hcju/isabelleps huggingface dataset:
induct: fixes x1 :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and x2 :: "('e \<Rightarrow> real) set set"   and x3 :: "'d"   and x4 :: "('e \<Rightarrow> real) set"   and x5 :: "'d"   and x6 :: "('e \<Rightarrow> real) set"   and P :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint) \<Rightarrow> ('e \<Rightarrow> real) set set \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> real) set \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> real) set \<Rightarrow> bool" assumes "x1,x2 \<turnstile> \<langle>x3, x4\<rangle> \<leadsto> \<langle>x5, x6\<rangle>"   and "\<And>\<R> X k A R R' l. \<lbrakk>\<R> = {region X I r |I r. valid_region X k I r}; valid_abstraction A X (\<lambda>x. real (k x)); R \<in> \<R>; R' \<in> Succ \<R> R; R \<subseteq> \<lbrace>inv_of A l\<rbrace>; R' \<subseteq> \<lbrace>inv_of A l\<rbrace>\<rbrakk> \<Longrightarrow> P A \<R> l R l R'"   and "\<And>\<R> X k A l g a r l' R. \<lbrakk>\<R> = {region X I r |I r. valid_region X k I r}; valid_abstraction A X (\<lambda>x. real (k x)); A \<turnstile> l \<longrightarrow>\<^bsup>g,a,r\<^esup> l'; R \<in> \<R>\<rbrakk> \<Longrightarrow> P A \<R> l R l' (region_set' (R \<inter> {u. u \<turnstile> g}) r 0 \<inter> {u. u \<turnstile> inv_of A l'})" shows "P x1 x2 x3 x4 x5 x6"
mt0: shows "t0 \<in> preflect t0 ` existence_ivl t0 x0"
Quotient3_ltln\<^sub>P: shows "Quotient3 (\<sim>\<^sub>P) abs_ltln\<^sub>P rep_ltln\<^sub>P"
not_eq_order_implies_strict: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "a \<noteq> b"   and "less_eq a b" shows "less a b"
N_eq: shows "N = Vd (d + 1) \<inter> E `` (Vd d - C)"
eq: shows "map (fst \<circ> (\<lambda>(F, y). ((F, C), y))) fs = map ((\<lambda>F. (F, C)) \<circ> fst) fs"
impl_of_cases: fixes y :: "('c, 'd) rbt" assumes "y \<in> {t. ord.is_rbt cless t \<or> ID ccompare = None}" obtains x :: "('c, 'd) mapping_rbt" where "y = impl_of x"
q_Mem_def: fixes x :: "hf"   and y :: "hf" shows "q_Mem x y \<equiv> \<langle>htuple 0, x, y\<rangle>"
irfd_irbd: shows "fd\<^sup>-\<^sub>\<R> = \<smile> \<circ> bd\<^sup>-\<^sub>\<R>"
rel_refl_strong: fixes x :: "'c list_R1"   and Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>z. z \<in> set_list_R1 x \<Longrightarrow> Ra z z" shows "rel_list_R1 Ra x x"
gcd_greatest_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times a (gcd b c) = (dvd.dvd times a b \<and> dvd.dvd times a c)"
bits_mod_0: 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 a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo zero a = zero"
trg_trg: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "trg\<^sub>D (trg\<^sub>D \<mu>) = trg\<^sub>D \<mu>"
fls_deriv_nth: fixes f :: "'a fls"   and n :: "int" shows "fls_nth (fls_deriv f) n = of_int (n + 1) * fls_nth f (n + 1)"
eval_simps_2: fixes f :: "'c local.term" assumes "Arr f" shows "\<lbrace>Src f\<rbrace> = src \<lbrace>f\<rbrace>"
undirected_nodes_set: fixes G :: "'b list_graph" shows "set (edgesL (FiniteListGraph.undirected G)) = set (edgesL G) \<union> {(e2, e1). (e1, e2) \<in> set (edgesL G)}"
P_def: shows "local.P = (SOME P. {} \<turnstile> P IFF local.H(X0::=\<guillemotleft>P\<guillemotright>))"
rem_effectless_works_5: fixes P :: "'c \<times> ('d, 'e) fmap \<Rightarrow> bool"   and as :: "('c \<times> ('d, 'e) fmap) list" shows "length (filter P (rem_effectless_act as)) \<le> length (filter P as)"
power_strict_increasing: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "nat"   and N :: "nat"   and a :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n < N"   and "less one a" shows "less (power.power one times a n) (power.power one times a N)"
hyps_3: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c\<^sub>1_,Normal s_\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c\<^sub>2_ \<down> Normal s' \<and> \<Gamma>\<turnstile>Language.normalize c\<^sub>2_ \<down> Normal s'"
supt_not_sym: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<rhd> t" shows "(t, s) \<notin> {\<rhd>}"
calcChain: fixes yvec :: "name list"   and xvec :: "name list" assumes "yvec \<sharp>* xvec"   and "length xvec = length yvec"   and "distinct xvec"   and "distinct yvec" shows "[xvec yvec] \<bullet>\<^sub>v xvec = yvec"
lincomb_insert: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "Module.module R M"   and "finite S"   and "S \<subseteq> carrier M"   and "a \<in> S \<union> {v} \<rightarrow> carrier R"   and "v \<notin> S"   and "v \<in> carrier M" shows "module.lincomb M a (S \<union> {v}) = a v \<odot>\<^bsub>M\<^esub> v \<oplus>\<^bsub>M\<^esub> module.lincomb M a S"
privateKey_image_eq: fixes b :: "bool"   and A :: "agent"   and c :: "bool"   and AS :: "agent set" shows "(invKey (publicKey b A) \<in> invKey ` publicKey c ` AS) = (b = c \<and> A \<in> AS)"
case_prodI: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'h"   and b :: "'i" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
assms_2: shows "\<forall>C\<in>H. atom s \<sharp> C"
univalent_def: fixes R :: "('d \<times> 'e) set" shows "univalent R = (\<forall>x y z. (x, y) \<in> R \<and> (x, z) \<in> R \<longrightarrow> z = y)"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and negation_type :: "'c negation_type" shows "h (case negation_type of Pos x \<Rightarrow> f1 x | Neg x \<Rightarrow> f2 x) = (case negation_type of Pos x \<Rightarrow> h (f1 x) | Neg x \<Rightarrow> h (f2 x))"
qEq: shows "pick X #[[y \<and> x]]_xs #= pick X' #[[y \<and> x']]_xs"
fp_cop_F_allocation: fixes ds :: "'b set" shows "inj_on Xd (Pds'.cop ds)"
sgn_inv: fixes dir :: "dir" shows "Grid_Point.sgn (Grid_Point.inv dir) = - Grid_Point.sgn dir"
select_convs_3: fixes statename :: "nat option"   and datastate :: "nat \<Rightarrow>f value option"   and action :: "String.literal \<times> value list"   and "output" :: "value option list"   and more :: "'a" shows "action \<lparr>statename = statename, datastate = datastate, action = action, output = output, \<dots> = more\<rparr> = action"
terminates_while: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and f :: "nat \<Rightarrow> 'e"   and k :: "nat" assumes "\<Gamma>\<turnstile>While b c \<down> Normal (f k)"   and "\<forall>i. \<Gamma>\<turnstile> \<langle>c,Normal (f i)\<rangle> \<Rightarrow> Normal (f (Suc i))" shows "\<exists>i. f i \<notin> b"
g: fixes x :: "'b" assumes "x \<in> T" shows "f (g x) = x"
assms_1: shows "y \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
quantale_closed_unital_maps_iff: fixes f :: "'c \<Rightarrow> 'd" shows "(f \<in> quantale_closed_unital_maps) = (\<forall>x y. f x \<cdot> f y \<le> f (x \<cdot> y) \<and> (1::'d) \<le> f (1::'c))"
cod_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g" shows "partial_magma.cod (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g) = free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C (partial_magma.cod (free_monoidal_category.comp C) f) (partial_magma.cod (free_monoidal_category.comp C) g)"
hyps: assumes "x \<le> y" shows "\<exists>stp. abc_steps_l (length A, xs @ m # (y - x) # x # anything) (A @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]) stp = (length A, xs @ m # y # 0 # anything)"
variety_of_ideal: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "\<V> (ideal F) = \<V> F"
Scoped_Proof_Graph_def: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and inPorts :: "'d \<Rightarrow> 'e fset"   and outPorts :: "'d \<Rightarrow> 'f fset"   and nodeOf :: "'g \<Rightarrow> 'd"   and hyps :: "'d \<Rightarrow> 'f \<Rightarrow> 'e option"   and nodes :: "'d stream"   and vertices :: "'g fset"   and labelsIn :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and labelsOut :: "'d \<Rightarrow> 'f \<Rightarrow> 'b"   and vidx :: "'g \<Rightarrow> nat"   and inst :: "'g \<Rightarrow> 'c"   and edges :: "(('g \<times> 'f) \<times> 'g \<times> 'e) set"   and local_vars :: "'d \<Rightarrow> 'e \<Rightarrow> 'a set" shows "Scoped_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP inPorts outPorts nodeOf hyps nodes vertices labelsIn labelsOut vidx inst edges local_vars \<equiv> (Instantiation inPorts outPorts hyps vertices freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx \<and> Well_Shaped_Graph nodes inPorts outPorts vertices nodeOf edges hyps) \<and> Solution inPorts outPorts nodeOf hyps vertices labelsIn labelsOut freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP vidx inst edges \<and> Well_Scoped_Instantiation freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP inPorts outPorts nodeOf hyps vertices vidx inst edges local_vars"
is_reducible_lit_remove1_Cons_iff: fixes L :: "'a literal"   and C' :: "'a literal list"   and Ds :: "'a literal list list"   and C :: "'a literal list"   and M :: "'a literal" assumes "L \<in> set C'" shows "is_reducible_lit Ds (C @ remove1 L (M # C')) L = is_reducible_lit Ds (M # C @ remove1 L C') L"
greaterThan_less_enum: fixes N :: "nat set"   and x :: "nat"   and i :: "nat" assumes "N \<subseteq> {x<..}"   and "infinite N" shows "x < enum N i"
diff_neq0_2: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "b - a \<noteq> (0::'b)"
lemma_4_3_ii_a: fixes a :: "'a"   and b :: "'a" shows "a \<squnion>2 b = (a r\<rightarrow> b) l\<rightarrow> b"
intro: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and C :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and G :: "'b \<Rightarrow> 'e" assumes "functor A B F"   and "functor B C G" shows "composite_functor A B C F G"
T1: fixes u :: "'a"   and \<omega> :: "'a" assumes "is_left_adjoint u"   and "\<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> r \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> g \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> ((r \<star> \<theta>) \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w)) \<cdot> \<nu> = \<omega>"
fls_prpart_def: shows "fls_prpart \<equiv> map_fun fls_nth id (\<lambda>f. Abs_poly (\<lambda>n. if n = 0 then 0::'a else f (- int n)))"
distr_merge: fixes I :: "'i set"   and J :: "'i set" assumes "I \<inter> J = {}"   and "finite I"   and "finite J" shows "distr (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M) (Pi\<^sub>M (I \<union> J) M) (merge I J) = Pi\<^sub>M (I \<union> J) M"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> VVV.arr f" shows "\<alpha> f = null"
filter_append: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" shows "filter P (xs @ ys) = filter P xs @ filter P ys"
discharge_is_Leaf: fixes a :: "'b tree" shows "(discharge a = Leaf) = (a = Leaf)"
SC1: fixes p :: "('b, 'c) Combinators list" assumes "singleCombinators p" shows "singleCombinators (removeShadowRules1 p)"
simps_2: fixes f :: "'c" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> \<^bold>\<I> = \<^bold>\<r>\<^bold>[\<^bold>\<langle>f\<^bold>\<rangle>\<^bold>]"
simps: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "rebalance_last_tree k ts t = (case last ts of (sub, sep) \<Rightarrow> rebalance_middle_tree k (butlast ts) sub sep [] t)"
R_lin_independentD_all_scalars_nth: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and rs :: "'a list"   and ms :: "'b list"   and k :: "nat" assumes "RModule R smult M"   and "set rs \<subseteq> R"   and "set ms \<subseteq> M"   and "R_scalar_mult.R_lin_independent R smult ms"   and "scalar_mult.lincomb smult rs ms = (0::'b)"   and "k < min (length rs) (length ms)" shows "rs ! k = (0::'a)"
lassocr_swap_sum_inverse: fixes x :: "'h + 'i + 'j" shows "lassocr_swap_sum (lassocr_swap_sum x) = x"
exhaust: fixes y :: "tm" obtains "y = Zero"   | x2 :: "name" where "y = Var x2"   | x31 :: "tm" and x32 :: "tm" where "y = Eats x31 x32"
Domain_Diff_subset: fixes A :: "('b \<times> 'c) set"   and B :: "('b \<times> 'c) set" shows "Domain A - Domain B \<subseteq> Domain (A - B)"
transitive: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "equivalence S E"   and "(a, b) \<in> E"   and "(b, c) \<in> E" shows "(a, c) \<in> E"
SplitGuards_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and c :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<Turnstile>\<^bsub>/F\<^esub> P c\<^sub>2 UNIV,UNIV"   and "(c\<^sub>1 \<inter>\<^sub>g c\<^sub>2) = Some c" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
ring_homomorphism_def: fixes \<eta> :: "'d \<Rightarrow> 'e"   and R :: "'d set"   and addition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and unit :: "'d"   and R' :: "'e set"   and addition' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and multiplication' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero' :: "'e"   and unit' :: "'e" shows "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' \<equiv> (Set_Theory.map \<eta> R R' \<and> ring R addition multiplication zero unit) \<and> ring R' addition' multiplication' zero' unit' \<and> group_homomorphism \<eta> R addition zero R' addition' zero' \<and> monoid_homomorphism \<eta> R multiplication unit R' multiplication' unit'"
Q_Arrow_strong_inversion: fixes x :: "name"   and \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and u :: "trm"   and T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and "x \<sharp> t"   and "x \<sharp> u"   and "\<Gamma> \<turnstile> t \<Leftrightarrow> u : T\<^sub>1\<rightarrow>T\<^sub>2" shows "(x, T\<^sub>1) # \<Gamma> \<turnstile> App t (Var x) \<Leftrightarrow> App u (Var x) : T\<^sub>2"
comp_red_monic_basis_pmdl: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "is_Groebner_basis (set xs)" shows "pmdl (set (comp_red_monic_basis xs)) = pmdl (set xs)"
InvariantCFalseAfterExplain: fixes conflictFlag :: "bool"   and M :: "(Literal \<times> bool) list"   and C :: "Literal list"   and literal :: "Literal"   and reason :: "Literal list"   and C' :: "Literal list" assumes "InvariantCFalse conflictFlag M C"   and "opposite literal el C"   and "isReason reason literal (elements M)"   and "C' = resolve C reason (opposite literal)" shows "InvariantCFalse conflictFlag M C'"
cong: fixes \<Delta> :: "('b, 'c) network_scheme" shows "flow_network.enum_v \<Delta> \<equiv> flow_network.enum_v \<Delta>"
assms: fixes s :: "'a" assumes "s \<in> S" shows "f s = g s"
independent_span_bound: fixes T :: "'b set"   and S :: "'b set" assumes "finite T"   and "independent S"   and "S \<subseteq> span T" shows "finite S \<and> card S \<le> card T"
HPhase0Read_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" shows "HPhase0Read s s' p d = (Phase0Read s s' p d \<and> HNextPart s s')"
inner_simps_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) \<bullet> z = x \<bullet> z + y \<bullet> z"
dma_\<C>_vars: fixes y :: "var\<^sub>C"   and mem\<^sub>1 :: "var\<^sub>C \<Rightarrow> nat"   and mem\<^sub>2 :: "var\<^sub>C \<Rightarrow> nat" assumes "\<forall>x\<in>\<C>_vars\<^sub>C y. mem\<^sub>1 x = mem\<^sub>2 x" shows "dma\<^sub>C mem\<^sub>1 y = dma\<^sub>C mem\<^sub>2 y"
l_new_character_data_get_child_nodes_axioms_def: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_child_nodes_axioms get_child_nodes get_child_nodes_locs \<equiv> (\<forall>ptr' new_character_data_ptr h h' r. ptr' \<noteq> cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr \<longrightarrow> h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr \<longrightarrow> h \<turnstile> new_character_data \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_child_nodes_locs ptr' \<longrightarrow> r h h') \<and> (\<forall>h new_character_data_ptr h'. h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr \<longrightarrow> h \<turnstile> new_character_data \<rightarrow>\<^sub>h h' \<longrightarrow> h' \<turnstile> get_child_nodes (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr) \<rightarrow>\<^sub>r [])"
cspan_butterfly_UNIV: fixes basisA :: "'c set"   and basisB :: "'d set" assumes "cspan basisA = UNIV"   and "cspan basisB = UNIV"   and "is_ortho_set basisB"   and "\<And>b. b \<in> basisB \<Longrightarrow> norm b = 1" shows "cspan {butterfly a b |a b. a \<in> basisA \<and> b \<in> basisB} = UNIV"
sigma_sets_eqI: fixes A :: "'c set set"   and M :: "'c set"   and B :: "'c set set" assumes "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A" shows "sigma_sets M A = sigma_sets M B"
FL_valid_Act_strong: fixes X :: "'b"   and P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('c, 'pred, 'act, 'effect) formula" assumes "finite (supp X)" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>\<alpha>'' x' P'. Act f \<alpha> x = Act f \<alpha>'' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x' \<and> bn \<alpha>'' \<sharp>* X)"
valid_call_list_def: fixes cs :: "'edge list"   and n :: "'node" shows "valid_call_list cs n \<equiv> \<forall>cs' c cs''. cs = cs' @ c # cs'' \<longrightarrow> valid_edge c \<and> (\<exists>Q r p fs. kind c = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<and> p = get_proc (case cs' of [] \<Rightarrow> n | a # list \<Rightarrow> last (sourcenodes cs')))"
\<R>\<^sub>1_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" where "c\<^sub>2 = c\<^sub>1" and "mds\<^sub>2 = mds" and "\<turnstile> \<Gamma>,\<S>,P {c\<^sub>1} \<Gamma>''__,\<S>''__,Pa__" and "tyenv_wellformed mds \<Gamma> \<S> P" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" and "pred P mem\<^sub>1" and "pred P mem\<^sub>2" and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
decomp: fixes ss :: "('c, 'd) Term.term list"   and ts :: "('c, 'd) Term.term list"   and f :: "'c"   and E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "length ss = length ts" shows "UNIF1 Var (add_mset (Fun f ss, Fun f ts) E) (E + mset (zip ss ts))"
eventually_nonneg_ident_nhds: fixes a :: "'a" assumes "(0::'a) < a" shows "eventually_nonneg (nhds a) (\<lambda>x. x)"
simps_2: fixes n :: "nat" shows "B (Suc n) = B n div A n"
addrow_mat_inv: fixes k :: "nat"   and n :: "nat"   and l :: "nat"   and a :: "'a" assumes "k < n"   and "l < n"   and "k \<noteq> l" shows "addrow_mat n a k l * addrow_mat n (- a) k l = 1\<^sub>m n"
flip_out_of_bounds: fixes X :: "bool list"   and y :: "nat" assumes "length X \<le> y" shows "flip y X = X"
subst_of_map_def: fixes d :: "'d \<Rightarrow> 'c"   and \<sigma> :: "'d \<Rightarrow> 'c option"   and x :: "'d" shows "subst_of_map d \<sigma> x = (case \<sigma> x of None \<Rightarrow> d x | Some t \<Rightarrow> t)"
circ_sup_9: fixes circ :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.left_conway_semiring_1 circ one times sup less_eq less bot" shows "circ (sup x y) = times (circ (times (circ x) y)) (circ x)"
add_0_r: fixes a :: "int"   and p :: "int point" shows "local.add a p Infinity = p"
keys_map_key: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "inj f" shows "keys (Poly_Mapping.map_key f p) = f -` keys p"
satisfies_cong: fixes \<phi> :: "WS1S_Nameful.ws1s"   and I1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and J1 :: "WS1S_Nameful.fo \<Rightarrow> nat"   and I2 :: "WS1S_Nameful.so \<Rightarrow> nat fset"   and J2 :: "WS1S_Nameful.so \<Rightarrow> nat fset" assumes "\<forall>x\<in>set (fos \<phi>). I1 x = J1 x"   and "\<forall>X\<in>set (sos \<phi>). I2 X = J2 X" shows "satisfies I1 I2 \<phi> = satisfies J1 J2 \<phi>"