_coeff_not_zero: fixes K :: "'a list set"   and a :: "'a list"   and p :: "'a list list" assumes "polynomial\<^bsub>poly_ring R\<^esub> K (a # p)" shows "a \<in> K - {\<zero>\<^bsub>poly_ring R\<^esub>}"
not_arg_cong_Inr: fixes x :: "'ag"   and y :: "'ag" assumes "x \<noteq> y" shows "Inr x \<noteq> Inr y"
1: shows "((\<lambda>x. f (g1 x) * vector_derivative g1 (at x)) has_integral i1) {0..1}"
not_all_zero: shows "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray A)"
transpose_transpose: fixes A :: "'a Matrix.mat" shows "A\<^sup>t\<^sup>t = A"
times_subset_iff: fixes A :: "'e set"   and C :: "'f set"   and B :: "'e set"   and D :: "'f set" shows "(A \<times> C \<subseteq> B \<times> D) = (A = {} \<or> C = {} \<or> A \<subseteq> B \<and> C \<subseteq> D)"
intros_2: fixes a :: "'a"   and \<alpha> :: "'b"   and b :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss :: "('b \<times> 'a) list" assumes "(a, \<alpha>, b) \<in> ars"   and "(b, ss) \<in> seq ars" shows "(a, (\<alpha>, b) # ss) \<in> seq ars"
lm114: fixes l :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<exists>n\<in>{0..<length l}. P (l ! n)" shows "concat (map (\<lambda>n. if P (l ! n) then [n] else []) [0..<length l]) \<noteq> []"
basis_left_null_space_iarrays_def: fixes A :: "'b iarray iarray" shows "basis_left_null_space_iarrays A = (let GJ = Gauss_Jordan_iarrays_PA A; rank_A = length (filter (\<lambda>x. \<not> is_zero_iarray x) (IArray.list_of (snd GJ))) in set (map (\<lambda>i. row_iarray i (fst GJ)) [rank_A..<nrows_iarray A]))"
extend_measure_caratheodory_pair: fixes M :: "'b measure"   and \<Omega> :: "'b set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and G :: "'c \<Rightarrow> 'd \<Rightarrow> 'b set"   and \<mu> :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal"   and i :: "'c"   and j :: "'d" assumes "M = extend_measure \<Omega> {(a, b). P a b} (\<lambda>(a, b). G a b) (\<lambda>(a, b). \<mu> a b)"   and "P i j"   and "semiring_of_sets \<Omega> {G a b |a b. P a b}"   and "\<And>i j. \<lbrakk>P i j; G i j = {}\<rbrakk> \<Longrightarrow> \<mu> i j = 0"   and "\<And>i j k l. \<lbrakk>P i j; P k l; G i j = G k l\<rbrakk> \<Longrightarrow> \<mu> i j = \<mu> k l"   and "\<And>i j. P i j \<Longrightarrow> 0 \<le> \<mu> i j"   and "\<And>A B j k. \<lbrakk>\<And>n. P (A n) (B n); P j k; disjoint_family (\<lambda>n. G (A n) (B n)); (\<Union>i. G (A i) (B i)) = G j k\<rbrakk> \<Longrightarrow> (\<Sum>n. \<mu> (A n) (B n)) = \<mu> j k" shows "emeasure M (G i j) = \<mu> i j"
not_and_lt: 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 D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "\<not> (Tarski_neutral_dimensionless.Lt Bet Cong A B C D \<and> Tarski_neutral_dimensionless.Lt Bet Cong C D A B)"
LeastI_ex: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"
adjoint_pair_def: fixes f :: "'a"   and g :: "'a" shows "adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
SUP_emeasure_spmf_neq_top: fixes A :: "'c set"   and Y :: "'c spmf set" shows "(\<Squnion>p\<in>Y. emeasure (measure_spmf p) A) \<noteq> \<top>"
fresh_star_def_1: fixes xs :: "'a list"   and c :: "'b" shows "xs \<sharp>* c \<equiv> \<forall>x\<in>set xs. x \<sharp> c"
preprm_unit_inaction: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<noteq> a"   and "x \<noteq> b" shows "preprm_apply (preprm_unit a b) x = x"
cr_vfsequence_def: fixes a :: "V"   and b :: "V list" shows "cr_vfsequence a b = (a = vfsequence_of_vlist b)"
cs_after_all_prerecording_events: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat" assumes "trace init t final"   and "\<forall>i'\<ge>i. \<not> prerecording_event t i'"   and "\<forall>j'<i. \<not> postrecording_event t j'"   and "i \<le> length t" shows "cs_equal_to_snapshot (S t i) final"
HEndPhase0_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
rtrancl_trancl_absorb: fixes R :: "('b \<times> 'b) set" shows "(R\<^sup>*)\<^sup>+ = R\<^sup>*"
DBDeq_visible: shows "\<turnstile> <DBDeq>_(inp, mid, out, q1, q2) = DBDeq"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Left_a.in_hom f a b = (Left_a.arr a \<and> Left_a.arr b \<and> Left_a.arr f \<and> \<guillemotleft>f : a \<Rightarrow> b\<guillemotright>)"
term_order_rel'_alt_def_lexord: fixes x :: "char list list"   and y :: "char list list" shows "term_order_rel' x y = ord_class.lexordp x y"
eq_fs: fixes x :: "nat" assumes "x < m" shows "gs2.gso x = gs1.gso x"
lemma_2_5_9_a: fixes a :: "'a"   and b :: "'a" shows "a \<le> b l\<rightarrow> a"
IH: fixes gpv :: "('e, 'a, 'c) gpv" shows "expectation_gpv' (left_gpv gpv) = expectation_gpv'' gpv"
POR_simps_1: fixes y :: "ValD" shows "(ValTT por y) = ValTT"
inf_top_left: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and top :: "'e"   and x :: "'e" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf top x = x"
simps_2: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm"   and at :: "('c, 'd) atm" shows "substL \<pi> (Neg at) = Neg (substA \<pi> at)"
baseClasses_repeats_or_shares: fixes D :: "char list"   and Bs :: "base list" assumes "D \<in> baseClasses Bs" shows "Repeats D \<in> set Bs \<or> Shares D \<in> set Bs"
callee_invariant_on_alt_def: shows "callee_invariant_on = (\<lambda>callee I \<I>. (\<forall>s\<in>Collect I. \<forall>x\<in>outs_\<I> \<I>. \<forall>(y, s')\<in>set_spmf (callee s x). I s') \<and> (\<forall>s\<in>Collect I. \<I> \<turnstile>c callee s \<surd>))"
hyps_6: shows "eucl_of_list ` take DIM('a) ` env \<subseteq> res"
walk_array_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c array"   and x :: "'b" shows "walk_array f a x = walk_array' f a x (array_length a) 0"
nneg_def: fixes P :: "'b \<Rightarrow> 'c" shows "nneg P = (\<forall>x. (0::'c) \<le> P x)"
rep_eq: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and x :: "'l word" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero x = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('l) (Word.rep x)))"
strict_trans2: fixes b :: "'b multiset"   and a :: "'b multiset"   and c :: "'b multiset" assumes "b \<subset># a"   and "c \<subseteq># b" shows "c \<subset># a"
effect_of_listE: fixes xs :: "'a list"   and h :: "heap"   and h' :: "heap"   and r :: "'a array" assumes "effect (Array.of_list xs) h h' r" obtains "r = fst (Array.alloc xs h)" and "h' = snd (Array.alloc xs h)" and "Array.get h' r = xs" and "Array.present h' r" and "\<not> Array.present h r"
c_A'_def: fixes n :: "nat" shows "c_A' n = int (index (swaps (paid_A' ! n) (s_A' n)) (qs ! n) + 1)"
PInit_def: fixes rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds" shows "PInit rs p \<equiv> rs!p = #NotAResult"
execute_operator_sas_plus_effect_i: fixes s :: "'c \<Rightarrow> 'd option"   and op :: "('c, 'd) sas_plus_operator"   and v :: "'c"   and a :: "'d" assumes "is_operator_applicable_in s op"   and "\<forall>(v, a)\<in>set (effect_of op). \<forall>(v', a')\<in>set (effect_of op). v \<noteq> v' \<or> a = a'"   and "(v, a) \<in> set (effect_of op)" shows "(s \<then>\<^sub>+ op) v = Some a"
HBL1_PP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "bprv (PP \<langle>\<phi>\<rangle>)"
ArityAnalysisHeapSafe_axioms: shows "ArityAnalysisHeapSafe Aexp Aheap"
append_exec_frags_is_exec_frag: fixes A :: "('b, 'c) ioa"   and e :: "'b \<times> ('c \<times> 'b) list"   and e' :: "'b \<times> ('c \<times> 'b) list" assumes "is_exec_frag_of A e"   and "last_state e = fst e'"   and "is_exec_frag_of A e'" shows "is_exec_frag_of A (append_exec e e')"
infsetsum_mono_neutral: fixes f :: "'d \<Rightarrow> real"   and A :: "'d set"   and g :: "'d \<Rightarrow> real"   and B :: "'d set" assumes "f abs_summable_on A"   and "g abs_summable_on B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<le> g x"   and "\<And>x. x \<in> A - B \<Longrightarrow> f x \<le> 0"   and "\<And>x. x \<in> B - A \<Longrightarrow> 0 \<le> g x" shows "infsetsum f A \<le> infsetsum g B"
le_of_nat_term_order_def: fixes to :: "'a nat_term_order" shows "le_of_nat_term_order to = le_of_key_order (key_order_of_nat_term_order to)"
sum_upto_moebius_mu_integrable': fixes a :: "real"   and b :: "real" assumes "1 \<le> a" shows "A integrable_on {a..b}"
distinct_41: fixes x61 :: "poscheckvariant"   and x62 :: "('c, 'd) Term.term"   and x63 :: "('c, 'd) Term.term"   and x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "\<langle>x61: x62 \<in> x63\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
fcompose_def: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" shows "fcompose p q r = fst (fold_coeffs (\<lambda>a (c, d). (d * [:a:] + q * c, r * d)) p (0, 1))"
eq_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(a = divide b c) = (if c \<noteq> zero then times a c = b else a = zero)"
single_nz_zero_singleton: fixes v :: "'a" shows "single_nz_val 1 0 v = [v]\<^sub>v"
assms_5: shows "h \<turnstile> get_element_by_id ptr idd \<rightarrow>\<^sub>h h'"
cop_perp2__col: fixes A :: "'p"   and B :: "'p"   and Y :: "'p"   and Z :: "'p"   and X :: "'p" assumes "Coplanar A B Y Z"   and "X Y Perp A B"   and "X Z Perp A B" shows "Col X Y Z"
cases: fixes x :: "('c \<times> nat) tree" obtains "x = \<langle>\<rangle>"   | uu_ :: "('c \<times> nat) tree" and uv_ :: "'c" and lv :: "nat" and uw_ :: "('c \<times> nat) tree" where "x = \<langle>uu_, (uv_, lv), uw_\<rangle>"
poly_eqI: fixes p :: "linear_poly"   and q :: "linear_poly" assumes "\<And>v. coeff p v = coeff q v" shows "p = q"
axioms_1: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_sgrp \<alpha> \<SS>" shows "\<Z>_sgrp_basis \<alpha> \<SS>"
L4_2a: fixes \<sigma> :: "nat \<Rightarrow> 'a node"   and \<mu> :: "'a ltlr"   and \<eta> :: "'a ltlr" assumes "ipath gba.E \<sigma>"   and "\<mu> U\<^sub>r \<eta> \<in> old (\<sigma> 0)" shows "(\<forall>i. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i) \<and> \<eta> \<notin> old (\<sigma> i)) \<or> (\<exists>j. (\<forall>i<j. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> i)) \<and> \<eta> \<in> old (\<sigma> j))"
null_comp_def: fixes a :: "ref \<Rightarrow> ref"   and b :: "ref \<Rightarrow> ref" shows "a \<propto> b = (\<lambda>p. if b p = Null then Null else (a \<circ> b) p)"
assms: shows "y * x \<le> x * y\<^sup>\<circ>"
Sigma_mset_plus_distrib1: fixes A :: "'c multiset"   and B :: "'c multiset"   and C :: "'c \<Rightarrow> 'd multiset" shows "Sigma_mset (A + B) C = Sigma_mset A C + Sigma_mset B C"
step_less: fixes t :: "nat \<Rightarrow> nat"   and x :: "nat" assumes "t \<in> set ts"   and "x\<^sub>1 \<le> x" shows "t x < x"
transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "pcr_mapping A B Map.empty Mapping.empty"
P1: shows "deg R (p \<ominus>\<^bsub>P\<^esub> q) \<le> deg R p"
simps: fixes a :: "'node SDG_node"   and n :: "'node SDG_node" shows "(a \<in> sum_SDG_slice2 n) = (a = n \<and> valid_SDG_node n \<or> (\<exists>n'' n'. a = n'' \<and> n'' s\<longrightarrow>\<^bsub>cd\<^esub> n' \<and> n' \<in> sum_SDG_slice2 n) \<or> (\<exists>n'' V n'. a = n'' \<and> n'' s-V\<rightarrow>\<^sub>d\<^sub>d n' \<and> n' \<in> sum_SDG_slice2 n) \<or> (\<exists>n'' p n'. a = n'' \<and> n'' s-p\<rightarrow>\<^bsub>ret\<^esub> n' \<and> n' \<in> sum_SDG_slice2 n) \<or> (\<exists>n'' p V n'. a = n'' \<and> n'' s-p:V\<rightarrow>\<^bsub>out\<^esub> n' \<and> n' \<in> sum_SDG_slice2 n) \<or> (\<exists>n'' p n'. a = n'' \<and> n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n' \<and> n' \<in> sum_SDG_slice2 n))"
0: shows "rawpsubst (rawpsubst (subst \<phi> (Var u) x) (zip (map Var us) xs)) (zip ts us) = rawpsubst (rawpsubst (subst \<phi> (Var u) x) (zip (map Var vs) xs)) (zip ts vs)"
Ioi_le_Ico: fixes a :: "'c multiset" shows "subset_mset.greaterThan a \<subseteq> subset_mset.atLeast a"
converse3pD: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c"   and b :: "'b"   and a :: "'a" assumes "converse3p r c b a" shows "r a b c"
EqOnUC_def: fixes p :: "pvar"   and p' :: "pvar"   and u :: "'U"   and c :: "'C" shows "EqOnUC p p' u c \<equiv> Eq (Atom (Last u c) p) (Atom (Last u c) p')"
is_le_relI_1: shows "ord_term_lin.is_le_rel (=)"
d_isotone_var: fixes x :: "'a"   and y :: "'a" shows "d x \<le> d (x \<squnion> y)"
linear_bounded_pos: fixes f :: "'c \<Rightarrow> 'd" assumes "linear f" obtains B :: "real" where "0 < B" and "\<And>x. norm (f x) \<le> B * norm x"
non_comm: shows "u\<^sub>1 \<cdot> u\<^sub>0 \<noteq> u\<^sub>0 \<cdot> u\<^sub>1"
butlast_split: fixes n :: "nat"   and q :: "'a list" assumes "n < length q" shows "butlast q = take n q @ drop n (butlast q)"
is_prefix_sublist: fixes ls :: "'a list"   and vs :: "'a list" assumes "is_prefix ls vs" shows "is_sublist ls vs"
simps_28: fixes x31 :: "'a local.term"   and x32 :: "'a local.term" shows "\<^bold>\<I> \<noteq> x31 \<^bold>\<otimes> x32"
card_UNIV_finite_2_def: shows "card_UNIV_class.card_UNIV = Phantom(Enum.finite_2) 2"
algebraic_inverse_iff: fixes y :: "'a" shows "algebraic (inverse y) = algebraic y"
ipresCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'e"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'f"   and MOD :: "('index, 'bindex, 'varSort, 'g, 'opSym, 'var, 'e, 'f, 'h) model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
EM_remainder'_def: fixes n :: "nat"   and f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" shows "EM_remainder' n f a b = ((- 1) ^ Suc n / fact n) *\<^sub>R integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t)"
numneg: fixes bs :: "real list"   and t :: "Ferrack.num" shows "Inum bs (numneg t) = Inum bs (Neg t)"
fs: shows "fs = map (\<lambda>(f, i). (normalize f, i)) fs'"
ab: fixes h :: "'a set" assumes "h \<in> F" shows "a h \<noteq> (0::'a) \<and> h = {x. a h \<bullet> x \<le> b h}"
rev_predicate1D: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and Q :: "'d \<Rightarrow> bool" assumes "P x"   and "P \<le> Q" shows "Q x"
word_bw_assocs_1: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "(x AND y) AND z = x AND y AND z"
obs_2: shows "exp (- a \<cdot> t) \<le> 1"
set_permutation_bij_restrict1: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "bij_betw f A A" shows "bij (restrict1 f A)"
pmf_set_map: fixes f :: "'b \<Rightarrow> 'c" shows "set_pmf \<circ> map_pmf f = (`) f \<circ> set_pmf"
gec_is_cbox: fixes L :: "'a set" assumes "content (L \<inter> {x. c \<le> x \<bullet> i}) \<noteq> 0"   and "L \<in> \<D>" shows "\<exists>a b. L \<inter> {x. c \<le> x \<bullet> i} = cbox a b"
klift_prop: fixes f :: "'d \<Rightarrow> 'c set"   and X :: "'d set" shows "bd\<^sub>\<F> f X = \<mu> (\<P> f X)"
continuous_real_polymonial_function: fixes f :: "'a \<Rightarrow> real"   and x :: "'a" assumes "real_polynomial_function f" shows "isCont f x"
case_prodI: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"